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title: 'Informational Neurobayesian Approach to Neural Networks Training. Opportunities and Prospects '
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Informational Neurobayesian Approach, text recognition, information quantity, classification, Sequence to Sequence model, Neurobayesian Approach, Gradient Descent, one-shot learning, information emergence, large neural networks, information neural models.
Introduction
============
The classification problem nowadays is usually solved by complex analytical algorithms requiring plenty of data. Another field of research is neurobayesian approach, which proposes optimisation based on Bayes rule with some tricks to cope with functions complexity and make them well-defined. The authors introduce a method based on the same concepts but they also take into consideration the informational side of the data - namely its meaning. The problem of determining the quantity of information until recently was based on Hartley and Shennon approaches. The first one represents just the quantity of information, the second is its generalisation which takes into account different probabilities for classes. Another fundamental concept is Harkevich formula which measures the informational value of event or message as the logarithm of ratio of probability of a class after and before the event. The System Theory of Information (STI) proposed by Lucenko [^1] merges these concepts, taking each formula with corresponding power, which illustrates the emergence of classes and features. The main idea of the article is to develop a method, that allows to calculate weight coefficients of artificial neural network’s synapses. We propose to move from uninterpreted weights (trial and error method) to the weight representing the amount of information that is contained in the feature regarding a possible consequence. Thus, a neural network is considered as a self-training system of transforming the input data into the amount of information about the possible consequences.The proposed approach allows to train big neuromodels with billions of neurons on ordinary CPUs.
Classical Gradient Descent
==========================
Description
-----------
Gradient Descent is a widely used approach to solve optimisation problems which are too complex to find solution analytically. As the approach proposed in the paper is applied in the context of neural networks, we will briefly show the usage of gradient descent for their training.
The problem is formulated in the following way: model’s performance is measured by the error function, which depends on $n$ neurons and $m$ observations. So, $n$ weights are to be optimised to achieve the minimum of error function. The gradient descent principle is to push the weights towards the direction opposite to the error function’s gradient (all weights are changed simultaneously).
Formulae
--------
$$\nabla Q(w_0, w_1, ..., w_n) =\Bigg[ \frac {\delta Q}{\delta w_0}, ..., \frac {\delta Q}{\delta w_n} \Bigg]$$ So, on each iteration the weights are updated according the following rule: $$w_i = w_i - \mu \frac {\delta Q}{\delta w_i},$$ where $\mu$ is an exogenous parameter, which determines the learning rate. The most popular type of error functions is sum of squares, so the aim is to minimise the expression: $$Q(w_0, w_1, ..., w_n) = \frac 12 \sum^m_{j =1} (y_j - f(x_0^j,x_1^j, ..., x_n^j ))^2 ,$$ where $f(x_0^j,x_1^j, ..., x_n^j ) = \sum^n_{i=1} w_i x_i^j $ is a linear function, giving the values on the output of the neuron and depending on the weights of the input vector (instead of linear various functions can be used, e.g. tanh, sigmoid, softmax, the general principle remains the same). The derivative by each weight will be equal: $$\frac{ \delta Q(\cdot)}{\delta w_i} =\frac 12 \sum^m_{j =1} \frac\delta {\delta w_i} (y_j - \sum^n_{i=1} w_i x_i^j )^2 =$$
$$= \sum^m_{j =1} (y_j - \sum^n_{i=1} w_i x_i^j )(-x_i^j)$$ So, the formula for weights updating is: $$w_i = w_i + \mu x_i^j \sum^m_{j =1} (y_j - \sum^n_{i=1} w_i x_i^j )$$ Iterations are performed until the weight’s changes are more than some exogenous determined parameter.
Advantages and disadvantages
----------------------------
The approach performs well for a very narrow class of problems, though it has advanced modifications like Adagrad, Adam, and Gradient Boosting which show outstanding results although require considerable computational power.
Computational Complexity
------------------------
Algorithm’s complexity for the described case is $O(n^2)$ as on each iteration the sum of $n$ terms is computed $n$ times. The complexity is linear relative to input data size.
Neurobayesian Approach
======================
Description
-----------
The bayesian approach means designing the forecast of distribution of variables and then updating it based on new knowledge about an object. Bayesian framework in comparison with traditional (frequentist) one considers all data random, makes great use of Bayes theorem and instead of ML-estimates maximise posterior probability. Though it works with any amount of data, even small,the more information about the object is obtained, the more precise the final knowledge will be (the Bayesian updating can be applied iteratively as new data becomes available).
The problem is formulated as follows: let $p(X,T,W) $ be a joint distribution of hidden variables $T$, observed variables $X$ and decision rule parameters $W$. If training data is $ (X_{tr}, T_{tr})$, then posterior distribution of W is $$p(W|T_{tr},X_{tr}) = \frac {p(T_{tr},X_{tr}|W) p(W)}{ \int p(T_{tr},X_{tr}|W) p(W) d W}$$ Analytical approach to that expression is to make use of delta-method and maximise likelihood function. But the bayesian approach proposes a less straightforward, yet fruitful step - to regularise ML estimate using prior distribution. It helps to cope with overfitting and gives a more thorough picture of hidden variables and also allows to learn from incomplete data. But the main advantage is the possibility of the algorithm to obtain high accuracy results with big data. Application of bayesian approach to neural networks is based on cross-entropy error function minimisation with updating of the weights using existing data.
Formulae
--------
In practice the optimisation procedure goes in the following way[@vetrov][^2]\[Vetrov D., 2017\]. If only $X_{tr}$ is known, $T$ is a hidden variables vector, $p(X), q(T) $ are their density functions respectively, and $W$ is to be determined, then: $$W_* = \arg \max p(W|X_{tr}) = \arg \max \log p(W|X_{tr}) =$$ $$= \arg \max (\log p(X_{tr}|W) + \log p(W)) =$$ $$= \arg \max \Big( \log \int p(X_{tr}, T| W) dT + \log p(W)\Big)$$ The problem is that the first term is not concave. To cope with that a trick is introduced: $$\log p(X_{tr}|W) = \int q(T) \log p(X_{tr}|W)d T =$$ $$= \int q(T) \log \frac{p (X_{tr}, T |W)}{p (T|X_{tr},W)} dT =$$ $$= \int q(T) \log \frac{p (X_{tr}, T |W) q(T)}{p (T|X_{tr},W) q(T)} dT =$$ $$= \int q(T) \log \frac{p (X_{tr}, T |W)}{ q(T)} dT +
\int q(T) \log \frac{q(T)}{p (T|X_{tr},W) } dT =$$ $$\mathcal{L} (q,W) + KL(q(T)|| p(T|X_{tr},W))$$ where $ KL(q||p)$ is Kullback-Leibler divergence, which can be interpreted as distance between distribution and is always non-negative. So we can iteratively maximize $\mathcal{L} (q,W)$ instead of original expression. Procedure is called EM-Algorithm and consists of two steps:
1. **E-step** $\mathcal{L} (q,W_{t-1}) \rightarrow \max_q$ - corresponds to KL-divergence minimization. $$q_t (T)= \arg \min_q KL(q(T) || p(T| X_{tr}, W_{t-1})) = p ( T| X_{tr}, W_{t-1})$$
2. **M-step** $\mathcal{L} (q,W_{t-1}) \rightarrow \max_W. $ $$W_t = \arg \max_w \mathcal{L} (q,W_{t-1}) =$$ $$= \arg \max_w \int q_t(T) \log \frac{p(X_{tr},T|W)}{q_t (T)} dT=$$ $$= \arg \max_w \int q_t(T) \log p(X_{tr} ,T|W) dT,$$ which is a concave function, so the maximisation is legal.
The procedure is repeated until convergence.
Advantages and disadvantages
----------------------------
The main advantage of the approach is its scalability to large datasets, and less complexity in comparison with analytical optimisation solutions, which are quadratic over data size $ O(n^2)$, while EM algorithm is linear on data $ O(n)$ and weight vector’s dimensions. The approach allows to combine models, trained on different data (for same consequences), and thus achieve better results. The drawback is that the approach does not take into consideration the data’s structure, its intrinsic features, which can be exploited to build more realistic a priori distributions and achieve better performance. Since 2012, a number of studies were published, in which a new mathematical apparatus that allows to scale Bayesian methods to big data is proposed. At the heart of it lies an interesting idea. First, the problem of Bayesian inference (that is, the process of application of the Bayes theorem to data) was formulated as an optimization problem, and then modern techniques of stochastic optimization were applied to it, which made it possible to solve extremely large optimization problems approximately. This allowed Bayesian methods to enter the field of neural networks.Over the past 5 years, a whole class of Neurobayes models, that can solve a wider range of problems than conventional deep-seated neural networks, has been developed.
Informational Neurobayesian Approach
====================================
Theoretical background
----------------------
Some facts from the information theory are required for understanding this article. At first, the number of characters that can be obtained using the alphabet consisting of n symbols is $N = m^n,$ where $n$ is the number of characters in the message. To avoid exponential dependency, it was proposed by Hartly to represent the quantity of information as a logarithm of number of all possible sequences: $$I = \log N = \log m^n = n \log m$$ Let there be $k$ characters, each has $m$ features and $p_i$ is a probability to get each feature. Shannon proposed a proportion for determining the average quantity of information in a message with random probabilities of character’s values: $$I = - k \sum_{i=1}^m p_i \log p_i$$ The more uncertain is the consequence, the more information can be obtained from receiving the information about the consequence.To measure the uncertainty we use entropy, which is the average quantity of information per character. $$H = \frac Ik = - \sum_{i=1}^m p_i \log p_i$$ The key task in this context is to determine the amount of valuable information in a message (cause) for classification of possible consequence. For its solution it is required to calculate the amount of information, contained in $i$ feature and in the fact, that the object with this feature belongs to class $j$.
The average amount of information, contained in all features on all classes: $$I (W, M) = \sum^W_{j=1} \sum^M_{i=1} p_{ij} \log_2 \frac{p_{ij}}{p_i p_j}$$ is precisely the average of “individual amounts of information" in each feature about each class. If there are $M$ characters in the message, then the information on belonging to class $j$ can be referenced as information’s density and is expressed via Pointwise Mutual Information $$PMI (w_j, m_i) = \log \frac{p_{ij}}{p_j p_i}= \log \frac{p_{ij}}{p_j} +\log \frac{1}{p_i} =$$ $$=\log p_{ij} - \log p_i - \log p_j$$ The expression above represents the sum of quantities of relevant information by Harkevich and Shennon, and is interpretation of Bayes’ theorem for information theory.
Foundations of System Theory of Information (STI)
-------------------------------------------------
In this paper the system generalisation of Hartley’s formula is used as $ I = \log W^\varphi,$ where $W$ is a number of pure conditions of the system, $\varphi$ – Hartley’s emergence coefficient (the level of system’s complexity, which consists of $W$ pure conditions). Lucenko \[Lucenko, 2002\]\] took as an axiom the statement that system generalisation of Hartley’s formula is $$\varphi=\frac{ \log \sum_{z=1} ^Z C^m_W}{\log_2 W}$$ For every number of system’s elements there is a maximal level of system’s synergy. According to STI, the amount of information should be evaluated as $$I (W, Z) = \log W + \log W^{\varphi-1},$$ in other words, it consists of classical and synergic parts. The system’s information that we obtain from an object via STI approach is actually information on all possible configurations of the system. Prof. Lucenko \[Lucenko, 2002\] discovered a tendency for share of synergic information to raise as the number of elements increases and he propose to name it as the **Law of emergence increase**, illustrated at Figure 1.
The classical Harkevich formula is $$I_{ij} = \log \frac{P_{ij}}{P_j},$$ where $P_{ij} $ is the probability of attaining target $J$ after getting message about feature $i$ and $P_j$ is the probability of attaining target (class) $j$ without any information. This concept does not take into account the cardinality of space of future configurations of the object, while they can be taken into consideration by involving the classical and synergic Hartley’s formulas, nevertheless it does not allow to obtain the quantity of information in bits. To solve that problem, we take for each feature $i, \ 1\leq i \leq M$ and class $j, \ 1\leq j \leq W$ approximate the probabilities $P_{ij}$ and $P_j$ through frequencies: $$P_{ij} = \frac{N_{ij}}{N_i}, \ \ P_i = \frac{N_i}{N}, \ \ P_j = \frac{N_j}{N},$$ $$N_i = \sum ^W _{i=1}N_{ij}; \ \ N_j= \sum^M _{j=1}N_{ij}; \ \ N = \sum ^W _{i=1}\sum ^M _{j=1} N_{ij}$$ where $N_{ij}$ is the total number of events like “condition $j$ was obtained after feature $i$ acting”, $N_j$ - total number of features, with which the condition $j$ was obtained, $N$ - total number of of various features for all final conditions. After placing into $ I_{ij} = \log \frac{P_{ij}}{P_j} $ values $P_{ij}$ and $P_j$ we get the expression of value of information measured as its quantity: $$I_{ij} (W,M) = \log_2 \frac{N_{ij}N}{N_i N_j}$$
As it is known, the classical Shennon formula for quantity of information for events with varied probabilities transforms into Hartley’s formula with the condition that all events have the same probability, i.e. satisfy the basic property of conformance. So the same can be applied to the Harkevich formula, which means that in marginal case it should become Hartley’s formula (marginal case means that there is unique feature for every class (object’s condition) and vice versa, and all thesade classes have the same probability. In that case the quantity of information in each feature about belonging to classes is maximal and is equal to information calculated through Hartley’s formula. So, in case of one-to-one correspondence of features and classes: General formula of Lutsenko formula and his emergence coefficient $\psi$ are: $$I_{ij} = \log_2 N^{\psi} = \log_2 W^\varphi,$$
$$\psi = \frac{\log_2 W^\varphi}{\log_2 N}$$
Taking it into account, authors suggest to add an emergence coefficient into the modified Harkevich formula. The resulting expression is named Lutsenko emergence information quantity formula: $$I_{ij} (W,M,Z)= \log_2 \Big(\frac{N_{ij}N}{N_i N_j}\Big)^\psi,$$ where $\psi$ is Harkevich’s emergence coefficient, which defines the degree of determination of the object with the system’s organisation level $\varphi$, having $W$ pure conditions and $M$ features, on which the final condition is dependent, and Z is the maximal complexity calculated for each factor separately. $N$ observations on system’s behaviour were made. That generalisation can be performed in the following way: $$I_{ij} = \log_2 \Big(\frac{N_{ij}N}{N_i N_j}\Big)^\psi = \log_2 \Big(\frac{N_{ij}N}{N_i N_j}\Big)^{\frac{\log_2 W^\varphi}{\log_2 N}} =$$ $$=\frac{\log_2 W^\varphi}{\log_2 N} \Big( \log_2 \Big(\frac{N_{ij}}{N_i N_j}\Big) +\log_2 N \Big) =$$ $$=\log_2 \Big(\frac{N_{ij}}{N_i N_j}\Big)^{\frac{\log_2 W^\varphi}{\log_2 N}} + \log_2 W^{\varphi}$$
The same calculations for the continuous case:
$$I = \log_2 \int^{Z}_{z=q} \Big( \frac{G(W)}{G(z)\cdot G(W-z) }\Big) =$$ $$= \log_2 \left \{ G (W) \cdot \int ^{Z}_{z=1} \frac{\,d z}{G(z)\cdot G(W-z)}\right \} =$$ $$= \log_2 G(W) + \log_2 \int^{Z}_{z=1} \frac{\,d z}{G(z)\cdot G(W-z)}$$ And finally: $$I = \log_2 G(W) + \log_2 \int^Z_{z=1}\frac{\,d z}{G(z)\cdot G(W-z)} = I(W) + I(W, Z)$$
Generalised Kharkevich [@khar2] formula also satisfies correspondence principle, i.e, transforms into Hartley’s formula: $\log \frac{P_{ij}}{P_j} = \log \frac{N_{ij}N}{N_i N_j}=\log N.$ In the marginal case, when every class (condition of the object) corresponds only one feature, and for every feature - one class, $\forall \ \ N_ij = N_i = N_i =1 $ these classes and features have the same probabilities.
System Emergence Coefficient
----------------------------
General formula of Lutsenko for Harkevich emergence coefficient: $$\psi = \frac{\log_2 W^\varphi}{\log_2 N}$$
where $\varphi = \frac{\log_2 \sum^Z_{m=1} C^m_W}{\log_2 W} $.\
$\psi$ varies from 0 to 1 and determines the degree of determination of the system:
1. $\psi=1$ corresponds to absolutely determined system, behaviour of which depends on the minimal number of features.
2. $\psi=0$ corresponds to totally random system.
3. $0<\psi<1$ corresponds to system, where there are more features than classes and none of them plays the key role in determining the class.
Professor Lutsenko[@luc2] proposed coefficients representing the degree of determination of the possible state of the object with a set of features at the system organization level $\varphi$. Nevertheless, as it seems hard to evaluate, it was decided to omit that concept and admit the coefficient $\varphi$ to be equal 1 (minimum level of complexity). $$\psi = \frac{\log_2 W}{\log_2 N}$$
Straightforward way to calculate it is to consider all possible of combinations and classes, and for each one to evaluate the amount of information. Evidently, it will require enormous calculations. $$\psi = \frac{\log_2 W^{\varphi}}{\log_2 N} =\frac{\log_2 W^{\frac{log_2 \sum_{m=1}^Z C^{W}_m}{\log_2 W}}}{\log_2 N}$$ The Lucenko-Artemov formula for emergence coefficient of system: $$\psi = \frac{log_2 \sum^Z_{m=1} C_W^m}{\log_2 N}$$ The value obtained is the emergence coefficient of the system for the real level of complexity. The optimal solution is to consider only relevant combinations of feature-classes, dividing them into groups by complexity, up to which the combinations will be evaluated. There are two approaches for calculation the complexity Z. The first one for each feature takes the number of classes the feature corresponds to (gives positive amount of information). The number of classes represents the complexity, up to which the combinations of classes will be calculated. The second method is to divide all features and classes into groups with the same number of features as well as classes. The idea is to take all classes to whom the feature corresponds and select other features which correspond to the same classes. The key point is that it helps to build one-to-one correspondence between features and classes within that group. Finally we obtain limited number of features $i$ or groups of features $i \in g(\sum_g i= |M|$, where $M$ is the number of features), which are acting collectively $N_g$ times in the same number of classes $W_g$. That means for given features maximal complexity is $Z_g = W_g$. So $ \sum ^{Z_g}_{m=1} W_g = 2^{Z_g}-1| Z_g = W_g, g \in M$, and we get the final expression for the coefficient of emergence for each group: $$\psi_g= \frac{log_2 (2^{W_g} -1)}{\log_2 N_g}$$
The coefficient is normalised by the following rule: $$\psi_g \ = \
\begin{cases}
1, & \text{if} \ W_g \textgreater \log_2 2N \\
\frac{log_2 (2^{W_g}-1)}{\log_2 N}, &\text{otherwise}
\end{cases}$$ Formula 21 is the Artemov-Lucenko emergence coefficient, for the system of features, aggregated by the maximal level of of system’s complexity.
Application to neural networks
------------------------------
The essence of the proposed approach shows its best application to neural networks. The main idea is that the quantity of information, calculated for all feature-class pairs can be expressed in network’s weights, so they gain a meaning instead of being simply empirical parametrs. It explains the given name “Informational Neurobayesian” for the approach. Every weight represents quantity of information - the object with the feature $i$ activates given neuron $j$: $$I_{ij} = \psi_{i} \ \log{\frac{P_{ij}}{P_j}} ,$$ where $\psi_i$ - coefficient of emergence of the system for feature $i$.
In the scheme above the $L_i$ is $i$-th object’s feature. $\vv{L}\{X_1, ...,X_k\} $– a vector of object features:
$$L_i \ = \
\begin{cases}
1, & \text{if} \ X_k \geq \epsilon \\
0, &\text{otherwise}
\end{cases}$$
For every object with a given set of features the neural network chooses over all classes the one with maximal value of the activation function, represented as an activated sum by all features. The process is illustrated in Figure 3. $$S_{j}=\sum^M_{j=1} I_{ij}\cdot L_{i}+\text{bias}, \ \text{bias} = I_0$$ Bias is a parameter for activation of class. It allows to solve two problems: 1. To set a minimum threshold of neuron activation. 2. Allows integration with other models. $$j^*= \arg \max_j f(S_j)$$ $$y = f(S_{j*})$$
The Y is a chosen class for the given features, $f$ is activation function. The whole workflow is presented in Figure 2.
### Generalised EM-Algorithm for INA
EM-algorithm consists of two steps iteration repetions. On the E-step expected hidden variables values are calculated using current estimate of parameters. On the M-step likeli- hood is maximised and new parameters vectors estimate is being obtained based on their current values and hidden variables vector.\
**Step 0**: Calculate a frequency matrix for input data.
Unique features $M$ and classes $W$ are determined in training data and their frequency table which reflects the distribution of parameters $N_{ij}$ - number of features $i \in M$ for each class $j \in W$. Also $Z_g$ parameter (number of classes, in which a features group $g$ is active) is determined during this step.\
**Step E**: Creating informational neurobayesian model.\
During the E-step expected value of hidden variables $x_i$ by current estimation of parameter vector $N_{ij}$ is evaluated. The data is represented in Figure 4. and the output is organised in the same way, as presented in Figure 5. $$I_{i,j} (W, M, Z) = \log \big( \frac{N_{i,j}N}{N_i N_j}\big)^{\psi_g}$$ $$\psi_{g}=\frac{log_2 (2^{W_g}-1)}{\log N_g}$$ $$I_{i,j} (W, M, Z) \rightarrow \max_Z$$
**Step M**: Model maximisation by Van Rijsbergen’s effectiveness measure (F-measure).
1. Calculate $E(F_{micro}):$
Calculate average micro $F$-measure for each observed class $j_l \in W$ based on training set, comparing the result with $j_t$ - test dataset for object’s features vector with the same set of features, $t$ stands for test data, $l$ - data the model has been trained on («model has learnt»). $$E (F_{\text{micro}}) = \frac{\sum_k F_{j_t}}{k}$$ $$F_{j_t}=(1+\beta)^2 \frac{\text{ precision}_{j_t} \times \text{ recall}_{j_t} }{\beta^2\text{ precision}_{j_t} + \text{ recall}_{j_t} }$$ $$0<\beta<1,$$
$$ j_t = \arg \max F(S_{j_t})$$
2. For each $j_t \in W,$ where $F_{j_t} < \ E(F_{j_t})$, i.e. $j_l \ne \arg \max F(S_{j_t})$ or $S_{j_t}<S_{j_l}.$\
2.1 Determine $j^*_t$ and $j^*_l$ - correct class for the set of features $L_i$, $i \in M$.\
2.2 Exclude all objects $L_i$ with the same set of features, but $j^*_t \ne j^*_l$ classes\
2.3 For remaining objects $L_i$ change quantity of information for each feature, acting in given class $I_{j_l } \rightarrow I_{j^*_l}$, so that $S_{j_l } \rightarrow \max S_{j^*_l}$ $$j_l = \arg \max F(S_{j^*_l})$$\
2.4 Calculate $$E (F^*_{\text{micro}}) = \frac{\sum_k F_{j_{t*}}}{k}$$ The updated model is checked for improvement on wrongly classified objects.\
2.5 Continue until $E(F^*_{\text{micro}}) \ \textgreater \
E(F_{\text{micro}})$
Experiments
===========
For conducting the tests we used Brain2 - a software platform implementing Informational Neurobayesian Approach. The biggest built neuromodel is aimed at word recognition by letter and bigrams in Russian language for 100 000 words – it consists of 1042 features and 5 099 300 classes, which makes the total number of parameters 6.3 bln (weight coefficients). An important factor here is that it took only 2 h 35 min to build the model on an average server without GPU (205 GFlops, 8 cores CPU). In comparison, the Google Brain biggest neural network consists of 137 billion parameters, but if we consider the needed resources – only for 4.3 bln parameters it took them 47 hours with 16-32 Tesla K40 GPUs (1220 GFlops) [@shaz]\[Shazeer N, 2017\]. On other basic machine learning tasks such as MNIST and House Prices our models have shown an acceptable result – 0.81 on MNIST (Kaggle task) with the best 1.0; and 0.42 on House Prices with the best one at 0.06628 (For one epoch). It is important to note that we did not set the task to achieve the best possible result and have trained the model as is. Instead, we have concentrated on a more difficult problem, connected with natural text – reconstruction of words with errors and finding answers on questions, where we have reached accuracy close to 1. Experiment results are presented in Table 1.
Conclusion
==========
In the paper the INA method aimed to cope with complex classification problems more intelligently than currently used analytical and numerical methods was described. Its ability to be applied to neural networks and provided experiments showed opportunities for the suggested method to simplify training process in various classification problems and achieve at least the same performance with less computational costs. The results of the experiments are presented in Table 1. And the summary of comparison with other method is presented in Table 2.
The additive properties of the amount of information allow not only to explain the correctness of addition of weights while choosing a class within a single model (layer), but also to link different models (layers), transferring values of the amount of information from one input to another. In addition, every information neuromodel can be trained independently. Eventually it makes possible to facilitate the construction of super-neural networks - multilevel (and multi-layer) neural networks, where each input of the neuron is a neural network of a lower order (from the position of information amount). The proposed information approach makes it possible to obtain an acceptable level of quality of the model (F1 $>$ 0.6, for 1 epoch), even on relatively small data. And given the low complexity $(O(n))$ of the learning algorithm , the INA opens new horizons to the training of superneuromodels on conventional (not super) computers.
In our opinion, the INA combines in itself a general philosophical view presented by such cognitive scientists as D. Chalmers[@chal][^3] (as each layer represents a level of cognition, influencing a higher cognition level), V. Nalimov (in the works on the [@nal][probabilistic model of language](https://www.tib.eu/en/search/id/BLCP%3ACN026838838/V-V-Nalimov-s-probabilistic-model-of-language-and/ ) \[Nalimov, 1990\]) and the pragmatic concept of meaning - the amount of information in the cause for a given consequence required for making a managerial decision [^4] [@kha] (Ashby[@ash] [^5], Wiener, U. Hubbard).
[^1]: Section 4.2 author,[@luc] \[Lutsenko E., 2002\]
[^2]: https://www.sdsj.ru/slides/Vetrov.pdf
[^3]: \[Chalmers,1996\]
[^4]: \[Kharkevich,2009\]
[^5]: \[Ashby,1961\]
|
---
abstract: 'RLCard is an open-source toolkit for reinforcement learning research in card games. It supports various card environments with easy-to-use interfaces, including Blackjack, Leduc Hold’em, Texas Hold’em, UNO, Dou Dizhu and Mahjong. The goal of RLCard is to bridge reinforcement learning and imperfect information games, and push forward the research of reinforcement learning in domains with multiple agents, large state and action space, and sparse reward. In this paper, we provide an overview of the key components in RLCard, a discussion of the design principles, a brief introduction of the interfaces, and comprehensive evaluations of the environments. The codes and documents are available at <https://github.com/datamllab/rlcard>.'
author:
- |
Daochen Zha$^1$, Kwei-Herng Lai$^1$, Yuanpu Cao$^1$, Songyi Huang$^2$, Ruzhe Wei$^1$, Junyu Guo$^1$, Xia Hu$^1$\
$^1$ Department of Computer Science and Engineering, Texas A&M University, College Station, USA\
$^2$ Simon Fraser University, BC, Canada\
Contact Emails: {daochen.zha, khlai037}@tamu.edu
bibliography:
- 'reference.bib'
title:
- 'RLCard: A toolkit for Reinforcement Learning in Card Games'
- 'RLCard: A Toolkit for Reinforcement Learning in Card Games'
---
Introduction {#sec:1}
============
Reinforcement learning (RL) is a promising paradigm in Artificial Intelligence for learning goal-oriented tasks. Through interactions with the environments, reinforcement learning agents learn to make decisions at each state in a trial-and-error fashion. With neural networks as function approximators, deep reinforcement learning has recently achieved breakthroughs in various domains: Atari games [@mnih2015human], Go game [@silver2017mastering], continuous control [@lillicrap2015continuous], and neural architecture search [@zoph2016neural], just to name a few. Out of these achievements, however, reinforcement learning is still immature and unstable in applications with multiple agents, large decision space or sparse reward [@li2017deep].
In this paper, we introduce various styles of card environments for reinforcement learning research. Card games are ideal testbeds with several desirable properties. **First**, card games are played by multiple agents who must learn to compete or collaborate with each other. For example, in Dou Dizhu, peasants need to work together to fight against the landlord in order to win the game. **Second**, card games have huge state space. For instance, the number of states in UNO can reach $10^{163}$. The cards of each player are hidden from the other players. A player not only needs to consider her own hand, but also has to reason about the other players’ cards from the signals of their actions. **Third**, card games may have large action space. For example, the possible number of actions in Dou Dizhu can reach $10^4$ with an explosion of card combinations. **Last**, card games may suffer from sparse reward. For example, in Mahjong, winning hands are scarce. We observe one winning hand every five hundreds of games if playing randomly. Moreover, card games are easy to understand with huge popularity. Games such as Texas Hold’em, UNO and Dou Dizhu are played by hundreds of millions of people. We usually do not need to spend efforts on learning the rules before we can dive into algorithm development.
{width="6cm"}
\[fig:1\]
**Environment** **InfoSet Number** **Avg. InfoSet Size** **Action Size**
------------------------ -------------------------- ----------------------- -----------------
Blackjack $10^3$ $10^1$ $10^0$
Leduc Hold’em $10^{2}$ $10^2$ $10^0$
Limit Texas Hold’em $10^{14}$ $10^3$ $10^0$
Dou Dizhu $10^{53} \sim 10^{83}$ $10^{23}$ $10^4$
Mahjong $10^{121}$ $10^{48}$ $10^2$
No-limit Texas Hold’em $10^{162}$ $10^3$ $10^4$
UNO $10^{163}$ $10^{10}$ $10^1$
Sheng Ji $10^{157} \sim 10^{165}$ $10^{61}$ $10^{11}$
To develop card environments with easy-to-use interfaces is a challenging task. First, the interfaces must be accessible to RL researchers who may or may not have a game theory background. In the extensive form games, the player will not observe her next state immediately after taking an action. The next state is exposed to the player only after all other players have chosen their actions. This makes it difficult to design environment interfaces. Second, the environments need to be configurable. The state representation, action abstraction, reward design, or even the game rules should be easily adjusted for research purposes.
We present RLCard, an opensource toolkit designed for reinforcement learning in card games. It supports various card environments, as summarized in Table \[tab:summary\]. The interfaces are straightforward for reinforcement learning. The transitions of each player and collected and well organized after a complete game in the multi-agent setting. We also provide a single-agent interface, where the other players are simulated using pre-trained models. The state and action encoding can be easily configured. The games are implemented under the same structure with clear logic. The evaluation tools are provided to measure the performance by winning rates of tournaments. Future versions will extend the toolkit to include more environments. The goal of RLCard is to bridge reinforcement learning and imperfect information games, and push forward the research of reinforcement learning in domains with multiple agents, large state space, large action space, and sparse reward.
Overview {#sec:2}
========
Figure 1 shows an overview of RLCard. Each game is wrapped by an environment class with easy-to-use interfaces. With RLCard, we can focus on algorithm development instead of engineering efforts on games. When developing the toolkit, we adopt the following design principles:
- **Reproducible.** Results on the environments can be reproduced. The same result should be obtained with the same random seed in different runs.
- **Accessible.** Experiences are collected and well organized after each game with straightforward interfaces. State representation, action encoding, reward design, or even the game rules, can be conveniently configured.
- **Scalable.** New card environments can be added conveniently into the toolkit with the above design principles. We try to minimize the dependencies in the toolkit so that the codes can be easily maintained.
Available Environments
----------------------
The toolkit provides various styles of card games that are popular among hundreds of millions of people, including betting games, Chinese Poker, and some boarding games. Table \[tab:summary\] summarizes the card games[^1] in RLCard and estimates the complexity of each game. The game size can be measured by the number of information sets, which are the observed states from the view of one player. The average size of the information set is defined as the average number of possible game states in each information set. For example, given the observation from the view of one player in Texas Hold’em, the other players could have many possible hands. Each possible hand corresponds to one game state in this information set. The size of the action space is also provided since large action space will greatly increase the difficulty.
Game Design {#sec:2:2}
-----------
Card games are usually played following similar procedures. We design several abstract base classes which are implemented in the specific games. In the toolkit, some common concepts are abstracted and defined as follows:
- **Player.** The person who plays the game. Each game is usually played by multiple players.
- **Game.** A game is a complete sequence starting from one of the non-terminal states to a terminal state. At the end of a game, each player will receive a payoff.
- **Round.** A round is a part of the sequence of a game. Most card games can be naturally divided into multiple rounds. For instance, Texas Hold’em consists of four rounds of betting. In Doudi Zhu a round is finished when two consecutive players pass.
- **Dealer.** Card games usually require shuffling and allocating a deck of cards to players. Dealer is responsible for deck management.
- **Judger.** A judger is responsible for making major decisions in a round or at the end of a game. For example, the next player in UNO is decided based on the type of the last card. In Texas Hold’em, the payoff is determined in the end of a game.
The games in the toolkit are implemented by associating a class with each of the above concepts. The common design principle makes the game logic easy to follow and understand. Other card games are usually compatible with the above structure so that can be easily added to the toolkit.
Environment Interfaces
======================
This section briefly introduces the interfaces of the toolkit. We begin with core interfaces that allow a quick start of training a reinforcement learning agent. Then we describe state representation, action encoding, and how we can modify them to customize the environments. After that, we show how to generate data with multiple processes. Finally, we introduce a single-agent interface, where the other players are simulated by pre-trained or rule-based models. More documents and examples can be found at the Github repository.
Core Interfaces
---------------
We provide a `run` function for quickly getting started. It directly generates payoffs and game data, which are organized as transitions, i.e., (state, action, reward, next\_state, done). This interface is designed for algorithms that do not need to traverse the game tree. An example of running Dou Dizhu with three random agents is as follows:
import rlcard
import RandomAgent
# Initialize the environment
env = rlcard.make('doudizhu')
# Initialize random agents
agent = RandomAgent()
env.set_agents([agent, agent, agent])
while True:
# Generate data from the environment
trajectories, payoffs = env.run()
# Train agent here
We also provide some advanced interfaces that operate upon the game tree. Following other RL toolkits [@brockman2016openai; @Marc2019openspiel], we define a `step` function which moves the environment to the next state given the current action. We also provide a `step_back` function, which traverses back to the previous state. This design enables flexible node visiting strategies of the game tree, such as external sampling MCCFR [@lanctot2009monte]. Note that the action of the current player will lead to the observation of the next player, and the current player can observe her next state only after all other players have chosen their actions. Thus, users need to be careful with `step` and `step_back` since the next state is “delayed”.
State Representation
--------------------
State is defined as all the information that can be observed from the view of one player in a specific timestep of the game. In the toolkit, each state is a dictionary consisting of two values. The first value is a list of legal actions. The second value is observation. There are various ways to encode the observation. For Blackjack, we directly use the player’s score and the dealer’s score as a representation. For other games in the toolkit, we encode the observed cards into several card planes. For example, in Dou Dizhu, the input of the policy is a matrix of $6$ card planes, including the current hand, the union of the other two players’ hands, the recent three actions, and the union of all the played cards.
Action Encoding
---------------
The specific actions in a game are all encoded into action indices, which are positive integers starting from $0$. Each action index corresponds to exactly one action in the game. The legal actions are also represented as a list of action indices. At each step, an agent should choose one of the action indices (i.e., integer values) among the legal actions instead of specific actions (such as “hit” or “stand” in Blackjack).
For some large games, action abstraction is adopted to reduce the action space. For example, Dou Dizhu suffers from the combinational explosion of the action space with more than $3 \times 10^4$ actions, where any trio, plane or quad can be combined with any individual card or pair (kicker). To reduce the action space, we only encode the major part of a combination and use rules to decide the kicker. In this way, the action space of Dou Dizhu is reduced to $309$.
Customization
-------------
In addition to the default state and action encoding, our design enables customization of state representation, action encoding, reward design, and even the game rules.
Each game is wrapped by an `Env` class, in which we can rewrite some key functions to customize the environments. The function of `extract_state` is to convert the original game state into representation. The function of `decode_action` is to map action indices to actions. One can implement his own abstraction of the actions by modifying this function. The function of `get_payoffs` will return the payoffs of the players in the end of the game. For each game, we provide a default setting for each of the above components. Users are encouraged to customize these settings to achieve better performance.
The parameters of each game can also be adjusted. For example, one can change the number of players or the fixed raise in Limit Texas Hold’em by modifying `__init__` function of the `LimitholdemGame` class. In this way, we may adjust the difficulty of the games for our purposes and design algorithms step by step.
Parallel Training
-----------------
The toolkit supports generating game data with multiple processes. Running in parallel will greatly accelerate the training in large environments. Specifically, we create duplicate environments in initialization. Then each worker will copy the model parameters from the main process, generate game data in a duplicate of the environment, and send the data to the main process. The main process will collect all the data to train the agent on either CPU or GPU. Example implementations of training agents with multiple processes can be found at the Github repository.
Single-Agent Interfaces
-----------------------
We provide interfaces to explore training single-agent reinforcement learning agents in card games. Specifically, we develop pre-trained or rule-based models to simulate other players so that the games essentially become single-agent environments from the view of one player. These single-agent environments are also challenging since they have large state and action space, and sparse reward. In the future, we plan to use different levels of simulating models to create environments with various difficulties. The single-agent interfaces follow OpenAI Gym [@brockman2016openai]. Specifically, in the single-agent mode, given an action, `step` function will return the next state, reward, and whether the game is done. The `reset` function will reset the game and return the initial state. Standard single-agent RL algorithms can be easily applied to the environments.
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Evaluation
==========
This section conducts experiments to evaluate the toolkit. We mainly focus on the following two questions: (1) How the state-of-the-art reinforcement learning algorithms perform on the introduced environments? (2) How many computation resources are required to generate game data?
Training Agents on Environments
-------------------------------
We apply Deep Q-Network (DQN) [@silver2016mastering], Neural Fictitious Self-Play (NFSP) [@heinrich2016deep], and Counterfactual Regret Minimization (CFR) [@zinkevich2008regret] to the environments. These algorithms belong to different categories. DQN is a standard single-agent reinforcement learning algorithm, NFSP is a deep reinforcement learning approach for extensive games with imperfect information, and CFR is a standard regret minimization method for extensive imperfect information games. For DQN, we fix other players as random agents so that DQN agent can be trained in single-agent setting. We only test CFR on Leduc Hold’em since it is computationally expensive, requiring complete traversal of the game tree. Blackjack is only tested by DQN because it is a single-agent environment.
To evaluate the agents is non-trivial. The performance of imperfect information game is usually measured by exploitablity [@zinkevich2008regret; @johanson2011accelerating], which searches for the best response against the trained policy. However, it is computationally expensive to obtain the best response for the large environments in the toolkit since it relies on traversal of the game tree. Thus, we evaluate the performance based on winning rates. In this paper, we adopt two methods to empirically evaluate the performance. First, we report the winning rates of the agents against random agents. Second, we compare the agents with tournaments.
In our experiments, the hyperparameters are lightly tuned. For DQN, the memory size is selected in $\{2000, 100000\}$, the discount factor is set to $0.99$, Adam optimizer is applied with learning rate $0.00005$, and the network structure is MLP with size 10-10 128-128, 512-512 or 512-1024-2048-1024-512 based on the size of the state and action space. For NFSP, the anticipatory parameter is chosen from $\{0.1, 0.5\}$. Memory size for supervised learning and reinforcement learning are $10^6$ and $3 \times 10^4$, respectively.
**Tournament** **NFSP** **DQN**
------------------------------ ---------- ---------
Leduc Hold’em 1.0691 -1.0691
Limit Texas Hold’em -0.0308 0.0308
Dou Dizhu with NFSP landlord 0.7049 0.2951
Dou Dizhu with DQN landlord 0.7303 0.2697
Mahjong -0.0090 -0.0104
No-limit Texas Hold’em 9.5610 -9.5610
UNO -0.0428 0.0428
: Average payoffs of NFSP and DQN by playing $10,000$ games. For Dou Dizhu, we switch roles of landlord and peasants and report the results separately. For Mahjong, two DQN agents and two NFSP agents randomly choose seats in each game, and the averaged results are reported.[]{data-label="tab:tournament"}
[**Results against random agents.**]{} The learning curves in terms of the performance against random agents are shown in Figure \[fig:curve\]. The rewards of the agents are obtained through competitions against random agents. Specifically, the rewards of betting games (Leduc Hold’em, Limit Texas Hold’em, No-limit Texas Holdem) are defined as the average winning big blinds per hand. The rewards of the other games are obtained directly from the winning rates. In Dou Dizhu, players are in different roles (landlord and peasants). We fix the role of the agent as the landlord in evaluation.
[l|C[1.2cm]{}|C[1.4cm]{}|C[1.2cm]{}|C[1.4cm]{}|C[1.2cm]{}|C[1.4cm]{}|C[1.2cm]{}|c]{} & & & &\
& total & per step & total & per step & total & per step & total & per step\
Blackjack & $156.1$ & $1.1 \times 10^{-4}$ & $45.6$ & $3.3 \times 10^{-5}$ & $23.0$ & $1.7 \times 10^{-5}$ & $12.8$ & $9.3 \times 10^{-6}$\
Leduc Hold’em & $204.6$ & $8.0 \times 10^{-5}$ & $58.7$ & $2.3 \times 10^{-5}$ & $28.7$ & $1.1 \times 10^{-5}$ & $17.9$ & $7.0 \times 10^{-6}$\
Limit Texas Hold’em & $324.6$ & $1.1 \times 10^{-4}$ & $83.3$ & $2.8 \times 10^{-5}$ & $45.9$ & $1.6 \times 10^{-5}$ & $24.6$ & $8.3 \times 10^{-6}$\
Dou Dizhu & $87270.8$ & $1.4 \times 10^{-3}$ & $22894.3$ & $3.6 \times 10^{-4}$ & $12753.9$ & $2.0 \times 10^{-4}$ & $7275.9$ & $1.1 \times 10^{-4}$\
Mahjong & $74786.0$ & $8.1^{-4}$ & $20825.9$ & $2.3^{-4}$ & $11059.7$ & $1.2^{-4}$ & $6169.6$ & $6.7^{-5}$\
No-limit Texas Hold’em & $597.7$ & $1.4 \times 10^{-4}$ & $160.6$ & $3.7 \times 10^{-5}$ & $81.9$ & $1.9 \times 10^{-5}$ & $48.4$ & $1.1 \times 10^{-5}$\
UNO & $4952.9$ & $1.1 \times 10^{-4}$ & $1366.0$ & $3.0 \times 10^{-5}$ & $696.5$ & $1.5 \times 10^{-5}$ & $440.7$ & $9.5 \times 10^{-6}$\
We make two observations. First, all the algorithms have similar results against random agents. DQN is slightly better than NFSP on Texas Hold’em and UNO, while NFSP is slightly better than DQN on Leduc Hold’em and Dou Dizhu. It is reasonable for DQN to achieve this result since DQN is trained to exploit the random agents. Second, NFSP and DQN are highly unstable in large games. Specifically, they only achieve minor improvements during the learning process on UNO, Mahjong and Dou Dizhu. These games are challenging due to their large state/action space and sparse reward. We believe there is a lot of room for improvement. More efforts are needed to study how we can stably train reinforcement learning agents in these large environments.
[**Tournament results.**]{} We report the average payoffs the agents achieved when playing against each other. The results between NFSP and DQN are shown in Table \[tab:tournament\]. We observe that NFSP is stronger than DQN on most of the environments. We also compare CFR with NFSP and DQN on Leduc Hold’em. CRF achieves better performance, winning $0.0776$ and $1.2493$ against NFSP and DQN, respectively.
[**Discussion**]{} We further analyze the trained agents. We find that the DQN agents play very aggressively in betting games. For example, in Leduc Hold’em environment, DQN agent tend to choose “raise” or “call” in almost every decision. Interestingly, this naive strategy works well when the opponent is a random agent since the random agent may easily choose “fold” so that that the DQN policy can win. However, DQN policy may be highly exploitable since one can easily find its weaknesses. Thus, performance against random agents can only be a way to get a sense of whether the agent is improving on the environment, but is not enough to be used to evaluate algorithms. For large games, we recommend evaluating algorithms by playing against existing models. To benchmark the evaluation, we will develop rule-based agents and stronger pre-trained models in the future.
Running Time Analysis
---------------------
We evaluate the efficiency of the implemented environments by running self-play on the games with random agents. Specifically, we report the running time in seconds of $1,000,000$ games using a single process and multiple processes. Since Dou Dizhu, UNO and Mahjong have long sequences in one game, we additionally report a normalized version of running time, i.e., the average running time per timestep. Our experiments are conducted on a server with 24 Intel(R) Xeon(R) Silver 4116 CPU @2.10GHz processors and 64.0 GB memory. Each experiment is run $3$ times with different random seeds. The average running time in seconds is reported in Table \[tab:efficiency\]. We observe that all the environments achieve higher throughputs with more processors.
Related work
============
There are a few open-source reinforcement learning libraries, most of which focus on single-agent environments [@brockman2016openai; @duan2016benchmarking; @shi2019virtual]. Recently, there have been some projects that support multi-agent environments [@vinyals2017starcraft; @zheng2018magent; @suarez2019neural]. However, they do not support card game environments. OpenSpiel [@Marc2019openspiel] includes a large collection of games, providing several simple card games. Our toolkit is specifically designed for card games with straightforward interfaces for reinforcement learning, supporting various styles of card games that are not included in existing toolkits.
The most popular techniques for solving poker games in literature are Counterfactual Regret Minimization (CFR) [@zinkevich2008regret] and its variants [@brown2018deep]. Achievements have been made on betting games such as Texas Hold’em [@moravvcik2017deepstack; @brown2017safe]. However, CFR is computationally expensive, since it relies on complete traversal of the game tree, and is infeasible for games with large state space such as Dou Dizhu [@deltadou].
Recent studies show that reinforcement learning strategies can perform well in betting games [@heinrich2015fictitious; @heinrich2016deep; @lanctot2017unified], and achieve satisfactory performance in Dou Dizhu [@deltadou]. The inspiring results and the flexibility of RL offer the opportunity to explore deep reinforcement learning in more difficult card games with large state and action space.
Conclusions and Future Directions
=================================
In this paper, we introduce RLCard, an open-source toolkit for reinforcement learning research in card games. RLCard supports multiple challenging card environments wrapped with common and easy-to-use interfaces. In the future, we plan to enhance the toolkit in several aspects. First, in order to benchmark the evaluation, we would like to design rule-based agents and provide more pre-trained models for evaluation. Second, we plan to develop visualization and analysis tools for the environments. Third, we will include more interesting games and more algorithms to enrich the toolkit.
Acknowledgements
================
We would like to thank JJ World Network Technology Co., LTD for the generous support.
[^1]: Sheng Ji is under development.
|
---
abstract: 'We consider photon emission from a supercritical accretion disk in which photons in the inner regions are trapped and advected in towards the center. Such disks are believed to be present in failed supernovae and gamma-ray bursts, and possibly also in ultra-luminous X-ray sources. We show that the luminosity from a supercritical accretion disk is proportional to the logarithm of the mass accretion rate when the vertical profile of the matter density is exponential. We obtain analytical results also for other density profiles, but the dependence of the luminosity on the accretion rate is no longer logarithmic.'
author:
- |
Kazunori Kohri,$^{1,2}$[^1] Ken Ohsuga$^{3}$ and Ramesh Narayan,$^{1}$\
$^{1}$Harvard-Smithsonian Center for Astrophysics, MS-51, 60 Garden Street, Cambridge, MA 02138, USA\
$^{2}$Physics Department, Lancaster University, Lancaster,LA1 4YB,UK,\
$^{3}$Department of Physics, Rikkyo University, Toshimaku, Tokyo 171-8501, Japan\
date: ' Accepted YYYY MMM DD. Received YYYY MMM DD; in original form 2006 Nov. 29 '
title: 'Multi-dimensional Treatment of Photon Emission from Accretion Disks around Black Holes'
---
\[firstpage\]
accretion, accretion disks — black hole physics — hydrodynamics — radiative transfer
Introduction
============
There is wide consensus that active galactic nuclei (AGNs), gamma-ray bursts (GRBs), X-ray binaries, etc., are powered by accretion flows on to compact relativistic objects, most often a black hole (BH). A well-known model of such accretion flows is the standard-disk model proposed by Shakura & Sunyaev (1973), which has been used to explain a variety of observations of AGNs and X-ray binaries.
Recently, it has been recognized that observations of many objects cannot be explained by the standard-disk model. For instance, this model cannot produce the very high temperature ($T > 10^{10}$ K) and broad-band spectrum (extending from $10^9$ to $>10^{18}$ Hz) observed in the Galactic Center source Sgr A$^*$ and in X-ray binaries in the hard state. This has led to the idea of an advection-dominated accretion flow (ADAF, Narayan & Yi 1995, Abramowicz et al. 1995 [^2]), which is also called a radiatively inefficient accretion flow (RIAF). The RIAF model explains the low luminosity, high temperature and optically-thin emission of these systems. [^3]
The standard disk cannot also be applied to very high-luminosity accretion disks in which the mass-accretion rate $\dot{M}$ exceeds the critical mass-accretion rate $\dot{M}_{\rm crit}$ $(\equiv L_{\rm
E}/c^{2}$, where $L_{\rm E}$ is the Eddington luminosity). In this regime, the disk becomes optically very thick and as a result photons are partially trapped inside the accreting gas. This is the regime of interest to us.
Photon-trapping is important at any radius where the accretion time scale is longer than the photon diffusion time scale from the disk midplane to the surface (Begelman 1978). The photons produced in this region of the disk are advected into the central BH and are unable to escape from the flow. Begelman considered only spherically symmetric accretion, but it is recognized that photon-trapping is important even for systems with a disk geometry.
Photon-trapping has been included approximately in the so-called “slim disk model” proposed by Abramowicz, Igumenshchev & Lasota (1998), and in recent calculations by Szuszkiewicz et al. (1996) and Watarai et al. (2000), as well as in numerical simulations by Honma et al. (1991), Szuszkiewicz & Miller (1997), and Watarai & Mineshige (2003). However, these studies do not model the effect fully since they consider vertically integrated quantities in the disk and reduce the problem to a one-dimensional model. Full two-dimensional radiation-hydrodynamical simulations (2D-RHD) were done for the first time by one of the current authors in Ohsuga et al. (2005) [^4].
A realistic model of supercritical accretion must include the following two effects: i) Even at disk radii inside the trapping radius, photons that are emitted near the surface of the disk can escape, though most of the photons emitted deeper inside are advected into the BH. ii) The vertical profiles of various quantities in the disk such as density, optical depth, etc., play a critical role in determining what fraction of the photons escape, and thereby influence the total luminosity of the disk.
The first effect was pointed out and studied by Ohsuga et al. (2002) using simple analytical models and numerical simulations. In this paper, we include the second effect and study the combined influence of both effects for a broad class of accretion-disk models. We develop an analytical model and show that it agrees well with the numerical results of Ohsuga et al. (2005).
The results obtained here may change the current understanding of models such as the slim-disk and convection-dominated accretion flow model (CDAF, Stone, Pringle & Begelman 1999; Narayan, Igumenshchev & Abramowicz 2000; Quataert & Gruzinov 2000; Igumenshchev, Abramowicz & Narayan 2000), as well as the neutrino-dominated accretion flow (NDAF) model (Popham, Woosley & Fryer. 1999; Narayan, Piran & Kumar 2001; Kohri & Mineshige 2002; Di Matteo, Perna & Narayan 2002; Kohri, Narayan & Piran 2005 Gu, Liu & Lu, 2006) which is believed to play an important role in the central engine of GRBs (Narayan, Paczynski & Piran 1992). [^5] In the following we use $R$ to represent the cylindrical radius of a point with respect to the central mass and $z$ to represent its vertical distance from the equatorial plane of the disk. We also assume that the temperature $T$ in the disk is lower than the electron mass ($kT < m_{e}c^2$), so that the electrons are nonrelativistic.
Standard one-dimensional approximations
=======================================
In this section we introduce the standard one-dimensional (1D) treatment of accretion disks, following the approach described in Ohsuga et al (2002). In this approach, we integrate along the $z$-axis and omit the $z$-dependences of various astrophysical quantities. For instance, we write the optical depth from the mid-plane of the disk ($z=0$) to the surface ($z=z_{\rm max}$) as $$\begin{aligned}
\label{eq:tau_1_1}
\tau &=& \int_{0}^{z_{\rm max}} dz~ \sigma_{\gamma e} n_{e} \\
\label{eq:tau_1_2}
&\simeq& \sigma_{\rm T} n_{e} H,\end{aligned}$$ where $\sigma_{\gamma e}$ is the scattering cross section of a photon off an electron. Although $\sigma_{\gamma e}$ depends in general on the photon energy $E_{\gamma}$, we limit ourselves to Thomson scattering, for which the cross-section is independent of energy. This is reasonable for the low energy photons $E_{\gamma} \ll
{{\cal O}}({\rm MeV})$ of interest to us. $H$ is the disk half-thickness and $n_{e}$ is the electron number density in the disk mid-plane. The diffusion time scale for a photon to escape from the disk mid-plane is given by $$\begin{aligned}
\label{eq:tdiff_1}
t_{\rm diff} \approx N_{\rm scatt} \ \lambda /c,\end{aligned}$$ where $N_{\rm scatt}$ is the number of scatterings, $\lambda =
1/(\sigma_{\rm T} n_{e})$ is the mean free path, and $c$ is the speed of light. Since $N_{\rm scatt}$ is given by $$\begin{aligned}
\label{eq:Nscatt_1}
N_{\rm scatt} \approx 3~\tau^{2},\end{aligned}$$ (assuming a random walk in three-dimensions, see Appendix \[sec:deriv\_tdiff\]), we write $$\begin{aligned}
\label{eq:t_diff_1}
t_{\rm diff} \approx 3 \tau H/ c.\end{aligned}$$ The accretion time scale is $$\begin{aligned}
\label{eq:tacc}
t_{\rm acc} = - \frac{R}{ v_{r}},\end{aligned}$$ where the radial velocity $v_{r}$ is related to the mass accretion rate $\dot{M}$ and the density $\rho$ by $$\begin{aligned}
\label{eq:vr}
v_{r} &=& - \displaystyle{\frac{\dot{M}}
{4 \pi R \int_{0}^{H}\rho dz }} \\
&=& - \displaystyle{\frac{\dot{M}}
{4 \pi R \rho H}}.\end{aligned}$$ Thus $$\begin{aligned}
\label{eq:tacc1}
t_{\rm acc} \approx \frac{2\tau}{\dot{m}}\frac{R^{2}}{c \ R_{g} },\end{aligned}$$ where $R_{g} = 2 G M/c^{2}$ is the Schwarzschild radius, and we have set $\rho \approx m_{p} n_{e}$ and introduced a dimensionless mass accretion rate $\dot{m}$, $$\begin{aligned}
\label{eq:smlmdot}
\dot{m} = \dot{M} / \dot{M}_{\rm crit}.\end{aligned}$$ The critical mass accretion rate $\dot{M}_{\rm crit}$ is defined in terms of the Eddington luminosity $L_{\rm E}$ as follows, $\dot{M}_{\rm crit} = L_{\rm E}/c^{2}$, where $L_{\rm E} = {4 \pi c G
m_{p}}M /{\sigma_{\rm T}} \simeq 1.3 \times 10^{38} {\rm erg}~{\rm
sec}^{-1} (M/M_{\odot})$ for a black hole of mass $M$.
Photons can escape from the disk only if the condition $t_{\rm diff}
< t_{\rm acc}$ is satisfied. Therefore, the disk radiates freely only from radii larger than a certain limit, $$\begin{aligned}
\label{eq:largeR}
R > R_{\rm trap},\end{aligned}$$ where the trapping radius $R_{\rm trap}$ is given by $$\begin{aligned}
\label{eq:Rtrap}
R_{\rm trap} = \frac32 h_{\rm in} \dot{m} R_{g}.\end{aligned}$$ and $$\begin{aligned}
\label{eq:small_h}
h_{\rm in} \left. \equiv \frac{H}{R} \right|_{R \le R_{\rm trap}},\end{aligned}$$ assuming that, when $R<R_{\rm trap}$ and radiation is trapped, the disk is geometrically thick with $h_{\rm in}\sim1$ and that $h$ here is independent of $R$. As we show in Appendix \[sec:disk-half-thickness\], $h$ is indeed constant ($\approx$ 0.5) for $R \le R_{\rm trap}$ and decreases $\propto 1/R$ for $R_{\rm trap} < R$. This validates Eq. (\[eq:Rtrap\]). Note that $R_{\rm trap}$ is proportional to $\dot{m}$.
The energy released in the disk per unit time by viscous dissipation is given by $$\begin{aligned}
\label{eq:lum_1}
L = 2 \int_{R_{\rm in}}^{R_{\rm out}} 2 \pi R Q_{\rm vis} (R) dR,\end{aligned}$$ where the viscous heating rate is $$\begin{aligned}
\label{eq:Qvis}
Q_{\rm vis } (R) = \frac{3}{8\pi} \frac{GM}{R^{3}} \dot{M} \left[1
- \left( \frac{R_{\rm in}}{R} \right)^{1/2} \right].\end{aligned}$$ Here $R_{\rm in}$ and $R_{\rm out}$ are the radii of the inner and outer edge of the disk. However, not all the released energy is radiated because of photon-trapping. We discuss two cases below, depending on whether $R_{\rm trap}$ is larger or smaller than $R_{\rm
in}$. For readers’ convenience we introduce a dimensionless inner radius normalized by the Schwarzschild radius. $$\begin{aligned}
\label{eq:smallrin}
r_{\rm in} \equiv R_{\rm in} /R_{\rm g}.\end{aligned}$$
$R_{\rm trap} < R_{\rm in}$ {#subsec:notrap}
---------------------------
When $R_{\rm trap} < R_{\rm in}$, there is no trapped region and we estimate the total luminosity to be $$\begin{aligned}
\label{eq:lum_2_1}
L_{{\rm 1D, tot},R_{\rm trap} < R_{\rm in}}
&=&
2 \int_{R_{\rm in}}^{\infty} 2 \pi R Q_{\rm vis} (R)
dR \nonumber \\
&=& \frac1{6 h_{\rm in}} L_{\rm E} \frac{R_{\rm trap}}{R_{\rm in}}
\nonumber \\
&=& \frac{L_{\rm E}}{4 r_{\rm in}} \dot{m},\end{aligned}$$ which is linearly proportional to $\dot{m}$. This is a well known result for the standard thin accretion disk model in Newtonian gravity.
$R_{\rm in} < R_{\rm trap}$ {#subsec:sometrap}
---------------------------
When $R_{\rm in} < R_{\rm trap}$, only the region of the disk outside $R_{\rm trap}$ radiates freely and the luminosity from this region of the disk is $$\begin{aligned}
\label{eq:lum_2_2_1}
L_{{\rm 1D}, R_{\rm trap}<R}
&=& 2 \int_{R_{\rm trap}}^{\infty}
2 \pi R Q_{\rm vis} (R) dR
\nonumber \\
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[1 - \frac23
\left(\frac{R_{\rm trap}}{R_{\rm in}}
\right)^{-1/2}\right]
\nonumber \\
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[1 -
\sqrt{\frac{8r_{\rm in}}{27h_{\rm in}}}~\dot{m}^{-1/2}\right].\end{aligned}$$ Within the one-dimensional approximation being discussed in this section, it is unclear how much luminosity is emitted from the region of the disk $R_{\rm in} < R < R_{\rm trap}$. One extreme assumption is that there is no luminosity at all from this region, as in Ohsuga et al. 2002. Alternatively, one might assume that the disk flux here is limited to the local Eddington flux $F \approx F_{\rm E}(R) =
L_{\rm E}/(4\pi R^{2}) = G M m_{p} c / (R^{2}\sigma_{\rm T} )$. The luminosity from $R_{\rm in} < R < R_{\rm trap}$ of the disk is then given by $$\begin{aligned}
\label{eq:lum_2_2_2}
L_{{\rm 1D}, R<R_{\rm trap}}
&=& 2 \int_{R_{\rm in}}^{R_{\rm trap}}
2 \pi R F_{\rm Edd}(R) dR
\nonumber\\
&=& L_{\rm E} \ln\left(\frac{R_{\rm trap}}{R_{\rm in}}
\right)
\nonumber\\
&=& L_{\rm E} \ln\left(\frac{3 h_{\rm in}}{2 r_{\rm in}}\dot{m}
\right).\end{aligned}$$ If we include the term given in Eq. (\[eq:lum\_2\_2\_2\]), keeping in mind that it is uncertain, then the “total luminosity” becomes $$\begin{aligned}
\label{eq:lum_2_2}
\lefteqn{L_{\rm 1D,tot,R_{\rm in} < R_{\rm trap}}\nonumber } \\
&=& L_{{\rm 1D}, R_{\rm trap}<R}
+ L_{{\rm 1D}, R<R_{\rm trap}}
\nonumber \\
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[1 + 2 h_{\rm in}\ln\left(\frac{R_{\rm trap}}{R_{\rm in}}
\right)
- \frac23
\left(\frac{R_{\rm trap} }{R_{\rm in}}
\right)^{-1/2}\right]
\\ \nonumber
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[1 + 2 h_{\rm in}\ln\left(\frac{3 h_{\rm in}}{2 r_{\rm in}}\dot{m}
\right)
- \sqrt{\frac{8r_{\rm in}}{27h_{\rm in}}}~\dot{m}^{-1/2}\right].\end{aligned}$$ Note that the luminosity is reduced relative to the result in Eq. (\[eq:lum\_2\_1\]) for the untrapped case, and increases only logarithmically with $\dot{m}$. This explains why disks with supercritical accretion ($\dot{m}\gg1$, “slim” disks) are radiatively inefficient.
Two-dimensional Treatment {#sec:2dim}
=========================
So far we have discussed a 1D approximation and assumed somewhat arbitrarily that the region $R < R_{\rm trap}$ either does not radiate at all or radiates at the local Eddington rate. In this section we carry out a more careful two-dimensional analysis. We focus on the region $R_{\rm in} \le R < R_{\rm trap}$ discussed in Subsection \[subsec:sometrap\], but now we carefully consider the two-dimensional distribution of quantities.
In the following, we consider most of the quantities in the disk to depend on both $R$ and $z$ (the vertical height). However, we assume that the thin-disk approximation is approximately valid in the sense that we omit the $z$-dependence when considering hydrostatic balance in the $z$-direction ($c_{S} = \Omega_{K} H $, with the sound speed $c_{S}$ and the Keplerian angular velocity $\Omega_{K} =
\sqrt{GM/R^{3}}$) and angular momentum conservation ($\nu \int \rho dz
= \dot{M}(1-\sqrt{R_{\rm in}/R})/(3\pi)$, with the kinetic viscosity $\nu$). Then the $z$-dependent optical depth at a given $R$ is given by $$\begin{aligned}
\label{eq:tauz}
\tau(z) &=& \int_{z}^{z_{\rm max}} dz~ \sigma_{\gamma e} n_{e} (z) \\
\label{eq:tauz_2}
&\simeq& \sigma_{\rm T} \int_{z}^{z_{\rm max}} dz~ n_{e} (z),\end{aligned}$$ where $n_{e}(z)$ is the electron number density at height $z$, and $z_{\rm max}$ corresponds to the upper surface of the disk. We see that the expression for $\tau$ given in Eq. (\[eq:tau\_1\_1\]) in the 1D approximation corresponds to $\tau(0)$ in Eq. (\[eq:tauz\]).
We now introduce a detailed model for the vertical electron density profile. As seen in Fig. \[fig:rho\_x\], we can fit the the density profile found in the numerical simulation in Ohsuga et al. (2005) quite well with the following power-law form, $$\begin{aligned}
\label{eq:power-law}
n_{e}(z) = n_{e}(0) \left( 1- \frac{z}{q H}
\right)^{q-1}
(q \ge 1,~{\rm and}~0 \le z \le q H),\end{aligned}$$ where $n_{e}(0)$ and the power-law index $q$ may depend on the radius $R$. In the limit as $q\to\infty$, this model takes the form of an exponential [^6] $$\begin{aligned}
\label{eq:exponential} n_{e}(z) = n_{e}(0) \exp\left(- \frac{z}{H}
\right) \qquad (0 \le z \le \infty).\end{aligned}$$ The optical depth from $z$ to the surface is $$\begin{aligned}
\label{eq:tauz_pow}
\tau(z) = \tau(0) \left(1 - \frac{z}{q H} \right)^{q},\end{aligned}$$ except for $q\to\infty$ when it takes an exponential form.
Because we adopt the numerical results corresponding to $\dot{m}
\gtrsim 100$ in Ohsuga et al. (2005), the radii $R$ of order tens of $
R_{g}$ that we consider are smaller than $R_{\rm trap} \sim 10^{2}
R_{g}$ and so we are in the radiation-trapped regime. Note that Fig. \[fig:rho\_x\] gives $H/R \approx 0.4$ for the numerical simulation of Ohsuga et al. (2005), which is approximately consistent with the approximation $h_{\rm in} = 0.5$ that we made in the previous section and in Appendix \[sec:disk-half-thickness\].
![ Variation of the matter density $\rho$ with height $z$ at a radius $R = 20 R_{\rm g}$. The solid line represents the result from the numerical simulation of Ohsuga et al. (2005). The data are fit well with either an exponential (dotted line) or a power-law (dashed) profile, with $H/R$ = 0.4 and (for the power-law profile) $q = 10$. The numerical results in Ohsuga et al. (2005) can be fit with the same value of $H/R$ = 0.4 for all radii up to $\sim 50 R_{\rm g}$.[]{data-label="fig:rho_x"}](x20_log.ps){width="90mm"}
The photon diffusion time scale from height $z$ to the surface of the disk at $z_{\rm max}$ is given by $$\begin{aligned}
\label{eq:tdiffz}
t_{\rm diff}(z) \approx N_{\rm scatt}(z,z_{\rm max})~\lambda(z)/c,\end{aligned}$$ where $N_{\rm scatt}(z,z_{\rm max})$ is the number of scatterings in the 3D random walk, $$\begin{aligned}
\label{eq:nscatt_2}
N_{\rm scatt}(z,z_{\rm max}) = 3~\left[\tau(z)\right]^{2},\end{aligned}$$ and $\lambda(z)$ is the mean free path which is estimated using the properties of the disk at the starting position $z$ of the photon $$\begin{aligned}
\label{eq:lambda_z}
\lambda(z) = \frac1{\sigma_{\rm T}~n_{e}(z)}.\end{aligned}$$ The detailed derivation of the above relation is given in Appendix \[sec:deriv\_tdiff\]. By using Eqs. (\[eq:nscatt\_2\]) and (\[eq:lambda\_z\]), we obtain an expression for the $z$-dependent diffusion time scale, $$\begin{aligned}
\label{eq:tdiffz2}
t_{\rm diff}(z) &=& 3 \tau(z)^{2} \lambda(z) /c \nonumber \\
&=& 3 \tau(0) \frac{H}{c}
\left[ \frac{\tau(z)}{\tau(0)} \right]^{(q+1)/q} \,.\end{aligned}$$ By requiring the diffusion time scale $t_{\rm diff}(z)$ to be shorter than the accretion time scale $t_{\rm acc}$ given in Eq. (\[eq:tacc1\]), we find that photons can escape only from the region where the following condition is satisfied, $$\begin{aligned}
\label{eq:cond_diff_0}
\frac{\tau(z)}{\tau(0)} <
\left(\frac{R}{R_{\rm trap}} \right)^{q/(q+1)},\end{aligned}$$ The photon-trapping radius $R_{\rm trap}$ is determined by Eq. (\[eq:Rtrap\]) and is proportional $\dot{m}$. A schematic picture indicating the region of the disk from which radiation can escape is shown in Fig. \[fig:diffusion\]. We see that trapping is not determined by simply a critical radius $R_{\rm trap}$, but is described in terms of a two-dimensional surface $z_{\rm trap}(R)$, where $$\begin{aligned}
\label{eq:z_trap}
\frac{\tau(z_{\rm trap})}{\tau(0)} \equiv \left(\frac{R}{R_{\rm
trap}}\right)^{q/(q +1)}.\end{aligned}$$ Note that $z_{\rm trap}$ is a function of $R$ in this case given by $$\begin{aligned}
\label{eq:ztrap_R}
z_{\rm trap} = qH \left[ 1
- \left(\frac{R}{R_{\rm trap}} \right)^{{1/(q+1)}}
\right],\end{aligned}$$ and its second-order derivative by $R$ is always negative $d^2z_{\rm
trap}/dR^{2} <0$ for $R<R_{\rm trap}$ because $$\begin{aligned}
\label{eq:dztrapdR}
\frac{dz_{\rm trap}}{dR} =
q h_{\rm in} \left[1 - \frac{q+2}{q+1}
\left(\frac{R}{R_{\rm trap}} \right)^{1/(q+1)} \right],\end{aligned}$$ and $$\begin{aligned}
\label{eq:d2ztrapdR2}
\frac{d^{2}z_{\rm trap}}{dR^{2}} =
- \frac{q(q+2)}{q+1} h_{\rm in} \frac1{R_{\rm trap}}
\left(\frac{R}{R_{\rm trap}} \right)^{-q/(q+1)},\end{aligned}$$ for $R<R_{\rm trap}$. The luminosity from the region $R_{\rm in} \le R \le R_{\rm trap}$ is then given by $$\begin{aligned}
\label{eq:L_add_pow}
L_{{\rm 2D}, R<R_{\rm trap}}
= 2 \int_{R_{\rm in}}^{R_{\rm trap}} 2 \pi R Q_{\rm
vis } (R) \frac{\tau(z_{\rm trap})}{\tau(0)}dR.\end{aligned}$$ Here we assume that all the energy released at $z \geq z_{\rm trap}$ is radiated, while the energy released at $z < z_{\rm trap}$ is completely trapped. Also, because $Q_{\rm vis}$ and $\tau(z)$ are both proportional to $\int \rho dz$, we express the vertical distribution of viscous dissipation directly by means of the factor $\tau(z_{\rm trap})/\tau(0)$.
![Schematic representation of a supercritical accretion disk around a BH. The shaded region shows the photon-trapping region from which no photons can escape. We see that, even for $R < R_{\rm trap}$, there exist regions near the surface of the disk from which photons can diffuse out (white region). []{data-label="fig:diffusion"}](diffusion_4.eps){width="85mm"}
Evaluating the above integral, we obtain the luminosity of the disk from the region $R < R_{\rm trap}$: $$\begin{aligned}
\label{eq:L_add_pow2_1_1}
\lefteqn{ L_{{\rm 2D},R<R_{\rm trap}} } \nonumber \\
\lefteqn{= 2\int_{R_{\rm in}}^{R_{\rm trap}} 2 \pi R Q_{\rm
vis } (R) \left(\frac{R}{R_{\rm trap}}
\right)^{q/(q+1)} dR } \nonumber \\
\label{eq:L_add_pow2_2}
\lefteqn{ = \frac1{2h_{\rm in}} L_{\rm E}
\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{1/(q+1)} }
\nonumber \\
\lefteqn{ \hspace{1cm} \times \int_{1}^{R_{\rm trap}/R_{\rm in}} dx
\frac{1}{x^{(q+2)/(q+1)}} \left(1 - \frac{1}{x^{1/2}}
\right) } \nonumber \\
\label{eq:L_add_pow2_3}
\lefteqn{=\frac1{2h_{\rm in}} L_{\rm E} \times} \nonumber \\
\lefteqn{ \left[
\frac{2(q+1)}{q+3}
\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{-1/2}
+ \frac{(q+1)^{2}}{q+3}
\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{1/(q+1)}
- (q + 1)
\right]} \nonumber
\\
\label{eq:L_add_pow2_3_2}
\lefteqn{=\frac1{2h_{\rm in}} L_{\rm E} \times} \\
\lefteqn{ \left[
\frac{2(q+1)}{q+3}
\left(\frac{3 h_{\rm in} \dot{m}}{2 r_{\rm in}} \right)^{-1/2}
+ \frac{(q+1)^{2}}{q+3}
\left(\frac{3 h_{\rm in} \dot{m}}{2 r_{\rm in}} \right)^{1/(q+1)}
- (q + 1)
\right].} \nonumber\end{aligned}$$ The dominant contribution comes from the second term, which is proportional to $\dot{m}^{1/(q+1)}$ for $\dot{m} \gg 1$. The luminosity from the region $R_{\rm trap} < R$ is, of course, the same as in the 1D case (Eq. \[eq:lum\_2\_2\_1\]), i.e., $L_{{\rm 2D},
R_{\rm trap}<R} = L_{{\rm 1D}, R_{\rm trap}<R}$, since there is no photon trapping. Therefore, we have the following expression for the total radiated luminosity from the entire disk, $$\begin{aligned}
\label{eq:L_add_pow2_1}
\lefteqn{L_{\rm 2D,tot}(q) } \nonumber \\
\lefteqn{= L_{{\rm 2D},R<R_{\rm trap}} + L_{{\rm 2D},R_{\rm
trap}<R} }
\nonumber\\
\lefteqn{= \frac1{2h_{\rm in}} L_{\rm E}\times} \nonumber \\
\lefteqn{ \left[
\frac{ 4 q}{3(q+3)}
\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{-1/2}
+ \frac{(q+1)^{2}}{q+3}
\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{1/(q+1)}
- q
\right] } \nonumber \\
\nonumber \\
\lefteqn{= \frac1{2h_{\rm in}} L_{\rm E}\times} \\ \nonumber
\lefteqn{ \left[
\frac{ 4 q}{3(q+3)}
\left(\frac{3 h_{\rm in} \dot{m}}{2 r_{\rm in}} \right)^{-1/2}
+ \frac{(q+1)^{2}}{q+3}
\left(\frac{3 h_{\rm in} \dot{m}}{2 r_{\rm in}}\right)^{1/(q+1)}
- q
\right]. } \nonumber\end{aligned}$$ In a similar fashion, we obtain the following result for the case of an exponential density profile ($q\to\infty$), $$\begin{aligned}
\label{eq:L_add_exp}
\lefteqn{ L_{\rm 2D,tot}(q=\infty)}
\nonumber \\
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[
\frac43 \left(\frac{R_{\rm trap}}{R_{\rm in}} \right)^{-1/2}
+ \ln\left(\frac{R_{\rm trap}}{R_{\rm in}} \right)
-1
\right]
\\ \nonumber
&=& \frac1{2h_{\rm in}} L_{\rm E}
\left[
\sqrt{\frac{32r_{\rm in}}{27h_{\rm in}}}~\dot{m}^{-1/2}
+ \ln\left(\frac{3 h_{\rm in}}{2 r_{\rm in}}\dot{m}
\right)
-1
\right].\end{aligned}$$ To verify that this result is consistent with the limit $q\to\infty$ of Eq. (\[eq:L\_add\_pow2\_3\_2\]), we make use of the following useful relation, $$\begin{aligned}
\label{eq:limit_eps0}
\lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \frac1{2\epsilon +
1} x^{\epsilon}-1 \right] = \ln x -2.\end{aligned}$$ This shows that the dominant terms in Eq. (\[eq:L\_add\_pow2\_3\_2\]) do not diverge as $q\to\infty$ but tend to a finite value that is proportional to $\ln(\dot{m})$ for $\dot{m} \gg 1$.
Comparison with numerical simulation
====================================
Here we compare the analytical formulae derived above for the luminosity $L$ of the disk with the results of multi-dimensional numerical simulations reported by Ohsuga et al. (2005).
In Fig. \[fig:lm\] we plot the disk luminosity in Eddington units as a function of the dimensionless mass-accretion rate $\dot{m}$. Analytical formulae are denoted by solid lines and correspond, in descending order, to the following cases: no photon-trapping (Eq. \[eq:lum\_2\_1\]), power-law density profile with $q$ = 1 and $q$ = 10 (Eq. \[eq:L\_add\_pow2\_1\]), exponential density profile (Eq. \[eq:L\_add\_exp\], i.e., $q\to\infty$), and the standard 1D model which ignores photons emitted from $R_{\rm in} < R < R_{\rm
trap}$ (Eq. \[eq:lum\_2\_2\_2\]). Triangles, circles, and squares connected by long-dashed lines indicate the numerical results from Fig. 7 of Ohsuga et al. (2005) corresponding to metallicity $Z$ = 0, $1 Z_{\odot}$, and 10 $Z_{\odot}$, respectively. [^7] The dotted line represents the results of the hydrodynamical simulations of the slim-disk model carried out by Watarai et al. (2000), which did not explicitly include the effects discussed here. In these calculations, we adopted $R_{\rm in}=3
R_{\rm g}$ and $h$ = 0.4.
![Variation of the disk luminosity as a function of the mass-accretion rate. Analytical formulae are denoted by solid lines. From above, the lines correspond to the case with photon-trapping ignored, power-law density profiles with $q$ = 1 (constant density), $q$ = 10, and $q\to \infty$ (exponential profile), and a standard 1D model in which photons are assumed to be fully trapped for $R_{\rm in} < R < R_{\rm trap}$. The triangles, circles, and squares connected by long-dashed lines indicate the numerical results shown in Fig. 7 in Ohsuga et al. (2005), for metallicity $Z$ = 0, $Z_{\odot}$, and 10 $Z_{\odot}$, respectively. The dotted line represents the case of the hydrodynamical slim-disk simulations of Watarai et al. (2001), without explicitly including 2D effects. The short-dashed line shows the profile of luminosity in an approximate analytical 1D model in which we assume that the disk emits at the local Eddington rate at radii $R_{\rm in} < R < R_{\rm trap}$. All results correspond to $R_{\rm in}=3 R_{\rm g}$ and $h$ = 0.4. []{data-label="fig:lm"}](lm_dec18.ps){width="95mm"}
From Fig. \[fig:lm\] we see that the luminosity is larger than with the standard analytical 1D model if we ignore radiation from the photon-trapped region. The increase is the result of the multi-dimensional nature of photon diffusion, which allows radiation to partially escape even from radii $R < R_{\rm trap}$.
For reference, we have also plotted by a short-dashed line the approximate formula Eq. (\[eq:lum\_2\_2\]) for the disk luminosity in the 1D model, which includes the photon luminosity emitted from $R_{\rm in} < R < R_{\rm trap}$ by assuming a simple local Eddington flux. Because of the logarithmic functional form of this approximation, this model resembles the exponential density model in the 2D treatment. Correspondingly, we conclude that this model provides a reasonable 1D approximation.
Ohsuga et al. (2005) pointed out that photons tend to have momenta pointed inwards because of advection by the accretion flow, which causes the radiation to spend a longer time in the disk before escaping relative to the simple diffusion estimate we have used in our model. However, we believe this introduces only a minor correction to our results because we are focusing mainly on photons for which $t_{\rm diff} < t_{\rm acc}$.
In addition, Ohsuga et al. (2002) obtained outflows driven by the super Eddington luminosity of their models. Although our analytical model does not include outflows, we believe our discussion is still applicable in a time-averaged mean sense.
Discussion and Conclusion
=========================
In this paper, we have studied photon emission from a supercritical accretion disk. In the usual one-dimensional treatment of disks, one integrates all quantities along the vertical ($z$) direction. This is reasonable for a thin accretion disk when the accretion rate is well below the Eddington rate. However, we show that, when the accretion rate is large, it is quite important to consider the $z$-dependence of fluid quantities and to detreat the vertical diffusion of photons carefully. If we omit the $z$-dependence of the diffusion process in the simplest version of the one-dimensional model, we tend to significantly underestimate the luminosity. On the other hand, if we omit the $z$-dependences of the matter density and assume uniform density, we overestimate the luminosity. Only when we consider the $z$-dependence of the matter density and adopt the appropriate density profile such as a power-law or exponential form as a function of $z$, as seen in 2D radiation-hydrodynamical simulations, are we able to reproduce the luminosity found in the numerical simulations.
Using the analytical approach developed here, we should be able to model how the peak energy of the photon spectrum decreases when the photons produced near the BH are trapped, as pointed out by Ohsuga et al. (2003). This will be discussed in a forthcoming paper (Kohri & Ohsuga, 2007).
When we simultaneously include radiative, advective, convective, and neutrino cooling processes in an analytical model of the accretion disk, the effects discussed here will play a crucial role. In neutrino dominated accretion flows (NDAFs), for instance, the neutrino luminosity and annihilation rate could be modified substantially relative to the predictions of one-dimensional models. The dominant cooling process could be changed, and the entire picture of the accretion might be dramatically modified, e.g., for the produced energy through the $\nu$ $\bar{\nu}$ annihilation (Di Matteo, Perna & Narayan 2002, Gu, Liu & Lu 2006, Chen & Beloborodov), $r$-process nucleosynthesis (Surman, McLaughlin, Hix 2006), and so on. We plan to discuss these effects in forthcoming papers.
Multi-dimensional hydrodynamic simulations of accretion disks including all of these cooling processes have not been done so far (except for the 2D numerical simulations of neutrino cooled disks by Lee & Ramirez-Ruiz, 2006). To clarify the role of multi-dimensionality on CDAFs, NDAFs, etc., it would be useful to compare the analytical estimates obtained in this paper with full multi-dimensional numerical simulations. It is hoped that these simulations will be done in the near future.
Acknowledgements {#acknowledgements .unnumbered}
================
K.K. would like to thank A. E. Broderick, J. C. McKinney and S. Mineshige for valuable discussions at an early stage of this work. This work was supported in part by NASA grant NNG04GL38G, PPARC grant, PP/D000394/1, EU grant MRTN-CT-2006-035863, the European Union through the Marie Curie Research and Training Network “UniverseNet” (MRTN-CT-2006-035863), and Research Grant from Japan Society for the Promotion of Science (17740111).
Analytical estimates of disk-half thickness $H$ {#sec:disk-half-thickness}
===============================================
In this section, we simply try to analytically estimate the $R$-dependence on disk-half thickness $H$. The force balance along $z$-axis is generally expressed by $$\begin{aligned}
\label{eq:z_force}
\frac{\sigma_{T}}{m_{p} c} F = \frac{G M}{R^{2}} \frac{H}{R},\end{aligned}$$ with a photon flux of $F$.
If $R_{\rm trap} < R$, then it would be reasonable to assume that the flux is equal to the viscous heating rate, $F \approx Q_{\rm vis}$. Then, we obtain $$\begin{aligned}
\label{eq:h_large}
\left. \frac{H}{R} \right|_{R_{\rm trap} < R}
&\approx&
\frac34 \frac{R_{\rm g}}{ R} \dot{m}.\end{aligned}$$ Therefore $H/R$ is approximately proportional to $\propto 1/R$ for $R_{\rm trap} < R$.
On the other hand, if $R < R_{\rm trap}$, we may assume that the flux would be approximately the order of the Eddington flux $F \approx
\frac12 F_{\rm E} = \frac12 L_{\rm E}/(4\pi R^{2})$. The factor 1/2 in front of $F_{\rm E}$ is attached as a matter of convenience for the consistency. That comes from the viewpoint of continuities of astrophysical quantities although physics does not change at all by this artificial factor. From Eq. (\[eq:z\_force\]) we find that $H/R$ is constant for $R < R_{\rm trap}$, $$\begin{aligned}
\label{eq:h_small}
\left. \frac{H}{R} \right|_{R<R_{\rm trap}} \approx \frac12.\end{aligned}$$ Then we see that $h = H/R$ is a smooth function of $R$, $$\begin{aligned}
\label{eq:h_func}
h =
\left\{
\begin{array}{cc}
1/2 & (R \le R_{\rm trap}),\\
\\
\frac12 R_{\rm trap}/R & (R_{\rm trap} < R ),
\end{array}
\right.\end{aligned}$$ if we take $R_{\rm trap} = 3/2 h_{\rm in} \dot{m}
R_{g}$ with $h_{\rm in} = 1/2$.
Derivation of the $z$-dependent/independent diffusion time scale {#sec:deriv_tdiff}
================================================================
Here we discuss the diffusion time scale along with $z$-axis in 3D random walk processes when the scattering length depends on the position, i.e., in the case that we consider the $z$-dependent number density of electron, the cross section, and so on in the electron-photon scattering processes.
In general, we can write the summed distance measured along the 3D random walk process from the stating point $z=z_{0}$ as $$\begin{aligned}
\label{eq:dz}
d(z_{0},z_{\rm max}) =
\sum_{i_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})~}
\int_{z_{i_z-1}}^{z_{i_z}} dz,\end{aligned}$$ where $N_{\rm scatt}(z_{0},z_{\rm max})$ is the number of the scattering from $z=z_{0}$ to a maximum of $z$ ($\equiv z_{\rm
max}$). The integral interval $[z_{i_z-1}, \ z_{i_z}]$ is determined by solving the following differential equation for the number of the scattering, $$\begin{aligned}
\label{eq:dNdz}
\frac{dN}{dz} = \sigma_{\gamma e} n_{e} (z).\end{aligned}$$ When we solve it, we may assume that the interval between the scatterings is determined by the approximate relation, $$\begin{aligned}
\label{eq:intdN}
\left| \int_{z_{i_z-1}}^{z_{i_z}} \sigma_{\gamma e} n_{e} (z) dz
\right| = \frac13,\end{aligned}$$ where the meaning of dividing by three comes from the contribution only to $z$-axis in the the 3D random walk. Of course the right hand side would not have to be one third if only it were the order of $\cal{O}$(1). For simplicity, here we have just took it one third. In addition, generally $z_{i_z}$ must not be larger than $z_{i_z-1}$.
As we will see later, the diffusion tends to be proceed outward with increasing diffusion length or mean free path. Then the time spent in inner regions tends to be much longer than that in outer ones. That means that phenomena of the diffusion which started from $z=z_{0}$ are mainly determined by the local physics at around $z=z_{0}$. Then Eq. (\[eq:intdN\]) might be rewritten as a definition of the $z$-dependent mean-free path $\lambda(z)$ through $$\begin{aligned}
\label{eq:def_lambda_1}
\int_{z_{i_z-1}}^{z_{i_z}} \sigma_{\gamma e} n_{e} (z) dz
\approx \sigma_{\rm T} n_{e} (z_{i_z-1})\int_{z_{i_z-1}}^{z_{i_z}}dz,
$$ by $$\begin{aligned}
\label{eq:def_lambda_0}
\lambda(z_{i_z-1}) \equiv \left| \int_{z_{i_z-1}}^{z_{i_z}} dz \right|
\approx \frac13 \frac1{ \sigma_{\rm T} \ n_{e}(z_{i_z-1})},\end{aligned}$$ where we assumed that $\sigma_{\gamma e} =\sigma_{\rm T}$. Although there would be a lot of ways to define the $z$-dependent mean-free path, the definition in Eq. (\[eq:def\_lambda\_0\]) would be relatively natural in the current context of the accretion disks because the phenomena are mainly locally determined.
Then $d(z_{0},z_{\rm max})$ in Eq. (\[eq:dz\]) is rewritten as $$\begin{aligned}
\label{eq:dz2}
d(z_{0},z_{\rm max}) =
\sum_{i_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})}
\ell(z_{i_z-1}),\end{aligned}$$ with $$\begin{aligned}
\label{eq:ell}
\ell(z_{i_z}) =
\left\{
\begin{array}{cc}
\lambda (z_{i_z}) & (z_{i_z+1} \ge z_{i_z}),\\
\\
- \lambda (z_{i_z}) & (z_{i_z+1} < z_{i_z}).
\end{array}
\right.\end{aligned}$$ Next let us consider the averaged value of $d(z)$ after sufficiently a lot of tries. $$\begin{aligned}
\label{eq:ave_dz}
\left\langle d(z_{0},z_{\rm max}) \right\rangle
= \sum_{i_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})}
\left\langle \ell(z_{i_z-1}) \right\rangle,\end{aligned}$$ where $\langle X \rangle$ means the average of $X$ after such trials. Here we can assume $\left\langle \ell(z_{i_z-1})
\right\rangle \approx 0$. That is because we have assumed that the local physics at around $z=z_{i_z-1}$ determines the phenomena, and surely then this approximation would not be so bad. Then we see that the averaged value of $d(z_{0},z_{\rm max})$ vanishes, $$\begin{aligned}
\label{eq:ave_dz2}
\left\langle d(z_{0},z_{\rm max}) \right\rangle
\approx 0.\end{aligned}$$ On the other hand, however, the averaged value of the square of $d(z)$ must be finite. $$\begin{aligned}
\label{eq:ave_dz_sq}
\lefteqn{ \left\langle d(z_{0},z_{\rm max})^{2} \right\rangle}
\nonumber \\
&=& \sum_{i_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})} \quad
\sum_{j_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})} \left\langle
\ell(z_{i_z-1}) \ell(z_{j_z-1}) \right\rangle \nonumber \\
&=& \sum_{i_{z}=j_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})}
\left\langle \ell(z_{i_z-1})^{2} \right\rangle +
\sum_{i_{z} \neq j_{z}} \left\langle
\ell(z_{i_z-1}) \ell(z_{j_z-1}) \right\rangle \nonumber \\
&\approx& \sum_{i_{z}=j_{z}=1}^{N_{\rm scatt}(z_{0},z_{\rm max})}
\left\langle \lambda(z_{i_z-1})^{2} \right\rangle,\end{aligned}$$ where we approximated $$\begin{aligned}
\label{eq:second_term}
\lefteqn{\sum_{i_{z} \neq j_{z}} \left\langle
\ell(z_{i_z-1}) \ell(z_{j_z-1}) \right\rangle }\nonumber \\
&\approx&
\sum_{i_{z} \neq j_{z}} \ell(z_{i_z-1}) \left\langle
\ell(z_{j_z-1}) \right\rangle
\left( \approx
\sum_{i_{z} \neq j_{z}} \left\langle
\ell(z_{i_z-1}) \right\rangle \ell(z_{j_z-1}) \right) \nonumber \\
&\approx& 0,\end{aligned}$$ because the step $i_{z}$ does not correlate with that of $j_{z}$ among the independent trials for $\left\langle \ell(z_{i_z-1})
\right\rangle = 0$. Now we can approximate $\left\langle
\lambda(z_{i_z-1})^{2} \right\rangle \approx \lambda(z_{0})^{2}$ because the scatterings at the outer regions do not frequently occur and hardly contribute to the summation at all.
Then from (\[eq:ave\_dz\_sq\]), we find that the number of the scattering of the photon diffused from $z$ to $z_{\rm max}$, i.e., $\langle d^{2}(z,z_{\rm max}) \rangle = (\int_{z}^{z_{\rm
max}}dz)^{2}$, is represented by [^8] $$\begin{aligned}
\label{eq:nscatt_z}
N_{\rm scatt}(z,z_{\rm max}) &\approx& 3 \left[ \sigma_{\rm T} n_{e}(z)
\right]^{2} \left(\int_{z}^{z_{\rm max}} dz \right)^{2}
\nonumber \\
&\approx&
3 \left[ \sigma_{\rm T} \int_{z}^{z_{\rm max}} dz~ n_{e} (z)\right]^{2}
\nonumber \\
&=& 3 \tau(z)^{2},\end{aligned}$$ where we have used the same logic in Eq. (\[eq:def\_lambda\_1\]) to transform the first line to the second.
Here we get the expression of the $z$-dependent diffusion time scale, $$\begin{aligned}
\label{eq:tdiff_z_a}
t_{\rm diff}(z) &\equiv& N_{\rm scatt} (z,z_{\rm max})
\lambda(z)/c \nonumber \\
&\approx& 3 \tau(z)^{2} \lambda(z) /c.\end{aligned}$$ Of course, if we omit the $z$-dependence, we immediately get the $z$-independent diffusion time scale shown in Eqs. (\[eq:tdiff\_1\]) and (\[eq:t\_diff\_1\]).
When we consider the power-law density profiles $\propto (1-z/q
H)^{q-1}$ (or the exponential one as their large-$q$ limit), surely $t_{\rm diff}(z)$ decreases rapidly as $\propto (1-z/q
H)^{q+1}$ as a function of $z$. This means that the time spent in the inner regions is much longer than that in the outer ones, that validates our assumption in the current context in the accretion disks.
[999]{} \[\] å[A&A]{}
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\[lastpage\]
[^1]: E-mail: k.kohri@lancaster.ac.uk
[^2]: We note that Ichimaru (1977) proposed similar ideas 20 years earlier.
[^3]: For recent developments on the RIAF model of Sgr. A$^{*}$, see Manmoto, Kusunose & Mineshige. (1997), Yuan, Cui & Narayan (2005) and references therein.
[^4]: Eggum et al. (1988) performed the first 2D numerical simulations of supercritical accretion disks, and their work was later improved by Okuda (2002). However, the computation time in these studies was short and the numerical models did not reach steady state. Ohsuga et al. (2005) were the first to successfully compute models approaching steady state.
[^5]: The relativistic magnetohydrodynamic (MHD) jet model is another attractive model of energetic accretion sources (e.g., McKinney 2005; 2006 and references therein). This model is less likely to be affected by our work.
[^6]: If on the other hand we have a constant $z$-independent density, as considered by Ohsuga et al. (2002), it would correspond to $q = 1$. The corresponding results can be recovered by setting $q=1$ in all the expressions in the current paper.
[^7]: The existence of metallicity has two conflicting effects which influence the disk structure, i) larger radiation coefficient induced by the sizable amount of metals helps to cool the gas more, which means that the gas can be easily accreted, and ii) the larger absorption coefficient by such a metallicity pushes the gas outside, which means that it is difficult to be accreted. However there is an uncertainty of metallicity in the system. Therefore, Ohsuga et al. (2005) studied three cases of the metallicity in a reasonable range, i.e., $Z / Z_{\odot}$=0, 1, and 10. The variation of the luminosity which comes from this uncertainty should be treated as an error in the numerical simulation. For details see Ohsuga et al. (2005).
[^8]: Note that we have removed the subscript “0” in $z$.
|
---
abstract: 'We introduce a data-driven method and shows its skills for spatiotemporal prediction of high-dimensional chaotic dynamics and turbulence. The method is based on a finite-dimensional approximation of the Koopman operator where the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The predictive capabilities of the method are demonstrated for well-known prototypes of chaos such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the data-driven predictions are accurate for several Lyapunov timescales. Similar performance is seen for two-dimensional lid-driven cavity flows at high Reynolds numbers.'
author:
- 'M. A. Khodkar$^1$'
- 'Pedram Hassanzadeh$^{1,2}$'
- 'Athanasios Antoulas$^{3, 4, 5}$'
bibliography:
- 'Main.bib'
title: 'A Koopman-based framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings'
---
[^1]
[^2]
[^1]: mkhodkar@rice.edu
[^2]: pedram@rice.edu
|
---
abstract: 'We investigate the interior nodal sets $\mathcal{N}_\lambda$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.'
address: |
Department of Mathematics\
Johns Hopkins University\
Baltimore, MD 21218, USA\
Emails: jzhu43@math.jhu.edu
author:
- Jiuyi Zhu
title: Interior nodal sets of Steklov eigenfunctions on surfaces
---
[^1]
Introduction
============
Let $(\mathcal{M}, g)$ be a smooth, connected and compact surface with smooth boundary $\partial \mathcal{M}$. The main goal of this paper is to obtain an upper bound of interior nodal sets $$\mathcal{N}_\lambda=\{ z\in\mathcal{M}|e_\lambda=0\}$$ for Steklov eigenfunctions $$\left\{
\begin{array}{lll}
\triangle_g e_\lambda=0,
\quad z\in\mathcal{M},
\medskip \\
\frac{\partial e_\lambda}{\partial \nu}(z)=\lambda e_\lambda(z),
\quad z\in
\partial\mathcal{M},
\end{array}
\right. \label{Stek}$$ where $\nu$ is a unit outward normal on $\partial\mathcal{M}$. The Steklov eigenfunctions were introduced by Steklov in 1902 for bounded domains in the plane. It interprets the steady state temperature distribution in the domain such that the heat flux on the boundary is proportional to the temperature. It is also found applications in a quite few physical fields, such as fluid mechanics, electromagnetism, elasticity, etc. Especially, the model (\[Stek\]) was studied by Calderón [@C] as it can be regarded as eigenfunctions of the Dirichlet-to-Neumann map. The interior nodal sets of Steklov eigenfunctions represent the stationary points in $\mathcal{M}$. In the context of quantum mechanics, nodal sets are the sets where a free particle is least likely to be found.
It is well-known that the spectrum $\lambda_j$ of Steklov eigenvalue problem is discrete with $$0=\lambda_0<\lambda_1\leq
\lambda_2\leq \lambda_3,\cdots, \
\ \mbox{and} \ \ \lim_{j\to\infty}\lambda_j=\infty.$$ There exists an orthonormal basis $\{ e_{\lambda_j}\}$ of eigenfunctions such that $$e_{\lambda_j}\in C^\infty(\mathcal{M}), \quad
\int_{\partial\mathcal{M}} e_{\lambda_j} e_{\lambda_k}\, dV_{
{g}}=\delta_{j}^{k}.$$ Estimating the Hausdorff measure of nodal sets has always been an important subject concerning the study of eigenfunctions. This subject centers around the famous Yau’s conjecture. Recently, much work has been devoted to the bounds of nodal sets $$Z_\lambda=\{z\in\partial\mathcal{M}| e_\lambda(z)=0\}$$ of Steklov eigenfunctions on the boundary. Bellova and Lin [@BL] proved the $H^{m-1}(Z_\lambda)\leq C\lambda^6$ with $C$ depending only on $\mathcal{M}$, if $\mathcal{M}$ is a $m+1$ dimensional analytic manifold. Zelditch [@Z1] improved their results and gave the optimal upper bound $H^{m-1}(Z_\lambda)\leq
C\lambda$ for analytic manifolds using microlocal analysis. For the smooth manifold $\mathcal{M}$, by assuming that $0$ is a regular value, Wang and the author in [@WZ] recently established a lower bound $$H^{m-1}(Z_\lambda)\geq C\lambda^{\frac{3-m}{2}}.$$
Before presenting our results for interior nodal sets, let’s briefly review the literature about the nodal sets of classical eigenfunctions. Interested reader may refer to the book [@HL] and survey [@Z] for detailed account about this subject. Let $e_\lambda$ be $L^2$ normalized eigenfunctions of Laplacian-Beltrami operator on compact manifolds $(\mathcal{M}, g)$ without boundary, $$-\triangle_g e_\lambda=\lambda^2 e_\lambda.
\label{class}$$ Yau’s conjecture states that for any smooth manifold, one should control the upper and lower bound of nodal sets of classical eigenfunctions as $$c\lambda\leq H^{n-1}(
\mathcal{N}_\lambda)\leq C\lambda \label{yau}$$ where $C, c$ depends only on the manifold $\mathcal{M}$. The conjecture is only verified for real analytic manifolds by Donnelly-Fefferman in [@DF]. Lin [@Lin] also showed the upper bound for the analytic manifolds by a different approach. For the smooth manifolds, the conjecture is still not settled. For the lower bound of nodal sets with $n\geq 3$, Colding and Minicozzi [@CM], Sogge and Zelditch [@SZ], [@SZ1] independently obtained that $$H^{n-1}(\mathcal{N}_\lambda)\geq C\lambda^{\frac{3-n}{2}}$$ for smooth manifolds. See also [@HSo] for deriving the same bound by adapting the idea in [@SZ]. For the upper bound, Hardt and Simon [@HS] gave an exponential upper bound $$H^{n-1}(\mathcal{N}_\lambda)\leq C e^{\lambda \ln \lambda}.$$ In surfaces, better results have been obtain. Brüning [@Br] and Yau (unpublished) derived the same lower bound as (\[yau\]). The best estimate to date for the upper bound is $$H^{1}(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$$ by Donnelly-Fefferman [@DF2] and Dong [@D] using different methods.
Let us return to Steklov eigenvalue problem (\[Stek\]). By the maximum principle, there exist nodal sets in the manifold $\mathcal{M}$ and those sets must intersect the boundary $\partial\mathcal{M}$. Thus, it is natural to study the size of interior nodal sets in $\mathcal{M}$. We can also ask Yau’s type questions about the Hausdorff measure of nodal sets. The natural and corresponding conjecture for Steklov eigenfunctions should states exactly the same as (\[yau\]). See also the open questions in the survey by Girouard and Polterovich in [@GP]. Recently, Sogge, Wang and the author [@SWZ] obtained a lower bound for interior nodal sets $$H^{n-1}({\mathcal{N}}_\lambda)\geq
C\lambda^{\frac{2-n}{2}}$$ for $n$-dimensional manifold $\mathcal{M}$. Very recently, Polterovich, Sher and Toth [@PST] can verify Yau’s type conjecture for (\[Stek\]) on real-analytic Riemannian surfaces.
An interesting topic related to the measure of nodal sets is about doubling inequality. Based on doubling inequalities, one can obtain the vanishing order of eigenfunctions, which characterizes how fast the eigenfunctions vanish. For the classical eigenfunctions of (\[class\]), Donnelly-Fefferman in [@DF], [@DF1] obtained that the maximal vanishing order of $e_\lambda$ is at most order $C\lambda$ everywhere. To achieve it, a doubling inequality $$\int_{\mathbb B (z_0,\, 2r)}e_\lambda^2\leq Ce^{\lambda}
\int_{\mathbb B(z_0,\, r)}e_\lambda^2 \label{doub}$$ is derived using Carleman estimates, where $\mathbb B(p, c)$ denotes as a ball centered at $p$ with radius $c$. The doubling estimate (\[doub\]) plays an important role in obtaining the bounds of nodal sets for analytic manifolds in [@DF] and upper bound of nodal sets for smooth surfaces in [@DF2]. For the Steklov eigenfunctions, we have obtained a doubling inequality on the boundary $\partial\mathcal{M}$ and derive that the sharp vanishing order is less than $C\lambda$ on the boundary $\partial\mathcal{M}$. For steklov eigenfunction in $\mathcal{M}$, we are also able to get the doubling inequality, see proposition \[proo\]. With aid of doubling estimates and Carleman inequalities, The following optimal vanishing order for Steklov eigenfunctions can be obtained.
The vanishing order of Steklov eigenfunction $e_\lambda$ of (\[Stek\]) in $\mathcal{M}$ is everywhere less than $C\lambda$. \[th3\]
It’s sharpness can be seen in the case that the manifold $
\mathcal{M}$ is a ball. Notice that the doubling estimates in proposition \[proo\] and the vanishing order in Theorem \[th3\] hold for any $n$-dimensional compact manifolds.
Singular sets $$\mathcal{S}_\lambda=\{ z\in\mathcal{M}|e_\lambda=0, \nabla e_\lambda=0\}$$ are contained in nodal sets. In Riemannian surfaces, those singular sets are finite points in the 1-dimensional nodal sets. It is interesting to count the number of those singular sets. Based on a Carleman inequality with singularities, we are able to show an upper bound of singular sets.
Let $(\mathcal{M}, g)$ be a smooth, compact surface with smooth boundary $\partial\mathcal{M}$. There holds $$H^0(\mathcal{S}_\lambda)\leq C\lambda^2$$ for Steklov eigenfunctions in (\[Stek\]). \[th2\]
For the nodal sets of Steklov eigenfunctions, we are able to build a similar type of Carleman inequality as [@DF2] and show the following result.
Let $(\mathcal{M}, g)$ be a smooth, compact surface with smooth boundary $\partial\mathcal{M}$. Then $$H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$$ holds for Steklov eigenfunctions in (\[Stek\]). \[th1\]
The outline of the paper is as follows. Section 2 is devoted to reducing the Steklov eigenvalue problem into an equivalent elliptic equation without boundary. Then we obtain the optimal doubling inequality and show theorem \[th3\]. In section 3, we establish the Carleman inequality with singularities at finite points. Under additional assumptions of those singular points, a stronger Carleman inequality is derived. We measure the singular sets in section 4. Section 5, 6 and 7 are devoted to obtaining the nodal length of Steklov eigenfunctions. Under the slow growth of $L^2$ norm condition, we find out the nodal length in section 6. Based on a similar type of Calderón and Zygmund decomposition procedure, we show the slow growth at almost every point. Then the measure of nodal sets is arrived by summing up the nodal length in each small square. The letter $c$, $C$, $C_i$, $d_i$ denote generic positive constants and do not depend on $\lambda$. They may vary in different lines and sections.
[**Acknowledgement.**]{} It is my pleasure to thank Professor Christopher D. Sogge for helpful discussions about this topic and guidance into the area of eigenfunctions. I also would like to thank X. Wang for many fruitful conversations.
Vanishing Order of Steklov Eigenfunctions
=========================================
In this section, we will reduce the Steklov eigenvalue problem to an equivalent model on a boundaryless manifold. The presence of eigenvalue on the boundary $\partial\mathcal{M}$ will be reflected on the coefficient functions of a second order elliptic equation. Let $d(z)= dist\{z,
\partial\mathcal{M}\}$ denote the geodesic distance function from $x\in\mathcal{M}$ to the boundary $\partial\mathcal{M}$. Since $\mathcal{M}$ is smooth, there exist a $\rho$-neighborhood of $\partial\mathcal{M}$ in $\mathcal{M}$ such that $d(x)$ is smooth in the neighborhood. Let’s denoted it as $\mathcal{M}_{\rho}$. We extend $d(z)$ smoothly in $\mathcal{M}$ by $$\delta(z)=\left \{\begin{array}{lll} d(z) \quad z\in \mathcal{M}_{\rho}, \medskip\nonumber\\
l(z) \quad z\in \mathcal{M}\backslash\mathcal{M}_{\rho},
\end{array}
\right.$$ where $l(z)$ is a smooth function in $\mathcal{M}\backslash\mathcal{M}_{\rho}$. Note that the extended function $\delta(z)$ is a smooth function in $\mathcal{M}$. We first reduce Steklov eigenvalue problem into an elliptic equation with Neumann boundary condition. Let $$v(z)=e_\lambda \exp\{\lambda \delta(z)\}.$$ It is known that $v(z)=e_\lambda(z)$ on $\partial \mathcal{M}$. For $z\in
\partial \mathcal{M}$, $\nabla_g \delta(z)=-\nu(z)$. Recall that $\nu(z)$ is the unit outer normal on $z\in\partial\mathcal{M}
$. We can check that the new function $v(z)$ satisfies $$\left \{ \begin{array}{lll} \triangle_g v+b(z)\cdot\nabla_g
v+q(z)v=0 &\quad \mbox{in}\ \mathcal{M},
\medskip\\
\frac{\partial v}{\partial \nu}=0 &\quad \mbox{on}\
\partial\mathcal{M},
\end{array}
\right.$$ with $$\left \{\begin{array}{lll}
b(z)=-2\lambda \nabla_g \delta(z), \medskip\\
q(z)=\lambda^2|\nabla_g \delta(z)|^2-\lambda\triangle_g \delta(z).
\end{array}
\right. \label{fffw}$$ In order to get rid of boundary condition, we attach two copies of $\mathcal{M}$ along the boundary and consider a double manifold $\overline{\mathcal{M}}=\mathcal{M}\cup \mathcal{M}$. The metric $g$ extends to $\overline{\mathcal{M}}$ with Lipschitz type singularity along $\partial\mathcal{M}$, since the lift metric $ {g'}$ of $g$ on $\mathcal{M}$ to the double manifold $\overline{\mathcal{M}}$ is Lipschitz. There also exists a canonical involutive isometry $\mathcal {F}: \overline{\mathcal{M}}\to \overline{\mathcal{M}}$ that interchanges the two copies of ${\mathcal{M}}$. Then the function $v(x)$ can be extended to $\overline{\mathcal{M}}$ by $v\circ \mathcal {F}= v$. Therefore, $v(z)$ satisfies $$\triangle_{g'} v+\bar{b}(z)\cdot\nabla_{g'} v+\bar{ q} (z)v=0 \quad
\mbox{in}\ \overline{\mathcal{M}}. \label{star}$$ From (\[fffw\]), one can see that $$\left \{\begin{array}{lll}
\|\bar b\|_{W^{1, \infty}(\overline{\mathcal{M}})}\leq C\lambda, \medskip\\
\|\bar q\|_{W^{1, \infty}(\overline{\mathcal{M}})}\leq C\lambda^2.
\label{core}
\end{array}
\right.$$ After this procedure, we can instead study the nodal sets for the second order elliptic equation (\[star\]) with assumption (\[core\]). Note that $\overline{\mathcal{M}}$ is a manifold without boundary.
We present a brief proof of Theorem \[th3\]. It is a small modification of the argument in [@Zh], where the sharp vanishing order of Steklov eigenfunctions on boundary $\partial\mathcal{M}$ is shown to be less than $C\lambda$.
By a standard regularity argument, we can still consider polar coordinate for (\[star\]). We are able to establish a similar Carleman inequality as [@Zh] for the general second order elliptic equation (\[star\]). See also e.g. [@BC].
Let $u\in C^{\infty}_{0}(\frac{1}{2}{\epsilon}_1<r<{\epsilon}_0)$. If $\tau>C_1(1+\|\bar b\|_{W^{1, \infty}}+\|\bar q\|^{1/2}_{W^{1,
\infty}})$. Then $$\int r^4 e^{2\tau \phi(r)}|\triangle_{g'} u+\bar b \cdot \nabla_{g'}
u+ \bar q u|^2\, drd\omega\geq C_2 \tau^3\int r^{{\epsilon}}
e^{2\tau\phi(r)} u^2 \,drd\omega, \label{cca}$$ where $\phi(r)=-\ln r+r^{\epsilon}$ and $r$ is the geodesic distance. $0<{\epsilon}_0, {\epsilon}_1, {\epsilon}<1$ are some fixed constants. Moreover, $(r,
\omega)$ are the standard polar coordinates. \[carl\]
Using this Carleman estimate and choosing suitable test functions, a Hadamard’s three-ball result can be obtained in $\overline{\mathcal{M}}$. There exist constants $r_0,$ $C$ and $0<\gamma<1$ depending only on $\overline{\mathcal{M}}$ such that for any solutions of (\[star\]), $0<r<r_0$, and $z_0\in\overline{\mathcal{M}}$, one has $$\int_{\mathbb B(z_0,\, r)}{ v}^2\leq e^{C(1+\|\bar b\|_{W^{1,
\infty}}+{\|\bar q\|}^{1/2}_{W^{1, \infty}})}(\int_{\mathbb B(z_0,\,
{2r})}{v}^2)^{1-\gamma}(\int_{\mathbb B(z_0,\, {r/2})}{
v}^2)^{\gamma}.$$ Based on a propagation of smallness argument using the three-ball result and Carleman estimates (\[cca\]), taking the assumptions (\[core\]) into account, we are able to obtain the doubling inequality in $\overline{\mathcal{M}}$.
There exist constants $r_0$ and $C$ depending only on $\overline{\mathcal{M}}$ such that for any $0<r<r_0$ and $z_0\in
\overline{\mathcal{M}}$, there holds $$\|v\|_{L^2(\mathbb B(z_0, \,{2r}))}\leq
e^{C\lambda}\|v\|_{L^2(\mathbb B(z_0,\, {r}))} \label{ddou}$$ for any solutions of (\[star\]). \[proo\]
One can see that the doubling estimate holds in $\mathcal{M}$ if $\mathbb B(z_0, {2r})\subset\mathcal{M}$. By standard elliptic estimates, one can have $L^\infty$ norm of doubling inequality $$\|v\|_{L^\infty(\mathbb B(z_0, \,{2r}))}\leq
e^{C\lambda}\|v\|_{L^\infty(\mathbb B(z_0,\, {r}))}.$$ Since $\overline{\mathcal{M}}$ is compact, we can derive that $$\|v\|_{L^\infty(\mathbb B(z_0, \,{r}))} \geq r^{C\lambda}$$ for any $z_0\in \overline{\mathcal{M}}$, which implies the vanishing order for $v$ is less than $C\lambda$. So is the vanishing order of $u$. This completes Theorem \[th3\].
Carleman estimates
==================
This section is to devoted to establishing Carleman inequalities involving weighted functions at finite points. We will consider the behavior of $v$ in a conformal coordinate patch. Since $\overline{\mathcal{M}}$ is a compact Riemannian surface. There exists a finite number $N$ of conformal charts $( \mathcal {U}_i,
\phi_i)$ with $\phi_i: \mathcal {U}_i\subset
\overline{\mathcal{M}}\to \mathcal{V}_i\subset \mathbb R^2$ and $i\in\{1, 2, \cdots N\}$. On each of these charts, the metric is conformally flat and there exists positive function $g_i$ such that $g'=g_i(x, y)(dx^2+dy^2)$. By the compactness of the surface, there is positive constants $c$ and $C$ such that $0<c<g_i<C$ for each $i$. Under this equivalent metric, the equation (\[star\]) can be written as $$\triangle v+{{\bar{b}}}(z)\cdot\nabla v+{ {\bar{q}}} (z)v=0 \quad
\mbox{in}\ \mathcal {V}_i, \label{star1}$$ where $ \triangle$ is Euclidian Laplacian, $\nabla$ is Euclidian gradient and $z=(x,\,y)$. We use the same notations $\bar{b}(z)$ and $\bar{q}(z)$ as that in (\[star1\]), since they satisfy the same conditions as (\[core\]). They only differ by some function about $g_i$.
By restricting into a small ball $\mathbb B(p, 3c)$ contained in the conformal chart, we consider $v$ in the small ball. Let $\bar
v(z)=v(cz)$. It follows from (\[star1\]) that $$\triangle \bar v+{\tilde{{b}}}(z)\cdot\nabla \bar v+{ \tilde{{q}}}
(z)\bar v=0 \quad \mbox{in}\ \mathbb {B}_3, \label{star2}$$ with $\tilde{{b}}=c\bar b$ and $\tilde{q}=c^2 \bar q$. If $c$ is sufficiently small, $\tilde{{b}}$ and $\tilde{q}$ are arbitrary small.
The crucial tool in [@DF2] is a Carleman inequality for classical eigenfunctions involving weighted functions with singularity at finite points. We will obtain the corresponding Carleman inequality for the second order elliptic equation (\[star2\]). We adapt the approach in [@DF2] to obtain the desirable Carleman estimate for (\[star2\]).
Let $\mathcal {D}\subset \mathcal {C}$ be an open set and $\psi\in
C^\infty_{0}(\mathcal {D})$ be a real valued unction. We introduce the following differential operators $$\mathcal{\partial} =\frac{1}{2}( \frac{\partial}{\partial x}-i \frac{\partial}{\partial
y}), \quad \quad \overline{\partial} =\frac{1}{2}(
\frac{\partial}{\partial x}+i \frac{\partial}{\partial y}).$$ Direct computation shows that $\overline{\partial}\partial\psi=\frac{1}{4}\triangle \psi$. By the Cauchy-Riemann equation, $u$ is holomorphic if and only if $\overline{\partial}u=0$. For the completeness, we present the elementary inequality in [@DF2].
let $\Phi$ be a smooth positive function in $\mathcal {D}$. Then $$\int_{\mathcal {D}}|\overline{\partial}u|^2\Phi\geq
\frac{1}{4}\int_{\mathcal {D}}(\triangle \ln \Phi)|u|^2\Phi.$$ Here the integral is taken with respect to the lebesgue measure. \[lem1\]
We want the weight function to involve those singular points. To specialize the choice of $\Phi $, we construct the following function $\psi_0$.
There exists a smooth function $\psi_0$ defined for $|z|>1-2a$ satisfying the following properties:\
(i) $a_1\leq \psi_0(z)\leq a_2$ with constant $a_1, a_2>0$.\
(ii) $\psi_0=1$ on $\{|z|>1\}$.\
(iii) $\triangle \ln \psi_0\geq 0$ on $\{|z|>(1-2a)\}$.\
(iv) If $1-2a<|z|<1-a$, then $\triangle \ln \psi_0\geq a_3>0$.
The existence of such $\psi_0$ follows from existence and unique theory of ordinary differential equations.
We assume that $$D_l=\{z|\, |z-z_l|\leq \delta\}.$$ Let $D_l$ be a finite collection of pairwise disjoint disks, which are contained in a unit disk centered at origin. Let $$D_l(a)=\{z|\, |z-z_l|\leq (1-2a)\delta\}$$ be the smaller concentric disk. We define a smooth weight function $\Psi_0(z)$ as $$\Psi_0(z)=\left \{ \begin{array}{lll} 1 \quad \mbox{if} \ z\not\in
\cup_l
D_l,\nonumber \medskip \\
\psi_0(\frac{z-z_l}{\delta}) \quad \mbox{if} \ z\in D_l. \nonumber
\end{array}
\right.$$ We also introduce the following domain $$A_l=\{ (1-2a)\delta\leq |z-z_l|\leq (1-a)\delta\}.$$ From the last lemma, $\Psi_0(z)$ satisfies those properties:\
(i) $a_1\leq \Psi_0(z)\leq a_2$.\
(ii) $\triangle \ln \Psi_0\geq 0$ for $z\in \mathbb R^2\backslash
\cup_l D_l(a)$.\
(iii) $\triangle \ln \Psi_0\geq a_3 \delta^{-2}$ for $z\in A_l$.\
Note that $a_i$ in above are positive constants independent of $\lambda$. Denote $$A=\cup_l A_l.$$ Suppose that $\tau$ is a nonnegative constant. We introduce $\Phi(z)=\Psi_0(z)e^{\tau
|z|^2}$. For $u\in C^{\infty}_0(\mathbb R^2\backslash \cup_l
D_l(a))$, we assume that $\mathcal {D}$ contains the support of $u$ and $A\subset \mathcal {D}\subset \mathbb R^2\backslash \cup_l
D_l(a)$. Obviously, $$\ln \Phi(z)= \ln\Psi_0(z) +\tau |z|^2.$$ Substituting $\Phi$ in lemma \[lem1\] gives that $$\int_{\mathcal{D}}|\overline{\partial} u|^2\Psi_0(z)e^{\tau
|z|^2}\geq C_1\tau \int_{\mathcal{D}}|u|^2\Psi_0(z)e^{\tau
|z|^2}+C_2 \delta^{-2} \int_{A}|u|^2e^{\tau |z|^2},$$ where we have used the properties (ii) and (iii) for $\Psi_0$. The boundedness of $\Psi_0(z)$ yields that $$\int_{\mathcal{D}}|\overline{\partial} u|^2 e^{\tau |z|^2}\geq
C_3\tau \int_{\mathcal{D}}|u|^2e^{\tau |z|^2}+C_4 \delta^{-2}
\int_{A}|u|^2e^{\tau |z|^2}. \label{impt}$$ Define a holomorphic function $$P(z)=\prod_l (z-z_l).$$ Then $\overline{\partial}({u}/{P})={\overline{\partial} u}/P$. Replacing $u$ by ${u}/{P}$ in (\[impt\]), it follows that $$\int_{\mathcal{D}}|\overline{\partial} u|^2 |P|^{-2}e^{\tau
|z|^2}\geq C_3\tau \int_{\mathcal{D}}|u|^2|P|^{-2}e^{\tau |z|^2}+C_4
\delta^{-2} \int_{A}|u|^2|P|^{-2}e^{\tau |z|^2}. \label{impt1}$$
We will establish a Carleman inequality for second order elliptic equations as (\[star2\]). Write $\tilde{b}(x)= (\tilde{b}_1(x),\,
\tilde{b}_2(x)).$ Let $$u=\partial f+\frac{1}{2}(\tilde{b}_1- i\tilde{b}_2)f,$$ where $f\in C_0^\infty(\mathbb R^2\backslash \cup_l D_l(a))$ is a real valued function. Then $$\overline{\partial} u=\frac{1}{4}[\triangle f+div\tilde{b} f+\tilde{b}\cdot\nabla f+i(\frac{\partial(\tilde{b}_1f)}{\partial y}
-\frac{\partial(\tilde{b}_2f)}{\partial x})].$$ Plugging above $u$ into (\[impt1\]), we obtain $$\begin{aligned}
\int_{\mathcal{D}}\big[|\triangle f &+&\tilde{b}\cdot\nabla
f|^2+|div\tilde{b} f|^2 +|\frac{\partial(\tilde{b}_1f)}{\partial y}
-\frac{\partial(\tilde{b}_2f)}{\partial x}|^2\big]|P|^{-2} e^{\tau
|z|^2} \nonumber\\
&\geq & C_3\tau\int_{\mathcal{D}}|\nabla f|^2 |P|^{-2}e^{\tau
|z|^2}-
C_3\tau\int_{\mathcal{D}}|\tilde{b}|^2|f|^2 |P|^{-2}e^{\tau |z|^2} \nonumber\\
&+ &C_4\delta^{-2}\int_{A}|\nabla f|^2|P|^{-2}e^{\tau |z|^2}-
C_4\delta^{-2}\int_{A}|\tilde{b}|^2|f|^2|P|^{-2}e^{\tau |z|^2}.
\label{many}\end{aligned}$$ If we choose $u=f$ in (\[impt1\]), we get $$\int_{\mathcal{D}}|\nabla f|^2 |P|^{-2}e^{\tau |z|^2}\geq C_3\tau
\int_{\mathcal{D}}|f|^2|P|^{-2}e^{\tau |z|^2}. \label{impt2}$$ Since the norm of $\tilde{b}$ is chosen small enough, it is smaller than $\tau$ which will be chosen large enough. With aid of (\[impt2\]), we can incorporate the terms about $\tilde{b}$ in the left hand side of (\[many\]) into the first term in the right hand side of (\[many\]), $$\begin{aligned}
\int_{\mathcal{D}}|\triangle f +\tilde{b}\cdot\nabla f|^2 |P|^{-2}
e^{\tau |z|^2}& \geq & C_5 \tau \int_{\mathcal{D}}|\nabla
f|^2|P|^{-2} e^{\tau |z|^2} +C_4\delta^{-2}\int_{A}|\nabla
f|^2|P|^{-2}e^{\tau |z|^2} \nonumber\\&-&
C_4\delta^{-2}\int_{A}|\tilde b|^2|f|^2|P|^{-2}e^{\tau |z|^2}.
\label{hao}\end{aligned}$$ Furthermore, if $u=f$, the inequality (\[impt1\]) implies that $$\int_{\mathcal{D}}|\nabla
f|^2 |P|^{-2}e^{\tau |z|^2}\geq C_4 \delta^{-2}
\int_{A}|f|^2|P|^{-2}e^{\tau |z|^2}. \label{any}$$ Applying (\[any\]) to the last term in the right hand side of (\[hao\]) gives that $$\begin{aligned}
\int_{\mathcal{D}}|\triangle f +\tilde{b}\cdot\nabla f|^2 |P|^{-2}
e^{\tau |z|^2}& \geq & C_6 \tau^2 \int_{\mathcal{D}}|
f|^2|P|^{-2} e^{\tau |z|^2} \nonumber \\
&+&C_7\delta^{-2}\int_{A}|\nabla f|^2|P|^{-2}e^{\tau |z|^2}.
\label{keys}\end{aligned}$$
We continue to get a refined estimate for the last term of (\[keys\]). In order to achieve this goal, we need the following hypotheses for the geometry of the disk $D_l$ and the parameter $\tau>1$:\
($R_{1}$) The radius $\delta$ for each disk $D_l$ is less than $a_4
\tau^{-1}$.\
($R_{2}$) The distance between any two distinct $z_l$ is at least $2a_5\tau^{\frac{1}{2}} \delta$.\
($R_{3}$) The total number of disk $D_l$ is at most $a_6\tau$.\
Under the those assumptions, we have those comparison estimates from [@DF2].
If $\bar z_1$ and $\bar z_2$ are any points in the same component $A_l$ of $A$, then\
(i) $a_7< e^{\tau |\bar z_1|^2}/e^{\tau |\bar z_2|^2}<a_8$.\
(ii) $a_9<|P(\bar z_1)|/|P(\bar z_2)|<a_{10}$. \[lemc\]
We also need the following Poincaré type inequality on each annulus. If $f\in C^{\infty}(A_l)$ and $f$ vanishes on the inner boundary of $A_l$, then $$\int_{A_l} |\nabla f|^2 \geq a_{11} \delta^{-2}\int_{A_l}|f|^2.
\label{poin}$$ The proof of (\[poin\]) can be seen in [@DF2]. Let $z_l\in
A_l$ be chosen arbitrarily. By lemma \[lemc\], it follows that $$\int_{A_l}|\nabla f|^2|P(z)|^{-2}e^{\tau |z|^2} \geq C_8\sum_{l}e^{\tau |z_l|^2}|P(z_l)|^{-2}\int_{A_l} |\nabla f|^2.$$ Since $f\in C_0^\infty(\mathbb R^2\backslash \cup_l D_l(a))$, the inequality (\[poin\]) yields that $$\int_{A_l}|\nabla f|^2|P(z)|^{-2}e^{\tau |z|^2} \geq
C_9\sum_{l}e^{\tau |z_l|^2}|P(z_l)|^{-2}\delta^{-2}\int_{A_l} |
f|^2.$$ Using lemma \[lemc\] again, we obtain $$\int_{A_l}|\nabla f|^2|P(z)|^{-2}e^{\tau |z|^2} \geq C_{10}
\delta^{-2}\int_{A}| f|^2|P(z)|^{-2}e^{\tau |z|^2}.$$ Substituting the last inequality into the last term in (\[keys\]) leads to $$\begin{aligned}
\int_{\mathcal{D}}|\triangle f +\tilde{b}\cdot\nabla f|^2 |P|^{-2}
e^{\tau |z|^2}& \geq & C_6 \tau^2 \int_{\mathcal{D}}|
f|^2|P|^{-2} e^{\tau |z|^2} \nonumber \\
&+&C_{11}\delta^{-4}\int_{A}| f|^2|P|^{-2}e^{\tau |z|^2}.
\label{keys1}\end{aligned}$$ We summarize the above arguments in the following proposition.
Assuming $f\in C_0^\infty(\mathbb R^2\backslash \cup_l D_l(a))$. Then\
(i) it holds that $$\int_{\mathcal{D}}|\triangle f +\tilde{b}\cdot\nabla f|^2 |P|^{-2}
e^{\tau |z|^2} \geq C \tau^2 \int_{\mathcal{D}}| f|^2|P|^{-2}
e^{\tau |z|^2}. \label{rec}$$ (ii) If the additional assumptions $(R_1)$-$(R_3)$ for $D_l$ hold, the stronger inequality (\[keys1\]) is satisfied. \[pro21\]
Measure of Singular Sets
========================
Let $\mathcal {M}$ be a compact smooth surface. In section 2, we have shown that the Steklov eigenfunction $e_\lambda$ vanishes at any points at most $C\lambda$. By the implicit function theorem, outside the singular sets, the nodal set is locally a 1-dimensional $C^1$ manifold. Adapting the arguments in [@DF2] for (\[star2\]), we can estimate those singular points in a quantitative way. We are able to obtain an upper bound for the singular points in term of eigenvalue $\lambda$.
Singular sets consist of at most finite many points.
Without loss of generality, we assume that $0\in \mathcal
{S}_\lambda$ and choose normal coordinate $(x, y)$ at the origin. Next we prove there are finite singular points in $\overline{\mathcal {M}}$. Using Taylor expansion, we expand $v$ locally at origin. Then $v(x, y)=F_j(x, y)+W_{j+1}(x, y)$, where $F_j(x,y)$ consists of the leading nonvanishing term with homogenous order $j\geq 2$. $W_{j+1}(x,y)$ is a higher order reminder term. Since $\triangle v+{{\bar{b}}}(z)\cdot\nabla v+{ {\bar{q}}} (z)v=0$ and the coordinate is normal, we obtain that $\triangle F_j=0$. Under polar coordinates, we find that $F_j=r^j\big(a_1
\cos(j\theta)+a_2\sin(j\theta)\big)$. Obviously, $r^{-1}\frac{\partial F_j}{\partial \theta}$ and $\frac{\partial
F_j}{\partial r}$ have no common zero if $r\not=0$. Since $$|\nabla F_j|^2=|\frac{\partial F_j}{\partial r}|^2 +\frac{1}{r^2}|\frac{\partial F_j}{\partial \theta}|^2,$$ there exists a small neighborhood of $\mathcal {U}$ of origin such that $\mathcal {U}\cap \mathcal {S}_\lambda=0 $. Since $\overline{\mathcal {M}}$ is compact, then the lemma follows.
We plan to count the number of singular points in a sufficiently small ball. Let $p\in \overline{\mathcal {M}}$. Consider a geodesic ball $\mathbb B(p, c\lambda^{-\frac{1}{2}})$. If $c$ is small enough, then this geodesic ball is contained in a conformal chart. If we choose $$w(z)= v(c {\lambda^{-\frac{1}{2}}z})$$ with $c$ sufficiently small. From equation (\[star1\]), $w$ satisfies $$\triangle w+\hat{{b}}(x)\cdot\nabla w+\hat{{q}}(x)w=0 \quad
\mbox{in}\ \mathbb B( 0, 4), \label{starr}$$ with $ \hat{b}(x)= c\lambda^{-\frac{1}{2}}\bar{b}(x)$ and $\hat{q}(x)= c^2\lambda^{-1}{ {\bar{q}}}(x).$ From (\[core\]), we obtain $$\left \{\begin{array}{lll}
\|\hat{b}\|_{W^{1, \infty}(\mathbb B( 0, 4))}\leq c\lambda^{\frac{1}{2}}, \medskip\\
\| \hat{q}\|_{W^{1, \infty}(\mathbb B( 0, 4))}\leq c^2\lambda
\label{core2}
\end{array}
\right.$$ with $c$ sufficiently small.
Next we will count the total order of vanishing of singular points for $w$ in the sufficiently small ball. We study $w$ in equation (\[starr\]),
Suppose $z_l\in \mathcal{ S}_\lambda\cap\mathbb B(p, c\lambda^{-\frac{1}{2}})$ where $v$ vanishes to order $n_l+1$. Then $\sum_l n_l\leq C\lambda$. \[pro2\]
It suffices to count the number of singular point of $w$ in a small Euclidean ball with radius $\frac{1}{10}$ centered at origin. Suppose that $w$ vanishes to order $n_l+1$. Let $n_l=m_l+1$. We first consider the case $n_l\geq 2$. Then $m_l\geq 1$. Define the polynomial $$P(z)=\prod(z-z_l)^{m_l}$$ with $|z_l|<\frac{1}{10}$. Let $\mathcal{D}=\mathbb B(0, \,2)$ and $D_l$ be small disjoint disks of radius $\delta$ centered at $z_l$. If $f\in
C_0^\infty(\mathbb R^2\backslash \cup_l D_l)$, the inequality (\[rec\]) in proposition \[pro21\] implies that $$\int_{\mathcal{D}}(|\triangle f|^2+|\hat{b}\cdot\nabla f|^2)
|P|^{-2} e^{d_1\lambda |z|^2} \geq C_2 \lambda^2\int_{\mathcal{D}}|
f|^2|P|^{-2} e^{d_1\lambda |z|^2}, \label{rec1}$$ where $\tau=d_1\lambda$. We choose a cut-off function $\theta(z)$ such that $\theta w$ is compact support in $\mathcal{D}$. We select the cut-off function $\theta\in C^{\infty}_{0}(\mathcal{D}\backslash \cup_l D_l
) $ with following properties:\
(i) $\theta(z)=1$, if $|z|<\frac{3}{2}$ and $|z-z_l|>2\delta$.\
(ii) $|\nabla \theta|<C_3$, $|\triangle\theta|<C_4$ if $|z|>\frac{3}{2}$.\
(iii) $|\nabla \theta|<C_5 \delta^{-1}$, $|\triangle\theta|<C_6\delta^{-2}$ if $|z-z_l|<2\delta$.
Substituting $f=\theta w$ into (\[rec1\]) yields that $$\begin{aligned}
\int_{(|z|<\frac{3}{2})\cup(\frac{3}{2}\leq |z|\leq 2 )}|\triangle
(\theta w)+\tilde{b}\cdot\nabla (\theta w)|^2 |P|^{-2} e^{d_1\lambda
|z|^2} \geq C_2 \lambda^2\int_{|z|<\frac{3}{2}}| w|^2|P|^{-2}
e^{d_1\lambda |z|^2}. \nonumber\end{aligned}$$ From the equation (\[starr\]), $$\triangle (\theta w) +\hat{b}\cdot\nabla (\theta
w)=-\hat{q}\theta w+\triangle\theta w+2\nabla \theta\cdot\nabla
w+\hat{b}\cdot\nabla \theta w.$$ By the assumption of $\theta$, we obtain $$|\triangle \theta w|+|\nabla \theta\cdot\nabla w|+|\nabla \theta w|\leq C_7 \delta^{m_l}, \
\mbox{if} \ |z-z_l|\leq
2\delta.$$ Taking $\delta\to 0$, by dominated convergence theorem, we have $$\begin{aligned}
c\lambda^2\int_{|z|<\frac{3}{2}} | w|^2 |P|^{-2} e^{d_1\lambda
|z|^2} &+&C_9(1+\lambda)^2\int_{\frac{3}{2}\leq |z|\leq 2} (|
w|^2+|\nabla w|^2) |P|^{-2} e^{d_1\lambda |z|^2} \nonumber \\&& \geq
C_{10} \lambda^2\int_{|z|<\frac{3}{2}}| w|^2|P|^{-2} e^{d_1\lambda
|z|^2}. \label{revi}\end{aligned}$$ Since $c$ is sufficiently small, we can absorb the first term in the left hand side of (\[revi\]) into the right hand side. Then $$\int_{\frac{3}{2}\leq |z|\leq 2} (| w|^2+|\nabla w|^2) |P|^{-2}
e^{d_1\lambda |z|^2} \geq C_{11}\int_{|z|\leq \frac{1}{2}}|
w|^2|P|^{-2} e^{d_1\lambda |z|^2}.$$ Obviously, it follows that $$\max_{|z|\geq \frac{3}{2}} |P|^{-2}\int_{\frac{3}{2}\leq |z|\leq 2}
(| w|^2+|\nabla w|^2) e^{d_1\lambda |z|^2} \geq C_{11}
(\min_{|z|\leq \frac{1}{2}}|P|^{-2})\int_{|z|\leq \frac{1}{2}}| w|^2.$$ By standard elliptic theory, the last inequality implies
$$\max_{|z|\geq \frac{3}{2}} |P|^{-2}e^{d_2\lambda} \int_{|z|\leq
\frac{5}{2}} | w|^2\geq C_{11} (\min_{|z|\leq
\frac{1}{2}}|P|^{-2})\int_{|z|\leq \frac{1}{2}}| w|^2.
\label{than}$$
We claim that $$e^{d_3 \sum m_l}\leq
\frac{\min_{|z|\leq \frac{1}{2}} |P|^{-2}} {\max_{|z|\geq
\frac{3}{2}} |P|^{-2}}. \label{more}$$ To prove (\[more\]), it reduces to verify $$e^{-d_4 \sum m_l}\min_{|z|\geq \frac{3}{2}} |P|\geq \max_{|z|\leq \frac{1}{2}}
|P|.
\label{mmore}$$ away from singular point $z_l$. Clearly, $$\max_{|z|\leq \frac{1}{2}} |P|\leq (\frac{1}{2})^{\sum m_l }.$$ Since $z_l\in \mathbb B (0, \frac{1}{10})$, we have $$(\frac{3}{4})^{\sum m_l }\leq\min_{|z|\geq \frac{3}{2}} |P|.$$ Combining the last two inequalities, we obtain (\[mmore\]). The claim is shown. Let’s return to (\[than\]), we get $$\begin{aligned}
\frac{\min_{|z|\leq \frac{1}{2}} |P|^{-2}} {\max_{|z|\geq
\frac{3}{2}}|P|^{-2}}&\leq& \frac{e^{d_5\lambda}\int_{|z|\leq
\frac{5}{2}}|w|^2 }{C_{11}\int_{|z|\leq \frac{1}{2}}|w|^2 }\nonumber
\\&\leq& e^{d_6\lambda},\end{aligned}$$ where we applied doubling estimates in the last inequality. Thanks to (\[more\]), we obtain $$\sum m_l\leq d_7\lambda.$$ Since $n_l=m_l+1\leq 2m_l$, we complete the lemma for $n_l\geq 2.$
If the vanishing order for the singular point is two, i.e. $n_l=1$. We consider $Q(z)=\prod(z-z_l)^{\frac{n_l}{2}}$ instead of $P(z)$. In this case, $Q(z)$ may not be defined as a single valued holmorphic function on $\mathcal {C}$. We pass to a finite branched cover of the disk $\mathcal {D}$ punctured at $z_l$. The Carleman estimates in previous sections still work. The same conclusion will follow.
Based on the vanishing order estimate in proposition \[pro2\], we are able to count the number of singular points.
We cover the double manifold $\overline{\mathcal{M}}$ by geodesic balls with radius $C\lambda^{-1/2}$. Since $\overline{\mathcal{M}}$ is compact, the order of those balls is $C\lambda$. From proposition \[pro2\], the conclusion in theorem \[th2\] is arrived.
Thanks to proposition \[pro2\], we can actually show a stronger result. Let $z_l\in \mathcal{M}$ be singular point with vanishing order $n_l+1$, Then $\sum_l n_l\leq C\lambda^2$.
Growth of eigenfunctions
========================
In this section, we will show that the eigenfunctions do not grow rapidly on too many small balls. We still restrict $v$ into the small geodesic ball $\mathbb B(p, c\lambda^{-\frac{1}{2}})$ in the conformal chart. Let $w(z)= v(c {\lambda^{-\frac{1}{2}}}z)$. Then $w$ satisfies the elliptic equation (\[starr\]) with assumptions (\[core2\]) in a Euclidean ball of radius four centered at origin. If we suppose that $w$ grow rapidly, that is, $$C_{1}\int_{(1-3a)\delta\leq|z-z_l|\leq (1-\frac{3a}{4})\delta}w^2
\leq \int_{(1-\frac{3a}{2})\delta\leq |z-z_l|\leq (1-a)\delta} w^2
\label{rap}$$ for all $l$ and some large $C_1$, then the following proposition is valid.
Suppose $D_l$ are disks contained in a Euclidean ball of radius $\frac{1}{30}$ centered at origin. Furthermore, assume that ($R_1$): $ \delta< d_1\lambda^{-1}$ and ($R_2$): $|z_l-z_k|>d_2\lambda^{\frac{1}{2}}\delta$, when $l\not=k$. If (\[rap\]) holds for all $l$, the number of disks $D_l$ is less than $d_3\lambda$. \[proh\]
We will use the stronger Carleman estimates in (\[keys1\]) in proposition \[pro21\]. We prove it by contradiction. Suppose that the collection $D_l=\{z| \,|z-z_l|\leq \delta\}$ are disjoint disks satisfying the hypotheses $(R_1)$-$(R_3)$ in section 3. Without loss of generality, we require all the $D_l$ are in a ball centered at origin with radius $\frac{1}{30}$. As before, $D_l(a)=\{z|\,
|z-z_l|\leq (1-2a)\delta\},$ where $a$ is a suitably small positive constant. Let $\mathcal {D}$ be a ball centered at origin with radius 2. We choose a cut-off function $\theta\in
C^{\infty}_{0}(\mathcal{D}\backslash
\cup_l D_l ) $ and assume $\theta(z)$ satisfies the following properties:\
(i) $\theta(z)=1$, if $|z|<1$ and $|z-z_l|>(1-\frac{3}{2}a)\delta$ for all $l$.\
(ii) $|\nabla \theta|+|\triangle\theta|<C_2$ if $|z|>1$.\
(iii) $|\nabla \theta|<C_3 \delta^{-1}$, $|\triangle\theta|<C_4\delta^{-2}$ if $|z-z_l|<(1-\frac{3}{2}a)\delta$.\
Substituting $f=\theta w$ into (\[keys1\]) gives that $$\begin{aligned}
\int_{\mathcal{D}}|\triangle (\theta w)+\hat{b}\cdot\nabla (\theta
w)|^2 |P|^{-2} e^{d_4\lambda |z|^2}& \geq & C_5 \lambda^2
\int_{\mathcal{D}}|
\theta w|^2|P|^{-2} e^{d_4\lambda|z|^2} \nonumber \\
&+&C_{6}\delta^{-4}\int_{A}| \theta w|^2|P|^{-2}e^{d_4\lambda|z|^2}.
\label{keyss}\end{aligned}$$ We also assume $\tau=d_4\lambda$. Recall that $A=\cup_l A_l$ and $A_l=\{z| (1-2a)\delta\leq
|z-z_l|\leq (1-a)\delta\}.$ We first consider the integral in the left hand side of the last inequality. Again, by (\[starr\]), $$\triangle (\theta w) +\hat{b}\cdot\nabla (\theta
w)=-\hat{q}\theta w+\triangle\theta w+2\nabla \theta\cdot\nabla
w+\hat{b}\cdot\nabla \theta w.$$ Thus,$$|\triangle (\theta w) +\hat{b}\cdot\nabla (\theta w)|^2 \leq
C(c\lambda^2\theta^2 w^2+|\triangle\theta|^2 w^2+|\nabla \theta|^2|\nabla w|^2 +c\lambda|\nabla
\theta|^2 w^2),$$ where $c$ is sufficiently small. We will absorb the term involving $\theta^2 w^2$ into the right hand side of (\[keyss\]). Since $c$ is small enough, we get $$\begin{aligned}
\int_{\mathcal{D}}(|\triangle \theta|^2 w^2+ c|\nabla \theta|^2
w^2+|\nabla \theta|^2|\nabla w|^2 ) |P|^{-2} e^{d_4\lambda |z|^2}&
\geq & C_7 \lambda^2 \int_{\mathcal{D}}|
\theta w|^2|P|^{-2} e^{d_4\lambda|z|^2} \nonumber \\
&+&C_{8}\delta^{-4}\int_{A}| \theta w|^2|P|^{-2}e^{d_4\lambda|z|^2}.\end{aligned}$$ Using the properties of $\theta(z)$ and taking into account that all $D_l$ lies in the ball centered at origin with radius $\frac{1}{30}$, we obtain $$\begin{aligned}
\int_{\mathcal{D}}(|\triangle \theta|^2 w^2 + c|\nabla \theta|^2 w^2
&+&|\nabla \theta|^2|\nabla w|^2 ) |P|^{-2} e^{d_4\lambda |z|^2}
\geq C_7 \lambda^2 \int_{\frac{1}{4}\leq |z|\leq\frac{1}{2} }|
w|^2|P|^{-2} e^{d_4\lambda|z|^2} \nonumber \\
&+&C_{9}\delta^{-4}\sum_{l} \int_{(1-\frac{3a}{2})\delta\leq
|z-z_l|\leq (1-a)\delta}| w|^2|P|^{-2}e^{d_4\lambda|z|^2}.
\label{cont}\end{aligned}$$ Next we want to control the left hand side of last inequality. Write $$\begin{aligned}
\int_{\mathcal{D}}|\triangle \theta|^2 w^2 + c|\nabla \theta|^2 w^2
+|\nabla \theta|^2|\nabla w|^2 ) |P|^{-2} e^{d_4\lambda |z|^2}
=I+\sum_l I_l,\end{aligned}$$ where $$I=\int_{1\leq|z|\leq 2}|\triangle \theta|^2 w^2 + c|\nabla \theta|^2 w^2
+|\nabla \theta|^2|\nabla w|^2 ) |P|^{-2} e^{d_4\lambda |z|^2},$$ $$I_l=\int_{(1-2a)\delta\leq|z-z_l|\leq (1-\frac{3a}{2})\delta}|\triangle \theta|^2 w^2 + c|\nabla \theta|^2 w^2
+|\nabla \theta|^2|\nabla w|^2 ) |P|^{-2} e^{d_4\lambda |z|^2}.$$ By standard elliptic estimates, $$I\leq e^{d_5\lambda}\max_{|z|\geq 1}|P|^{-2}\int_{\frac{3}{4}\leq
|z|\leq \frac{5}{2}}w^2.$$ Similarly, via elliptic estimates, $$I_l\leq
C_{10}\delta^{-4}(\max_{A_l}|P|^{-2}e^{d_4\lambda|z|})\int_{(1-3a)\delta\leq|z-z_l|\leq
(1-\frac{3a}{4})\delta}w^2.$$ Thanks to lemma \[lemc\], $$I_l\leq
C_{11}\delta^{-4}(\min_{A_l}|P|^{-2}e^{d_4\lambda|z|})\int_{(1-3a)\delta\leq|z-z_l|\leq
(1-\frac{3a}{4})\delta}w^2.$$ Combining those inequalities together in (\[cont\]) leads to $$\begin{aligned}
e^{d_5\lambda}\max_{|z|\geq 1}|P|^{-2}\int_{\frac{3}{4}\leq |z|\leq
\frac{5}{2}}w^2&+&C_{11}\delta^{-4}\sum_l(\min_{A_l}|P|^{-2}e^{d_4\lambda|z|})\int_{(1-3a)\delta\leq|z-z_l|\leq
(1-\frac{3a}{4})\delta}w^2 \nonumber \\
&\geq& C_{12} \min_{|z|\leq \frac{1}{2}}|P|^{-2}
\int_{\frac{1}{4}\leq |z|\leq\frac{1}{2} }|
w|^2 \nonumber \\
&+&C_{13}\delta^{-4}\sum_{l} \min_{A_l}(|P|^{-2}e^{d_4\lambda|z|^2})
\int_{(1-\frac{3a}{2})\delta\leq |z-z_l|\leq (1-a)\delta} w^2.
\label{end}\end{aligned}$$ Performing the similar arguments as (\[more\]) shows that $$\min_{|z|\leq \frac{1}{2}}|P|^{-2}>\max_{|z|\geq 1}|P|^{-2} e^{d_5\sum_l m_l}.$$ If the number of $D_l$ is $d_3\lambda$, then $$\min_{|z|\leq \frac{1}{2}}|P|^{-2}>\max_{|z|\geq 1}|P|^{-2}
e^{d_6\lambda}. \label{ppp}$$ We claim that $$e^{C_{14}\lambda}\int_{\frac{1}{4}\leq |z|\leq \frac{1}{2}}w^2\geq
\int_{\frac{3}{4}\leq |z|\leq \frac{5}{2}}w^2. \label{cla}$$ We prove the claim by doubling estimates shown in proposition \[proo\]. We choose a ball $\mathbb B(x_0, \frac{1}{8})\subset
\{z|\frac{1}{4}\leq |z|\leq \frac{1}{2}\}$. It is clear that $$\int_{\frac{1}{4}\leq |z|\leq \frac{1}{2}}w^2\geq \int_{\mathbb B(x_0, \frac{1}{8})}w^2.$$ Using doubling estimates, we have $$e^{C_{15}\lambda} \int_{\mathbb B(x_0, \frac{1}{8})}w^2\geq \int_{\mathbb B(x_0, \frac{2}{8})}w^2.$$ By finite iterations, we can find a large ball $\mathbb B(x_0, 3)$ that contains $\{z|\frac{3}{4}\leq |z|\leq \frac{5}{2}\}$. It yields that $$\int_{ \mathbb B(x_0, 3)}w^2\geq \int_{\frac{3}{4}\leq |z|\leq \frac{5}{2}}w^2.$$ Then the combination of those inequalities verify the claim.
If we choose $d_3$ is suitably large, since the number disk $D_l$ is $d_3\lambda$, then $d_6$ is suitably large. From the inequalities (\[ppp\]) and (\[cla\]), it follows that $$e^{d_5\lambda}\max_{|z|\geq 1}|P|^{-2}\int_{\frac{3}{4}\leq |z|\leq
\frac{5}{2}}w^2< C_{12}
\min_{|z|\leq\frac{1}{2}}|P|^{-2}\int_{\frac{1}{4}\leq |z|\leq
\frac{1}{2}}w^2 . \label{get}$$ It contradicts the estimates (\[rap\]) and (\[end\]). The proposition is arrived.
Growth Estimates and Nodal Length
=================================
This section is to find the connection between growth of eigenfunctions and nodal length. A suitable small growth in $L^2$ norm implies an upper bound of nodal length. We consider the second order elliptic equations $$\triangle \bar w+{b}^\ast \cdot \nabla \bar w +{q}^\ast \bar w=0
\quad \mbox{in} \ \mathbb B(0,\, 4). \label{lll}$$ Assume that there exist a positive constant $C$ such that $\|{b}^\ast\|_{W^{1,\infty}}\leq C $ and $\|{q}^\ast\|_{W^{1,\infty}}\leq C$. The following lemma relies on the Carleman estimates in lemma \[carl\]. Suppose ${\epsilon}_1$ is a sufficiently small positive constant.
Suppose that $w$ satisfies the growth estimate $$\int_{(1-\frac{3a}{2}){\epsilon}_0<r<(1-{a}){\epsilon}_0}\bar w^2\leq C_3
\int_{(1-{3a}){\epsilon}_0<r<(1-\frac{4a}{3}){\epsilon}_0} \bar w^2, \label{hyp}$$ where $a$ and ${\epsilon}_0$ are fixed small constants. Then for $0<{\epsilon}_1<\frac{{\epsilon}_0}{100}$, we have $$\max_{r\leq
{\epsilon}_1}|\bar w|\geq C_4(\frac{{\epsilon}_1}{{\epsilon}_0})^{C_5}
\big(\dashint_{\mathbb B(0, (1-\frac{4}{3}a){\epsilon}_0)}\bar w^2\big)^{1/2},
\label{god}$$ where $\dashint$ denotes the average of the integration. \[prof\]
We select a radial cut-off function $\theta\in
C^\infty_0(\frac{{\epsilon}_1}{2}<r<(1-\frac{11a}{10}){\epsilon}_0)$ satisfies the properties:\
(i) $\theta(r)=1$ for $\frac{3{\epsilon}_1}{4}<r<(1-\frac{10a}{9}){\epsilon}_0$.\
(ii) $|\nabla\theta|+|\triangle \theta|\leq C_6 $ for $r>(1-\frac{10a}{9}){\epsilon}_0$.\
(iii) $|\nabla \theta|\leq C_7{\epsilon}_1^{-1}$, $|\triangle
\theta|<C_8{\epsilon}_1^{-2}$ for $r\leq \frac{3{\epsilon}_1}{4}$.\
From the equation (\[lll\]), we get $$\triangle (\theta \bar w)+{b}^\ast\cdot \nabla (\theta \bar w)+{q}^\ast\theta \bar w=\triangle \theta \bar w+2\nabla \theta \cdot\nabla \bar w+
{b}^\ast\cdot\nabla \theta \bar w.$$ Assume that $\tau>C$ is large enough. Substituting $u=\theta \bar w$ in lemma \[carl\] yields that $$C_2 \tau^3\int r^{{\epsilon}} e^{2\tau\phi(r)} \theta^2 \bar w^2 \,drd\omega\leq I,
\label{nea}$$ where $$I=\int r^4 e^{2\tau\phi(r)}|\triangle \theta \bar w+2\nabla \theta \cdot\nabla \bar w+
{b}^\ast\cdot\nabla \theta \bar w|^2\,drd\omega.$$ Note that $\phi(r)$ is a decreasing function. Furthermore, by the assumptions of $\theta(z)$, we obtain $$\begin{aligned}
I&\leq&
e^{2\tau\phi(\frac{{\epsilon}_1}{2})}\int_{\frac{{\epsilon}_1}{2}<r<\frac{3{\epsilon}_1}{4}}|\triangle
\theta \bar w+2\nabla \theta \cdot\nabla \bar w+ {b}^\ast\cdot\nabla
\theta \bar
w|^2r\,drd\omega \nonumber \\
&+&
e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{(1-\frac{10a}{9}){\epsilon}_0<r<(1-\frac{11a}{10}){\epsilon}_0}|\triangle
\theta \bar w+2\nabla \theta \cdot\nabla \bar w+ {b}^\ast\cdot\nabla
\theta \bar w|^2r\,drd\omega. \nonumber\end{aligned}$$ By standard elliptic estimates, we derive that $$I\leq
C_9e^{2\tau\phi(\frac{{\epsilon}_1}{2})}\int_{\frac{{\epsilon}_1}{4}<r<{{\epsilon}_1}}\bar
w^2 r\,drd\omega+
C_{10}e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{(1-\frac{3a}{2}){\epsilon}_0<r<(1-a){\epsilon}_0}
\bar w^2 r\,drd\omega.$$ Taking the inequality (\[nea\]) and assumptions of $\theta$ into account, we have $$\begin{aligned}
&C_{10}&e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{(1-\frac{3a}{2}){\epsilon}_0<r<(1-a){\epsilon}_0}\bar
w^2
r\,drd\omega+C_9e^{2\tau\phi(\frac{{\epsilon}_1}{2})}\int_{\frac{{\epsilon}_1}{4}<r<{{\epsilon}_1}}\bar
w^2 r\,drd\omega \nonumber \\ &\geq& C_2
\tau^3\int_{\frac{3{\epsilon}_1}{4}<r<(1-\frac{10a}{9}){\epsilon}_0} r^{{\epsilon}}
e^{2\tau\phi(r)} \bar w^2\,drd\omega \nonumber \\
&\geq& C_2
\tau^3\big((1-\frac{10a}{9}){\epsilon}_0\big)^{{\epsilon}-1}
\int_{\frac{3{\epsilon}_1}{4}<r<(1-\frac{10a}{9}){\epsilon}_0}e^{2\tau\phi(r)}\bar
w^2r\,drd\omega. \label{nnn}\end{aligned}$$ Since ${\epsilon}, {\epsilon}_0$ are fixed positive constants, Taking $\tau$ large enough, we obtain $$\frac{1}{2}C_2
\tau^3\big((1-\frac{10a}{9}){\epsilon}_0\big)^{{\epsilon}-1}>C_{10}.$$ Taking the hypothesis (\[hyp\]) into consideration, we can incorporate the first term in the left hand side of inequality (\[nnn\]) into the right hand side. It follows that $$C_9e^{2\tau\phi(\frac{{\epsilon}_1}{2})}\int_{\frac{{\epsilon}_1}{4}<r<{{\epsilon}_1}}\bar
w^2 r\,drd\omega\geq C_{10}
e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{\frac{3{\epsilon}_1}{4}<r<(1-\frac{10a}{9}){\epsilon}_0}\bar
w^2r\,drd\omega.$$ Fix such $\tau$, adding the following term $$e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{r<\frac{3{\epsilon}_1}{4}}\bar
w^2r\,drd\omega$$ to both sides of last inequality yields that $$e^{2\tau\phi(\frac{{\epsilon}_1}{2})}\int_{r<{{\epsilon}_1}}\bar w^2
r\,drd\omega\geq C_{11}
e^{2\tau\phi((1-\frac{10a}{9}){\epsilon}_0)}\int_{r<(1-\frac{4a}{3}){\epsilon}_0}\bar
w^2r\,drd\omega,$$ where we have used the fact that $\phi$ is decreasing. Straightforward calculations show that $$e^{2\tau\big(\phi((1-\frac{10a}{9}){\epsilon}_0)-\phi(\frac{{\epsilon}_{1}}{2})\big)} \geq C_{13} (\frac{{\epsilon}_1}{{\epsilon}_0})^{C_{12}}.$$ Thus, $$\int_{r<{{\epsilon}_1}}\bar w^2 r\,drd\omega\geq
C_{13}(\frac{{\epsilon}_1}{{\epsilon}_0})^{C_{12}}\int_{r<(1-\frac{4a}{3}){\epsilon}_0}\bar
w^2r\,drd\omega.$$ This completes the lemma.
Our next goal is to find the relation of lemma \[prof\] and nodal length. We assume that the estimate (\[hyp\]) exists. Then the conclusion (\[god\]) in lemma \[prof\] holds. For ${\epsilon}_1\leq
\frac{{\epsilon}}{100}$, if $|z|<{\epsilon}_1$, using Taylor’s expansion gives that $$|\bar w(z)-\sum_{|\alpha|\leq C_{5}}\frac{1}{\alpha !}\frac{\partial^{\alpha} \bar w}{\partial z^{\alpha}}(0)z^\alpha
|\leq \sup_{|z|\leq
{\epsilon}_1}\sup_{|\alpha|=C_{5}+1}d_1|\frac{\partial^{\alpha} \bar
w}{\partial z^{\alpha}}(z)|{{\epsilon}_1}^{C_{5}+1},$$ where $\alpha=(\alpha_1, \alpha_2)$ and $\frac{\partial}{\partial
z^\alpha}=\frac{\partial}{\partial
{z_1}^{\alpha_1}}\frac{\partial}{\partial {z_2}^{\alpha_2}}$. To control the right hand side of the last inequality, by elliptic estimates and rescaling argument, we have $$|\bar w(z)-\sum_{|\alpha|\leq C_{5}}\frac{1}{\alpha !}\frac{\partial^{\alpha} \bar w}{\partial z^{\alpha}}(0)z^\alpha
|\leq d_2(\dashint_{\mathbb B(0, (1-\frac{4}{3}a){\epsilon}_0)}\bar
w^2)^{1/2}(\frac{{\epsilon}_1}{{\epsilon}_0})^{C_{5}+1}.$$ Using the estimate (\[god\]) in lemma \[prof\], we get $$|\bar w(z)-\sum_{|\alpha|\leq C_{5}}\frac{1}{\alpha !}\frac{\partial^{\alpha} \bar w}{\partial z^{\alpha}}(0)z^\alpha
|\leq d_3(\frac{{\epsilon}_1}{{\epsilon}_0}) \max_{|z|\leq {\epsilon}_1}|\bar w|.$$ Choosing ${{\epsilon}_1}/{{\epsilon}_0}$ sufficiently small, by the triangle inequality, we obtain $$\sup_{|\alpha|\leq C_{5}}\mid\frac{\partial^{\alpha} \bar w}{\partial
z^{\alpha}}(0)\mid{{\epsilon}_1}^{|\alpha|}\geq d_4 \max_{|z|\leq
{\epsilon}_1}|\bar w|.$$ Applying again the estimate (\[god\]) to the right hand side of the last inequality yields that $$\sup_{|\alpha|\leq C_{5}} \mid \frac{\partial^{\alpha} \bar w}{\partial
z^{\alpha}}(0)\mid{{\epsilon}_0}^{|\alpha|}\geq d_5 \big(\dashint_{\mathbb B(0,
(1-\frac{4}{3}a){\epsilon}_0)}\bar w^2\big)^{1/2}. \label{stan}$$ By standard elliptic estimates, we also have $$\sup_{|z|\leq \frac{{\epsilon}_0}{2}}\sup_{|\alpha|\leq C_{5}+1}
\mid \frac{\partial^{\alpha} \bar w}{\partial
z^{\alpha}}(z)\mid {{\epsilon}_0}^{|\alpha|}\leq d_6 \big(\dashint_{\mathbb B(0,
(1-\frac{4}{3}a){\epsilon}_0)}\bar w^2\big)^{1/2}. \label{stan1}$$
The basic relationship between derivatives and nodal length in two dimensions is shown in [@DF2].
Suppose that $\bar w$ satisfies (\[stan\]) and (\[stan1\]). Then $$H^1(z| \, |z|\leq d_7\bar{\epsilon}\ \mbox{and} \ \bar w(z)=0)\leq d_8\bar{\epsilon}.$$
With aid of the last lemma, we can readily obtain an upper nodal length estimate.
Let $\bar w$ be the solution of (\[lll\]). Suppose that $\bar
{\epsilon}\leq {\epsilon}_0$ and $w$ satisfies the growth condition $$\int_{(1-\frac{3a}{2})\bar {\epsilon}<r<(1-{a})\bar {\epsilon}}\bar w^2\leq C_3
\int_{(1-{3a})\bar {\epsilon}<r<(1-\frac{4a}{3})\bar {\epsilon}} \bar w^2.
\label{aid}$$ Then $$H^1(z|\, |z|\leq d_9\bar{\epsilon}\ \mbox{and} \ \bar w(z)=0)\leq d_{10}\bar{\epsilon}.$$ \[leng\]
Since the inequalities (\[stan\]) and (\[stan1\]) can be derived from (\[aid\]) by lemma \[prof\], the proposition follows from last lemma.
Total Nodal Length
==================
As the proposition \[proh\] indicates that the eigenfunctions can not grow rapidly on too many small balls. If it grows slowly, we have an upper bound on the local length of nodal sets by proposition \[leng\]. In this section, we will link these two arguments together. To achieve it, we will employ a process of repeated subdivision and selection squares. The idea is inspired by [@DF2].
Assume that $\mathbb B(p, c\lambda^{-\frac{1}{2}})$ is a geodesic ball of double manifold $\overline{\mathcal{M}}$. Choosing $c$ to be small, then it is contained in a conformal chart. Let $w(z)= v(c
{\lambda^{-\frac{1}{2}}}z)$ with $c$ sufficiently small. We know that $w$ satisfies $$\triangle w+\hat{{b}}(x)\cdot\nabla w+\hat{{q}}(x)w=0 \quad
\mbox{in}\ \mathbb B( 0, 4).$$
We consider the square $P=\{ (x,
y)| \max(|x|, |y|)\leq \frac{1}{60}\}$ in $\mathbb B (0, 4)$ and divide it into a grid of closed square $P_l$ with side $ \delta\leq
a_1\lambda^{-1}$. If (\[rap\]) holds for some point $z_l\in P_l$ and for some sufficiently large $C_{1}$. We call $P_l$ as a square of rapid growth. With aid of proposition \[proh\], we are able to obtain the following result.
There are at most $C\lambda^2$ squares with side $ \delta$ where $w$ is of rapid growth. \[fina\]
Let $I_1$ be the collection of those indices $l$ for which $P_l$ is a square of rapid growth. For each $l\in I_1$, there exists some point $z_l\in P_l$ such that (\[rap\]) holds. Let $|I_1|$ denote the cardinality of $I_1$. Define $$P_l^{\ast}=\{z|\, |z-z_l|<d_1\delta \lambda^{\frac{1}{2}}\}.$$ The collection of disks $P_l^{\ast}$ covers the collection of square $P_l$ for $l\in I_1$. We choose a maximal collection of disjoint disks of $P_l^{\ast}$ and denote it as $I_2$. If $l\in I_2$, we define $$P_l^{\ast\ast}=\{z| \,|z-z_l|<4d_1\delta \lambda^{\frac{1}{2}}\}.$$ Since the collection of disks in $I_2$ are disjoint and maximal, we obtain that $$\bigcup_{l\in I_2}P_l^{\ast\ast}\supseteq \bigcup_{l\in I_1}P_l^{\ast} \supseteq \bigcup_{l\in I_1}P_l.$$ Thus, $$|I_2|\times 16d_1^2\delta^2\lambda\geq |I_1|\delta^2,$$ which implies $$|I_2|\lambda\geq d_2 |I_1|.$$ Recall from proposition \[proh\] that $|I_2|\leq d_3\lambda$. Therefore, we obtain the desirable estimate $|I_1|\leq
d_4\lambda^2$.
Now we introduce a iterative process of bisecting squares. We begin by dividing the square into a grid of square $P_l(1)$ with side $\delta(1)=a_1\lambda^{-1}$, then separate them into two categories $R_l(1)$ and $S_l(1)$. $R_l(1)$ are those where $w$ is of rapid growth and $S_l(1)$ are those where (\[rap\]) fails for $w$. We continue to bisect each square $R_l(1)$ to obtain square $P_{l}(2)$ with side $\delta(2)=\frac{\delta(1)}{2}$. Again, we split $P_i(2)$ into the subcollection $R_l(2)$ with rapid growth and $S_l(2)$ with slow growth. We repeat the process at each step $k$. Then there are squares $R_l(k)$ and $S_l(k)$ with $\delta(k)=\frac{\delta(1)}{2^k}$. We count the number of $R_l(k)$ and $S_l(k)$ at step $k$.
\(i) The number of squares $R_l(k)$ is at most $C_2\lambda^2$.\
(ii) The number of squares $S_l(k)$ is at most $C_3\lambda^2$. \[mei\]
The conclusion (i) follows directly from the lemma \[fina\]. We only need to show (ii). If $k=1$, the conclusion (ii) follows because the total number of squares is at most the order of $\lambda^2$. If $k\geq 2$, by construction of those squares, $$|S_l(k)|\leq 4|R_l(k-1)|\leq C_4\lambda^2,$$ where we have used (i) in the last inequality. The lemma is done.
Next lemma tells that almost every point lies in some $R_l(k)$ with slow growth. It is the lemma 6.3 in [@DF2].
$\bigcup_{k,l} S_l(k)$ covers the square $P$ except for singular points $\mathcal {S}=\{z\in P| w(x)=0, \nabla w=0\}$. \[singular\]
We are ready to give the proof of Theorem \[th1\].
Consider $\bar w(z)= w(z_l+{\epsilon}_0^{-1}\delta(k)z)$. Then $\bar w(z)$ satisfies the equation (\[lll\]). Choosing a finite collection of $z_l\in S_l(k) $ and applying proposition \[leng\], we have $$H^1(z| \, w(z)=0 \ \mbox{and} \ z\in S_l(k) )\leq C_5 2^{-k}
\lambda^{-1}. \label{final}$$ Furthermore, thanks to lemma \[singular\], $$\begin{aligned}
H^1(z| \, w(z)=0 \ \mbox{and} \ \max(|x|, |y|)\leq \frac{1}{60}
)&\leq & \sum_{l, k} H^1(z|
\, w(z)=0 \ \mbox{and} \ z\in S_l(k) ) \nonumber \\
&\leq & \lambda^2 \sum_k C_5 2^{-k} \lambda^{-1} \nonumber\\
&\leq & C_6\lambda,\end{aligned}$$ where we have used (ii) in lemma \[mei\] and (\[final\]). Since $w(z)=v(c\lambda^{-\frac{1}{2}}z)$, by the rescaling argument, we obtain $$H^1( \{v(z)=0\}\cap \mathbb B(p, c\lambda^{-\frac{1}{2}}))\leq C_6\lambda^{\frac{1}{2}}.$$ Finally, covering $\overline{\mathcal{M}}$ with order $\lambda$ of geodesic balls with radius $c\lambda^{-\frac{1}{2}}$, we readily deduce that $$H^1(z\in \overline{\mathcal{M}} |\, v(z)=0) \leq C_7\lambda^{\frac{3}{2}}.$$ So is $H^1(\mathcal{N}_\lambda)$.
[CL]{}
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[^1]: Research is partially supported by the NSF grant DMS 1500468.
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---
abstract: 'We propose a deep neural network for supervised learning on neuroanatomical shapes. The network directly operates on raw point clouds without the need for mesh processing or the identification of point correspondences, as spatial transformer networks map the data to a canonical space. Instead of relying on *hand-crafted* shape descriptors, an optimal representation is learned in the end-to-end training stage of the network. The proposed network consists of multiple branches, so that features for multiple structures are learned simultaneously. We demonstrate the performance of our method on two applications: (i) the prediction of Alzheimer’s disease and mild cognitive impairment and (ii) the regression of the brain age. Finally, we visualize the important parts of the anatomy for the prediction by adapting the occlusion method to point clouds.'
author:
- 'Benjamín Gutiérrez-Becker'
- Christian Wachinger
bibliography:
- 'biblio.bib'
title: ' Deep Multi-Structural Shape Analysis: Application to Neuroanatomy'
---
Introduction
============
Shape analysis of anatomical structures is of core importance for many tasks in medical imaging, not only as a regularization prior for segmentation tasks, but also as a powerful tool to assess differences between subjects and populations. A fundamental question when operating on shapes is to find a suitable numerical representation for a given task. Hence, many different types of parameterizations have been proposed in the past including point distribution models [@Cootes1995], spectral signatures [@Wachinger2015], spherical harmonics [@gerardin2009multidimensional], medial representations [@Gorczowski2007], and diffeomorphisms [@miller2014diffeomorphometry]. Even though these representations have proven their utility for the analysis of shapes in the medical domain, they might not be optimal for a particular task. In recent years, deep networks have had ample success for many medical imaging tasks by learning complex, hierarchical feature representations from images. These representations have proven to outperform *hand-crafted* features in a variety of medical imaging applications [@Litjens2017]. One of the main reasons for the success of these methods is the use of convolutional layers, which take advantage of the shift-invariance properties of images [@Bronstein2017]. However, the use of deep networks in medical shape analysis is still largely unexplored; mainly because typical shape representations such as point clouds and meshes do not possess an underlying Euclidean or grid-like structure.
In this work, we propose an alternative approach to perform supervised learning on medical shape data. Our method is based on PointNet [@Qi2017], a deep neural network architecture, which operates directly on a point cloud and predicts a label in an end-to-end fashion. Point clouds present a raw and simple parameterization that avoids complexities involved with meshes and that is trivial to obtain given a segmented surface. The network does not require the alignment of point clouds, as a spatial transformer network maps the data to a canonical space before further processing. PointNet has been proposed for object classification, where the category of a single shape is predicted. For many medical applications however, not just a single anatomical structure is important for the prediction but a simultaneous view of multiple structures is required for a more comprehensive analysis of a subject’s anatomy. Hence, we propose the Multi-Structure PointNet (MSPNet), which is able to simultaneously predict a label given the shape of multiple structures. We evaluate MSPNet in two neuroimaging applications, neurodegenerative disease prediction and age regression.
Related Work
------------
Several shape representations have previously been used for supervised learning tasks. Spherical harmonics for approximating the hippocampal shape have been proposed in [@gerardin2009multidimensional]. Shape information has been derived from thickness measurements of the hippocampus from a medial representation [@costafreda2011automated]. Statistical shape models to detect hippocampal shape changes were proposed by [@shen2012detecting]. Multi-resolution shape features with non-Euclidean wavelets were employed for the analysis of cortical thickness [@Kim2014107]. The use of medial axis shape representations was used to compare the brain morphology of autistic and normal populations [@Gorczowski2007]. Recently, shape representation based on spectral signatures have been introduced to perform age regression and disease prediction [@Wachinger2015; @wachinger2016domain].
All the mentioned approaches rely on computing pre-defined shape features. Alternatively, a variational auto-encoder was proposed to automatically extract features from 3D surfaces, which can in turn be used in a classification task [@Shakeri2016]. However different to our approach, this is not an end-to-end learning since the variational encoder is not directly linked to the classification task. Consequently, the learned features capture overall variation but are not directly optimized for the given task. In addition, this approach relies on computing point correspondences between meshes and focuses on a single structure, while we simultaneously model multiple structures.
Method
======
We propose a method for multiple structure shape analysis that is divided into two main stages: the extraction of point clouds representing the anatomy of different structures from medical images (section \[sec:pointcloud\]), and a Multi-Structure PointNet () (section \[sec:architecture\]). Figure \[fig:network\] illustrates the architecture of MSPNet, which is based on PointNet [@Qi2017], and extends on it to allow the simultaneous processing of multiple structures.
{width="90.00000%"}
Point Cloud Extraction {#sec:pointcloud}
----------------------
We extract point clouds from MRI T1-weighted images of the brain. We process the images with the FreeSurfer pipeline [@Fischl2012] and obtain segmentations of multiple neuroanatomical regions. From the resulting segmentations, point clouds are created by uniformly sampling the boundary of each brain structure. After this process, the anatomy of a subject is represented by a collection of $m$ point clouds $S=\{ P_0, P_1, \hdots P_m \}$, where each point cloud represents a structure. A point cloud is defined as a set of $n$ points $P = [\mathbf{p}_0, \mathbf{p}_1, \hdots , \mathbf{p}_n] $, where each point is a vector of Cartesian coordinates $\mathbf{p}_i = (x,y,z)$.
MSPNet Architecture {#sec:architecture}
-------------------
We aim at finding a network architecture corresponding to a function $f : S \mapsto y$, mapping a collection of shapes described by $S$ to a prediction $y$. An overview of the network is shown in figure \[fig:network\]. MSPNet consists of multiple branches, where each branch processes the point cloud of one structure independently. This ensures that an optimal feature representation is learned per structure. At the end, the features of all branches are merged to perform a joint prediction. Each branch can be divided into the following stages: 1) point cloud alignment using a transformation network, 2) feature extraction, 3) feature alignment with a second transformation net, 4) dropout and 5) prediction. The first three stages of the architecture of each branch resemble that of a single PointNet architecture. The last two stages are particular to .
**Point Transformation Network:** In contrast to previous approaches in deep medical shape analysis [@Shakeri2016], MSPNet does not require point correspondences across shapes, i.e., the i-$th$ points of two shapes, $\mathbf{p}_i^1$ and $\mathbf{p}_i^2$, respectively, do not need to represent the same anatomical position. We obtain the invariance to rigid transformations in MSPNet by (i) augmenting the training dataset by applying a random rigid transformation to each shape during training time and by (ii) introducing a transformation network (T-Net). This network estimates a $3 \times 3$ transformation matrix, which is applied to the input as a first step. One can think of the T-Net as a transformation into a canonical space to roughly align point clouds before any processing is done. The T-Net is shown in figure \[fig:tnet\] and is composed of a multilayer perceptron (MLP), a max pooling operator and two fully connected layers.
**Feature Extractions:** The transformed points are fed into a MLP with shared weights among points. This MLP layer can be thought of as the feature extraction stage of the network. At this stage of the network, each point has access to the position of all the remaining points of the point cloud, and therefore as the output of the network, we obtain a $k$-dimensional feature vector for each point (in our case $k$ = 64). Although each point is assigned a single feature vector, in practice each feature vector point contains a global signature of the input point cloud.
**Feature Transformation:** A second T-Net is applied to the computed features. This network has the same properties as the first transformation network, but its output corresponds to a $k \times k$ transformation matrix. This transformation matrix has a much higher dimension than the previous spatial transformation, which makes the optimization more challenging. To facilitate the optimization of this larger feature transformation matrix $T$, we constrain it to be close to an orthogonal matrix $C_\text{reg} = \| I - T T^\top \|^2_F$, similar to [@Qi2017]. The regularization term ensures a more stable convergence of the network. After the points are transformed they are fed to a MLP layer.
**Dropout and prediction:** Up to this point, the architecture of each branch mirrors that of the PointNet. However the final dropout and prediction stage is particular to . In PointNet, the last stage corresponds to a max-pooling layer performed across $n$ points, so that the output is a vector with size corresponding to the feature dimensionality. Instead of performing max-pooling, which leads to a strong shrinkage in feature space, we propose to keep the localized information per point. This leads to an increase in the network capacity, which may lend itself to overfitting. Hence, we introduce a dropout layer (keep probability = 0.3) for regularization. The main advantage of the new design is that more localized information is retained in the network, which we hypothesize may boost the predictive power of our network. Finally, the individual features from each branch are concatenated and fed into a last MLP to perform prediction. Batch normalization is used for all MLP layers and ReLU activations are used. The last MLP perceptron counts with intermediate dropout layers with 0.7 keep probabilities as in PointNet. To facilitate the exposition, we assumed that each structure per branch is described by the same number of $n$ points, but in practice each structure can be represented by point clouds of different dimensions.
{width="80.00000%"}
Results
=======
We evaluate the performance of MSPNet in two supervised learning tasks, classification and regression. For the classification task, we aim at using shape descriptors to discriminate between healthy controls (HC), and patients diagnosed with mild cognitive impairment (MCI) or Alzheimer’s disease (AD). For the regression task, we perform age estimation of a subject based on shape information. In all our experiments, we compare to the standard PointNet architecture and spectral shape descriptors in BrainPrint [@Wachinger2015], which achieved high performance in a competition for Alzheimer’s disease classification [@wachinger2016domain]. For PointNet, the multi-structure input corresponds to a concatenation of the point clouds of all structures. We use image data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu) [@Jack2008]. We work with a total of 7,974 images (2,423 HC, 978 AD, and 4,625 MCI).
AD and MCI Classification on Shape Data
---------------------------------------
For this experiment, we perform classification based on the shape of the left and right hippocampus and the left and right lateral ventricles, due to their key importance in Alzheimer’s disease [@thompson2004mapping]. Each structure is represented by a Pointcloud of 512 points. For our experiments the dataset is split in a training, validation and test set (75%,15%,15%). Splitting is done on a per subject basis, to guarantee that the same subject does not appear in different sets. Table \[table:classification\] reports the results of the classification experiment, where we report average classification precision, recall and F1-score. In both classification scenarios, PointNet shows a higher accuracy than BrainPrint, illustrating the potential of learning feature representations. Further, MSPNet showed the best performance, highlighting the benefit of individual feature learning in each branch of the network.
------------ -------------- ----------- ------------- -------------- ----------- -------------
Precision Recall F1-score Precision Recall F1-score
BrainPrint 0.57 0.59 0.57 0.76 0.77 0.78
PointNet 0.60 0.61 0.59 0.77 0.77 0.78
0.62 0.60 **0.61** 0.78 0.79 **0.80**
------------ -------------- ----------- ------------- -------------- ----------- -------------
\[table:classification\]
Age Prediction on Shape Data
----------------------------
For the age estimation task, we perform two different evaluations. In the first one, we perform age estimation only on the healthy controls of the ADNI database. For the second evaluation, we also include patients diagnosed with MCI and AD. The evaluations are done again on the same brain structures used for the classification task. The results of these two experiments are summarized in the mean absolute error plots of figure \[fig:regression\]. For the experiment on HC MSPNet significantly outperformed BrainPrint (p-value ) and PointNet (p-value 0.03). In the experiment on all subjects both PointNet and MSPNet presented comparable performance, both outperforming BrainPrint (p-value 0.01).
![Mean absolute error for the age prediction experiment on healthy subjects (left) and on all subjects, including MCI and AD (right)[]{data-label="fig:regression"}](figures/boxplot_healthy.pdf){width="\textwidth"}
![Mean absolute error for the age prediction experiment on healthy subjects (left) and on all subjects, including MCI and AD (right)[]{data-label="fig:regression"}](figures/boxplot_dx.pdf){width="\textwidth"}
Visualizing Point Importance
----------------------------
Of key importance for making predictions with shapes is the ability to visualize the part of the anatomy that is driving the decision. This holds in particular in the clinical context. In , we introduce a simple yet effective method to visualize the importance that each point has in the prediction. Our visualization is inspired by the commonly used occlusion method [@Grun2016], which consists of occluding parts of a test image and observing differences in the network response. We apply a similar concept to visualize the response of . In our case, we assess the importance of each point in the classification task by occluding this point (making the point coordinates equal to 0) together with its nearest neighbors. Then the occluded point cloud is passed through the network and the response of the output ReLU is compared to that obtained when the full point cloud is evaluated. The difference between these responses can then be assigned as the importance of this particular point. In figure \[fig:visualization\], we can observe the result of using this visualization technique for one of the AD test subjects in the HC-AD classification experiment. If a point tends towards the red side of the scale, it indicates that by occluding this particular point, the network increases the activation of the AD class. This means that the region around this point is used by the network to predict AD. The exact opposite is true for points on the blue side of the scale. White points indicate that the network response was not largely affected by occluding this point. In the particular case of the example in figure \[fig:visualization\], the decision of the network to give this subject a AD label is mainly driven by the left hippocampus.
![Visualization of point importance for the HC-AD classification task for an AD subject. Figure on the left illustrates ventricles and hippocampi, while figures on the right illustrate single structures of the left hemisphere. []{data-label="fig:visualization"}](figures/visualization.pdf){width="85.00000%"}
Conclusion
==========
We introduced MSPNet, a deep neural network for shape analysis on multiple brain structures. To the best of our knowledge, this is the first time that a neural network for shape analysis on point clouds is proposed in medical applications. We have shown that our method is able to achieve high accuracy in both classification and regression tasks, when compared to shape descriptors based on spectral signatures. This performance is achieved without relying on point correspondences or meshes. MSPNet learns feature representations from multiple structures simultaneously. Finally, we illustrated point-wise importance for the prediction by adapting the occlusion method.\
**Acknowledgments.** This work was supported in part by SAP SE and the Bavarian State Ministry of Education, Science and the Arts in the framework of the Centre Digitisation.Bavaria (ZD.B).
|
---
abstract: 'We have measured the current-voltage characteristics of a Josephson junction with tunable Josephson energy $E_J$ embedded in an inductive environment provided by a chain of SQUIDs. Such an environment induces localization of the charge on the junction, which results in an enhancement of the zero-bias resistance of the circuit. We explain this result quantitatively in terms of the Bloch band dynamics of the localized charge. This dynamics is governed by charge diffusion in the lowest Bloch band of the Josephson junction as well as by Landau-Zener transitions out of the lowest band into the higher bands. In addition, the frequencies corresponding to the self-resonant modes of the SQUID array exceed the Josephson energy $E_J$ of the tunable junction, which results in a renormalization of $E_J$, and, as a consequence, an increase of the effective bandwidth of the lowest Bloch band.'
author:
- 'T. Weißl'
- 'G. Rastelli'
- 'I. Matei'
- 'I. M. Pop'
- 'O. Buisson'
- 'F. W. J. Hekking'
- 'W. Guichard'
bibliography:
- 'Bloch\_band\_dynamics\_v2.bib'
- 'revtex-custm.bib'
title: Bloch band dynamics of a Josephson junction in an inductive environment
---
Introduction {#intro}
============
Superconductors provide the unique possibility to create dissipationless macroscopic electrical quantum circuits that are characterized by the dynamics of a well-defined single degree of freedom, the superconducting phase. This is due to the macroscopic coherence of the superconducting wave function [@Tinkham]. In Josephson junction (JJ) circuits, the phase difference $\hat{\varphi}$ between the two superconductors forming the junction and $\hat{N}$ the number of Cooper pairs that tunnel through the junction are quantum conjugate variables satisfying $\left[ \hat{\varphi},\hat{N}\right] = i$. The relative strength of the fluctuations is proportional to the square root of the ratio of the charging energy $E_C$ and the Josephson energy $E_J$, $\Delta\varphi / \Delta N \sim \sqrt{E_C/E_J}$. Therefore, the Josephson effect in circuits containing large Josephson junctions ($E_J\gg E_C$) enables the control of a well-defined dissipationless phase state. This feature allowed the realization of the metrological Volt standard using networks of JJs [@Kautz1996].\
Decreasing the junction size into the submicron region decreases the ratio $E_J/E_C$ and, as a consequence the quantum fluctuations of the phase, $\Delta{\varphi}$, start to play a dominant role. Large quantum phase fluctuations can produce $\Delta{\varphi}\approx 2\pi$ windings, a phenomenon known as quantum phase-slips. Since increasing phase fluctuations implies decreasing fluctuations of its conjugate variable, the charge, the resulting well-defined charge state is expected to yield insulating behavior of the junction. However, such a state is not easily observed for a single Josephson junction, which is typically measured in a superconducting low-impedance environment. The associated charge relaxation time $\tau_q$ is too short to preserve the well-defined charge state on the measurement time scales.\
The relevance of Josephson junction circuits with a well-defined charge state has been pointed out in pioneering articles by Averin, Likharev and Zorin [@Likharev1985; @Averin1985]. Indeed, Josephson junctions with a well-defined charge should sustain Bloch oscillations, consisting of voltage oscillations[@Nguyen2007] on the junction due to a periodic motion of the charge in the lowest Bloch band. These Bloch oscillations are dual to the standard Josephson oscillations. Therefore, they could be used in principle in quantum metrology to realize a current standard, analogous to the way Josephson oscillations are used to realize a voltage standard.\
A possible strategy to obtain long charge relaxation times consists of integrating a highly resistive element with resistance $R$ near the junction in order to increase the charge relaxation time $\tau_q= RC$ enabling the realization of a Coulomb blockade state. However, the introduction of a dissipative element in the circuit introduces heating effects along with Johnson-Nyquist noise making it difficult to reach the quantum limit of minimum charge fluctuations [@Webster2013].\
The first experiments used single Josephson junctions with on-chip, highly resistive environments [@Kuzmin1991; @Likharev1985] in order to observe Bloch oscillations. Later, Josephson junction chains in the resistive state have been used to create an environment resistive enough to observe the so-called Bloch nose [@Watanabe2001; @Corlevi2006a]. More recently, longer chains have been studied [@Ergul2013] and the zero-bias resistance has been interpreted in terms of quantum phase-slips. In the limit of dominating charging energy, the zero bias resistance can be understood in terms of depinning of charges in the chain [@Vogt2014].\
Alternatively, nanowires have been suggested as superconducting elements sustaining phase-slips. Due to the low-dimensionality of these wires, phase-slips occur easily [@Arutyunov2008], thereby reducing the fluctuations of the charge. Inspired by Ref. , several experiments were performed using superconducting nanowires probing the dual physics of these systems such as coherent quantum phase-slips in a device dual to the Cooper pair box [@Astafiev2012], a quantum phase-slip interference device dual to a Superconducting Quantum Interference Device (SQUID) [@Hongisto2012; @Pop2012], and attempts to measure Bloch oscillations and dual Shapiro steps [@Lehtinen2012; @Webster2013].\
Another possible strategy to obtain long charge relaxation times is the additional integration of a large inductance close to the junction [@Guichard2010; @Manucharyan2009; @Masluk2012; @Gershenson2012]. Indeed, in the resulting $RLC$ circuit, the charge relaxation time will be dominated by $\tau_q=L/R$, as soon as $L/R \gg RC$. Hence the use of a highly inductive environment enables one to achieve long charge relaxation times with small resistance $R$, thus avoiding the aforementioned heating effects. Experiments performed in the group of M. Devoret at Yale, probing the quantum states of the so-called fluxonium qubit, demonstrated that charge fluctuations on a small junction can be reduced by embedding it in a highly inductive environment [@Manucharyan2009; @Pop2014].\
In this article we present an experimental study and a quantitative analysis of the zero bias resistance induced by Bloch band dynamics of the charge on a tunable Josephson junction included in an inductive environment. The Bloch bands result from coherent quantum phase-slips occuring on the tunable junctions. The dynamics of the system is related with the presence of the series inductance that plays the role of an effective mass. We demonstrate that this configuration allows the observation of charge localization.\
From the measurements we infer that the dynamics of the charge is a combination of thermal hopping in the lowest Bloch band and Landau-Zener processes causing interband transitions. In addition, we account for the effects of the zero point motion of the electromagnetic modes in the JJ chain and show that they result in a renormalization of the Josephson coupling energy $E_J$, and hence in an increase of the bandwidth of the lowest Bloch band of the tunable junction.
Single junction in an inductive environment {#singjunc}
===========================================
![a) Circuit diagram of a voltage-biased Josephson junction in an inductive environment. b) Wave-function localized in the effective potential $U_{0}\left(q \right)$ of the lowest Bloch band (black continuous line). The second band is indicated by the black dashed line. The red horizontal line corresponds to the lowest energy level. The graph corresponds to the parameters used in the experiment with a flux frustration of f=0.494.[]{data-label="fig:1"}](figure1.eps){width="40.00000%"}
Before presenting and discussing the experimental results, it is useful to briefly recall the theory of a single Josephson junction in an inductive environment [@Likharev1985; @Guichard2010]. The circuit is presented in Fig. \[fig:1\]a. It contains a junction with a capacitance $C$, such that the charging energy is given by $E_C = e^2/2C$, and a Josephson energy $E_J$. The junction is coupled to a series inductance $L$. The single junction is described by the standard Hamiltonian $$H = \frac{Q^2}{2C} - E_J \cos \varphi.
\label{ham}$$ The eigenfunctions are Bloch states and the energy spectrum is described by Bloch bands as a function of the corresponding quasi-charge $q$. If the ratio $E_J/E_C$ is small, the bands are almost parabolic with a width $$\Delta_0 \sim E_C
\label{eq:Delta0}$$ and with gaps $E_{gap}\sim E_J$. If $E_J/E_C$ is large, the bands are sinusoidal with a width $\Delta_0 \approx (E_J^3 E_C)^{1/4} e^{- \sqrt{8E_J/E_C}}$ and with gaps $E_{gap}\sim \hbar \omega_p$, where $\omega_p = \sqrt{8E_J E_C}/\hbar$ is the junction’s plasma frequency. As long as LZ transitions between the bands can be ignored, the behavior of the junction is completely governed by the properties of the lowest band. Outside the parameter range $E_J \gg E_C$, the bands are not sinusoidal and the dependence of the lowest band $U_0$ on the quasi-charge $q$ is given by $$U_{0}\left(q\right) =\sum_{k=1}^{\infty} \nu_k \cos{\left(k\pi q/e\right)} \,\label{eq:Ham_qps_2}.$$ The non-sinusoidal nature of the bands is reflected by the summation over higher harmonics with index $k$, which corresponds to the quantum phase slip processes in the junction where the phase winds by an amount $2\pi k$. The energies $\nu_k$ are the amplitudes for these multiple windings to occur. By embedding the Josephson junction in an inductive environment, it is possible to induce dynamics of the quasi-charge $q$ in the lowest band. The characteristic kinetic energy is then given by $E_{L}={[\Phi_{0}/(2\pi)]}^{2}/2L$, where $\Phi_0$ is the superconducting flux quantum. The full Hamiltonian for the corresponding circuit, shown in Fig. \[fig:1\]a for a voltage-biased configuration, is given by [@Mooij2005; @Mooij2006; @Guichard2010] $$H=-\left(\frac{\hbar^{2}}{2L}\right)\frac{\partial^{2}}{\partial q^{2}}+U_{0}\left(q \right)-V_{bias} q\,.\label{eq:Ham_qps_3}$$ The inductance plays the role of the mass of a fictitious particle with coordinate $q$, moving within the potential energy $U_0(q)$[@Guichard2010]. We denote $\Delta_0$ the barrier height separating the minima of the potential $U_0(q)$. In the tight-binding limit, $E_L\ll \Delta_0$, at vanishing voltage bias $V_{bias}=0$, we can use the harmonic approximation for the potential $U_{0}\left(q\right)\simeq q^{2}/(2C_{q})$ where the effective capacitance is defined as $C_{q}^{-1}={(\partial^{2}U_{0}/\partial q^{2})}_{q=0}$ [@Likharev1985]. In this case, the ground-state wave function is a Gaussian whose width equals to $\Delta q^{2}=(e^{2}/\pi)R_{Q}/\sqrt{L/C_{q}}$ with $R_{Q}=h/(4e^{2})\simeq$ $6.45$ k$\Omega$. The ground-state energy equals $\hbar\omega_{q}/2$, where $$\omega_{q}=1/\sqrt{LC_{q}}
\label{eq:wq}$$ is the associated dual plasma frequency.\
In our experiment we can change the ratio $E_{J}/E_{C}$ of the Josephson junction [*in situ*]{} for a given inductance $L$, thus changing the barrier height $\Delta_{0}$, and explore in particular the regime $\Delta_{0}>\hbar\omega_{q}/2$ where a localized charge-state is expected to appear, see Fig. \[fig:1\]b. We have measured the zero-bias resistance as a function of the ratio $E_{J}/E_{C}$ and analyze the result in terms of a possible charge localization.
System
======
![Experimental circuit showing the voltage-biased tunable Josephson junction (printed in red) connected to a Josephson junction chain with $N$ junctions. Each junction is a SQUID loop. The tunable junction and the chain are protected from spurious high-frequency noise by the parallel arrangement of the shunt capacitance $C_{sh}=\unit{200}{pF}$ and the resistance $R=\unit{3}{k\Omega}$.[]{data-label="fig:2"}](figure2.eps){width="50.00000%"}
The experiments were performed on the circuit shown in Fig. \[fig:2\]. In order to realize a Josephson junction with tunable $\Delta_0$, we designed it in the form of a SQUID such that the ratio $E_J/E_C \propto |\cos( \pi \Phi/\Phi_0)|$, where $\Phi$ is the magnetic flux threading the SQUID loop. For future use we define the flux frustration parameter $f = \Phi/\Phi_0$.\
The $Al/AlO_x/Al$ SQUIDs are fabricated using two-angle shados evaporation of aluminum on a silicon substrate with of silicon oxide. The two aluminum layers have a thickness of $\unit{20}{nm}$ and $\unit{40}{nm}$. The junction area inferred from SEM-images is $\unit{0.07}{\mu m^2}$ and the lattice parameter of the SQUID chain is $\unit{350}{nm}$.\
The junction capacitance is estimated from the design to be $C=\unit{6.9}{fF}$ such that $E_{C}/h=\unit{2.8}{GHz}$. At $f=0$ the Josephson energy of the tunable junction is $E_{J}/h=\unit{29}{GHz}$.\
The series inductance consists of a linear chain of $N$ SQUIDs, each characterized by a Josephson inductance $L_{SQ}=\unit{10}{nH}$ for an applied magnetic field that corresponds to $f=0.5$ for the tunable junction. The inductance $L_{SQ}$ was estimated from resistance measurements at $T=\unit{1.5}{K}$. This chain provides therefore a total inductance of $L= N L_{SQ}$. This inductance is also flux-dependent. The SQUID loops forming the chain are 1.6 times smaller than the SQUID loop forming the tunable junction. The inductance of the chain $L$, changes only by $\unit{10}{\%}$ in the small flux frustration range $f=[0.46,0.5]$ where the tunable junction is probed, so that in first order it can be considered as constant. We fabricated chains containing a different number $N$ of junctions $N=28, 38, 48, 68, 88$ and $108$ [^1]. The capacitance of the SQUIDs in the chain is equal to the one forming the tunable junction. The capacitance to ground of the islands between the SQUIDs is estimated to be $C_{0}\simeq C_{SQ}/75$. The sample is shunted by a home-made $NbTi/Al_{2}O_{3}/NbTi$ parallel capacitor $C_{sh}=\unit{200}{pF}$. This shunt capacitor, together with a resistance $R =\unit{3}{k\Omega}$, provides a low-pass filter with a cut-off frequency of about 60 MHz, thus protecting the junction and its inductive environment from spurious noise above this frequency. The measurements were carried out in a two-point configuration using high-frequency filters in form of thermocoax cables. We measure the I-V characteristics of the array comprising the single junction and the inductive Josephson chain, as depicted in Fig.\[fig:2\]. In the small region at $f=[0.46,0.5]$, where we will analyze our data, the circuit is effectively voltage biased. The sample is kept at the base temperature of about of our dilution refrigerator[@Balestro].
Experimental results and qualitative discussion {#qualitative}
===============================================
We start our discussion by focusing on the results obtained for the single junction in series with the 48 junction chain as an example to illustrate the results. Figure \[fig:3\]a shows the zero-bias resistance of this circuit as a function of the flux frustration parameter $f$.\
First of all we note that the resistance is a few k$\Omega$ for zero frustration. We attribute this finite resistance value to the occurrence of residual incoherent quantum phase-slips in the junctions forming the chain. We can estimate the rate[@Caldeira1983] for a phase winding by $\pm 2\pi$ for a given junction as $\Gamma_{qps}^{\pm} = A e^{-B^{\pm}}$, where $$A= 12 \sqrt{6 \pi}\frac{\omega_p}{2\pi} \sqrt{\frac{\Delta U}{\hbar \omega_p}} \mbox{, } B^{\pm} = \frac{36}{5}\frac{\Delta U^{\pm}}{\hbar \omega_p}$$ and where we neglected the change of the barrier height due to the current bias in the prefactor. Here, $\Delta U \sim 2 E_J-\hbar\omega_p/2$ and $\Delta U^{\pm} \sim 2 E_J-\hbar\omega_p/2\pm \pi I \hbar/2e $ is the effective barrier for tunneling of the phase of the junction by $\mp 2 \pi$. Since the temperature is lower than the plasma frequency $\omega_p/2\pi=\unit{25.4}{GHz}$, thermal activation can be ignored. Quantum phase-slips give rise to a voltage $$V=h (\Gamma_{qps}^+-\Gamma_{qps}^{-})/2e.
\label{eq:volt}$$ Linearizing (\[eq:volt\]) with respect to the bias current $I$ we find $V = R_{qps} I$ with the following estimate for the zero-bias resistance: $$R_{qps} \simeq R_Q \frac{36\pi}{5} \frac{\Gamma_{qps}}{\omega_p},$$ where $\Gamma_{qps}= 2 A \exp{(\frac{36}{5}\frac{\Delta U}{\hbar \omega_p})}$. Using the parameters of the experiment, we find a residual resistance per junction of $R_{qps}\approx\unit{10}{\Omega}$.\
This zero-bias resistance at small values of the frustration changes linearly with the number $N$ of the junctions (see inset of Fig \[fig:3\]a). From the experimental N dependence we extract a resistance of $\unit{59}{\Omega}$ per junction which is of the same order of magnitude as the theoretical estimate.\
The Josephson junction chain is a highly refractive material so that photons can only propagate slowly. This arises the question whether or not the processes ocurring on the single junction are influenced by all the junctions in the Josephson junction chain or not. We therefore estimate the horizon, that is the distance which photons can propagate within the timescales of phase and charge tunneling processes on the single junction. The relevant time scales on which this processes happen are given by $\Delta t_\varphi = \hbar/(2eV_{bias})$ for the phase tunneling and $\Delta t_q = 2e/ I $ for charge tunneling[@Devoret1990]. We use $V_{bias} =\unit{20}{\mu V}$ and $I=\unit{10}{pA}$ (see figure \[fig:3\] b) which yields $\Delta t_\varphi=\unit{16}{ps}$ and $\Delta t_q = \unit{32}{ns}$.\
The phase velocity (in units of junctions per second) can be estimated as $v_p= \omega / k = N/ (2\pi \sqrt{L_{SQ} C_0}) \approx 10^{13}$ junctions/s. This results in a horizon for the phase tunneling of $\approx 160$ junctions and $\approx 3\ 10^5$ junctions for the charge tunneling so that in both cases the entire Josephson junction chain contributes to the electromagnetic environment of the junction.
![a) Measured zero-bias resistance as a function of flux frustration parameter $f$ for a tunable junction connected to a 48-junction Josephson junction chain (black). The gray curve corresponds to a Josephson junction chain with 49 junctions but without the tunable junction. The inset shows the dependance of the zero-bias resistance of chains with tunable junctions as a function of N for $f=0$. b) Current-voltage characteristics of the tunable junction taken at $f=0.42$ and $f=0.5$. These flux values are marked in a) by the two arrows. The inset shows the corresponding I-V characteristics for the uniform chain. c) Zero-bias resistance as a function of temperature for $f=0.5$. []{data-label="fig:3"}](figure3.eps){width="50.00000%"}
\
Upon increasing the frustration $f$ on the tunable junction, we observe a significant increase of the zero-bias resistance $R_m$ up to 4 M$\Omega$ reached for the maximal frustration $f=0.5$ (black curve in figure \[fig:3\] a). As the SQUIDs in the Josephson junction chain have smaller surface their resistance remains almost unchanged. For comparison we plot the zero bias resistance of a uniform Josephson junction chain (gray curve in figure \[fig:3\]a). In this work we concentrate on the resistance peak around $f=0.5$ of the larger SQUID. This flux bias corresponds to a situation where phase-slips predominantly occur on the single junction and the juctions chains act like an inductance. The second peak in the zero bias resistance at higher flux frustraton corresponds to the situation where phase-slips occur on the SQUIDs in the junction chain. This regime will not be discussed in the paper. Figure \[fig:3\]b shows the current-voltage characteristics, taken at the two flux frustrations corresponding to the onset of the resistance increase ($f=0.42$, left arrow in Fig. \[fig:3\]a) and at its maximum ($f=0.5$, right arrow in Fig. \[fig:3\]a). The inset show the I-V characteristics of the uniform chain at the same flux biases. We understand this resistance increase by three orders of magnitude as a result of the enhanced charge localization within the first Bloch band. As the flux frustration $f$ is increased, the ratio $E_J/E_C$ of the junction decreases, thereby increasing the width $\Delta_0$ of the lowest Bloch band. As soon as the bandwidth $\Delta_0$ exceeds the kinetic energy $E_L$ due to the inductance, the charge becomes more and more localized. More precisely, in this limit the dual plasma frequency becomes $\omega_q /2 < \Delta_0/\hbar$ and localized states can form in the minima of the the lowest band $U_0(q)$. We estimate the barrier height of the potential $U_0(q)$ to be $\Delta_0/h \approx 2.4$ GHz and the dual plasma frequency to be $\omega_q /2\pi=\unit{4}{GHz}$. An estimation of the ratio $\Delta_{0}/E_{L}=0.6$ for our quantum phase-slip junction results in a situation where only one level, with energy $\hbar\omega_{q}/2$, is located in the potential $U_{0}(q)$, see Fig. \[fig:1\]b.\
At higher voltages, the I-V characteristic for $f=0.5$ shows a current peak that we attribute to the existence of electromagnetic modes in the chain at higher frequencies[@Hofheinz2011; @Ingold]. The effect of these modes on the behavior found at low bias will be discussed in detail in Sec. \[effcirc\]. For voltage biases much larger than the plasma frequency of the JJ chain $V_{bias}\gg \hbar \omega_{SQ} /2e$ photons can no longer be emitted to the environment so that the incoherent charge tunneling is suppressed and a zero current state is observed.\
In Fig. \[fig:3\]c, the temperature dependence of the zero-bias resistance is shown for $f=0.5$. We see that with decreasing temperature $T$, the zero bias resistance increases, down to a temperature of about 80 mK. At lower temperatures, $R_0$ saturates. The effect of finite temperatures is to induce thermal fluctuations kicking the charge particle out of the minima. The gradual decrease of $R_0$ with increasing $T$ is therefore expected: charge localization is more pronounced at lower temperatures. The low-temperature saturation is most probably a result of the existence of a finite environment noise-temperature of about 80 mK. We estimated that quantum tunneling can still be ignored at this temperature. Similar findings were reported in Ref. .
Charge diffusion {#chargediff}
================
In order to account for the dependence of the measured zero-bias resistance on the flux frustration parameter $f$, we start by analyzing the simplest possible model describing the dynamics of the quasi-charge $q$, assuming its dynamics to be restricted to the lowest charge band. This amounts to ignoring Landau-Zener transitions. We will come back to this assumption in section \[zener\] below. For now we consider the limit of low voltage bias and low temperature, $eV, k_BT < \Delta_0$. We include the effect of a small tilt due to finite bias, see Fig. \[fig:4\]. Classically, the charge will be localized in one of the minima of the band. This would give rise to a zero current state. However, driven by thermal or quantum fluctuations, the particle can hop between neighboring minima. In the presence of a finite bias voltage, the rate $\Gamma^-$ for hopping from right to left differs from $\Gamma^+$ corresponding to hopping from left to right, giving rise to a finite current $I = 2e (\Gamma^+ - \Gamma^-)$.
![Escape rates $\Gamma^+$ and $\Gamma^-$ for the fictitious quasi-charge particle moving in a tilted washboard potential. $P_Z$ denotes the Landau-Zener probability for the quasi-charge particle to undergo an interband transition.[]{data-label="fig:4"}](figure4.eps){width="35.00000%"}
We calculate the rates $\Gamma^\pm$ using Kramers’s classical result[@Kramers1940] for the escape of a particle from a potential well. We assume thermal activation to be dominant as the temperature is in the same orders of magnitude as the dual plasma frequency $\omega_q /2\pi=\unit{4}{GHz}$ so that the rates $\Gamma^\pm$ can be expressed as $$\Gamma^\pm = \frac{\omega_q}{2 \pi}e^{- E^\pm/k_BT}.$$ Here we used the attempt frequency $\omega_A=\omega_q $, and $E^\pm = {\Delta_0} \mp eV$ denote the barrier heights for tunneling to the left and the right of the well in the presence of the tilt $V$ (see also Fig. \[fig:4\]). We recall that the parameter ${\Delta_0}$ depends on the ratio $E_J/E_C$ of the tunable junction that is varied [*in situ*]{} through the flux frustration parameter $f$.\
The current flowing through the system is $I = V/R_0$, and linearizing the rates with respect to a small bias voltage $V$ results in $$R_0 = R_Q\frac{k_B T}{\hbar \omega_q}e^{{\Delta_0}/k_BT}.
\label{R0fit}$$
![Zero-bias resistance $R_0$ as a function of the parameter $E_J/E_C$ for the chain with $N=48$: experimental results (data points) and fit (solid line) based on Eq. (\[R0fit\])[]{data-label="fig:5"}](figure5.eps){width="50.00000%"}
N $\alpha$ $\omega_q^{fit} / E_C $ $ \omega_q / E_C $
----- --------------- ------------------------- ---------------------
28 $1.7 \pm 0.1$ $0.61 \pm 0.02$ 1.17
38 $4.1 \pm 0.1$ $7.0 \pm 0.4$ 1.01
48 $4.6 \pm 0.2$ $1.7 \pm 0.5$ 0.99
68 $5.4 \pm 0.2$ $17 \pm 4$ 0.84
88 $6.3 \pm 0.2$ $52\pm 12$ 0.74
108 $5.5 \pm 0.2$ $22 \pm 7$ 0.67
: Fitting parameters and errors for all measured chains. $\alpha$ is the multiplicative factor in front of $\Delta_0$ used when fitting with Eq. (\[R0fit\]).[]{data-label="tab:1"}
In Fig. \[fig:5\], the measured zero-bias resistance $R_m$ is shown as a function of frustration and the corresponding ratio $E_J/E_C$ for the $N=48$ junction chain, together with a fit based on Eq. (\[R0fit\]). Similar fits have been performed for all the measured chains. These fits enable us to compare the behavior of the fitted exponent $\Delta_0^{fit}$ as a function of the ratio $E_J/E_C$ with the theoretically expected one $\Delta_0$, determined by solving the Mathieu equation corresponding to Hamiltonian (\[eq:Ham\_qps\_3\]) using the system’s parameters. From the fits we find that the actual values of the exponent and the prefactor differ greatly from the expected one. Specifically, we find that good fits can be obtained using a barrier height $\Delta_0^{fit}=\alpha \Delta_0$ that is $\alpha$ times higher than the expected value $\Delta_0$. In Table \[tab:1\], we list the multiplicative factors $\alpha$ for the other samples together with the prefactor $\omega_q^{fit}$ obtained by fitting and the corresponding prefactor expected from theory $\omega_q$. Note that the factor $\alpha$ is an almost monotonically increasing function of the chain length $N$. We obtain values ranging from $\alpha = 1.7$ to $\alpha = 6.3$. As will be argued in the following, we can attribute this discrepancy between the experimentally found exponent and the theoretically expected one to two phenomena: (i) the renormalization of the Josephson coupling energy of the tunable junction due to electromagnetic modes propagating along the chain and (ii) the effect of interband transitions (Landau-Zener processes) that dominate the charge dynamics whenever the gap $\sim E_J$ separating the lowest two charge bands becomes too small compared to the characteristic energy of the dynamics of the quasi-charge. This will happen when the flux bias on the single junction approaches $f=0.5$.
Renormalization of the bandwidth {#effcirc}
================================
![Dispersion of the propagating modes on Josephson junction chains with a finite capacitance to ground $C_0/C_{SQ} = 1/50$.[]{data-label="fig:6"}](figure6.eps){width="50.00000%"}
It is well-known that the Josephson coupling energy $E_J$ of a Josephson junction connected to an external circuit is suppressed down to a value $E_J^*$ by the quantum fluctuations induced by the corresponding environment [@Schmid1983; @Schoen1990; @Weiss2008]. As a consequence, the bandwidth $\Delta_0^*$ found for the lowest charge band will be larger than the bare width $\Delta_0$. Referring again to the measurement circuit as shown in Fig. \[fig:2\], we first note that the junction is well-protected from high-frequency noise originating from the external leads by thermo coax and the RC filter down to 60 MHz[@Balestro]. Since the resistance measurements were performed at flux frustration parameters $f$ corresponding to a bare Josephson coupling energy in the GHz range, we suppose that the external noise does not account for any renormalization of $E_J$. However, the chain providing the inductance is directly coupled to the tunable junction. The fact that the islands realize a finite capacitance $C_0$ to ground leads to the appearance of dispersive electromagnetic modes in the chain [@Rastelli2013; @Masluk2012; @Pop2011], with a dispersion relation given by $$\omega_k = \omega_p \sqrt{\frac{1 - \cos k}{1-\cos k +C_0/2C_{SQ}}},$$ where $k = 2 \pi n/N$ are the discrete wave-vectors of these modes. Here we assumed periodic boundary conditions as we do not know the exact boundary conditions in our experiment. The dispersion relation is shown in Fig. \[fig:6\]. It consists of a linear part, $\omega_k = \omega_0 k $ for small $k$-vectors, with $k < \sqrt{C_0/C_{SQ}}$. Here $\omega_0 = 1/\sqrt{L C_0}$. The frequency $\omega_l$ of the lowest mode is inversely proportional to the chain length $N$. For the longest chains measured, this frequency is estimated to be about 40 $\%$ of the chain’s plasma frequency $\omega_p$. For larger wave-vectors, the dispersion relation saturates at the plasma frequency $\omega_p$. In units of temperature, the frequency range between $\omega_l$ and $\omega_p$ covered by the modes corresponds to a range between $\unit{300}{mK}$ and $\unit{1}{K}$. Since the temperature at which the experiment is performed is lower than $\unit{300}{mK}$, a zero-temperature treatment of the modes is adequate. The equivalent voltage range is between $\unit{30}{\mu V}$ and $\unit{100}{\mu V}$, which corresponds to the range where the current peak is found in the I-V characteristic, see Fig. \[fig:3\]b, thereby providing indirect evidence for the existence of the modes. In this relatively high bias voltage range, Cooper pair transfer in the small junction is possible as the modes of the chain provide the necessary dissipation.\
![Dependence of the ratio $\Delta_0^*/\Delta_0$ of the tunable junction’s renormalized bandwidth and the bare bandwidth as a function of a) frustration for a chain length of $N=48$, b) chain length for a frustration $f=0.46$. []{data-label="fig:7"}](figure7.eps){width="50.00000%"}
In the limit of small voltage biases, as it is argued in Appendix \[harmodes\], the modes induce zero point quantum phase fluctuations at the end of the chain that couple to the small junction. They add to the phase difference $\varphi$ across the junction, thereby renormalizing the Josephson coupling energy $E_J$ of the tunable junction [@Hekking1997] down to a value $E_J^*$, given by $$E_J^* = E_J \exp\left\{-\frac{1}{N}\sum _k \frac{e^2}{\hbar}\sqrt{\frac{L}{2C(k)[1-\cos k]}} \right\}.
\label{ejstar}$$ Using this result, we calculate the renormalized effective bandwidth $\Delta_0^*$, corresponding to the spectrum of Hamiltonian (\[ham\]). Figure \[fig:7\]a shows the dependence of the ratio $\Delta_0^*/\Delta_0$ as a function of $f$, for the $N= 48$ junction chain. We see that this ratio is larger than unity and a monotonically decreasing function of $f$, reaching unity at $f= 1/2$, where the Josephson coupling energy vanishes and the bandwidth attains its largest possible value $E_C$. The $N$-dependence of the ratio $\Delta_0^*/\Delta_0$ is plotted in Fig. \[fig:7\]b, for a fixed value of $f= 0.46$. Indeed, it is a monotonically increasing function of $N$, however, even for $N=108$ it does not exceed a value of about $1.7$, and therefore can account only in part for the observed discrepancy discussed in section \[chargediff\] above. In the next Section we will show that Landau-Zener processes between the lowest charge band and the higher ones may account for the rest of the discrepancy.
Charge diffusion in the presence of Landau-Zener processes {#zener}
==========================================================
In this section we extend the charge diffusion model presented in Sec. \[chargediff\] and include the effect of possible interband transitions ignored so far. The probability to pass from the lowest charge band to the next one is given by [@Schoen1990] $$P_Z = \exp \left\{- \frac{\pi ^2} {4} \frac{E_{gap}^2} {E_C \hbar \omega_x}\right\},
\label{PZ}$$ where $E_{gap}\sim E_J$ is the gap separating the two bands and $\omega_x$ the relevant frequency associated with the dynamics of the quasi-charge. At zero voltage bias, this frequency will be approximately given by the attempt frequency $\omega_q$ at which the quasi-charge tries to escape by thermal activation from the well formed by the minima of the lowest band. In presence of dissipation, the Landau-Zener probability is determined by coupling of the quasi-charge with the external enviroment. This can give rise to an effective gap appearing in Eq.(\[PZ\]). [@Ingold; @Schoen1990] We observe that Landau-Zener processes are flux-dependent. We take into account the dissipative corrections by allowing $\omega_x$ in Eq.(12) to be an independent fit parameter. The probability to remain in the lowest band is given by $1-P_Z$. We then expect the measured resistance $R_m$ to be given by the weighted sum $$R_m = (1-P_Z) R_0 + P_Z R_Z,
\label{fitform}$$ where $R_0$, the resistance associated with charge diffusion in the lowest band (Eq. (\[R0fit\])) is now calculated with a multiplicative factor $\alpha=1$ but taking into account the renormalized barrier height $\Delta_0^*$. $R_Z$ is a constant fit parameter, representing the resistance associated with charge dynamics in the higher bands.
![Zero-bias resistance $R_0$ as a function of the parameter f for all the chains discussed in this work: experimental results (data points), fit (solid line) based on Eq. (\[fitform\]). The dashed line shows the contribution of Landau-Zener interband transitions alone $ P_Z R_Z$. []{data-label="fig:8"}](figure8.eps){width="50.00000%"}
We assume $R_Z$ to be independent of the flux frustration parameter. When calculating $E_{gap}$ in Eq. (\[PZ\]), we solve the Mathieu equation associated with Hamiltonian (\[ham\]) for the single junction, using the renormalized Josephson energy $E_J^*$, Eq. (\[ejstar\]). In Fig. \[fig:8\] we show a fit of the data for all our chains with Eq. (\[fitform\]). Table \[tab:2\] shows the parameters used to obtain the best fits.\
N $R_{Z}\, (k\Omega)$ $\omega_x / E_C $ $\omega_q^{fit} / E_C $
----- --------------------- -------------------- -------------------------
28 $84 \pm 2$ $0.046 \pm 0.001 $ $0.535 \pm 0.005$
38 $350 \pm 10$ $0.020 \pm 0.001 $ $0.400 \pm 0.010$
48 $3600 \pm 200$ $0.005 \pm 0.002 $ $0.025 \pm 0.005$
68 $170 \pm 100 $ $0.010 \pm 0.003 $ $0.120 \pm 0.010$
88 $3000 \pm 200$ $0.005 \pm 0.001 $ $0.100 \pm 0.020$
108 $800 \pm 50$ $0.017 \pm 0.002 $ $0.170 \pm 0.030$
: Fitting parameters and errors for all measured chains for fits using Eq. (\[fitform\]).[]{data-label="tab:2"}
A few remarks are in order at this point. Note that $\omega_q^{fit}$ has the tendency to decrease with the chain length, a fact that is expected as the attempt frequency of the escaping particle decreases with increasing particle mass, the mass of the charge being given by the chain’s inductance. On the other hand, the frequency $\omega_q^{fit}$ and the fitted LZ frequency $\omega_x$, are systematically smaller than the frequency $\omega_q$ (see Table \[tab:1\] and \[tab:2\]) associated with the curvature of the lowest Bloch band. This indicates that the charge motion is possibly overdamped [@Wubs2006]. Such overdamped motion could result from a finite quality factor of the electromagnetic modes. Indeed, microwave transmission experiments[@Weissl] done on a 200 Josephson junction chain with a similar ratio $E_J/E_C\approx10 $ as in the experiment here have shown an internal quality factor of about 100. We note that the fitting parameter $R_Z$ increases as a function of $N$ and takes a maximum value of $R_Z=\unit{3500}{\Omega}$ for $N=48$. Above $N=48$ it is difficult to conclude a systematic behavior of $R_Z$. A more detailed understanding of the behavior of $R_Z$ requires a more detailed understanding of the processes responsible for charge relaxation in the higher bands, which is beyond the scope of the present paper.\
In summary, as it can be seen in Fig. \[fig:8\], the measured zero-bias resistance as a function of the flux frustration parameter shows three different behaviours. Close to $f=0.5$, where $E_J \ll E_C$, Landau-Zener processes dominate. The measured resistance has a peak, the form of which is entirely dominated by the flux-dependence of the Landau-Zener probability $P_Z$. Away from $f=0.5$, the decrease of the resistance with $f$ becomes slower than the one predicted by $P_Z$ alone: the system enters the regime where charge diffusion within the lowest band dominates. Even farther away from $f=0.5$, the charge diffusion model breaks down. Here, the bandwidth becomes smaller than the residual noise temperature of the system that we estimate to be about $80$ mK. This happens at $f \approx 0.47$. The charge is no longer localized and our model based on simple escape rates underestimates the actual charge transfer rates. Hence the fit over-estimates the measured resistance.
Conclusions
===========
Our transport measurements suggest the existence of a localized charge state on a Josephson junction due to an inductive environment. We could explain the measured zero-bias resistance with a model combining charge diffusion within the lowest Bloch band and Landau-Zener processes between bands. In order to reduce the charge dynamics to the lowest Bloch band, future experiments should study single Josephson junctions with a larger Josephson coupling $E_J$ hence avoiding Landau-Zener transitions. At the same time the inductance of the environment should be increased to ensure a localized charge state. Increasing the SQUID chain length to obtain a larger inductance reduces the frequencies of the electromagnetic modes and renormalisation effects will start to play a dominant role. Future experiments using the combination of a tunable junction with a controllable SQUID chain constitute an experimental test bed to explore the renormalization effects of the electromagnetic environment on a small junction in a more systematic way. The understanding of the interplay between the charge dynamics and the electromagnetic environment is also relevant for future applications where an inductive environment could play an important role such as current Shapiro steps in a superconducting phase-slip circuit[@Guichard2010; @Webster2013] or a quantum phase-slip qubit[@Mooij2005].
acknowledgements
================
T.W. acknowledges support from the Grenoble Nanoscience Foundation. G.R acknowledges support from the EU FP7 Marie Curie Zukunftskolleg Incoming Fellowship Programme, University of Konstanz (grant no. 291784). F.H. and W.G. are supported by Institut universitaire de France. W.G. also acknowledges support from the European Research council (grant no. 306731). The authors thank N. Roch for fruitful discussions and M. Vanevi[ć]{} for useful comments on the manuscript.
Harmonic modes of a Josephson junction chain {#harmodes}
============================================
![Josephson junction chain.[]{data-label="fig:A"}](figure9.eps){width="50.00000%"}
In this Appendix we briefly review the quantum theory of a harmonic Josephson junction chain. We consider a Josephson junction chain, consisting of $N$ junctions, each with a capacitance $C_{SQ}$ and a Josephson coupling energy $E_{J,ch}$. We denote the capacitance of the islands between the junctions to ground by $C_0$. In the harmonic limit, valid when $E_{J,ch} \gg E_C = e^2/2 C$, the junctions forming the chain behave as inductances with inductance $L_{SQ} = (\hbar/2e)^2/E_{J,ch}$ each. Then the chain is described by the circuit model shown in Fig. \[fig:A\]. Its Hamiltonian can be written as $$H_{ch} = \frac{1}{2} \sum _{n,m} Q_n C_{nm}^{-1} Q_m + \frac{1}{2L_{SQ}} \left(\frac{\hbar}{2e}\right)^2\sum_n (\phi_n - \phi_{n+1})^2,
\label{eq:A_ham}$$ where $Q_n$ and $\phi_n$ denote the charge and the phase of the $n^{th}$ island, respectively. These variables satisfy the canonical commutation relation $[Q_n,\phi_m] = -2ie \delta_{n,m}$. The matrix $C^{-1}_{nm}$ is the inverse of the chain’s capacitance matrix $$C_{nm} = (C_0 + 2C_{SQ})\delta_{n,m} - C_{SQ} \delta_{n+1,m} - C_{SQ} \delta_{n-1,m}.$$ We diagonalize the Hamiltonian (\[eq:A\_ham\]) with the help of the following mode expansions for $Q$ and $\phi$ $$\begin{aligned}
\phi_n = \frac{1}{\sqrt{N}}\sum _k \sqrt{\frac{2 e^2}{C(k) \hbar \omega_k}}(a_k + a^\dagger_{-k})e^{i k n},\label{phiexp}\\
Q_n = \frac{-ie}{\sqrt{N}}\sum _k \sqrt{\frac{C(k) \hbar \omega_k}{2 e^2}}(a_k - a^\dagger_{-k})e^{i k n}.\end{aligned}$$ Here, $C(k) = C_0 + 2C_{SQ}(1-\cos k)$ is the discrete Fourier transform of the capacitance matrix, $C(k) = (1/N)\sum_k e^{ik(n-m} C_{nm}$. The dispersion relation is given by $$\omega_k = \omega_p \sqrt{\frac{2(1 - \cos k)}{C_0/C_{SQ} + 2(1 - \cos k)}},$$ with the plasma frequency $\omega_p = 1/\sqrt{L_{SQ} C_{SQ}}$. We use periodic boundary conditions, which implies that $k = 2 \pi m/N$ with $m = \pm 1, 2, \ldots , \pm N/2$. The diagonal form of $H_{ch}$ reads $$H_{ch} = \sum _k \hbar \omega_k (a^\dagger_k a_k + 1/2).$$ The small Josephson junction is connected to one of the ends of the chain, say the one corresponding to $n=0$. As a result, the phase difference $\phi$ across the the junction acquires a fluctuating part, $\phi_0$, and its Josephson coupling energy can be written as $-E_J \cos (\phi + \phi_0)$. Upon averaging over the fluctuations $\phi_0$, we obtain the junction’s effective Josephson energy $U(\phi)$ with a renormalized Josephson coupling energy $E_J^*$, $$U(\phi) = -E_J \langle \cos (\phi + \phi_0)\rangle_{H_{ch}} = -E_J^*\cos \phi,$$ where $E_J^* = E_J \langle \cos \phi_0\rangle_{H_{ch}}$ and the average $\langle \ \rangle_{H_{ch}}$ is taken with respect to the Hamiltonian $H_{ch}$ at zero temperature. Using the fact that $\langle \cos \phi_0\rangle_{H_{ch}} = \exp\{- \langle\phi_0^2\rangle_{H_{ch}}/2\}$ and the mode expansion (\[phiexp\]) to calculate $\langle\phi_0^2\rangle_{H_{ch}}$ we finally obtain $$E_J^*=E_J \exp\left\{-\frac{1}{N} \sum _k \frac{e^2}{\hbar} \sqrt{\frac{L}{2C(k)[1-\cos k]}}\right\}.$$
[^1]: The measured samples comprise two sets of four chains (8, 18, 28, 38) and (48, 68, 88, 108). The two sets have slightly different parameters. As the chains with 8 and 18 squids do not show charge localization we did not include them in the discussion. The chains are actually measured in a configuration where one chain is connected in series with the parallel combination of the other three chains on the chip. We corrected the zero bias resistance for the presence of the other three chains.
|
---
abstract: 'We perform an in-depth analysis on the inequality in $863$ Wikimedia projects. We take the complete editing history of $267\,304\,095$ Wikimedia items until 2016, which not only covers every language edition of Wikipedia, but also embraces the complete versions of Wiktionary, Wikisource, Wikivoyage, etc. Our findings of common growth pattern described by the interrelations between four characteristic growth yardsticks suggest a universal law of communal data formation. In this encyclopaedic data set, we observe the interplay between the number of edits and the degree of inequality. In particular, the rapid increase in the Gini coefficient suggests that this entrenched inequality stems from the nature of such open-editing communal data sets, namely the abiogenesis of the supereditors’ oligopoly. We show that these supereditor groups were created at the early stages of these open-editing media and are still active. Furthermore, our model considers both short-term and long-term memories to successfully elucidate the underlying mechanism of the establishment of oligarchy in Wikipedia. Our results anticipate a noticeable prospect of such communal databases in the future: the disparity will not be resolved spontaneously.'
address:
- 'Future Technology Analysis Center, Korea Institute of Science and Technology Information, Seoul 02456, Korea'
- 'Department of Liberal Arts, Gyeongnam National University of Science and Technology, Jinju 52725, Korea'
- 'Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea; Institute for the BioCentury, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea'
author:
- Jinhyuk Yun
- Sang Hoon Lee
- Hawoong Jeong
date: This manuscript was compiled on
title: 'Early onset of structural inequality in the formation of collaborative knowledge, Wikipedia'
---
The world we are living in today is a result of an enormous amount of cumulated human knowledge. It is therefore essential to understand the process of knowledge creation, and, perhaps more importantly, the collaborative human behaviours behind it, to maintain and enhance the development of our society. Contrary to this, the quantified data for analysing human history have been mostly far from satisfactory, obviously because of the lack of systematically preserved records describing the details of human knowledge development. As a result, investigation of human knowledge creation and collaboration has long been anecdotal and it was the job of the historians and anthropologists to fill the gaps based on fragmented evidences found all over the place. However, the situation completely changed at the turn of the century. The emergence of information technology in this century has offered environments to share up-to-date information generated by everyone online. The segregating lines between information producers and consumers have blurred in modern society, which is called *produsage* [@Bruns2008]. People believe that such a new environment will bring in “democratisation” of knowledge [@Lemke2009]. At the same time, this accumulation of an inconceivable volume of information produced by everyone who is online every second has generated an unprecedented scale of exhaustive records of digital footprints harbouring human knowledge creation processes. Wikipedia, a representative open editing knowledge, may be referred to as the department store for contributors who generate information [@Wikipedia]. Of course, due to the nature of information online, there has been a perpetual doubt regarding its credibility, i.e., it is sometimes considered unstable, imprecise, and even misleading. On the other hand, studies have proved that the current state of accuracy of Wikipedia is remarkable; the accuracy of its contents surpasses that of traditional encyclopaedias [@Chesney2006; @Giles2005]. Nevertheless, researchers found significant heterogeneities in the editing processes, based on the monopoly or oligopoly of a few supereditors’ who govern the contents; thus, it is still far from being the Elysium of communal knowledge that we desire it to be [@Gandica2015; @Yun2016; @Heaberlin2016; @Zha2016; @Ortega2008].
However, the majority of such studies, including our own, focused on a few language editions, mostly the English edition of Wikipedia to examine the dynamics and the properties of the communal data set (an editable data set shared within a community to build collective knowledge) [@Chesney2006; @Yun2016; @Kittur2008; @Adler2008; @Yasseri2012]. Although they successfully warned of the potential risks behind the current social structure in English editions, it is clear that cultural background affects the behaviours of individuals. For instance, people belonging to different cultural backgrounds tend to use different symbols in their Web usage [@Barber1998], and the design of Web pages is also affected by their backgrounds [@Marcus2000; @Schmid-Isler2000]. Similarly, the users of Wikipedia must be affected by their social norms or cultures. It is also reported that editors in different language editions edit Wikipedia in distinctive patterns [@Pfeil2006]. Moreover, the linguistic complexity of English Wikipedia differs from those of the German and Spanish editions [@SKim2016]. Therefore, the results based on cultural differences appear to deny the generality in establishing such heterogeneity. On the other hand, those studies are based on small samples obtained from non-identically sized data sets, e.g., the number of articles in the English Wikipedia and the number of articles in Wikipedia in other languages are of different orders of magnitude; the English Wikipedia has at least five times more edits than others. Thus, the result might be caused by their relative sizes, and it is impossible to accept or to deny the lack of generality. As a result, the discussion is limited to a vague regional generality among the English users; however, the panhuman-scale behaviour remains unexplained.
To investigate the aforementioned generality, we extend our previous analysis on the English Wikipedia [@Yun2016] to all $863$ Wikimedia projects [@Wikimedia], which are composed of various types of communal data sets served by the non-profit organisation called Wikimedia Foundation to encourage worldwide knowledge collection and sharing. For this purpose, we investigate the heterogeneity in contribution and supereditors’ share in the Wikimedia projects to understand the socio-psychology behind open-editing communal data sets. In particular, we inspect the complete editing history of every Wikimedia project to assess Wikipedia’s growth. We mainly focus on the number of edits, the number of editors, the number of articles, the article sizes, and the level of heterogeneity displayed in the inequality index. We demonstrate that there exist typical growth patterns of such open-editing communal data sets that eventually establish drastic heterogeneity among the editors’ contribution. In addition, to comprehend the mechanism behind such disparity, we introduce an agent-based model that replicates the interactions between communal data sets and editors. Our model takes into account the competition between the editors’ natural decrement in motivation over time, the editors’ stronger memory on more recent activities, and the psychological attachment to their articles. The model reproduces the actual universal growth patterns, which are consistent with real data.
Results {#results .unnumbered}
=======
Data Set {#sec:data_set .unnumbered}
--------
For our analysis, we use the March 2016 dump of the entire Wikimedia projects [@WikimediaDownloads]. This data set not only includes the well-known Wikipedia, but also covers its sibling projects such as Wiktionary, Wikibooks, Wikiquote, Wikisource, Wikinews, Wikiversity, Wikivoyage, etc., in different languages (see Supplementary Table \[table01\] for its detailed composition). Each of these open-editing projects has a distinct subject and object. For example, each language edition of Wiktionary aims to describe all words from all the other languages listed in the main language of the edition, e.g., the English edition of Wikitionary has descriptions of the words of all languages in English. The differences between the objects may yield the gaps in the editing behaviours of editors belonging to each project, caused by the difference in demographic pools, accessibility, degrees of interests, etc. This dump contains the complete copy of Wikimedia articles from the beginning of 15 January 2001 to 5 March 2016, including the raw text and metadata sources in the Extensible Markup Language (XML) format.
In this data set, there are a total of $267\,304\,095$ articles across all Wikimedia projects with their complete edit histories. Each article documents either the Wikipedia account identification (ID) or the Internet protocol (IP) address of the editor for each edit, the article size, the timestamp for each edit, etc. A single character takes one byte, except for a few cases such as Korean (two bytes) and Chinese (two or three bytes), so the article size in bytes is a direct measure of the article length [@Utf8]. Each data set contains a number of articles ranging from $43\,124\,816$ (Wikimedia Commons: a database of freely usable audiovisual media) to $3$ (Wikipedia Login: a database used for administrative purposes), the number of editors range from $44\,349\,908$ (English Wikipedia) to $5$ (Nostalgia Wikipedia: read-only copy of the old English Wikipedia), the number of edits range from $654\,163\,757$ (English Wikipedia) to $14$ (Wikipedia Login), and the total article size ranges from $99\,519\,138\,751$ bytes (English Wikipedia) to $1\,206$ bytes (Wikipedia Login). See Supplementary Fig. \[fig:wiki\_statistics\] for the distributions of various characteristic measures.
Previous studies, including our own, used a few language editions, commonly restricted to the English edition of Wikipedia, which is the largest [@Chesney2006; @Yun2016; @Kittur2008; @Adler2008; @Yasseri2012]. In addition, Wikimedia projects other than Wikipedia are not usually analysed, even though several language editions of Wikipedia were considered [@SKim2016; @Hale2014]. It is true that most other Wikimedia projects are not as large as the English Wikipedia, as shown by the fat-tailed distributions for different Wikimedia projects in Supplementary Fig. \[fig:wiki\_statistics\]. However, the properties of everyday phenomena vary by their sizes [@WSong2003]; thus, the characteristics of such communal databases may vary by their sizes and categories. Therefore, these smaller editions should not be neglected because of their being much smaller than the English editions, if we aim to comprehend the omnidirectional properties of communal data sets. Accordingly, we consider all 863 editions of the Wikimedia projects for our analysis.
Our main goal is to find the underlying principle in the development of communal data sets. The Wikimedia project, as the representative player among such communal data sets, consists of various types of data sets operated by the Wikimedia Foundation. This massive record of knowledge spans $273$ different languages and $12$ different types of subjects (see Supplementary Table \[table01\] for details). This variety allows us to explore the innate nature of human behaviour based on each of their written languages and purposes of use. We consider a single Wikimedia project as a sample of such communal data sets. In order to proceed with the in-depth analysis on the evolution of communal data sets, we stress the fact that most data sets are aged approximately $3.5 \times 10^8$ seconds (about eleven years; see Supplementary Fig. \[fig:wiki\_statistics\_pdf\]). Thus, most Wikimedia projects are of similar ages, suggesting that raw characteristic measures without time-rescaling are appropriate, as they are free from the age effect.
Universality and Disparity for Communal Data Sets {#sec:universality .unnumbered}
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In this section, we present the evidence of a universal growth pattern shared by all Wikimedia projects, as displayed by characteristic measures based on the current snapshot of the communal data set, such as the total number of edits $N_e$, the total number of editors $N_p$, the total number of articles $N_a$, and the current sizes $S$ (in bytes). Our primary interest is to identify the generality in growth of the communal data set, not individual articles. Thus, we use the total sum of the above values for all articles in a Wikimedia project, without considering the individualistic properties of its constituent articles. For example, $N_e$ is the total number of edits for a given edition of a Wikimedia project, or the sum of the number of edits of individual articles for that edition. Our first analysis on the interplay between such measures indicates their regularity regardless of age, language, and the type of data sets.
### Growth Scale of Communal Data Sets {#sec:growth_scale_of_communal_data sets .unnumbered}
We begin our analysis with the inspection of the inter-correlations between $N_e$, $N_a$, $N_p$, and $S$ in the current Wikimedia projects. One may speculate the absence of a general rule between measures due to the excessive heterogeneity of the current status (as shown in Supplementary Fig. \[fig:wiki\_statistics\]) compared to its age distribution (as shown in Supplementary Fig. \[fig:wiki\_statistics\_pdf\]). As an example of the difference in different language editions of Wikipedia projects, it has been reported that the levels of language proficiency among editors in English Wikipedia are qualitatively different from those in the other language editions [@SKim2016]. Despite such differences, we find common positive correlations between the measures. First, we observe a clear tendency that the number of editors, the number of articles, and the sizes of data sets gradually vary as the functions of the number of edits. The growing patterns are characterised by a simple sublinear growth of the form $y \sim x^\lambda$, where $x$ is the number of edits, and the exponent $\lambda \simeq 0.70$ (as shown in Fig. \[fig:wiki\_statistics2\]**a**), $\lambda \simeq 0.85$ (as shown in Fig. \[fig:wiki\_statistics2\]**b**), and $\lambda \simeq 0.87$ (as shown in Fig. \[fig:wiki\_statistics2\]**c**), respectively. In other words, the frequency of appearance of new editors, that of brand new articles, and the increase in the amount of text slows down when more edits have taken place; from the perspective of editability, larger data set are more inefficient than smaller ones.
To find the reason behind this stagnation in terms of the number of edits, we also track the interrelations between the other measures. The number of editors increases with the number of articles with the exponent $\lambda \simeq 0.78$ (as shown in Fig. \[fig:wiki\_statistics2\]**d**). Meanwhile, the size of the article roughly linearly increases with the number of articles and the number of editors, with the exponents $\lambda \simeq 1.02$ and $\lambda \simeq 1.06$, respectively. In short, the rate of accumulation of the text remains almost constant regardless of the number of articles and the number of editors. One should note that our previous study on English Wikipedia suggests that i) the inter-event times between two consecutive edits in an individual article follows a universal distribution regardless of its age in real time and ii) the size difference between two consecutive edits also follows a universally right-skewed distribution regardless of the size [@Yun2016]. Accompanied by the findings of our previous study, the result implies that the stagnation is caused by the decelerated appearance of new editors, and not the decreased productivity of the existing editors. In addition, in contrast to the interrelations between the measures we report here, the measures are not correlated with the age of the data sets (as shown in Supplementary Fig. \[fig:wiki\_statistics3\]), indicating that the raw number of edits is a proper measure of time for comparison of various data sets rather than the actual time. As we observe in Supplementary Fig. \[fig:wiki\_statistics\_pdf\], most of the Wikimedia projects are of similar ages; therefore, our analysis implies that the rate of growth per unit time also decreases as its size increases, as we have revealed in our previous study on English Wikipedia [@Yun2016]. This universal growth scale is observed for all Wikimedia projects, regardless of their institutional aim, which implies that the common growth patterns is caused by the mutual nature of communal data sets (See Supplementary Fig. \[fig:wiki\_type\]).
Along with the common scales observed in the different measures of a certain Wikimedia project, i.e., $N_e$, $N_a$, $N_p$, and $S$, it is also worthwhile to find possible scales between different types of Wikimedia projects in the same language. As we show in Supplementary Figs. \[fig:wiki\_type\_ne\_size\], \[fig:wiki\_type\_np\_size\], \[fig:wiki\_type\_na\_size\], and \[fig:wiki\_type\_S\_size\], the bulks of different types of Wikimedia projects show almost identical tendency with respect to the size ratio. Each of the four measures for Wikipedia is considerably larger than that of any other types, and Wiktionary has the second largest size; other types of Wikimedia projects do not show great differences between them.
The next question is whether languages in Wikimedia projects can be categorised into distinct clusters that share growth patterns. To answer this question, we perform a cluster analysis by constructing a simple feature vector for a language, which is made up of characteristic measures from different types of Wikimedia projects (see Supplementary Methods for details). We present our results from two modern machine-learning techniques. In particular, we use the Dirichlet Process Gaussian Mixture Model [@Blie2006], which is known for its efficiency in partitioning the vectors in the case of unknown numbers of groups. Then, we use the t-SNE algorithm to visualise the higher dimensional feature vectors in the two-dimensional space, while preserving their original degrees of separations [@Maaten2008]. As a result of the analysis, we could not observe any clear-cut clustering structures for different dimensions of feature vectors and different clustering parameters (Supplementary Figs. \[fig:wiki\_cluster\_3vec\], \[fig:wiki\_cluster\_4vec\], and \[fig:wiki\_cluster\_5vec\]). Therefore, our clustering results also support the existence of universal rules governing the growth of a communal data set, regardless of its language.
### Wikimedia projects and Their Corresponding Socio-economic Indicators {#sec:wiki_socioeconomic .unnumbered}
We would like to check if there are any possible external factors that affect the current status of different Wikimedia projects in terms of the volume indicated by $N_e$, $N_a$, $N_p$, and $S$. The volume of Wikimedia projects is not simply determined by the total number of speakers of a language. As an illustrative example, the Spanish Wikipedia is approximately ten times larger than the Hindi Wikipedia, despite the fact that both Spanish and Hindi have around half a billion speakers each [@Ethnologue2017]. Considering the fact that the ages of both Wikipedia language editions are comparable (16 years for Spanish Wikipedia and 14 years for Hindi Wikipedia), the growth of Hindi Wikipedia has been much slower than the Spanish Wikipedia up to this point. To elucidate the reason behind such a big difference, we inspect the factors affecting the current statuses of Wikimedia projects. The simplest factor is the ratio between the number of people using the language as a native language and those as a secondary language (see the Supplementary Methods for details). One may presume that the people using a particular language as a secondary language have less impact on the formation of communal data set written in that specific language, compared to its native language users. However, we find that the volumes of the Wikimedia project are more correlated with the volume of the secondary language users than the total volume of language users or the volume of users who are native speakers of the language (see Supplementary Fig. \[fig:languser\]). Other linguistic properties may also influence the growth rate of Wikimedia projects, so we tried categorising Wikimedia projects according to their written scripts. Rather surprisingly, there is no notable differences between the scripts (see Supplementary Fig. \[fig:wiki\_script\]) because a single character in each script takes a different size in bytes, as we mentioned earlier.
In addition to the elementary linguistic factors, we try to consider more convoluted factors by cross-correlating the language editions of Wikimedia projects with the socio-economic statuses of countries to which the Wikimedia projects belong. We assign the dominant country of a certain language edition according to the following criterion: i) a country using the language as a primary or official language and ii) the first country in terms of the pageview share of the Wikipedia in that language (see Supplementary Methods for details). First, education levels of the countries show positive correlations with the volume of the corresponding Wikimedia projects, but not in a statistically significant manner (see Supplementary Fig. \[fig:education\]). In addition, the total population does not impact much on the statuses of the Wikimedia projects, whereas the total population of the Internet users shows a strong positive correlation with the volume of the Wikimedia projects (see Supplementary Fig. \[fig:economy\]). The Gross Domestic Product (GDP) is also well correlated with the growth of the Wikimedia projects (see Supplementary Fig. \[fig:economy\]). On the other hand, the GDP per capita is not correlated with the volume of the Wikimedia projects (see Supplementary Fig. \[fig:economy\]). In summary, the scale of the economy, which is partly reflected in the number of Internet users, affects the growth of Wikipedia projects.
Apart from the scale of the overall economy, we also observe that the national expenditures and products for research and development (R&D) show a significant correlation with the current volume of the Wikimedia projects. Larger investors of research also tend to have larger Wikimedia projects compared to their smaller counterparts (see Supplementary Fig. \[fig:rndinvest\]). Consequently, the number of patents and the number of academic papers are also strongly correlated with the sizes of the Wikimedia projects (See Supplementary Figs. \[fig:patent\] and \[fig:paper\], and Supplementary Methods for the details of the patent and academic paper data set). Such an R&D scale is determined by the national economic scale. Together with the results given above, the sizes of the Wikimedia projects are closely connected with the economic scale of a country in terms of total economic size, yet the per capita levels do not impact much on the current sizes of the Wikimedia projects. In other words, Wikimedia projects of richer countries grow faster and larger.
### Disparity in Contributions {#sec:gini_editnumber .unnumbered}
The general growth patterns of the characteristic measures, $N_e$, $N_a$, $N_p$, and $S$, trigger an interesting proposition: could there also be a universal rule in the formation of recently reported structural heterogeneity [@Yun2016; @Heaberlin2016]? To examine the validity of the proposition, we employ the Gini coefficient, which is a conventional measure for inequality [@Gini1912]. In our analysis, the Gini coefficient quantifies *how the number of edits is distributed among different editors* involved in a certain Wikimedia project of interest, i.e., who have edited an article in the project at least once. The Gini coefficient ranges from $0$ for the minimal heterogeneity (or the maximal homogeneity—when every editor contributes equally) to $1$ for the maximal heterogeneity (when only a single editor contributes everything). We consider the number of edits and the data size for individual editors as the variables of interest in the Gini coefficient; these variables are referred to as “wealth”, unless specified otherwise (as the Gini coefficient is usually used to quantify the inequality in economic wealth).
The trend of the Gini coefficient as an increasing function of $N_e$ displayed in Supplementary Figs. \[fig:wiki\_gini\] connotes that the disparity is intensified as the communal data set grows. Larger values of $N_e$ induce intenser disparity not only for the number of edits performed by the editors (as shown in Supplementary Fig. \[fig:wiki\_gini\], but also for the total data changes (in bytes) made by the editors (as shown in Supplementary Fig. \[fig:wiki\_gini\]). This increasing trend is still valid when we perform addition (as shown in Supplementary Fig. \[fig:wiki\_gini\]) and subtraction separately (as shown in Supplementary Fig. \[fig:wiki\_gini\]). In addition, because the age of the article does not severely affect the heterogeneity (as shown in Supplementary Fig. \[fig:wiki\_gini\]), the observation of the Gini coefficient is consistent with our observation that the age of the article does not affect the current state of the communal data sets. We predict that the heterogeneity will become severer if a given data set is edited more frequently. There is no notable distinction between written scripts of Wikimedia projects (see Supplementary Fig. \[fig:wiki\_script\_gini\]) and the institutional objectives for different Wikimedia projects (see Supplementary Figs. \[fig:wiki\_type\_gini\] and \[fig:wiki\_type\_gini2\]). To sum up, we observe that the universal pattern of heterogeneity increased with the number of edits based on the current snapshot of the communal data set.
Evidences for the Establishment of the Supereditors’ Dominance {#sec:editorscartel .unnumbered}
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In the previous section, we have shown the current snapshots displaying the high level of heterogeneity and the increase in the Gini coefficient with the number of edits (see Supplementary Fig. \[fig:wiki\_gini\]). Although the current statuses of all Wikimedia projects appear to follow a specific function of $N_e$, this snapshot could be coincidental. Thus, we further track the actual history to confirm or reject the hypothesis of possible coincidence so that we can judge if the increasing trend is actually the inherent nature of the formation of communal data sets. To check the hypothesis, we set the initial number of edits of all $863$ data sets to the same value \[$N_e(t = 0) = 0$\] and record the trajectories of the Gini coefficient as functions of $N_e$ (see Fig. \[fig:wiki\_gini2\]**a** for the curve averaged over data sets with the deviation). Similar to the conventional usage of the Gini coefficient for wealth distributions, we use the cumulated number of edits up to $N_e$ (note that as discussed previously, the unit of time in this case is $N_e$) for each editor, which is a wealth variable. Note that, technically, the Gini coefficient is undefined when a single editor has edited a data set (as we define the set of editors as the editors who have contributed at least once), but we take the Gini coefficient as $1$ for that case because it well describes the completely monopolised state. Our result shows that the average Gini coefficient is coterminous with the current states of the Wikimedia projects (see Fig. \[fig:wiki\_gini2\]**a**); thus the current status of a specific data set can be taken as a certain midpoint of a single master curve described as a function of $N_e$. For example, a history of the Cebuano Wikipedia clearly follows the typical growth pattern for $N_e > 10^4$ (see Fig. \[fig:wiki\_gini2\]**b**), except for the initial fluctuations for small values of $N_e$.
Although we employ the Gini coefficient as the inequality measure in accumulated wealth distributions, an alternative approach of the index for an income is also widely accepted. In economics, the income is defined as the value gained within a specific time frame [@Mankiw2014]. Alternatively, we also consider the number of edits for individual editors per unit time frame as the “income” variable in the Gini coefficient; this is called income unless otherwise specified. In other words, the wealth analysed before is the accumulated income from the onset of an individual editor’s first activity. In this study, we use the time window of $10^4$ edits, but the different values of time frame do not affect the result meaningfully. The Gini coefficient in terms of the income distribution for the communal data set as a function of $N_e$ indicates that the larger $N_e$ values induce less severe heterogeneity in the contributions (see Fig. \[fig:wiki\_gini2\]**c**). It indeed suggests that the income distribution becomes more homogeneous with time (see Fig. \[fig:wiki\_gini2\]**c**), whereas the disparity in the wealth distribution is maintained (see Fig. \[fig:wiki\_gini2\]**a**).
Therefore, the heterogeneity in the wealth distributions are intensified over time, whereas the disparity in the contributions per time frame become less severe with time for all editors. To consolidate the two results, we examine how the rich-get-richer concept affects the communal data set in details. Figures \[fig:wiki\_revisit\]**a, b** suggest that the editors tend not only to keep their short-term social positions but also to maintain their long-term social positions. For instance, $58.1\%$ of the editors remain in the rank in the range of $[0\%,10\%]$ ($10\%$ from the top) for next $10^4$ edits if the editors were ranked in the range of $[0\%,10\%]$ ($10\%$ from the top) in the time window of $990\,000 \le N_e < 1\,000\,000$; meanwhile, only $32.6\%$ of the editors ranked in the range of $[0\%,90\%]$, i.e., except for the bottom $10\%$, retain their positions (see Fig. \[fig:wiki\_revisit\]**a**). In other words, the editors who edit more often within a specific time window tend to edit more often later as well. Although the exact proportion and the number of edits for each percentile vary over time, the distinction between the social classes is preserved. As a result, a hierarchical structure between the editors is gradually becoming concrete.
The trend is even clearer for the accumulated number of edits (see Fig. \[fig:wiki\_revisit\]**b**). At the early stage, only highly ranked editors, whose amount of contributions are much larger than the median, maintain the positions represented by their cumulative number of edits up to that moment. Meanwhile, the rest of the lowly ranked editors, whose amount of contributions is much smaller than the median, change their positions more frequently. For every percentile, the percentile of revisiting editors gets more associated with the previous class over time, which eventually groups most editors under a stratified percentile. Therefore, rather solid classes are established at a very early stage and remain for a long time. The oligopoly of supereditors is thus visible [@Yun2016; @Heaberlin2016]. We have not only revisited the existence of such oligopoly, but also observed how its degree of influence changes as more edits are performed. The territories of these conglomerates extend beyond single articles and span the entire Wikimedia project level, and their leverage on Wikipedia is still growing (see Fig. \[fig:wiki\_revisit\]**b**).
To comprehend the formation of such supereditors’ massive share, we further discuss the interrelationship between the number of edits in two consecutive edit sequences in various time windows from the onset of the data set (as shown in Fig. \[fig:wiki\_revisit\]**c**). We calculate the Pearson correlation coefficient between the lists of number of edits in two successive frame windows for an editor. Initially, two consecutive sequences of number of edits are highly correlated across various lengths of time windows, but the short-term correlation values are gradually decreased as more edits are performed. In addition, a boundary between the high-correlated domain (correlation $\gtrsim 0.7$) and low-correlated domain (correlation $\lesssim 0.7$) goes upwards with time; consequently, only long-term correlation is maintained.
In the light of the above information, the results shown in Fig. \[fig:wiki\_revisit\] explain the results shown in Figs. \[fig:wiki\_gini2\]**a, c**; the disparity in the wealth distributions is preserved by the long-term correlations, whereas the disparity in the income distribution is steadily resolved due to the diminution of the short-term correlations. Although the short-term activities of editors may vary, the existence and dominance of a few editors are not resolved in the long run because of the dominance constructed at a very early stage of the communal data set.
Other collaborative knowledge creations: the case of patents and academic papers {#sec:paper_patent .unnumbered}
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One clear advantage of investigating online-based data such as Wikimedia projects is that we can identify individual contributions in the formation of the collective knowledge. On the other hand, the observation of ubiquitous growth scales and formation of strong heterogeneity also prompt an essential question: is early onset of heterogeneity caused by the inherent nature of online-based communal data set? In other words, is it also possible to find similar growth patterns in conventional knowledge formation processes? Although the Internet revolution leads online-based media to play an important role in constructing and spreading knowledge in this century, conventional platforms are a major route for disseminating expertise. To explore such a larger landscape of the collective knowledge formation, we extend our analysis to academic papers and patents, which are two pivotal media of traditional knowledge formations.
For our analysis, we use patent data from the spring 2017 edition of European Patent Office (EPO) Worldwide Patent Statistical Database (PATSTAT) and academic paper data from 22 August 2017 dump of the entire SCOPUS CUSTOM XML DATA (see Supplementary Methods for details). For the patent data set, we assume that 91 distinct patent offices play roles analogous to the different editions of the Wikimedia projects. Similarly, we also considered each author’s affiliated country as a unit of knowledge formation, analogous to an edition of the Wikimedia projects. Naturally, a single patent or a single scholarly article can be considered as an article in the Wikimedia projects. Unfortunately, it is impossible to trace the entire editing process during the composition of a single patent or a single article. Accordingly, we only use the information of the number of patents/articles and number of participants for each country. In addition, in contrast to the Wikimedia project, the time frame of our patent/paper data set does not cover the very beginning of the platform. Considering the long history of patents and academic papers, we could examine only a small contemporary subset of them, specifically, from 2000 for the patents and from 1996 for the papers.
Based on our analysis of the Wikimedia projects, one may expect the existence of a general rule between the number of participants and the number of patents/articles. We find strong positive correlations between the measures for both patents and scholarly articles (Supplementary Figs. \[fig:patent\_gini\] and \[fig:paper\_gini\]). Specifically, we find the Pearson correlation coefficient to be $\rho = 0.85$ between the number of patents for the number of inventors (who originally designed the technology), whereas it is $\rho = 0.74$ for the number of applicants (who originally filed the patent for the intellectual property rights). Statistics of academic papers show larger Pearson correlation coefficient $\rho = 0.97$ between the number of articles and the number of authors.
Our finding of general growth patterns across conventional knowledge platforms and online-based platforms encourage us to seek possible inequalities among participants of conventional knowledge formation platforms as well. We employ the Gini coefficient again to measure the degrees of inequality between the players in conventional knowledge media. Similar to the results of the Wikimedia projects, the heterogeneity levels of both patents and papers grow as increasing functions of the number of participants and number of research outputs. In contrast to the steep increment observed in the Wikimedia projects, patent and paper data sets show more gradual rising (compare Supplementary Figs. \[fig:wiki\_type\_gini2\], \[fig:patent\_gini\], and \[fig:paper\_gini\]), yet the Gini coefficient already reaches a high level ($\simeq 0.8$) for the leading countries in terms of the number of research output. In summary, an early rise in the disparity between the participants is a unique phenomenon of online-based communal data sets, whereas growing disparity seems to be a common nature of human knowledge formation.
Agent-based Model of Heterogeneity Formation {#sec:memoryeffect .unnumbered}
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To elucidate the dynamics in the formation of the supereditors’ oligopoly, we introduce an agent-based model by importing different types of editors’ “memory” affecting the motivation for edits. We assume that there are two fundamental and inherent motivations decaying over time, which govern the short-term and long-term behaviours of the editors. Our primary purpose is to examine the separate effects of these two memories governing the current state of Wikimedia projects. Besides these two decaying factors, the editors are also engaged in certain articles when they have already put in more effort in editing the articles [@George2004], which represents their psychological attachment to the articles. In the following, we describe in detail how we implement the socio-psychological effects into our mathematical model.
### Model Description {#sec:model_description .unnumbered}
Our observation is mainly based on the indicators as functions of $N_e$ in the previous section, as we have already observed its validity with respect to the real data. Accordingly, we set a single edit as the unit of time $t$. The model begins with a single agent. Each agent represents a single editor who participates in the editing processes. We considered a single media representing the communal data set, or a single language edition of a certain Wikimedia project. In our model, we consider the action of editors to be motivated by their inherent natures, and introduce the parameters for the editors to describe their activities. First, for an editor $i$, we denote the accumulated number of edits as $N_i(t)$ at time $t$. The time of the first edit by the editor $t_{b;i}$ and time when the last edit occurred $t_{e;i}$ are specified. The dynamic rules are as follows. For each simulation step, the debut of a new agent and the revisit (or re-edit) by an already existing agent occur in turns. For every simulation step, a new agent appears with a constant probability $b$ and begins to participate in the editing process. Once a new agent appears in the data set at time $N_e$, the agent edits the data set at the time of inauguration so that $t_{b;i}$ and $t_{e;i}$ are assigned as $N_e$, and the time unit $t$ is increased by $1$ (the unit of the edit number). One should note that the time scales of $N_e$ for model and data are not identical, because model time scale can vary with the system size and parameters.
In the second step, an editor is chosen uniformly at random attempts to edit the data set. There are many factors affecting the motivation for the edits, but we take three: the long-term decay of motivations, the short-term motivation of ownership, and the psychological engagement of editors. In general, editors are highly motivated at the beginning of the participation, but their motivation fades steadily [@Crane2008; @FWu2007]. Thus, participants lose their attention as time goes by, which is modelled by the power-law decay as the factor $(t - t_{b;i})^{-k}$, where $k$ is the characteristic exponent representing the decay in motivation, which is observed in many temporally varying systems [@Karsai2012; @Karsai2014]. In addition, a fat-tailed distribution is observed for the time between the consecutive edits [@Yun2016], which suggests that the editing time scale of Wikipedia shows “bursty” behaviours, meaning that there is a short-term stimulation of edit motivation affected by the interval between an editor’s latest editing $t_{e;i}$ and the current time $t$ [@Karsai2012; @HJo2015]. This short-term stimulation of motivation is modelled as the factor $[1 + e^{-(t - t_{e;i})/\tau}]$, where $\tau$ is the characteristic time of this stimulation. Finally, there is a tendency for editors to be engaged when they have already participated more frequently [@George2004; @Yun2016]. The number of edits is assigned as $1$ at the time of first participation of the editor and increases by unity every time an agent participates in the edit process so that it is equivalent to the number of edits $N_i(t)$ up to the time point $t$.
Taking these factors together, in our model, when an agent $i$ is chosen for editing, she participates in the editing with the probability $$P_i [t; N_i(t), t_{b;i}, t_{e;i} ] = \min\left\{ 1, N_i (t) (t - t_{b;i})^{-k} [1 + e^{-(t- t_{e;i})/\tau}] \right\} \,.$$
Once she decides to participate, $t_{e;i}$ is newly set as $t+1$ and $N_i(t+1) = N_i(t) + 1$. In addition, we also include the possibility for an agent to leave the editing process indefinitely. We consider that this departure is based on the loss of motivation to edit [@Crane2008; @FWu2007]. Therefore, in our model, an agent leaves the system when she chooses not to edit and $P_i [t; N_i(t), t_{b;i}, t_{e;i} ] < r$, where $r$ is a preassigned cut-off parameter common to all the editors. In the following section, we give some evidences that the formation of the current inequality is because of the above factors, regardless of the innate nature of an individual editor.
### Model Results {#sec:model_results .unnumbered}
Previously, we have shown the increasing trend of the Gini coefficient as the number of edits is increased, which is, in particular, rapidly increased at the early stage of the data set and stabilised at the high level (the Gini coefficient $\gtrsim 0.8$ for $N_e \gtrsim 10^4$, see Fig. \[fig:wiki\_gini2\]). Our model result is consistent with the empirical observations. The Gini coefficient of the model data set is rapidly increased until the high level is reached at $N_e \simeq 10^5$ for $k = 0.8$ (compare Fig \[fig:wiki\_model\]**a** with Fig. \[fig:wiki\_gini2\]**a**). Smaller $k$ values yield a slower increment in the Gini coefficient, whereas the $\tau$ value does not affect it significantly. The Gini coefficient does not reach the high level (the Gini coefficient $\simeq$ 1) if we assign $k \gtrsim 1$, which suggests that a moderate decaying of motivation is essential to reproduce the current state of the communal data sets. The Gini coefficient of the income distribution also displays results from our model that are similar to that from the data. For $k = 0.8$, the Gini coefficient for the income is steadily decreased from $N_e \simeq 10^5$ (see Fig \[fig:wiki\_model\]**b**), which is observed in the data for $N_e \gtrsim 10^5$ (see Fig. \[fig:wiki\_gini2\]**c**).
One may claim that the early tendency of our model ($N_e \lesssim 10^5$) for the Gini coefficient for income disagrees with the data. In addition, one may observe undulation points, which is absent in the data, at $N_e \simeq 2\times10^4$ for the Gini coefficient for wealth in cases other than $k = 0.8$. Although our model reproduces the patterns in the data at the later stage ($N_e \gtrsim 10^5$ for $k = 0.8$), we would like to remark on this seemingly different growth pattern at the early stage. First, as a minimalistic model, we do not mean to reproduce the inherent disparity from the agents’ attributes, e.g., social class, education level, and language fluency. Furthermore, we do not mean to explain the data all the way from the complicated early procedure when people launch a new project which is exposed exclusively for limited users, such as i) *language proposal* and ii) *incubator*. Naturally, the dynamics of this early stage is very different from the public launch. Despite such difficulty, our model starting with a “regular” dynamics from the very first agent also shows such an early stabilizing period indeed. The relation between the number of editors and the number of edits in the model also displays two different stages (see Supplementary Figs. \[fig:model\_npne\_0\_4\]– \[fig:model\_npne\_0\_8\]). Although the transition point between two stages varies by the specific values of $k$, we assume that this point may correspond to the undulation points for the Gini coefficients for wealth and income (compare Fig. \[fig:wiki\_model\] with Supplementary Figs. \[fig:model\_npne\_0\_4\]– \[fig:model\_npne\_0\_8\]). To conclude, the apparent mismatch for the early stage between model and data is caused by the simplified early stabilizing stage for the model, which is difficult to capturing the very specific details in the real incubating stage for the Wikimedia projects.
In addition to the Gini coefficient, our model also reproduces the trend of reduced short-term correlations for the number of edits between time windows reported in Fig. \[fig:wiki\_revisit\]**c**. As shown in Fig. \[fig:wiki\_model\]**c**, the interrelationship between the number of edits in two consecutive sequences in various time frames from the onset of the data set gives a similar result. In the model and the real data, we observe a significant correlation between two consecutive sequences regardless of the lengths of the sequences. With time, the short-term correlation is steadily reduced, whereas the long-term correlation is sustained. Similar to the data, the border between large-correlation (correlation $\gtrsim 0.7$) and small-correlated domains (correlation $\lesssim 0.7$) rises as more edits are performed (see Fig. \[fig:wiki\_model\]). The slope of a border is different for different $k$ values, but $\tau$ does not affect the slope.
In short, the parameter $k$ mainly governs the overall dynamics despite the fact that the rapid increment in wealth inequality happens at the early stage and the gradual decrement in the income inequality always occurs. In other words, the loss of long-term motivation induces the inequality, whereas the short-term memory does not affect the system much. Therefore, the rich-gets-richer effect is mainly driven by the accumulated engagement induced by previous edits, and such a long-term engagement leads to the formation of the supereditors’ oligopoly lasting to date. In addition, our model indicates that the supereditors’ oligopoly can be formed without the direct communication between editors or, in other words, any direct pressure from the society.
Discussion {#sec:conclusion .unnumbered}
==========
In this study, we have examined the common patterns in the communal data sets displayed in all language editions of different types of Wikimedia projects. Although some studies have uncovered the general patterns before, it is usually based on partial observations of data sets of specific type or specific languages, which have left many unanswered questions and speculations [@Chesney2006; @Yun2016; @Kittur2008; @Adler2008; @Yasseri2012]. However, the extensive data set formed from all Wikimedia projects, which record large-scale collaboration for the creation of collective knowledge, has given us an unprecedented opportunity to explore collaborative behaviours of human beings quantitatively. In this data set, we have observed the universal interplays between the number of editors, the number of articles, the number of edits, and the total length of the articles, which are characterised by the power-law scaling form with a single set of exponents. The existence of the universal growth rules in all the $273$ languages and $12$ types of Wikimedia projects suggests the panhuman-scale behaviour with regard to collaboration.
This universal pattern is seen not only in the external appearances of the data sets, but also in their heterogeneity quantified by the Gini coefficient; the disparity is formed at a very early stage of the communal data sets and continues forward. It was widely hoped that the communal data sets will bring democratisation of knowledge [@Lemke2009], yet studies reveal that the current Wikimedia projects are hampered by strong heterogeneity in editing [@Yun2016; @Heaberlin2016]. We have demonstrated that the heterogeneity between the editors can be more deep-rooted than expected. The existence of the supereditors’ massive share is a universal phenomenon across all communal data sets, i.e., Wikimedia projects, regardless of their sizes and activities. We have also observed the universal trend of intensified disparity for all types of data sets, which suggests that the vast share of a few dedicated editors will be intensified further. The value of such dedicated editors should be honoured because their voluntary dedication has indeed archived the current level of accuracy in Wikipedia [@Chesney2006; @Kittur2008a]. However, on the other hand, biased narrations on the topics were indeed reported and lost diversity may intensify the issues of systematic biases [@Hube2017; @Callahan2011; @Reagle2011], notwithstanding Wikipedia’s continuing efforts toward neutral point of view. In addition, we have shown that a social stratum of such communal data sets can be formed at the very early stage and the polarisation of editors is already in process.
Our study is not limited to the diagnosis of the current state of Wikimedia projects, but provides a general insight on the future direction of communal data sets. For instance, our simulation suggests that the inequality can be formulated without direct interactions between editors. Indeed, it is also observed that the editors tend to obey pre-established authorities [@Heaberlin2016]. Again, undoubtedly, we acknowledge the dedication of such supereditors for maintaining the high-standard quality of the current state of Wikipedia by their (by definition) large amount of contribution to it [@Chesney2006; @Kittur2008a]. However, the total productivity of each editor is decreased as the number of edits is increased, which may result in less productivity and even less accuracy in the future. It was already reported that the growth of Wikipedia has slowed down [@Suh2009], and our analysis also warns that the inequality will not be easily resolved without active efforts.
Since the turn of the century, Wikipedia has served as a spearhead of the international open knowledge market. However, strategic actions considering the nature of such a social structure are required to sustain the abundant playground for worldwide collaborations. Giving incentives to new editors may help this situation, but a suitable tutorial system that prevents vandalism and encourages decent editing activities is also needed. Fork and merge system commonly used in open-source communities also improve the editability of Wikimedia projects by serving as a secondary method of talk pages where editors can debate. With the Fork system, new editors fork their own versions to show their ideas, and it can be merged with the original article with debating.
Our finding displays abiogenesis imbalances in the formation of a particular set of communal data, but the result and implication of our study can be applied outside the Wikimedia projects. With the Internet penetrating our life deeply, online-based collaboration environments have become a mainstream platform. Therefore, interests on the contribution patterns in various communal datasets such as open-source and free software movements attract public attention [@Gherardi2013; @Tsay2014; @Padhye2014; @Benkler2006]. Compared to traditional (offline) collaboration system, the fruit of online collaboration system is immediately released as products in a collaborative fashion. Analysing millions of the outcomes in GitHub, Apache, GNU & Free softwares, copyleft movements will display more detailed landscape of collaborative knowledge creations, which we will leave for future study. There will be extensive applications for understanding the collective behaviours of humankind based on this type of analysis, which, we hope, could give clues to solve social inequalities of even larger scale.
Methods {#sec:methods .unnumbered}
=======
Data Description {#data-description .unnumbered}
----------------
The data were obtained by downloading the March 2016 dump of all Wikimedia projects [@WikimediaDownloads]. This data set include Wikipedia and its sibling projects such as Wiktionary, Wikibooks, Wikiquote, Wikisource, Wikinews, Wikiversity, Wikivoyage, etc., in different languages. Our data contain $267\,304\,095$ articles across all Wikimedia projects with the complete history of edits and the Wikipedia account identification (ID) or Internet protocol (IP) address of the editor for each edit, the article size, the timestamp for each edit, etc.
Estimation of power exponent for the correlation between the measures $N_e$, $N_p$, $N_a$, and $S$ {#estimation-of-power-exponent-for-the-correlation-between-the-measures-n_e-n_p-n_a-and-s .unnumbered}
--------------------------------------------------------------------------------------------------
To estimate the power-law scaling relations between the measures, we apply a simple linear regression method to the logarithm of the values of interest with the assumption of the simple power-law scaling $y = Cx^ \lambda$. Inevitably, there are various types of noises and fluctuations in empirical observations, so the distribution should be in fact written as $ y = k(x/x_{\min})^{\lambda} + \kappa + \eta$, where $x_{\min}$ is the minimum value of $x$ from which the power law is observed, $\kappa$ is the constant background offset, and $\eta$ is the term for random fluctuations. We neglect those impacts to obtain overall collective tendency for the entire Wikimedia projects. This simple method has clear advantages over the complex multivariate regression: it is less sensitive to the heterogeneous disparity in empirical distribution. The aforementioned power-law scaling can be transformed as $\ln y = \lambda \ln x + \ln C$, and we perform the simple linear regression on these logarithmic values $\ln x$ and $\ln y$ to yield the exponent $\lambda$ and the proportionality constant $C$ (see Supplementary Tables \[table-nenp\]–\[table-nps\] for the statistical details).
Model Construction {#model-construction .unnumbered}
------------------
Our model begins with a single agent accompanied with a single media representing the communal data set. For each simulation step, the entry of a new agent occurs (with the probability $b$). Then, an editor is chosen uniformly at random attempts to edit the data set. We consider three factors to determine the probability of editing. First, the motivation for performing the edit decays slowly by the power-law decay as the factor $(t - t_{b;i})^{-k}$, where $k$ is the characteristic exponent representing the motivation decay [@Karsai2012; @Karsai2014]. In addition, we include the short-term stimulation of motivation, which is modelled as the factor $[1 + e^{-(t - t_{e;i})/\tau}]$, due to consideration of “bursty” behaviours [@Yun2016]. Finally, the attachment to the edited articles increased by unity from an initial value of one every time an agent participated in the editing process, so that it is equivalent to the number of edits $N_i(t)$ at the time point $t$.
Considering these factors together in our model, when an agent $i$ is chosen for editing, she participates in the editing with the probability $$P_i [t; N_i(t), t_{b;i}, t_{e;i} ] = \min\left\{ 1, N_i (t) (t - t_{b;i})^{-k} [1 + e^{-(t- t_{e;i})/\tau}] \right\} \,.$$
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Acknowledgements {#acknowledgements .unnumbered}
================
This work received institutional supports by Korea Institute of Science and Technology Information, and was supported by Gyeongnam National University of Science and Technology Grant in 2018–2019 (S.H.L.). The National Research Foundation (NRF) of Korea Grant funded by the Korean Government also supported this work through Grant NRF-2017R1E1A1A03070975 (J.Y.) and No. NRF-2017R1A2B3006930 (H.J.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Contributions {#contributions .unnumbered}
=============
All authors designed the experiment and wrote the manuscript. J.Y. collected and analysed the data.
Competing interests {#competing-interests .unnumbered}
===================
The authors declare no competing interests.
Corresponding author {#corresponding-author .unnumbered}
====================
Correspondence to Sang Hoon Lee and Hawoong Jeong.
[Supplementary Information]{}
Early onset of structural inequality in the formation of collaborative knowledge, Wikipedia
supplementary Figures {#sec:sup_figs}
=====================
![The age distribution of Wikimedia projects, where we bin the data in the uniform length of $2\times10^7$ seconds (the resolution of the horizontal axis).[]{data-label="fig:wiki_statistics_pdf"}](fig001_1_wiki_stats_pdf.pdf)
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supplementary Tables {#sec:sup_tables}
====================
Project Editions Description/Notes
------------- ---------- --------------------------------------------------- --
Wikipedia 273 Encyclopedia articles
Wiktionary 172 Dictionary
Wikibooks 121 Educational textbooks and learning materials
Wikiquote 89 Collection of quotations
Wikisource 65 Library of source documents and translations
Wikinews 33 News source
Wikiversity 16 Educational and research materials and activities
Wikivoyage 17 Travel guide
etc. 77 Deactivated (not editable) ones are included
Total 863
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_e$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nenp"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_e$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nenp"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_e$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nenp"}
\
All & 0.704249 & 0.738648 & $0$ & 0.0110146 & 0.895056 & 0.908861\
Wikipedia & 0.842443 & 0.155088 & $8.82\times10^{-138}$ & 0.0192937 & 0.895667 & 0.924524\
Wiktionary & 0.539973 & 1.830600 & $4.72\times10^{-63}$ & 0.0201075 & 0.871989 & 0.899576\
Wikibooks & 0.800070 & 0.593431 & $2.52\times10^{-85}$ & 0.0148827 & 0.993453 & 0.980026\
Wikiquote & 0.800656 & 0.552054 & $9.37\times10^{-49}$ & 0.0258568 & 0.976789 & 0.957503\
Wikisource & 0.629638 & 1.278470 & $1.53\times10^{-23}$ & 0.0399435 & 0.926182 & 0.893163\
Wikinews & 0.622501 & 1.116610 & $2.68\times10^{-8}$ & 0.0844640 & 0.773823 & 0.797903\
Wikivoyage & 1.078590 & 0.017792 & $9.86\times10^{-8}$ & 0.1136140 & 0.979092 & 0.925912\
Wikiversity & 0.971437 & 0.068704 & $3.74\times10^{-8}$ & 0.0902719 & 0.989328 & 0.944534
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_e$–$N_a$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nena"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_e$–$N_a$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nena"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_e$–$N_a$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nena"}
\
All & 0.851328 & 0.953075 & 0 & 0.0063968 & 0.851105 & 0.976546\
Wikipedia & 0.937205 & 0.256999 & $4.78\times10^{-217}$ & 0.0115751 & 0.849992 & 0.976168\
Wiktionary & 0.886418 & 0.876804 & $4.55\times10^{-133}$ & 0.0116775 & 0.971678 & 0.985567\
Wikibooks & 0.793356 & 1.773870 & $5.16\times10^{-104}$ & 0.0101733 & 0.979341 & 0.990358\
Wikiquote & 0.769828 & 1.939820 & $1.33\times10^{-75}$ & 0.0118208 & 0.949461 & 0.989899\
Wikisource & 0.971340 & 0.472484 & $4.70\times10^{-51}$ & 0.0204457 & 0.980636 & 0.986329\
Wikinews & 0.780295 & 1.999770 & $1.35\times10^{-13}$ & 0.0626789 & 0.789431 & 0.912861\
Wikivoyage & 0.809312 & 0.853837 & $1.04\times10^{-10}$ & 0.0516414 & 0.982997 & 0.970794\
Wikiversity & 0.928145 & 0.316529 & $3.56\times10^{-9}$ & 0.0717772 & 0.984704 & 0.960594
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_e$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nes"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_e$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nes"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_e$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nes"}
\
All & 0.874691 & 1060.2 & 0 & 0.0102172 & 0.945518 & 0.945977\
Wikipedia & 0.987816 & 194.4 & $1.68\times10^{-186}$ & 0.0153652 & 0.946319 & 0.962968\
Wiktionary & 0.802352 & 1652.4 & $1.59\times10^{-103}$ & 0.0160625 & 0.892158 & 0.967582\
Wikibooks & 0.976815 & 728.3 & $6.50\times10^{-61}$ & 0.0301235 & 0.966832 & 0.947805\
Wikiquote & 0.891003 & 830.3 & $1.22\times10^{-57}$ & 0.0223718 & 0.977249 & 0.973655\
Wikisource & 0.989188 & 1610.7 & $7.95\times10^{-34}$ & 0.0405094 & 0.951270 & 0.951021\
Wikinews & 0.814366 & 1697.1 & $1.79\times10^{-13}$ & 0.0661310 & 0.955948 & 0.911193\
Wikivoyage & 0.846150 & 1612.5 & $3.33\times10^{-7}$ & 0.0980059 & 0.956239 & 0.912402\
Wikiversity & 0.894525 & 1310.5 & $1.21\times10^{-8}$ & 0.0760369 & 0.981032 & 0.952962
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_a$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nanp"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_a$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nanp"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_a$–$N_p$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nanp"}
\
All & 0.778094 & 1.199350 & $4.75\times10^{-274}$ & 0.0146429 & 0.649975 & 0.875402\
Wikipedia & 0.855545 & 0.817538 & $7.79\times10^{-120}$ & 0.0228277 & 0.651000 & 0.901426\
Wiktionary & 0.585252 & 2.473690 & $5.63\times10^{-56}$ & 0.0246021 & 0.773866 & 0.876922\
Wikibooks & 0.980838 & 0.403144 & $3.04\times10^{-69}$ & 0.0253558 & 0.956404 & 0.962462\
Wikiquote & 1.012750 & 0.338911 & $5.48\times10^{-43}$ & 0.0387238 & 0.878766 & 0.941891\
Wikisource & 0.616042 & 2.848200 & $4.00\times10^{-20}$ & 0.0459310 & 0.884531 & 0.860595\
Wikinews & 0.797727 & 0.642675 & $3.09\times10^{-11}$ & 0.0796528 & 0.955328 & 0.874015\
Wikivoyage & 1.188170 & 0.082902 & $1.53\times10^{-5}$ & 0.1898720 & 0.929965 & 0.850316\
Wikiversity & 0.940047 & 0.609241 & $5.83\times10^{-6}$ & 0.1334550 & 0.966194 & 0.883137
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_a$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nas"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_a$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nas"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_a$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nas"}
\
All & 1.021710 & 1173.3 & 0 & 0.0097043 & 0.837651 & 0.963288\
Wikipedia & 1.050380 & 844.92 & $6.06\times10^{-241}$ & 0.0108699 & 0.838299 & 0.983090\
Wiktionary & 0.900004 & 1952.1 & $1.04\times10^{-114}$ & 0.0153493 & 0.916696 & 0.976157\
Wikibooks & 1.231870 & 358.1 & $4.13\times10^{-66}$ & 0.0340094 & 0.958276 & 0.957518\
Wikiquote & 1.154880 & 392.9 & $3.45\times10^{-64}$ & 0.0241913 & 0.938157 & 0.981443\
Wikisource & 0.999274 & 4167.2 & $1.49\times10^{-32}$ & 0.0430899 & 0.944978 & 0.946118\
Wikinews & 0.981187 & 1486.8 & $7.42\times10^{-16}$ & 0.0648819 & 0.910617 & 0.938418\
Wikivoyage & 1.022660 & 2346.8 & $1.83\times10^{-7}$ & 0.1130380 & 0.970173 & 0.919303\
Wikiversity & 0.944827 & 4725.6 & $2.93\times10^{-10}$ & 0.0604193 & 0.980323 & 0.972548
[lrrrrrr]{} Project &
----------
Power
exponent
----------
: Statistical details about power-law fitting for the $N_p$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nps"}
&
-----------------
Proportionality
coefficiant
-----------------
: Statistical details about power-law fitting for the $N_p$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nps"}
& $P$-value & Standard Error & Pearson $\rho$ &
----------------
Pearson $\rho$
(log-log)
----------------
: Statistical details about power-law fitting for the $N_p$–$S$ relation. When we write that a $P$-value is 0, it means that this value is smaller than the minimum (approximately $2.23 \times 10^{-308}$) of the floating-point variables in Python.[]{data-label="table-nps"}
\
All & 1.059150 & 5791.2 & $4.34\times10^{-292}$ & 0.0187328 & 0.726656 & 0.887589\
Wikipedia & 1.022530 & 6194.5 & $1.11\times10^{-124}$ & 0.0261606 & 0.725107 & 0.908313\
Wiktionary & 1.223030 & 3538.5 & $2.02\times10^{-58}$ & 0.0492685 & 0.747880 & 0.885312\
Wikibooks & 1.160450 & 1966.0 & $5.32\times10^{-50}$ & 0.0455632 & 0.965096 & 0.919231\
Wikiquote & 1.029180 & 2743.8 & $1.57\times10^{-42}$ & 0.039893 & 0.967549 & 0.940423\
Wikisource & 1.304610 & 7218.7 & $1.68\times10^{-22}$ & 0.0868298 & 0.927855 & 0.884204\
Wikinews & 0.968762 & 16377.6 & $5.85\times10^{-10}$ & 0.1098120 & 0.896203 & 0.845663\
Wikivoyage & 0.593184 & 190739 & $5.99\times10^{-4}$ & 0.1370960 & 0.906938 & 0.745099\
Wikiversity & 0.809554 & 38139.5 & $4.66\times10^{-6}$ & 0.1126350 & 0.978002 & 0.887005
Supplementary Methods {#sec:sup_methods}
=====================
Introduction to the Patent Data Set {#sec:sup_patent .unnumbered}
-----------------------------------
For our analysis, we use the spring 2017 edition of European Patent Office (EPO) Worldwide Patent Statistical Database (PATSTAT), which contains the metadata of patents from 91 national and international patent offices. Inventors and applicants are identified by the EPO using their person IDs and we neglected the persons who were not identified by the EPO. It contains various types of intellectual properties, but we only consider patent applications. The timestamps of patent applications are preferentially extracted from the application year field. Before 2000, the United States Patent and Trademark Office (USPTO) only published granted patents. We thus used patent applications after 2000, which was the year USPTO began to publish all patent applications. Furthermore, a certain publication is dated after 2017, which we truncated for data consistency. One should note that the person identification provided by the EPO is automatic; therefore, it is not perfect and may have unexplored errors.
Introduction to the Research Paper Data Set {#sec:sup_paper .unnumbered}
-------------------------------------------
For our analysis on paper metadata, we use the dump of the entire SCOPUS CUSTOM XML DATA for 22 August 2017. This custom data contains the complete copy of data from the SCOPUS website from the very beginning, i.e., January 1996 to August 2017, and includes the title, journal, abstract, author information, and citation records in the XML format. Each type of document plays different roles for knowledge formation. For example, conference proceedings are a conventional method for presenting new research in the fields of computer science, whereas journal articles are the main method for many other disciplines. Some disciplines in social science also consider books and reports as important archives of knowledge. Therefore, to prevent a possible bias towards specific disciplines, we use the entire metadata regardless of the citation type in SCOPUS.
The timestamps of the publications are preferentially extracted from the `publicationdate` element. It is occasionally replaced by the `xocs:sort-year` element only if the `publicationdate` element is missing or broken. If the timestamp of a certain publication is not between 1996 and 2017, we consider the data as erroneous and remove it. In addition, users were identified as `auid` elements, along with its affiliation element of `afid`. We only count the authors with a clear identification, and the nationalities of the authors is assumed as the country to which the author’s affiliation belonged at the time of publication.
Matching Languages of Wikimedia Projects and Their Dominant Countries {#sec:sup_matching .unnumbered}
---------------------------------------------------------------------
One may ask how the current statuses of the Wikimedia projects are related to the empirical socio-economic status of each society. In Supplementary Figs. \[fig:languser\]–\[fig:paper\], we present several examples in response to the above question. An essential prerequisite is the designation of a specific language in the Wikimedia project to its dominant country. We assume that a certain language is mainly spoken in countries wherein people use the language as their official or dominant language. Therefore, we obtained language usage statistics of the countries from CIA World Factbook [@CIA16] and remove the subsidiary languages to retain the official or dominant languages alone. If a country has more than three official languages, we neglect the third and lower-ranked languages based on the order of the populations that speak the languages. Subsequently, we collect the share of pageviews per Wikipedia language from a certain country based on the IP address [@Wikitraffic]. Using these two data sets, we match each language used in each Wikimedia project with a specific country as follows: first, if a language is used by only one country in the CIA database, we consider the country to be substantially in possession of the Wikimedia projects written in that language. Second, if there is more than one country that uses the language as the primary language, we consider the country with the largest share of Wikipedia page view traffic as substantially occupying the Wikimedia projects written in the language. Note that the *de facto* owner country of a Wikimedia project does not always match the country of origin of the language. For example, the Spanish Wikipedia is mainly used by Mexicans, and English Wikipedia is mainly used by people in the United States. We only use countries with ISO2 and ISO3 as the main country code. In total, 383 Wikimedia projects correspond to a language in the 80 distinct languages in the data set.
Calculating Representative Values of Socio-economic Indicators for Countries {#sec:sup_repval .unnumbered}
----------------------------------------------------------------------------
For capturing the socio-economic statuses of countries, we use various socio-economic indicators offered by the Central Intelligence Agency (CIA) [@CIA16] and UNESCO Institute for Statistics (UIS) [@UIS]. The CIA provides basic statistics such as population and the number of Internet users in different countries. UIS provides various statistics collected worldwide for four categories: 1) education and literacy, 2) science, technology, and innovation, 3) culture, and 4) communication and information. Some survey data are collected irregularly by different organisations; it is thus futile to choose a specific reference year to use. To circumvent this issue, we extract all the existing data from 2000 to 2016, and averaged them over the entire duration for a certain country and its index value. Note that the economic indicators such as Gross Domestic Product (GDP) and Gross Domestic Expenditure on R&D (GERD) are usually collected yearly, whereas survey data such as attainment rate of school is collected irregularly.
Feature Vector Construction for Clustering of Languages in Wikimedia projects {#sec:sup_vector .unnumbered}
-----------------------------------------------------------------------------
To explore the relations between different languages in Wikimedia projects, we construct an $n$-dimensional vector for each language. Each vector element represents a characteristic measure for a specific type of Wikimedia project based on the number of its language editions (see Table \[table01\]). Specifically, it follows the order 1) Wikipedia, 2) Wiktionary, 3) Wikibooks, 4) Wikiquote, and 5) Wikisource. As an illustrative example, the 3-vector representation of $N_e$ for English Wikimedia Projects is $[N_e(\textrm{English Wikipedia}), N_e(\textrm{English Wiktionary}), N_e(\textrm{English Wikibooks})]
= (654\,163\,757, \\36\,453\,984, 2\,572\,276)$.
[11]{}
Paolillo, J. C. and Anupam. D. Evaluating language statistics: The Ethnologue and beyond, *Contract report for UNESCO Institute for Statistics* (2006).
CIA. *CIA World Factbook 2016-2017*. Available at: <https://www.cia.gov/library/publications/the-world-factbook/>.
*Wikimedia Traffic Analysis Report - Page Views Per Wikipedia Language - Breakdown, Monthly requests or daily averages, for period: 1 Sep 2017 - 30 Sep 2017*, Available at: <https://stats.wikimedia.org/wikimedia/squids/SquidReportPageViewsPerLanguageBreakdown.htm>
*UNESCO Institute for Statistics*, Available at: <http://uis.unesco.org/>.
|
---
abstract: 'We present high time-resolution multicolour observations of the quiescent soft X-ray transient [XTEJ1118+480]{} obtained with [ULTRACAM]{}. Superimposed on the double-humped continuum [$\it g''$]{} and [$\it i''$]{}-band lightcurves are rapid flare events which typically last a few minutes. The power density spectrum of the lightcurves can be described by a broken power–law model with a break frequency at $\sim$2mHz or a power–law model plus a broad quasi-periodic oscillation (QPO) at $\sim$2mHz. In the context of the cellular-automaton we estimate the size of the quiescent advection-dominated flow (ADAF) region to be $\sim 10^{4}$ Schwarzschild radii, similar to that observed in other quiescent black hole X-ray transients, suggesting the same underlying physics. The similarites between the low/hard and quiescent state PDS suggest a similar origin for the optical and X-ray variability, most likely from regions at/near the ADAF.'
title: '[ULTRACAM]{} observations of the black hole X-ray transient XTEJ1118+480 in quiescence.'
---
accretion, accretion disc – binaries: close – stars: individual: XTEJ1118+480
Introduction {#INTRODUCTION}
============
X-ray transients (XRTs) are a subset of low-mass X-ray binaries (LMXBs) that display episodic, dramatic X-ray and optical outbursts, usually lasting for several months. More than 70 percent of XRTs are thought to contain black holes [@Charles03].
The XRT [XTEJ1118+480]{} was discovered by the [*RXTE*]{} All Sky Monitor on 2000 March 29 [@Remillard00]. Given its very high Galactic latitude ($\rm b$ = +62 degrees) and correspondingly low interstellar absorption ($\rm N_{\rm H} \sim 1.2 \times 10^{20}\,cm^{-2}$), it was possible to observe the multi-wavelength spectrum including the soft X-ray and extreme ultraviolet during outburst (@Hynes00; @McClintock01b; @Chaty03). Throughout this time, [XTEJ1118+480]{} remained in the low/hard state, characteristic of an accreting black hole binary. A power–law spectrum with an index of -1.7 was seen extending out to 120keV (@Wilson00; @Frontera01) with a similar spectral index to CygX–1 in the low/hard state [@Remillard00]. Faint associated radio emission was detected at 6.2mJy and interpreted in terms of a steady radio jet [@Fender01].
A 13th magnitude optical counterpart was promptly identified [@Uemura01] and its optical spectrum was typical of a black hole X-ray transient in outburst [@Garcia00]. A 4.1hr weak optical photometric modulation was discovered [@Cook00], most likely due to superhumps, i.e. disc precession [@Uemura02]. By late November 2000, [XTEJ1118+480]{} was nearly in quiescence and optical radial velocity studies led to the determination of the large binary mass function, [*f(M)*]{}=6.1$\pm$0.3[$\rm M_{\odot}$]{}, thereby establishing that the compact object is a black hole (@Wagner01; @McClintock01a; @Torres04). [XTEJ1118+480]{} has the shortest known orbital period for an XRT of 4.08hr [@Zurita02], the spectral type of the secondary star is approximately K5V (@Wagner01; @McClintock01a; @McClintock03a) and the binary inclination of the system is high, $i\sim$80degrees (@Wagner01; @Zurita02).
Black hole X-ray transients are known to exhibit five distinct X-ray spectral states, distinguished by the presence or absence of a soft blackbody component at 1keV and the luminosity and spectral slope of emission at harder energies; the quiescent, low, intermediate, high and very high state [@Tanaka96]. Four of them are successfully explained with the advection dominated accretion flow (ADAF) model (Narayan, McClintock & Yi 1996; Esin, McClintock & Narayan 1997). In the context of the ADAF model, properties similar to the low/hard state are expected for the quiescent state, as there is no distinction between the two except that the mass accretion rate is much higher and the size of the ADAF region is smaller for the former state.
Unlike the transition between the low/hard and high/soft (thermal-dominant) state, where there is a reconfiguration of the accretion flow [@Esin97], there is no observational evidence for a transition between the low/hard and quiescent states. In both these states, the ADAF model predicts that the inner edge of the disc is truncated at some large radius, with the interior region filled by an ADAF. Strong evidence for such a truncated disc is provided by observations of [XTEJ1118+480]{} in the low/hard state during outburst (@Hynes00; @McClintock01b; @Esin01; @Chaty03), where the disc has an inner radius of $>$55 Schwarzschild radii ($\rm R_{\rm sch}$) and a hot optically-thin plasma in the inner regions. With [XTEJ1118+480]{} now in quiescence, the ADAF model predicts that the inner disc edge will move outward to larger radii [@Esin97]. Indeed, @McClintock03a recently fit the X-ray/UV/optical quiescent spectrum of [XTEJ1118+480]{} and found that it has a hard X-ray spectrum with a spectral photon index of $\sim$2, and an optical/UV continuum that resembles a 13,000K blackbody disc spectrum with several strong emission lines superimposed. They presented a two-component accretion flow model, an interior region where the flow is advection-dominated and emits in X-rays, and an exterior accretion disc truncated at a transition radius of $\rm R_{\rm
tr}\sim 10^{4}\,R_{\rm sch}$ that is responsible for most of the optical/UV spectrum.
Here we report on our high-time resolution multi-colour optical observations of [XTEJ1118+480]{} in quiescence. We determine the quiescent PDS, compare it to the X-ray low/hard state PDS and determine the size of the ADAF region in quiescence. These observations are part of a continuous campaign with [ULTRACAM]{} to obtain high-time resolution photometry of X-ray binaries.
Observations and Data Reduction {#OBSERVATION}
===============================
Multi-colour photometric observations of [XTEJ1118+480]{} were obtained with [ULTRACAM]{} on the 4.2-m William Herschel Telescope atop La Palma during the period 2003 June 6 to 8. [ULTRACAM]{} is an ultra-fast, triple-beam CCD camera, where the light is split into three broad-band colours (blue, green and red) by two dichroics. The detectors are back-illuminated, thinned, E2V frame-transfer 1024$\times$1024 CCDs with a pixel scale of 0.3arcsecs/pixel. Due to the architecture of the CCDs the dead-time is essentially zero (for further details see @Dhillon01).
Our observations were taken using the Sloan [$\it u'$]{}, [$\it g'$]{} and [$\it i'$]{}filters with effective wavelengths of 3550Å, 4750Å and 7650Årespectively. For the first 2 nights (June 6 to 7), we used an exposure time of 11.65s, which was sufficient to give reasonable count rates in the [$\it g'$]{}and [$\it i'$]{} bands. Given the faintness of the object ([*V*]{}=19.5) and the short exposure times, few counts were obtained in the [$\it u'$]{} band. On the last night (June 8) we decreased the exposure time to 1.65s, but only data in the [$\it i'$]{} band were usable. Hereafter, we will refer to the data taken with exposure times of 11.65s and 1.65s as the [“slow”]{} and [“fast”]{} data, respectively.
The [ULTRACAM]{} pipeline reduction procedures were used to debias and flat-field the data. The same pipeline was also used to obtain lightcurves for [XTEJ1118+480]{} and several comparison stars by extracting the counts using aperture photometry. The most reliable results were obtained using a large aperture with a radius of 1.8arcsec. The count ratio of [XTEJ1118+480]{} with respect to the local standard (2.28North 3.75East of [XTEJ1118+480]{} with similar colour to our target) was then determined. The magnitude of [XTEJ1118+480]{} was then obtained using the calibrated magnitude of the local standard. As a check of the photometry and systematics in the reduction procedure, we also extracted lightcurves of a comparison star similar in brightness to the target. The mean [$\it g'$]{} and [$\it i'$]{} band magnitudes of [XTEJ1118+480]{} are 21.00 and 19.04 respectively and the rms are 14% and 11% respectively. We estimate the photometric accuracy to be 7.1 and 5.5 percent for the [“slow”]{}[$\it g'$]{} and [$\it i'$]{} band respectively and 6.0 percent for the [“fast”]{}[$\it i'$]{} band data.
[lcccccc]{} band & $v_{obs}$ & $v^{'}_{d}$ & $\bar{z}_{f}$ & $\sigma_z$ & $\sigma^{*}_z$ & $\eta$\
[$\it g'$]{}& 40%$^1$ & 39.9% & 0.0007 & 0.0011 & 0.0027 & 0.2%\
[$\it i'$]{}& - & - & 0.0036 & 0.0050 & - & -\
$^1$ @Torres04
The short-term variability {#VAR}
==========================
It is clear that the optical lightcurve of [XTEJ1118+480]{} (Figure\[FIG:SLOWLC\], \[FIG:FASTLC\]) is dominated by the secondary star’s ellipsoidal, which is not linked to the short-term variability/flares (Zurita, Casares & Shahbaz 2003, @Hynes03a, @Shahbaz03b). Therefore if we want to determine the flux of the flares, these “steady” contributions must first be removed from the lightcurves. In order to isolate the short-term variability in each band. we first de-reddened the observed magnitudes using a colour excess of E(B-V)=0.017 [@Chaty03] and the ratio $A_{\rm V}/E(B-V)$=3.1 [@Cardelli89], giving [$\it g'$]{} and [$\it i'$]{} extinction values of 0.06 and 0.04 mags respectively and then converted the Sloan AB magnitudes to flux density [@Fukugita96]. We then fitted a double sinusoid to the lower-envelope of the lightcurve with periods equal to the orbital period and its first harmonic, where the phasing was allowed to float free. We rejected points more than 3-$\sigma$ above the fit, then refitted, repeating the procedure until no new points were rejected (@Zurita03; @Hynes03a). The resulting lightcurves did not show any long-term structure, suggesting that any contamination from a superhump modulation is weak; at the $<$0.50% level. As one can see from Figure\[FIG:FLARES\], there are numerous rapid flare events, which typically last 5min or less. The parameters of the flares as defined by @Zurita03 are given in Table\[Table:Flares\]. Using these flare lightcurves we determined the flux density ratio $\rm f_{g'}/f_{i'}$ and the equivalent blackbody radius $\rm R_{\rm BB}$. To determine the colour temperature corresponding to a given flux density ratio, we integrated blackbody functions with the CCD and Sloan filter response functions and then determined the $\rm
f_{g'}/f_{i'}$ flux density ratio. Given this blackbody temperature we can then determine the corresponding radius of the region that produces the observed de-reddened flux at a distance of 1.7kpc [@Chaty03]. Typically, the flares have $\rm f_{g'}/f_{i'} \sim 0.20$, similar to the flares in V404Cyg [@Shahbaz03b] and $\rm R_{\rm BB}\sim 0.10\,R_{\odot}$. (see Figure\[FIG:FRATIO\]). Following the method described in @Shahbaz03b, we estimate the ratio $\rm f_{g'}/f_{i'} \sim 0.20\pm
0.10 $ corresponds to a LTE slab of hydrogen with a temperature of $\sim$3500$\pm$500K.
The power density spectrum {#PDS}
==========================
To compute the power density spectrum (PDS) of the [“slow”]{} and [“fast”]{} data, we detrended the data using the double sinusoid fit described in the previous section and then added the mean flux level of the data. Although the [ULTRACAM]{}sampling is perfectly uniform, we use the Lomb-Scargle method to compute the periodograms [@Press92] with the same normalization method as is commonly used in X-ray astronomy, where the power is normalized to the fractional root mean amplitude squared per hertz [@Klis94]. We used the constraints imposed by the Nyquist frequency and the typical duration of each observation and binned and fitted the PDS in logarithmic space [@Papadakis93], where the errors in each bin are determined from the standard deviation of the points within each bin. The white noise level was subtracted by fitting the highest frequencies with a white-noise (constant) plus red-noise (power–law) model.
[lcccccc]{} PDS & Slope & Break-freq. & QPO & FWHM & rms & Q\
& & (mHz) & (mHz) & (mHz) & (%) &\
[“slow”]{} [$\it i'$]{}& & & & & &\
PLB & 1.8$\pm$0.2 & 1.9$\pm$0.5 & - & - & - & -\
PL+QPO & 1.4$\pm$0.2 & - & 2.2$\pm$0.3 & 0.7$\pm$0.8 & 3.1 & 3.1\
[“slow”]{} [$\it g'$]{}& & & & & &\
PLB & 2.9$\pm$0.3 & 1.0$\pm$0.7 & - & - & - & -\
PL+QPO & 1.9$\pm$0.2 & - & 2.1$\pm$0.2 & 0.8$\pm$0.3 & 5.4 & 2.6\
[“fast”]{} [$\it i'$]{}& & & & & &\
PLB & 2.3$\pm$0.2 & 1.9$\pm$0.3 & - & - & - & -\
PL+QPO & 1.7$\pm$0.2 & - & 2.4$\pm$0.4 & 0.5$\pm$1.3 & 1.9 & 4.8\
-----------------------------------------------------------------------------------------------
$^*$PL refers to power–law fit of the form $P\propto\nu^{\alpha}$;
PLB refers to a power–law break and broad QPO refers to a quasi-periodic oscillation;
”rms” refers to the fractional root-mean squared amplitude; ”Q” refers to the quality factor.
-----------------------------------------------------------------------------------------------
In Figure\[FIG:PDS\_OP\], we show the PDS which can be described by a broken power–law model or a power–law model + quasi-periodic oscillation (QPO). One can see suggestions of a broad QPO in all the PDS, at $\sim$2mHz and/or a break at $\sim$2mHz. Note that the [“slow”]{}[$\it g'$]{} and [$\it i'$]{} PDS are not independent, being simultaneous, but the [“fast”]{}[$\it i'$]{} is independent, and shows the same (even stronger) feature. Table\[Table:PDS\] gives the fitted properties of the PDS. The Q-factor, defined as the centroid frequency divided by the FWHM of the peak in the PDS. Given that the QPO has Q$>$3, in the following sections we will refer to the possible QPO as a “broad QPO”
We tested the significance of the models to describe the PDS using a Monte Carlo simulation similar to @Hynes03a. We generated lightcurves with exactly the same sampling and integration times as the real data. We started with a model for the ellipsoidal modulation, calculated using the X-ray binary model described in @Shahbaz03a with the parameters given in section\[VAR\]. To this we added a model noise lightcurve generated using a power–law index of –1.0, a break-frequency at 1mHz, calculated using the method of @Timmer95, or a broad QPO at 2mHz with similar strength as observed in the [“slow”]{} PDS. We then added Gaussian noise using the errors derived from the photometry. We calculated 1000 simulated lightcurves and analyzed them in exactly the same way as for the real data. We created individual PDS with the same logarithmic frequency binning used for the data. With the 1-$\sigma$ confidence levels, we found that the broken power–law and the power–law + QPO model produced a PDS that matched the observations equally (see Figure\[FIG:PDS\_MCQPO\]). We therefore conclude that given the uncertainties in the data, the PDS can be described by either the broken power–law or a power–law + broad QPO model.
The break frequency {#BMODELS}
===================
Most models for the observed optical and X-ray variability from an accretion disc around a black hole \[see @Wallinder92 for a review\] can produce periodic or quasi-periodic time variations. To reproduce the observed X-ray fluctuations, the physical quantities of the disc must change abruptly and in order to explain the observed 1/f-like fluctuations, a smooth distribution of flares on a variety of time-scales is required. Here we outline the two most attractive mechanisms to produce the 1/f-like fluctuations in X-ray binaries and cataclysmic variables.
The self-organized criticality model {#SOC}
------------------------------------
@Mineshige94a proposed a cellular-automaton model using the concept of self-organized criticality of @Bak88 to explain the observed 1/f-like X-ray fluctuations in X-ray binaries. In their model, the accretion disc comprises of two parts: an outer disc where the disc material smoothly drifts inwards, and an inner disc which suffers an instability. Gas particles are randomly injected into the inner regions of the accretion disc around a black hole. The system then evolves to and stays in a self-organized critical state in spite of random mass injections. Mass accretion occurs either by an avalanche, which is triggered when the mass density of the disc exceeds some critical value, or by gradual gas diffusion [@Mineshige94b].
@Manmoto96 and @Takeuchi97 took this work further and investigated the response of an advection-dominated disc to an assumed thermal perturbation, producing models in which such X-ray shots could be continuously created. Since the standard-disc cannot explain the long time-scale of the observed X-ray fluctuations, an advection-dominated disc is favoured, since it is characterized by low-emissivity and a large infall velocity. The main energy release occurs by sporadic reconnection events [@Haswell92], leading to magnetic eruptions, similar to solar flares. Magnetic energy is stored and released in an avalanche. They found that the optically thin solutions, dominated by advection (@Ichimaru77; @Narayan94), could produce persistent, but fluctuating hard X-ray emission. Assuming that a disturbance is produced by some critical behavior in the disc, they successfully reproduce the 1/f-like fluctuations at high frequencies and also the break in the PDS at low frequencies, as seen in the PDS of X-ray binaries. The range in frequency over which the 1/f-like fluctuations are observed, depends on the size of the ADAF region of the disc. Beyond the ADAF region (in the outer disc regions), matter accretes smoothly via the usual viscous diffusion process, thus producing white-noise.
The break-frequency is determined by the maximum peak intensity of the X-ray shots which is on the order of the size of the advection-dominated region, and corresponds to the inverse of the free-fall time-scale of the largest avalanches (Takeuchi, Mineshige & Negoro 1995) at the radius of the ADAF region ($\rm R_{\rm crit}$). As shown in equation 13 of @Takeuchi95
$$\frac{R_{\rm crit}}{R_{\rm sch}} \sim 10^{3.2} \left( \frac{f_{\rm break}}{0.1} \right)^{-2/3}
\left( \frac{M_{\rm X}}{10M_{\odot}} \right)^{-2/3}$$
where $\rm R_{\rm sch} = 2GM_{\rm X}/c^{2}$ is the Schwarzschild radius for a black hole with mass $\rm M_{\rm X}$. However, it should be noted that the break frequency depends not only on the size of the ADAF region but also on the propagation speed of the perturbation (Mineshige priv. comm.). Since the perturbation velocity should be less than the free-fall velocity, the free-fall velocity gives an upper limit to the size of the ADAF. Using $\rm M_{\rm X}$=7.2[$\rm M_{\odot}$]{}and the $\rm f_{\rm break}\sim$2mHz observed in quiescence (section\[PDS\]), we find $\rm R_{\rm crit} < 3 \times 10^{4}\,R_{\rm sch}$ (=0.9[$\rm R_{\odot}$]{}).
The fluctuating viscosity model {#VISCOSITY}
-------------------------------
@Lyubarskii97 considered an ADAF disc and showed that the 1/f-like fluctuations in the accretion rate (and thus luminosity) near the inner radius of the disc can in principle be caused by the viscosity fluctuating independently (most likely due to a magnetic dynamo) at different radii on the local viscous time-scale. @Lyubarskii97 was able to reproduce a power–law PDS with a frequency break given by the Keplerian accretion time-scale $\rm \tau = (\alpha \Omega_{\rm K}^{-1}$), where $\Omega_{\rm K}$ is the Keplerian angular velocity and $\alpha$ is the dimensionless viscosity parameter [@Shakura73]. For an observed break-frequency of 2mHz, this corresponds to an ADAF with an outer disc radius of $\rm \sim 10^{4}\,R_{\rm sch}$.
Recently, @King04 have shown how the local magnetic dynamo in the disc can also affect the disc evolution. Angular momentum losses due to a disc wind/jet can significantly drive the accretion rate in systems which are in a jet-dominated state [@Fender03]. @King04 consider an accretion disc in which a magnetic dynamo generates the viscosity, but also occasionally produces a well-ordered poloidal field which affects the accretion rate by driving a wind/jet [@Livio03]. The luminosity arises from viscous dissipation in the disc after allowing for the energy removed by a wind/jet, and produces a 1/f-like PDS and a break-frequency given by the magnetic alignment time-scale at the inner disc edge. However, it should be noted that in their models the disc extends to the last stable orbit around a black hole, as is the case for the X-ray high/soft thermal dominant state. Although this is an attractive model, because in principle the same model could be used to explain the flickering in cataclysmic variables, it is clear that it needs to be adapted to include an ADAF in the low/hard or quiescent states.
The low-frequency $\sim$2mHz broad QPO {#LFQPO}
======================================
During the outburst of [XTEJ1118+480]{} a $\sim$0.08Hz QPO was reported in the X-rays [@Revnivtsev00] and subsequently confirmed by [*ASCA*]{} data [@Yamaoka00] as well as [*RXTE*]{} observations [@Wood00]. Simultaneous [*RXTE/HST*]{} observations showed a similar QPO [@Haswell00] and that the variability in the X-rays and optical/UV bands were correlated [@Hynes03b]. In Figure\[FIG:PDS\_OP\] there is a suggestion of a broad QPO in the quiescent simultaneous [“slow”]{} [$\it g'$]{} and [$\it i'$]{} PDS [*and*]{} the independent [“fast”]{} [$\it i'$]{} PDS at $\sim$2mHz.
In the context of a two-component accretion flow model (@Narayan96; Narayan, Barret & McClintock 1997) an ADAF has turbulent gas at all radii, with a variety of time-scales, ranging from a slow time-scale at the transition radius down to nearly the free-fall time close to the black hole. Interactions between the hot inner ADAF and the cool, outer disc, at or near the transition radius, can be a source of optical QPO variability, due to synchrotron emission by the hot electrons in the ADAF, and would have a characteristic time-scale given by a multiple of the Keplerian rotation period at the transition radius. If we assume that the 2mHz broad QPO observed in the PDS is the dynamical time-scale at the transition radius, then the transition radius lies at $\sim$8000$\rm R_{\rm sch}$.
Discussion {#DISCUSSION}
==========
Band-limited noise and QPO’s are a common feature in the low/hard state XRTs and many LMXBs [@McClintock03b]. Indeed, observations of [XTEJ1118+480]{} in the low/hard state reveal the presence of band-limited noise at X-ray and optical/UV wavelengths [@Hynes03b]. For comparison, in Figure\[FIG:PDS\_OPX\] we show the low/hard state X-ray PDS of [XTEJ1118+480]{}along with our optical quiescent PDS (section\[PDS\]). The low/hard state X-ray PDS shows a low-frequency break at 23mHz and a QPO at $\sim$80mHz, whereas the optical quiescent PDS shows either a break at a much lower frequency of $\sim$ 2mHz or a broad QPO at $\sim$2mHz. It is interesting to compare the optical and X-ray PDS assuming that the optical PDS can be described by either a break-frequency or a broad QPO model. The position of the quiescent optical broad QPO is a factor of $\sim$40 lower than the low/hard state QPO, and the quiescent optical break-frequency is $\sim$12 lower than the low/hard state break-frequency. The presence of the possible break-frequency or a broad QPO (possibly with a multiple frequency ratio) provides evidence that we are seeing the same phenomenon in outburst. The similarity between the low/hard and quiescent state PDS suggest that the optical variability could have a similar origin to the X-ray PDS and might be associated with the size of the direct emission from the self-absorbed synchrotron emission arising from an advective-dominated flow (see @Narayan94 and references therein) or from optically thin synchrotron emission directly from a jet (@Markoff01b; @Fender03). The quiescent state would have a larger ADAF region than in the low/hard state.
The slope of the [$\it g'$]{} PDS is steeper than the slope of the [$\it i'$]{} PDS, in contrast to what is observed in V404Cyg [@Shahbaz03b]. This difference could be due to either more low-frequency disc variability in [$\it g'$]{} compared to [$\it i'$]{} or more high-frequency variability in [$\it i'$]{}compared to [$\it g'$]{}. The latter would arise if there were more high frequency variability in [$\it i'$]{} from the inner disk, such as from the synchrotron emission arising from the ADAF.
The low/hard and quiescent states of the XRTs are often interpreted as having a truncated disc with an ADAF central region. An ADAF model has been successfully applied to [XTEJ1118+480]{} (@McClintock01b; @Esin01; @Chaty03). With spectral coverage which includes X-rays, Extreme Ultraviolet (EUV) and radio, model fits to the spectral energy distribution in the low/hard state suggests an ADAF inner radius of 350$\rm R_{\rm sch}$ [@Chaty03]. Recently @McClintock03a have determined the multi-wavelength X-ray/UV/optical quiescent spectrum of [XTEJ1118+480]{} and find that the spectrum has two components explained by an ADAF in the disc interior with a radius of $\rm \sim 10^{4}\,R_{\rm sch}$, and an optical/UV continuum that resembles a 13,000K disc blackbody model spectrum with a radius of 1500$\rm R_{\rm sch}$ arising from the disc/stream impact region.
The exterior accretion disc truncated at a radius of $\rm R_{\rm tr} \sim
10^{4}\,R_{\rm sch}$ is responsible for a substantial fraction of the optical/UV spectrum. Although an ADAF of this large size fits the X-ray data it does not produce a significant UV/optical component [@Narayan97]. Furthermore, a significant ADAF contribution to the optical/UV emission is ruled out because of (1) the presence of broad emission lines; (2) the Planckian shape of the optical/UV continuum spectrum, and (3) the large orbital modulation of the NUV continuum which cannot be attributed to synchrotron emission from an ADAF. For the models outlined in sections\[BMODELS\] and \[LFQPO\] we also find a quiescent ADAF radius of $\rm R_{\rm crit} \sim 10^{4}\,R_{\rm sch}$, comparable to the size of the quiescent ADAF region in the black hole A0620–00 and V404Cyg, suggesting that the size of the ADAF region is determined by the same underlying physical mechanism, such as viscosity and inner disc temperature.
In section\[VAR\] we determined the blackbody colour and radius of the short-term rapid flares. We find that the flares have a blackbody temperature of $\sim$3500K and a radius of $\sim$0.10[$\rm R_{\odot}$]{}(=3200$\rm R_{\rm sch}$), which is larger and cooler than the size and temperature of the gas stream/disc impact region determined by @McClintock03a. Most probably the short-term flares arise from [*all*]{} regions in the outer parts of the disc. However, it should be noted that @Torres04 find no evidence for a bright spot (disc/stream impact region) from their optical spectroscopy taken in Jan 2003; a 13,000K spectrum [@McClintock03a] is not enough to produce Balmer emission lines. It could be that the mass transfer from the secondary star has reduced considerably and so the parameters for the short-term rapid flares we derive could be consistent with an origin from the bright spot. Only high-time resolution spectroscopic observations can resolve this issue.
Quiescent state models
----------------------
Although the quiescent thermal ADAF model, which predicts a curvature in the X-ray spectrum primarily because of the assumption of a thermal (power–law) energy distribution for the electrons (@Esin97; @Esin98), is consistent with the observed X-ray data [@McClintock03a], it should be noted that given the quality of the data, a power–law model as predicted by an ADAF with non-thermal electrons cannot be ruled out. The quiescent optical/UV data suggests a 13,000K multi-colour blackbody spectrum, arising from the accretion disc and/or where the gas stream impacts the disc. In principle a jet model, such as the one presented by @Markoff01b for [XTEJ1118+480]{} in the low/hard state, could be modified and applied to the quiescent optical/X-ray data shown in @McClintock03a. Since the jet model is a non-thermal model, one would expect it to predict a power–law spectrum in the X-ray band, which would be consistent with the data.
In the context of a jet model, it could be that the quiescent infrared/optical spectrum of [XTEJ1118+480]{} is an extension of the radio spectrum. If the radio spectrum is due to self-absorbed synchrotron emission from a conical jet, then above some frequency (at which the whole jet is optically thin) there should be a break to an optically thin synchrotron spectrum. Observations of the low/hard state X-ray source GX339–4 appears to have identified such a break in the near-infrared $I$-band [@Corbel02]. For GX339–4 the X-ray spectrum seems to lie on an extrapolation of the non-thermal component. Looking at quiescent optical/X-ray SED of [XTEJ1118+480]{}[@McClintock03a], it seems difficult to describe the X-ray as a continuation of the optical data. For [XTEJ1118+480]{} a similar two-component quiescent SED overlapping in the optical/infrared, with an additional Planckian component due to an accretion disc may be expected, as is indeed predicted by the jet models [@Markoff01b], but it is clear that quiescent data at radio and infrared wavelengths are needed before any firm conclusions can be drawn.
Conclusion
==========
We present [ULTRACAM]{} observations of the quiescent soft X-ray transient [XTEJ1118+480]{}. Superimposed on the [$\it g'$]{} and [$\it i'$]{}-band ellipsoidal lightcurves are rapid flare events, the PDS of which can be described by either a broken power–law model with a break frequency at $\sim$2mHz or a power–law model plus a broad quasi-periodic oscillation (QPO) at $\sim$2mHz. In either caase, the size of the quiescent ADAF region is esimated to be $\sim 10^{4}$ Schwarzschild radii, similar to that observed in other quiescent black hole X-ray transients. The similarities between that the low/hard state PDS (80mHz QPO and a $\sim$23mHz break-frequency) with the quiescent state optical PDS suggest a similar origin for the optical and X-ray variability, most likely from regions at/near the ADAF region.
Acknowledgments {#acknowledgments .unnumbered}
===============
TS and JC acknowledge support from the Spanish Ministry of Science and Technology under the grant AYA200203570 and the programme Ramón y Cajal. TRM acknowledges the support of a PPARC Senior Research Fellowship. ULTRACAM is supported by PPARC grant PPA/G/S/2002/00092. Based on observations made with the William Herschel Telescope operated on the island of La Palma by the Issac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.
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---
abstract: 'We introduce a new multilevel domain decomposition method (MDD) for electronic structure calculations within semi-empirical and Density Functional Theory (DFT) frameworks. This method iterates between local fine solvers and global coarse solvers, in the spirit of domain decomposition methods. Using this approach, calculations have been successfully performed on several linear polymer chains containing up to 40,000 atoms and 200,000 atomic orbitals. Both the computational cost and the memory requirement scale linearly with the number of atoms. Additional speed-up can easily be obtained by parallelization. We show that this domain decomposition method outperforms the Density Matrix Minimization (DMM) method for poor initial guesses. Our method provides an efficient preconditioner for DMM and other linear scaling methods, variational in nature, such as the Orbital Minimization (OM) procedure.'
author:
- |
M. Barrault$^1$, E. Cancès$^2$, W. W. Hager $^3$ and C. Le Bris$^2$\
[$^1$ EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart Cedex, France]{}\
[ ]{}\
[$^2$ CERMICS, École Nationale des Ponts et Chaussées,]{}\
[6 & 8, avenue Blaise Pascal, Cité Descartes,]{}\
[77455 Marne-La-Vallée Cedex 2, France ]{}\
[$^3$ Department of Mathematics, University of Florida,]{}\
[ Gainesville FL 32611-8105, USA, ]{}
title: Multilevel domain decomposition for electronic structure calculations
---
\[section\] \[section\] \[section\] \[section\] \[section\]
Introduction and motivation
===========================
A central issue in computational quantum chemistry is the determination of the electronic ground state of a molecular system. For completeness and self-consistency, we now briefly introduce the problem. In particular, we present it in a mathematical way.
Standard electronic structure calculations
------------------------------------------
A molecular system is composed of $N$ electrons, modelled quantum mechanically, and a given number of nuclei, the latter being considered as classical point-like particles clamped at known positions (Born-Oppenheimer approximation). We refer to [@cances-handbook] for a general mathematical exposition and to [@Hehre; @McWeeny1] for the chemical background. Determining the electronic ground state amounts to solving a time-independent Schrödinger equation in $\RR^{3N}$. This goal is out of reach for large values of $N$. In fact it is already infeasible for values of $N$ exceeding three or four, unless dedicated techniques are employed. Examples are stochastic-like techniques such as Diffusion Monte-Carlo approaches, or emerging techniques, such as sparse tensor products techniques [@acta]. Approximations of the Schrödinger equation have been developed, such as the widely used *tight-binding*, *Hartree-Fock* and *Kohn-Sham* models. For these three models, the numerical resolution of a problem of the following type is required: given $H$ and $S$, respectivement an $N_b \times N_b$ symmetric matrix and an $N_b \times N_b$ symmetric positive definite matrix (with $N_b > N$), compute a solution $D_\star$ of the problem $$\label{eq:euler}
\left\{
\begin{array}{l}
\displaystyle H c_i = \epsilon_i S c_i,
\qquad \qquad
\displaystyle \epsilon_1 \leq \ldots \leq\epsilon_N \le \epsilon_{N+1}\leq
\ldots \leq \epsilon_{N_b},\\
\\
\displaystyle c_i^t Sc_j = \delta_{ij},\\
\\
D_\star = {\displaystyle }\sum_{i=1}^N c_ic_i^t.
\end{array} \right.$$ Let us mention that most electronic structure calculations are performed with closed shell models [@Hehre], and that, consequently, the integer $N$ in (\[eq:euler\]) then is the number of electron pairs. We remark that when $S$ is the identity matrix, a solution $D_\star$ to (\[eq:euler\]) is a solution to the problem $$\label{eq:pb}
\left\{
\begin{array}{l}
\hbox{\rm Find the {\em orthogonal projector} on
the space spanned by the}\, N\,\hbox{\rm
eigenvectors}\\
\hbox{\rm associated with the lowest}\, N\,\hbox{\rm eigenvalues of $H$.}
\end{array}
\right.$$ In (\[eq:pb\]), and throughout this article, the eigenvalues are counted with their multiplicities. The $N$ eigenvectors $c_i$, called *generalized* eigenvectors in order to emphasize the presence of the matrix $S$, represent the expansion in a given Galerkin basis $\left\{ \chi_i \right\}_{1 \le i \le N_b}$ of the $N$ one-electron wavefunctions. The matrix $H$ is a mean-field Hamiltonian matrix. For instance, for the Kohn-Sham model, we have $$\label{eq:defH}
H_{ij} = \frac 1 2 \int_{\RR^3} \nabla \chi_i \cdot \nabla
\chi_j + \int_{\RR^3} V \chi_i \chi_j$$ where $V$ is a mean-field local potential. The matrix $S$ is the overlap matrix associated with the basis $\left\{ \chi_i \right\}_{1 \le i \le N_b}$: $$\label{eq:defS}
S_{ij} = {\displaystyle }\int_{\RR^3} \chi_i \chi_j.$$ In this article, we focus on the *Linear Combination of Atomic Orbitals* (LCAO) approach. This is a very efficient discretization technique, using localized basis functions $\left\{ \chi_i
\right\}$, compactly supported [@SIESTA] or exhibiting a gaussian fall-off [@Hehre].
It is important to emphasize what makes the electronic structure problem, discretized with the LCAO approach, specific as compared to other linear eigenvalue problems encountered in other fields of the engineering sciences (see [@Lehoucq; @Lehoucq2] for instance). First, $N_b$ is proportional to $N$, and not much larger than it (say $N_b \sim 2N$ to fix the ideas). Hence, the problem is not finding a few eigenvectors of the generalized eigenvalue problem (\[eq:euler\]). Second, although the matrices $H$ and $S$ are sparse for large molecular systems (see section \[sec:linearscaling\] for details), they are not as sparse as the stiffness and mass matrices usually encountered when using finite difference or finite element methods. For example, the bandwith of $H$ and $S$ is of the order of $10^2$ in the numerical examples reported in section \[sec:numerical\]. Note that, in contrast, for plane wave basis set discretizations (which will not be discussed here), the parameter $N_b$ is much larger than $N$ (say $N_b \sim\, 100\, N$), the matrix $S$ is the identity matrix and the matrix $H$ est full. Third, and this is a crucial point, the output of the calculation is the matrix $D_\star$ and not the generalized eigenvectors $c_i$ themselves. This is the fundamental remark allowing the construction of linear scaling methods (see section \[sec:linearscaling\]).
A solution $D_\star$ of (\[eq:euler\]) is $$\label{eq:D}
\displaystyle D_\star = C_\star C_\star^t$$ where $C_\star$ is a solution to the minimization problem $$\label{eq:infF}
\inf \biggl\{\mbox{\textrm{Tr}}\Big(HCC^{t}\Big), \quad C\in
{{\mathcal M}}^{N_{b},N}(\RR), \; C^{t}SC=I_{N} \biggr\}.$$ Note that the energy functional $\mbox{\textrm{Tr}}\Big(HCC^{t}\Big)$ can be given the more symmetric form $\mbox{\textrm{Tr}}\Big(C^{t}HC\Big)$. Here and below, ${{\mathcal M}}^{k,l}$ denotes the vector space of the $k \times l$ real matrices. Notice that (\[eq:infF\]) has many minimizers: if $C_\star$ is a minimizer, so is $C_\star U$ for any orthogonal $N \times N$ matrix $U$. However, under the standard assumption that the $N$-th eigenvalue of $H$ is strictly lower than the $(N+1)$-th one, the matrix $D^\ast$ defined by (\[eq:D\]) does not in fact depend on the choice of the minimizer $C_\star$ of (\[eq:infF\]). Notice also that (\[eq:euler\]) are not the Euler-Lagrange equations of (\[eq:infF\]) but that any critical point of (\[eq:infF\]) is obtained from a solution of (\[eq:euler\]) by an orthogonal transformation of the columns of $C_\star = {\displaystyle }\left( c_1 | \cdots |c_N \right)$.
The standard approach to compute $D_\star$ is to solve the generalized eigenvalue problem (\[eq:euler\]) and then construct $C_\star$ thus $D_\star$ by collecting the lowest $N$ generalized eigenvectors of $H$. This approach is employed when the number $N$ of electrons (or electron pairs) is not too large, say smaller than $10^3$.
Linear scaling methods {#sec:linearscaling}
----------------------
One of the current challenges of Computational Chemistry is to lower the computational complexity $N^3$ of this solution procedure. A linear complexity $N$ is the holy grail. There are various existing methods designed for this purpose. Surveys on such methods are [@revON; @FOE]. Our purpose here is to introduce a new method, based on the *domain decomposition* paradigm. We remark that the method introduced here is not the first occurrence of a method based on a decomposition of the matrix $H$ [@DCchem], but a significant methodological improvement is fulfilled with the present method. To the best of our knowledge, such methods only consist of local solvers complemented by a crude global step. The method introduced below seems to be the first one really exhibiting the local/global paradigm in the spirit of methods used in other fields of the engineering sciences. Numerical obervations confirm the major practical interest methodological improvement.
Why is a *linear scaling* plausible for computing $D_\star$? To justify the fact that the cubic scaling is an estimate by excess of the computational task required to solve (\[eq:euler\]), we argue that the matrix does not need to be diagonalized. As mentioned above, only the *orthogonal projector* on the subspace generated by the lowest $N$ eigenvectors is to be determined and *not* the *explicit* values of these lowest $N$ eigenvectors. But in order to reach a linear complexity, appropriate assumptions are necessary, both on the form of the matrices $H$ and $S$, and on the matrix $D_\star$ solution to (\[eq:euler\]):
- (H1). The matrices $H$ and $S$ are assumed sparse, in the sense that, for large systems, the number of non-zero coefficients scales as $N$. This assumption is not restrictive. In particular, it follows from (\[eq:defH\]) and (\[eq:defS\]) that it is automatically satisfied for Kohn-Sham models as soon as the basis functions are localized in real space, which is in particular the case for the widely used atomic orbital basis sets [@cances-handbook];
- (H2). A second assumption is that the matrix $D_\star$ built from the solution to (\[eq:euler\]) is also sparse. This condition seems to be fulfilled as soon as the relative gap $$\label{eq:gam}
\displaystyle \gamma = \frac{\epsilon_{N+1} - \epsilon_{N}}{\epsilon_{N_b}-\epsilon_1}.$$ deduced from the solution of (\[eq:euler\]) is large enough. As explained in section \[sec:localization\] below, this observation can be supported by qualitative physical arguments. On the other hand, we are not aware of any mathematical argument of linear algebra that would justify assumption (H2) in a general setting.
We assume (H1)-(H2) in the following. Current efforts aim at treating cases when the second assumption is not fulfilled, which in particular corresponds to the case of conducting materials. The problem (\[eq:pb\]) is then extremely difficult because the gap $\gamma$ in (\[eq:gam\]) being very small, the matrix $D$ is likely to be dense. Reaching linear complexity is then a challenging issue, unsatisfactorily solved to date. State of the art linear scaling methods presented in the literature experience tremendous difficulties (to say the least) in such cases. It is therefore reasonable to improve in a first step the existing methods in the setting of assumption (H2), before turning to more challenging issues.
Before we get to the heart of the matter, we would like to point out the following feature of the problem under consideration.
In practice, Problem (\[eq:euler\]) has to be solved *repeatedly*. For instance, it is the inner loop in a nonlinear minimization problem where $H$ depends self-consistently on $D_\star$. We refer to [@outperform; @EDIIS] for efficient algorithms to iterate on this nonlinearity and to [@cances-handbook] for a review on the subject. Alternatively, or in addition to the above, problem (\[eq:euler\]) is parametrized by the positions of the nuclei (both the mean-field operator $H$ and the overlap matrix $S$ indeed depend on these positions), and these positions may vary. This is the case in molecular mechanics (find the optimal configuration of nuclei that gives the lowest possible energy to the molecular system), and in molecular dynamics as well (the positions of nuclei follow the Newton law of motion in the mean-field created by the electrons). In either case, problem (\[eq:euler\]) is not be solved *from scratch*. Because of previous calculations, we may consider we have at our disposal a good initial guess for the solution. The latter comes from e.g. previous positions of nuclei, or previous iterations in the outer loop of determination of $H$. In difficult cases it may even come from a previous computation with a coarse grained model. In other words, the question addressed reads [*solving Problem (\[eq:euler\]) for some $H+\delta H$ and $S +
\delta S$ that are small perturbations of previous $H$ and $S$ for which the solution is known*]{}. This specific context allows for a speed up of the algorithm when the initial guess is sufficiently good. This is the reason why, in the following, we shall frequently make distinctions between bad and good initial guesses.
Localization in Quantum Chemistry {#sec:localization}
=================================
The physical system we consider is a long linear molecule (for instance a one-dimensional polymer or a nanotube). Let us emphasize that we do not claim a particular physical relevance of this system. This is for the purpose of illustration. We believe the system considered to be a good representative of a broad class of large molecular systems that may be encountered practically. Each atomic orbital $\chi_i$ is centered on one nucleus. Either it is supported in a ball of small radius [@Siesta] (in comparison to the size of the macromolecule under study), or it has a rapid exponential-like or Gaussian-like [@gill] fall-off. The atomic orbitals are numbered following the orientation of the molecule. Then, the mean-field Hamiltonian matrix $H$ whose entries are defined by (\[eq:defH\]) has the band structure shown in Figure \[fig:Fband\].
Although the eigenvectors of $H$ are [*a priori*]{} delocalized (most of their coefficients do not vanish), it seems to be possible to build a $S$-orthonormal basis of the subspace generated by the lowest $N$ eigenvectors of $H$, consisting of [*localized*]{} vectors (only a few consecutive coefficients are non zero). This is motivated by a physical argument of locality of the interactions [@Kohn]. For periodic systems, the localized vectors correspond to the so-called Wannier orbitals [@Wannier]. It can be proven that in this case, the larger the band gap, the better the localization of the Wannier orbitals [@Kohn_crystals]. For insulators, the Wannier orbitals indeed enjoy an exponential fall-off rate proportional to the band gap. For conductors, the fall-off is only algebraic. As mentioned in the introduction, we only consider here the former case. This allows us to assume that there exists some integer $q \ll N_b$, such that $N_b/q$ is an integer, for which all of these localized functions can be essentially expanded on $q$ consecutive atomic orbitals. Denoting by $n=2q$, we can therefore assume a good approximation of a solution $C_\star$ to (\[eq:infF\]) exists, with the block structure displayed on Figure \[fig:Cblock\]. Note that each block $C_i$ only overlaps with its nearest neighbors. Correspondingly, we introduce the block structure of $H$ displayed on Figure \[fig:Fblock\]. The matrix $D$ constructed from a block matrix $C$ using (\[eq:D\]) has the structure represented in Figure \[fig:D\] and satisfies the constraints $D=D^t$, $D^2=D$, Tr($D)$ = $N$.
Let us point out that the integers $q$ and $n=2q$ depend on the band gap, *not* on the size of the molecule. The condition $n=2q$ is only valid for $S=I_{N_b}$. For $S\not=I_{N_b}$, it is replaced by $n=2q+nbs$ where $2 \, nbs - 1$ is the bandwidth of the matrix $S$.
The domain decomposition algorithm we propose aims at searching an approximate solution to (\[eq:infF\]) that has the block structure described above.
For simplicity, we now present our method assuming that $S=I_{N_b}$, i.e. that the Galerkin basis $\left\{
\chi_i \right\}_{1 \le i \le N_b}$ is orthonormal. The extension of the method to the case when $S\neq I_{N_b}$ is straightforward. Problem (\[eq:infF\]) then reads $$\label{eq:infF2}
\inf \biggl\{\mbox{\textrm{Tr}}\Big(HCC^{t}\Big), \quad C\in
{{\mathcal M}}^{N_{b},N}(\RR), \; C^{t}C=I_{N} \biggr\}.$$
Our approach consists in solving an approximation of problem (\[eq:infF2\]) obtained by minimizing the exact energy ${\displaystyle }\mbox{\textrm{Tr}}\Big(HCC^{t}\Big)$ on the set of the matrices $C$ which have the block structure displayed on Figure \[fig:Cblock\] and satisfy the constraint $C^{t}C=I_{N}$. The resulting minimization problem can be recast as $$\begin{aligned}
\mbox{\textrm{inf}}\bigg\{
\sum_{i=1}^{p}\mbox{\textrm{Tr}}\left(H_iC_iC_i^{t}\right), & & C_i\in
{{\mathcal M}}^{n,m_i}(\RR),\quad m_i \in \NN, \quad C_i^{t}C_i=I_{m_i} \quad \forall \; 1\leq i\leq p,
\nonumber \\
& & C_{i}^{t}TC_{i+1}=0 \quad \forall \; 1\leq i\leq p-1, \quad \sum_{i=1}^p m_i = N \bigg\}.
\label{eq:infF-blocks}\end{aligned}$$ In the above formula, $T \in {\cal M}^{n,n}(\RR)$ is the matrix defined by $$\label{eq:matrices_t}
T_{kl} =
\left\{ \begin{array}{ll}
1 & \mbox{ if } k - l = q \\
0 & \mbox{ otherwise}
\end{array} \right.$$ and $H_i \in {\cal M}^{n,n}(\RR)$ is a symmetric submatrix of $H$ (see Figure \[fig:Fblock\]). Indeed,
and
In this way, we replace the $\frac{N(N+1)}2$ [*global*]{} scalar constraints $C^tC=I_N$ involving vectors of size $N_b$, by the $\sum_{i=1}^p \frac{m_i (m_i+1)}2$ [*local*]{} scalar constraints $C_i^t C_{i}=I_{m_i}$ and the $\sum_{i=1}^{p-1} m_i m_{i+1}$ [*local*]{} scalar constraints $C_i^t T
C_{i+1}=0$, involving vectors of size $n$. We would like to emphasize that we can obtain in this way a basis of the vector space generated by the lowest $N$ eigenvectors of $H$, but not the eigenvectors themselves. This method is therefore not directly applicable to standard diagonalization problems.
Our algorithm searches for the solution to (\[eq:infF-blocks\]), not to (\[eq:infF2\]). More rigorously stated, we search for the solution to the Euler-Lagrange equations of (\[eq:infF-blocks\]): $$\label{eq:euler-blocks}
\left\{
\begin{array}{ccll}
\displaystyle H_iC_i &=& C_iE_i + T^t C_{i-1}\Lambda_{i-1,i} +
T C_{i+1}\Lambda_{i,i+1}^t & \qquad 1 \le i \le p,\\
\displaystyle C_i^{t}C_i &=&I_{m_i} & \qquad 1 \le i \le p,\\
\displaystyle C_{i}^{t}T C_{i+1} &=&0& \qquad 1 \le i \le p-1,
\end{array}\right.$$ where by convention $$\label{eq:convention}
C_0 = C_{p+1} = 0.$$ The matrices $(E_i)_{1 \le i \le p}$ and $(\Lambda_{i,i+1})_{1
\le i \le p-1}$ respectively denote the matrices of Lagrange multipliers associated with the orthonormality constraints $C_i^{t}C_i=I_{m_i}$ and $C_{i}^{t}T
C_{i+1} = 0$. The $m_i \times m_i$ matrix $E_i$ is symmetric. The matrix $\Lambda_{i,i+1}$ is of size $m_i \times m_{i+1}$. The above equations can be easily derived by considering the Lagrangian $$\begin{aligned}
{\cal L}\left(\left\{ C_i \right\}, \left\{ E_i \right\}, \left\{
\Lambda_{i,i+1} \right\} \right) & = & \sum_{i=1}^p {{\rm Tr}}\left( H_iC_iC_i^t
\right) + \sum_{i=1}^p {{\rm Tr}}\left( \left(C_i^tC_i-I_{m_i}\right) E_i
\right) \\ & & + \sum_{i=1}^{p-1} {{\rm Tr}}\left( C_i^tTC_{i+1}\Lambda_{i,i+1}^t
\right).\end{aligned}$$
The block structure imposed on the matrices clearly lowers the dimension of the search space we have to explore. However, this simplification comes at a price. First, problem (\[eq:infF-blocks\]) only [*approximates*]{} problem (\[eq:infF2\]). Second, (\[eq:infF-blocks\]) may have local, non global, minimizers, whereas all the local minimizers of (\[eq:infF2\]) are global. There are thus [*a priori*]{} many spurious solutions of the Euler Lagrange equations (\[eq:euler-blocks\]) associated with (\[eq:infF-blocks\]).
A point is that the sizes $(m_i)_{1 \le i \le p}$ are not [*a priori*]{} prescribed. In our approach, they are ajusted during the iterations. We shall see how in the sequel.
Description of the domain decomposition algorithm
=================================================
Description of a simplified form {#sec:simdesc}
--------------------------------
For pedagogic purpose, we first consider the following problem $$\label{eq:inf-vector-biblock}
\inf \left\{ \langle H_1Z_1,Z_1\rangle+\langle H_2Z_2,Z_2\rangle,
\quad Z_i \in{\RR}^{N_b}, \;
\langle Z_i,Z_i\rangle = 1, \; \langle
Z_1,Z_2\rangle=0 \right\}.$$ Problem (\[eq:inf-vector-biblock\]) is a particular occurence of (\[eq:infF-blocks\]). We have denoted by $\langle\cdot,\cdot\rangle$ the standard Euclidean scalar product on ${\RR}^{N_b}$.
For (\[eq:inf-vector-biblock\]), the algorithm is defined in the following simplified form. Choose $(Z_1^0,Z_2^0)$ satisfying the constraints and construct the sequence $(Z_1^k,Z_2^k)_{k \in \NN}$ by the following iteration procedure. Assume $(Z_1^k,Z_2^k)$ is known, then
- [ Local step]{}. Solve $$\label{eq:s11}
\left\{\begin{array}{lll}
\widetilde Z_1^k = \mbox{\textrm{arginf}}\big\{\langle H_1Z_1,Z_1\rangle,
\,\,Z_1\in{\RR}^{N_b},\,\,\,\langle Z_1,Z_1\rangle=1\,\,\langle Z_1,Z_2^k\rangle=0\big\},\\
\widetilde Z_2^k = \mbox{\textrm{arginf}}\big\{\langle H_2Z_2,Z_2\rangle,
\,\,Z_2\in{\RR}^{N_b},\,\,\,\langle Z_2,Z_2\rangle=1\,\,\langle
\widetilde Z_1^k ,Z_2\rangle=0\big\} \; ;
\end{array}
\right.$$
- [ Global step]{}. Solve $$\label{eq:s21}
\alpha^\ast= \mbox{\textrm{arginf}} \big\{\langle
H_1Z_1,Z_1\rangle+\langle H_2Z_2,Z_2\rangle, \; \alpha\in{\RR}\big\}$$ where $$\label{eq:s21bis}
Z_1=\frac{\widetilde Z_1^k+\alpha\widetilde
Z_2^k}{\sqrt{1+\alpha^2}},\,\,Z_2=\frac{-\alpha\widetilde Z_1^k+\widetilde
Z_2^k}{\sqrt{1+\alpha^2}},$$ and set $$\label{eq:newit}
Z_1^{k+1}=\frac{\widetilde Z_1^k+\alpha^\ast\widetilde
Z_2^k}{\sqrt{1+(\alpha^\ast)^2}},\quad
Z_2^{k+1}=\frac{-\alpha^\ast\widetilde Z_1^k+\widetilde Z_2^k}
{\sqrt{1+(\alpha^\ast)^2}}.$$
In the $k$-th iteration of the local step, we first fix $Z_2 = Z_2^k$ and optimize over $Z_1$ to obtain $\tilde{Z}_1^k$. Then we fix $Z_1 =
\tilde{Z}_1^k$ and optimize over $Z_2$ to obtain $\tilde{Z}_2^k$. This local step monotonically reduces the objective function, however, it may not converge to the global optimum. The technical problem is that the Lagrange multipliers associated with the constraint $\langle Z_1,Z_2\rangle = 0$ may converge to different values in the two subproblems associated with the local step. In the global step, we optimize the [*sum*]{} $\langle H_1
Z_1,Z_1\rangle + \langle H_2 Z_2,Z_2\rangle$ over the subspace spanned by $\tilde{Z}_1^k$ and $\tilde{Z}_2^k$, subject to the constraints in (\[eq:inf-vector-biblock\]). The global step again reduces the value of the objective function since $\tilde{Z}_1^k$ and $\tilde{Z}_2^k$ are feasible in the global step. It can be shown that the combined algorithm (local step + global step) monotonically decreases the objective function and globally converges to an optimal solution of (\[eq:inf-vector-biblock\]).
This algorithm operates at two levels: a fine level where we solve two problems of dimension $N_b$ rather than one problem of dimension $2N_b$; a coarse level where we solve a problem of dimension $2$. Left by itself, the fine step converges to a suboptimal solution of (\[eq:inf-vector-biblock\]). Combining the fine step with the global step yields convergence to a global optimum.
In addition to providing a pedagogic view on the general algorithm presented in the following section, the simplified form (\[eq:s11\])-(\[eq:newit\]) has a theoretical interest. In contrast to the general algorithm for which we cannot provide a convergence analysis, the simplified form (\[eq:s11\])-(\[eq:newit\]) may be analyzed mathematically, at least in the particular situation when $H_1=H_2=H$. Then solving (\[eq:inf-vector-biblock\]) amounts to searching for the lowest two eigenelements of the matrix $H$. Notice that the global step (\[eq:s21\])-(\[eq:newit\]) is then unnecessary because the functional to minimize in (\[eq:s21\]) does not depend on $\alpha$.
However, we can show that the iterations (\[eq:s11\]) converge in the following sense. The 2-dimensional vector space spanned by the lowest two eigenvalues of $H$ is reached asymptotically. This occurs under an appropriate condition on the matrix $H$. The latter is a condition of separation of the eigenvalues, namely $\displaystyle\epsilon_2-\epsilon_1 < \epsilon_3 - \epsilon_2$ with obvious notation. The gap $\epsilon_3-\epsilon_2$ gives the speed of convergence. For brevity, we do not detail the proof here (see [@theseMax]). Future work on the numerical analysis of more general cases is in progress.
Description of the algorithm {#sec:gendesc}
----------------------------
We define, for all $p$-tuple $(C_i)_{1\leq i\leq p}$, $$\displaystyle {{\mathcal E}}\Big((C_i)_{1\leq i\leq p}\Big) =
\sum_{i=1}^p\mbox{\textrm{Tr}}\Big(H_iC_iC_i^t\Big),$$ and set by convention $$U_{0} = U_{p} = 0.$$ We introduce an integer $\epsilon$, initialized to one, that will alternate between the values zero and one during the iterations.
At iteration $k$, we have at hand a set of block sizes $(m_i^k)_{1 \le i \le p}$ and a set of matrices $(C_i^k)_{1 \le i \le
p}$ such that $C_i^k \in {\cal M}^{n,m_i^k}(\RR)$, $[C_i^k]^tC_i^k=I_{m_i^k}$, $[C_i^k]^t T C_{i+1}^k = 0$. We now explain how to compute the new iterate $(m_i^{k+1})_{1 \le i \le p}$, $(C_i^{k+1})_{1 \le i \le
p}$.
[**Multilevel Domain Decomposition (MDD) algorithm**]{}
1. [**$\bullet$ Step 1: Local fine solver**]{}.
1. For each $i$, diagonalize the matrix $H_{2i+\epsilon}$ in the subspace $$V_{2i+\epsilon}^k = \left\{ x \in \RR^{n}, \quad
\left[C_{2 i + \epsilon - 1}^k\right]^t T x = 0, \quad x^t T C_{2i + \epsilon + 1}^k
= 0 \right\},$$ i.e. diagonalize $P_{2i+\epsilon}^k H_{2i+\epsilon} P_{2i+\epsilon}^k$ where $P_{2i+\epsilon}^k$ is the orthogonal projector on $V_{2i+\epsilon}^k$. This provides (at least) $n - m_{2i +\epsilon - 1}^k - m_{2i +\epsilon+
1}^k$ real eigenvalues $\lambda_{2i+\epsilon,1}^k \le \lambda_{2i+\epsilon,2}^k \le \cdots $ and associated orthonormal vectors $x_{2i+\epsilon,j}^k$. The latter are $T$-orthogonal to the column vectors of $C_{i-1}^k$ and $C_{i+1}^k$.
2. Sort the eigenvalues $(\lambda_{2i+\epsilon,j}^k)_{i,j}$ in increasing order, and select the lowest ${\displaystyle }\sum_i
m_{2i+\epsilon}$ of them. For each $i$, collect in block $\#2i+\epsilon$ the eigenvalues $\lambda^k_{2i+\epsilon,j}$ selected. New intermediate block sizes $\bar m_{2i+\epsilon}^{k}$ are defined.
3. For each $i$, collect the lowest $\bar m_{2i+\epsilon}^{k}$ vectors $x_{2i+\epsilon,j}^k$ in the $n \times
\bar m_{2i+\epsilon}^{k}$ matrix $\overline C_{2i+\epsilon}^{k}$.
4. For each $i$, diagonalize the matrix $H_{2i+\epsilon+1}$ in the subspace $$V_{2i+\epsilon+1}^k = \left\{ x \in \RR^{n}, \quad
\left[\overline C_{2i + \epsilon}^k\right]^t T x = 0, \quad x^t T
\overline C_{2i + \epsilon +2}^k = 0 \right\}$$ in order to get eigenvalues $\lambda_{2i+\epsilon+1,1}^k \le
\lambda_{2i+\epsilon+1,2}^k \le \cdots $ and associated orthonormal vectors $x_{2i+\epsilon+1,j}^k$. The latter are $T$-orthogonal to the column vectors of $ \overline C_{2i +
\epsilon}^k$ and $\overline C_{2i + \epsilon +2}^k$.
5. Sort all the eigenvalues $\displaystyle\bigl\{(\lambda_{2i+\epsilon+1,j}^k)_{i,j},
(\lambda_{2i+\epsilon,j}^k)_{i,j}\bigr\}$ in increasing order. Select the lowest $N$. For each $l$, collect in block $\#l$ the eigenvalues $\lambda^k_{l,j}$ selected. New intermediate block sizes $(m_{l}^{k+1})_{1 \le l \le p}$ are thus defined.
6. Set $\widetilde C_l^{k} = {\displaystyle }\left[
x_{l,1}^k | \cdots |x_{l,m_{l}^{k+1}}^k
\right]$.
7. Replace $\epsilon$ by $1-\epsilon$ and proceed to step 2 below.
2. [**$\bullet$ Step 2: global coarse solver**]{}. Solve $$\label{eq:global-manyblock}
\displaystyle {{\mathcal U}}^\ast = \mbox{\textrm{arginf}}\Big\{f({{\mathcal U}}),\; {{\mathcal U}}=
(U_{i})_i,\;\forall 1 \le i
\le p-1 \;\; U_i \in
{{\mathcal M}}^{m_{i+1},m_{i}}(\RR)\Big\},$$ where $$\label{eq:global-manyblock1}
\displaystyle f({{\mathcal U}}) = {{\mathcal E}}\bigg(\Big(C_i({{\mathcal U}})\big(C_i({{\mathcal U}})^tC_i({{\mathcal U}})\big)^{-\frac{1}{2}}\Big)_i\bigg),$$ and $$\label{eq:global-manyblock2}
C_i({{\mathcal U}}) = \widetilde{C}_i^k + T
\widetilde{C}_{i+1}^k U_i\Big( [\widetilde{C}_{i}^k]^t T T^t
\widetilde{C}_{i}^k \Big) - T^t
\widetilde{C}_{i-1}^k U_{i-1}^t\Big([\widetilde{C}_{i}^k]^tT^t T
\widetilde{C}_{i}^k\Big).$$ Next set, for all $1 \le i \le p$, $$\label{eq:defCkp1}
C_i^{k+1} = C_i \big({{\mathcal U}}^\ast\big) \; \left( C_i \big({{\mathcal U}}^\ast\big)^t
\, C_i \big({{\mathcal U}}^\ast\big) \right)^{-1/2} \; .$$ Note that ${\displaystyle }\left[ C_i^{k+1} \right]^t T C_{i+1}^{k+1}=0$ (this follows from $T^2=0$).
We think of the even indexed unknowns $C_{2i}$ as the black variables and the odd indexed unknowns $C_{2i+1}$ as the white variables. In the first phase of the local fine solver, we optimize over the white variables while holding the black variables fixed. In the second phase of the local fine solver, we optimize over the black variables while holding the white variables fixed. In the global step, we perturb each variable by a linear combination of the adjacent variables. The matrices ${{{\mathcal U}}} = (U_{i})_i$ in (\[eq:global-manyblock\]) play the same role as the real parameter $\alpha$ in (\[eq:s21\]). The perturbation is designed so that the constraints are satisfied. The optimization is performed over the matrices generating the linear combinations. In the next iteration, we interchange the order of the optimizations: first optimize over the black variables while holding the white variables fixed, then optimize over the white variables while holding the black variables fixed.
Let us point out that an accurate solution to (\[eq:global-manyblock\]) is not needed. In practice, we reduce the computational cost of the global step, by using again a domain decomposition method. The blocks $(C_i)_{1 \le
i \le p}$ are collected in $r$ overlapping groups $(G_l)_{1 \le l \le
r}$ as shown in Figure \[fig:groupe\]. Problem (\[eq:global-manyblock\]) is solved first for the blocks $(G_{2l+1})$, next for the blocks $(G_{2l})$. Possibly, this procedure is repeated a few times. The advantage of this strategy is that the computational time of the global step scales linearly with $N$. In addition, it is parallel in nature. The solution of (\[eq:global-manyblock\]) for a given group is performed by a few steps of a Newton-type algorithm. Other preconditioned iterative methods could also be considered.
Comments on the local step
--------------------------
The local step is based on a checkerboard iteration technique.
When $\epsilon=1$, steps 1a-1c search for a solution $(\bar m_{2i+1}^{k},\overline C_{2i+1}^{k})_i$ to the problem $$\begin{aligned}
\mbox{\textrm{inf}}\bigg\{
\sum_{i} \mbox{\textrm{Tr}}\left(H_{2i+1}C_{2i+1}C_{2i+1}^{t}\right),
& & C_{2i+1} \in
{{\mathcal M}}^{n,m_{2i+1}}(\RR),\quad C_{2i+1}^{t}C_{2i+1}=I_{m_{2i+1}}, \\
& &
[C_{2i}^k]^{t}TC_{2i+1}=0, \quad C_{2i+1}^{t}TC_{2i+2}^k=0, \\
& &
m_{2i+1} \in \NN, \quad \sum_{i} m_{2i+1} =
\sum_{i} m_{2i+1}^k \bigg\} .\end{aligned}$$ During steps 1a-1c, the “white” blocks $C_{2i}^k$ are kept fixed. The “black” blocks $C_{2i+1}^k$ are optimized under the orthogonality constraints imposed by the “white” blocks. A point is that most of the computational effort can be done [*in parallel*]{}. Indeed, for $p$ even, say, performing step 1a amounts to solving $p/2$ [*independent*]{} diagonalisation problems of size $n$.
Likewise, steps 1d-1f solve $$\begin{aligned}
\mbox{\textrm{inf}}\bigg\{
\sum_{i=1}^p \mbox{\textrm{Tr}}\left(H_{i}C_{i}C_{i}^{t}\right),
& & C_{i} \in
{{\mathcal M}}^{n,m_{i}}(\RR),\quad C_{i}^{t}C_{i}=I_{m_{i}}, \quad m_{i} \in
\NN, \quad \sum_{i} m_{i} = N
\\ & &
[\overline
C_{2j-1}^k]^{t}TC_{2j}=0 , \quad
[C_{2j}]^{t}T [\overline C_{2j+1}]^k=0, \\ & &
0 \le m_{2j+1} \le \bar m_{2j+1}^k, \quad C_{2j+1} \subset \overline
C_{2j+1}^k
\bigg\},\end{aligned}$$ where the notation $C_{2j+1} \subset \overline
C_{2j+1}^k$ means that each column of $C_{2j+1}$ is a column of $\overline C_{2j+1}^k$. Here again, most of the computational effort can be performed in parallel.
When $\epsilon=1$, “black” vectors (i.e. vectors belonging to blocks with odd indices) are allowed to become “white” vectors, but the reverse is forbidden. In order to symmetrize the process, $\epsilon$ is replaced by $1-\epsilon$ in the next iteration.
We wish to emphasize that, although called [*local*]{}, this step already accounts for some global concern. Indeed, and it is a key point of the local step, substeps (b) and (e) sort the *complete* set of eigenvalues generated locally. This, together with the update of the size $m_i$ of the blocks, allows for a preliminary propagation of the information throughout the whole system. The global step will complement this.
Finally, let us mention that in the local steps, (approximate) $T$-orthogonality is obtained by a Householder orthonormalization process. The required orthonormality criterion is $$\label{eq:epsilonL}
\forall \; 1 \le i \le p-1, \quad
\big\|[\tilde{C}^k_i]^tT\tilde{C}^k_{i+1} \big\| \leq \epsilon_L,$$ where $\epsilon_L > 0$ is a threshold to be chosen by the user.
Comments on the global step {#sec:global}
---------------------------
Let us briefly illustrate the role played by the global step. For simplicity, we consider the case of two blocks of same initial size $m_1=m_2=m$ and we assume that $m_1$ and $m_2$ do not vary during the iterations. If only the local step is performed, then the new iterate $$(C^{k+1}_1,C_2^{k+1})=(\widetilde{C}_1^{k},\widetilde{C}_2^{k})$$ does not necessarily satisfies (\[eq:euler-blocks\]). Indeed, there is no reason why the Lagrange multipliers corresponding to the two contraints $C^t
T C_2^k = 0$ (step 1a when $\epsilon=1$) on the one hand and $[\widetilde{C}_1^k]^t T C = 0$ (step 1d when $\epsilon=1$) on the other hand should be the same. The global step *asymptotically* enforces the equality of Lagrange multipliers. This is a way to account for a global feature of the problem.
Let us emphasize this specific point. Assume $U^\ast = 0$ in the global step of the $k$-th iteration of the algorithm, or in other words that the global step is not effective at the $k$-th iteration. Then it implies that the output $(\widetilde{C}_1, \widetilde{C}_2) = (\widetilde{C}_1^k,
\widetilde{C}_2^k)$ of the local step already satisfies (\[eq:euler-blocks\]). Indeed, $$\displaystyle f(U) = \mbox{\textrm{Tr}}\Big(J_1(U)C_1(U)^tH_1C_1(U)\Big) + \mbox{\textrm{Tr}}\Big(J_2(U)C_2(U)^tH_2C_2(U)\Big)$$ with $J_i(U) = \Big(C_i(U)^tC_i(U)\Big)^{-1}$ for $i=1,2$. Since $$\begin{aligned}
\Big(J_1(U)\Big)^{-1} &=& I_m +
\Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big) U^t
\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big) U
\Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big),\\
\Big(J_2(U)\Big)^{-1} &=& I_m +
\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big) U
\Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big) U^t
\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big),\end{aligned}$$ we have $\nabla J_1(0) = \nabla J_2(0) = 0$. The matrix $U$ being a square matrix of dimension $m$, for all $1 \le i,j \le m$, $$\begin{aligned}
\displaystyle \frac{1}{2}\frac{\partial f}{\partial U_{ij}}(0) &=&
\displaystyle
\mbox{\textrm{Tr}}\bigg( \left[\frac{\partial C_1}{\partial
U_{ij}}(0) \right]^tH_1\widetilde{C}_1\bigg) +
\mbox{\textrm{Tr}}\bigg( \left[\frac{\partial C_2}{\partial
U_{ij}}(0) \right]^tH_2\widetilde{C}_2\bigg) \nonumber\\
&=& \displaystyle
\bigg( \Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big)
\widetilde{C}_1^t H_1 T \widetilde{C}_2
\bigg)_{ji} -
\bigg(\widetilde{C}_1^tTH_2\widetilde{C}_2
\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big)\bigg)_{ji}
\nonumber\\
&=& \displaystyle \bigg(\Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big)(\Lambda_{1}-\Lambda_{2})\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big)\bigg)_{ji},\end{aligned}$$ where $\Lambda_1$ and $\Lambda_2$ are defined by $$\left\{
\begin{array}{ccc}
\displaystyle H_1\widetilde{C}_1 = \widetilde{C}_1E_1 + T\widetilde{C}_{2}\Lambda_{1}^t,\\
\displaystyle H_2\widetilde{C}_2 = \widetilde{C}_2E_2 +
T^t\widetilde{C}_{1}\Lambda_{2}.\\
\end{array}\right.$$ As $U^\ast = 0$ implies $$\displaystyle \forall \; 1 \leq i,j \leq m \quad \frac{\partial f}{\partial
U_{ij}} (0) = 0,$$ we conclude that $ \Lambda_1 = \Lambda_2$ if the matrices $\Big(\widetilde{C}_1^tTT^t\widetilde{C}_1\Big)$ and $\Big(\widetilde{C}_2^tT^tT\widetilde{C}_2\Big)$ are invertible, which is generally the case when $n \gg 2m$. Consequently, (\[eq:euler-blocks\]) is satisfied by $(\widetilde{C}_1, \widetilde{C}_2)$.
On the other hand, when $n$ is not much larger that $2m$, the above matrices are not invertible and (\[eq:euler-blocks\]) is usually not satisfied. In this case, the global step is slightly modified in order to recover (\[eq:euler-blocks\]) and thus improve the efficiency of the global step. We replace (\[eq:global-manyblock2\]) by $$\label{eq:global-manyblock2-modified}
\forall \; 1 \le i \le p, \quad C_i({{\mathcal U}}) = \tilde{C}_i^k + T
\widehat{C}_{i+1}^k U_i\Big( [\widehat{C}_{i}^k]^t T T^t
\widehat{C}_{i}^k \Big) - T^t
\widehat{C}_{i-1}^k U_{i-1}^t\Big([\widehat{C}_{i}^k]^tT^t T
\widehat{C}_{i}^k\Big)$$ where $\widehat{C}_i^k$ is a block formed by vectors collected in the vector space defined by $\widetilde{C}_i^k$. These vectors are selected using a modified Gram-Schmidt orthonormalization process. The size of the blocks $\widehat{C}_i^k$ is appropriately chosen. The larger the blocks $\widehat{C}_i^k$, the more precise the global step but the worse the conditioning of the optimization problem. In addition, since the global step is the most demanding step of the algorithm, considerations both on the computational time and in terms of memory are accounted for when fixing the sizes of the blocks $\widehat{C}_i^k$.
Our numerical experiments show that when the global step is performed (using (\[eq:global-manyblock2\]) or (\[eq:global-manyblock2-modified\]), depending on $n$ and $m$), the blocks $(C^{k+1}_i)_i$ do not exactly satisfy the orthonormality constraint, owing to evident round-off errors. All the linear scaling algorithms have difficulties in ensuring this constraint and our MDD approach is no exception. The tests performed however show that the constraint remains satisfied throughout the iterations within a good degree of accuracy.
Numerical tests {#sec:numerical}
===============
An extensive set of numerical tests was performed to illustrate the important features of the domain decomposition algorithm introduced above, and to compare it with a standard scheme, commonly used in large scale electronic structure calculations.
Setting of the algorithm and of the tests
-----------------------------------------
#### Molecular systems used for the tests
Numerical tests on the algorithm presented above were performed on three chemical systems. The first two systems both have formula COH-(CO)$_{n_m}$-COH. They differ in their Carbon-Carbon interatomic distances. For system $\mathcal{P}_1$, this distance is fixed to 5 atomic units, while it is fixed to 4 for system $\mathcal{P}_2$. On the other hand, our third system, denoted by $\mathcal{P}_3$ has formula CH$_3$-(CH$_2$)$_{n_m}$-CH$_3$.
For each of the three systems $\mathcal{P}_1$, $\mathcal{P}_2$, $\mathcal{P}_3$, several numbers $n_m$ of monomers were considered. A geometry optimization was performed using the GAUSSIAN package [@GAUSSIAN98] in order to fix the internal geometrical parameters of the system. The only exception to this is the Carbon-Carbon distance for $\mathcal{P}_1$ and $\mathcal{P}_2$, which, as said above, is fixed [*a priori*]{}. Imposing the Carbon-Carbon distance allows to control the sparsity of the matrices $H$ and $S$ (the larger the distance, the sparser the matrices). Although not physically relevant, fixing the Carbon-Carbon distance is therefore useful for the purpose of numerical tests.
#### Data, parameters and initialization
For an extremely large number $n_m$ of monomers, the matrices $H$, $S$, and $D_\star$ cannot be generated directly with the GAUSSIAN package. We therefore make a periodicity assumption. For large values of $n_m$, these matrices approach a periodic pattern (leaving apart, of course, the “boundary layer”, that is the terms involving orbitals close to one end of the linear molecule). So, we first fix some $n_m$ sufficiently large, but for which a direct calculation with Gaussian is feasible, and construct $H$, $S$. The matrices $H$ and $S$, as well as the ground-state density matrix $D_\star$, and the ground-state energy $E_0$, are then obtained for arbitrary large $n_m$ assuming periodicity out of the “boundary layer”. Likewise, the gap $\gamma$ in the eigenvalues of $H$ is observed to be constant, for each system, irrespective of the number $n_m$ of polymers, supposedly large. Proceeding so, the gap for systems $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ is respectively evaluated to 0.00104, 0.00357, and 0.0281.
For our MDD approach, localization parameters are needed. They are shown in Table \[table\] below. Additionally, we need to provide the algorithm with an initial guess on the size $m_i$ of the blocks. Based on physical considerations on the expected repartition of the electrons in the molecule and on the expected localization of the orbitals, the sizes were fixed to values indicated in Table \[table\]. The specific block $C_i$ is then initialized in one of the following three manners:
- strategy ${\mathcal I}_1$: the entries of $C$ are generated randomly, which of course generically yields a bad initial guess way;
- strategy ${\mathcal I}_2$: each block $C_i$ consists of the lowest $m_i$ (generalized) eigenvectors associated to the corresponding block matrices $H_i$ and $S_i$ in the matrices $H$ and $S$, respectively. This provides with an initial guess, depending on the matrices $H$ and $S$, thus of better quality than the random one provided by strategy ${\mathcal I}_1$;
- strategy ${\mathcal I}_3$: the initial guess provided by ${\mathcal I}_2$ is optimized with the local fine solver described in section \[sec:gendesc\].
\[table\]
$\mathcal{P}_1$ $\mathcal{P}_2$ $\mathcal{P}_3$
---------------------------- ----------------- ----------------- -----------------
n 130 200 308
q 50 80 126
Bandwith of $S$ 59 79 111
Bandwith of $H$ 99 159 255
Cut-off for entries of $H$ $10^{-12}$ $10^{-12}$ $10^{-10}$
Cut-off for entries of $D$ $10^{-11}$ $10^{-11}$ $10^{-7}$
Size of first block $m_1=67$ $m_1=105$ $m_1=136$
Size of last block $m_p=67$ $m_p=106$ $m_p=137$
Size of a generic block $m_i=56$ $m_i=84$ $m_i=104$
: Localization parameters and initial size of the blocks used in the tests
#### Implementation details
Exact diagonalizations in the local steps are performed with the routine [*dsbgv.f*]{} from the LAPACK package [@LAPACK]. In the global step, the resolution of the linear system involving the Hessian matrix is performed iteratively, using SYMMLQ [@paige]. Diagonal preconditionning is used to speed up the resolution.
The calculations have been performed using only one processor of a bi-processor Intel Pentium IV-2.8 GHz.
#### Criteria for comparison of results
For assesment of the quality of the results, we have used two criteria, regarding the ground-state energy and the ground-state density matrix, respectively. For either quantity, the reference calculation is the calculation using the Gaussian package [@GAUSSIAN98]. The quality of the energy is measured using the relative error ${\displaystyle }e_E= \frac{|E -
E_0|}{|E_0|}$. For evaluation of the quality of the density matrix, we use the $L^\infty$ matrix norm $$\label{norme-matrix}
e_\infty = \sup_{(i,j)\,\hbox{\small \rm s.t.}\,|H_{ij}|
\leq \varepsilon } \left| D_{ij}-\left[D_\star\right]_{ij}
\right|,$$ where we fix $\varepsilon=10^{-10}$. The introduction of the norm (\[norme-matrix\]) is consistent with the cut-off performed on the entries of $H$ (thus the exact value of $\varepsilon$ chosen). Indeed, in practice, the matrix $D$ is only used for the calculations of various observables (for instance electronic energy and Hellman-Feynman forces), all of the form ${{\rm Tr}}(AD)$ where the symmetric matrix $A$ shares the same pattern as the matrix $H$ (see [@cances-handbook] for details). The result is therefore not sensitive to entries with indices $(i,j)$ such that $|H_{ij}|$ is below the cut-off value.
Illustration of the role of the local and global steps
------------------------------------------------------
Our MDD method consists in three ingredients:
- the local optimization of each block performed in the local step;
- the transfer of vectors from some blocks to other blocks, along with the modification of the block sizes $m_i$, again in the local step;
- the optimization performed in the global step.
To highlight the necessity of each of the ingredients, and their impacts on the final result, we compare our MDD algorithm with three simplified variants. Let us denote by
- strategy ${\mathcal S}_1$: local optimization of the blocks, without allowing variations of the block sizes, and no global step;
- strategy ${\mathcal S}_2$: full local step (as defined in Section \[sec:gendesc\]), no global step;
- strategy ${\mathcal S}_3$: local optimization of the blocks, without allowing for variations of the block sizes, and global step;
- strategy ${\mathcal S}_4$: full algorithm.
We compare the rate of convergence for the above four strategies. Two categories of tests are performed, depending on the quality of the initial guess. The results displayed on Figures \[fig:res1\] to \[fig:res4\] concern polymer $\mathcal{P}_1$ with $n_m=801$ monomers. This corresponds to $N_b = 8050$ and $N=5622$. Analogous tests were performed on $\mathcal{P}_2$ and $\mathcal{P}_3$, but we do not present them here, for brevity.
The energy of the ground state of this matrix (i.e. the minimum of (\[eq:infF\])) is $E_0 =
-27663.484$. The number of blocks considered is $p=100$. For the global step, we have collected these 100 blocks in $99$ overlapping groups of $2$ blocks. Interestingly, such a partition provides with optimal results regarding CPU time and memory requirement. It is observed on Fig. \[fig:res1\]-\[fig:res4\] that ${\mathcal
S}_1$, ${\mathcal S}_2$ and ${\mathcal S}_3$ are not satisfactory for they converge towards some local, non global, minima of (\[eq:infF-blocks\]) whatever the initial guess. The failure of the strategy $S_3$ performed on the initial guess ${\mathcal I}_2$ is surprising: this initial guess is not good enough. Indeed, if the initial guess is ${\mathcal I}_3$, we check numerically that the strategies ${\mathcal S}_3$ and ${\mathcal S}_4$ behave identically. Notice that the strategy ${\mathcal I}_3$ is identical to ${\mathcal S}_2$ applied to the initial guess ${\mathcal I}_2$.
We also remark that the strategy ${\mathcal S}_4$ performs very well whatever the initial guess (see Fig. \[fig:res2\] and Fig. \[fig:res4\]). The same behavior is observed for the polymers ${\mathcal P}_2$ and ${\mathcal P}_3$. Finally, after orthonormalization, the Density Matrix Minimization (DMM) method [@DMM] failed with the random initial guess and reveals very slow with the initial guess ${\mathcal I}_2$. That is the reason why we consider the initial guess ${\mathcal I}_3$ to compare these methods.
Comparison with two other methods
---------------------------------
Having emphasized the usefulness of all the ingredients of our MDD algorithm, we now compare it to two other algorithms:
- the diagonalization routine [*dsbgv.f*]{} from the LAPACK library;
- the Density Matrix Minimization (DMM) method [@DMM].
These two algorithms are seen as prototypical approaches for standard diagonalization algorithms and linear scaling techniques respectively. They are only used here for comparison purposes. Regarding linear scaling methods, two other popular approaches are the Fermi Operator method [@FOE] and the McWeeny iteration method [@McWeeny]. We have observed that, at least in our own implementation, based on the literature, they are outperformed by the DMM method for the actual chemical systems we have considered. We therefore take DMM as a reference method for our comparison.
Recall that the routine [*dsbgv.f*]{} consists in the three-step procedure
- transform the generalized eigenvalue problem into a standard eigenvalue problem by applying a Cholesky factorization to $S$;
- reduce the new matrix to be diagonalized to a tridiagonal form;
- compute its eigenelements by using the implicit $QR$ method.
The algorithmic complexity of this approach is in $N_b^3$ and the required memory scales as $N_b^2$.
For the description of DMM method, we refer to [@DMM]. Let us only mention here that this approach consists in a minimization procedure, applied to the energy expressed in terms of the density matrix. Both the algorithmic complexity and the memory needed for performing the DMM approach scale linearly with respect to the size $N_b$ of the matrix. The DMM method is initialized with the density matrix $D=CC^t$ computed with the initial guess $C$ of the domain decomposition method. Two important points for the tests shown below are the following.
First, we perform a cut-off on the coefficients on the various matrices manipulated throughout the calculation: only the terms of the density matrices within the frame defined in Figure \[fig:D\] are taken into account. Such a cut-off has some impact on the qualities of the results obtained with the DMM method. We are however not able to design a better comparison.
Second, the DMM method requires the knowledge of the Fermi level (as is the case for the linear scaling methods commonly used in practice to date). The determination of the Fermi level is the purpose of an outer optimization loop. In contrast, the MDD approach computes an approximation of the Fermi level at each iteration. Here, for the purpose of comparison, we *provide* DMM with the exact value of the Fermi level. Consequently, the CPU times for the DMM method displayed in the sequel are underestimated.
We emphasize that the routine [*dsbgv.f*]{} computes the entire spectrum of the matrix, both eigenvalues and eigenvectors. In contrast, the MDD approach only provides with the lowest $N$ eigenvalues, among $N_b$, and the projector on the vector space spanned by the corresponding eigenvectors, not the eigenvectors themselves.
### Comparison with Direct diagonalization and DMM
We have computed the ground states of the polymers ${\mathcal P}_1$, ${\mathcal P}_2$ and ${\mathcal P}_3$ with the three methods (direct diagonalization, DMM and MDD) and for various numbers $n_m$ of monomers, corresponding to matrix sizes $N_b$ in the range $10^3$-$10^5$.
For DMM and MDD, the initial guess is generated following the strategy ${\mathcal I}_3$. The results regarding the CPU time at convergence and the memory requirement are displayed on Figures \[fig:res5\] to \[fig:res9\] for the polymers $\mathcal P_1$, $\mathcal P_2$, and $\mathcal P_3$ respectively.
For small values of $N_b$, i.e. up to around $10^4$, the results observed for the direct diagonalization, DMM and MDD agree. The CPU times for our MDD approach scale linearly with $N_b$.
For larger values of $N_b$, the limited memory prevented us from either performing an exact diagonalization or from implementing DMM. So, we extrapolate the CPU time and memory requirement according to the scaling observed for smaller $N_b$.
The data for the DMM method are not plotted in Figure \[fig:res9\] as the DMM method does not converge for the polymer ${\mathcal P}_3$ when the number of monomers exceeds $10^3$. From our point of view, it comes from the truncation errors which cause the divergence of the method (note that the truncation strategy we consider here is very simple).
### Comparison with DMM and a hybrid strategy
We now concentrate on the two approaches that scale linearly, namely DMM and MDD. We consider
- ${\mathcal P}_1$ with $4001$ monomers, corresponding to $N_b=40050$,
- ${\mathcal P}_2$ with $2404$ monomers, corresponding to $N_b=24080$,
- ${\mathcal P}_3$ with $208$ monomers, corresponding to $N_b=854$.
These particular values have been chosen for the purpose of having simple values for the numbers of blocks. For each of the three polymers, we compare the DMM and MDD methods initialized by the strategy ${\mathcal I}_3$ and a hybrid strategy. The hybrid strategy consists of a certain number of iterations performed with MDD, until convergence is reached for this method, followed by iterations with DMM. We use the following stopping criterion for MDD: $$\label{eq:stopping}
{\displaystyle }\|D_n-D_{n-1}\| \geq \|D_{n-1}-D_{n-2}\| \quad
\mbox{\textrm{and}} \quad \|D_n-D_{n-1}\| \leq \epsilon_a$$ where $\epsilon_a$ is a threshold parameter. We take $\epsilon_a=10^{-4}$, respectively $\epsilon_a=10^{-3}$, for the polymer ${\mathcal P}_1$, respectively ${\mathcal P}_2$ and ${\mathcal P}_3$.
The Figures \[fig:res11\] to \[fig:res13\] show the evolution of the error in density versus CPU time. The hybrid version is demonstrated to be a very efficient combination of the two algorithms.
For completeness, let us highlight the temporary increase for the error in density appearing in Fig. \[fig:res12\] when MDD is used on $\mathcal
P_2$. Analogously, the energy of the current solution, which is actually below the reference energy, also increases. In fact, this is due to a loss of precision in the orthonormality constraints. In MDD, these constraints are not imposed exactly at each iteration, but only approximately (see equation \[eq:epsilonL\]).
Finally, we report in figures \[fig:res14\] to \[fig:res16\] the results obtained with MDD for the largest possible case that can be perfomed on our platform, owing to memory limitation. We used the initial guesses obtained with the strategy ${\mathcal I}_3$. Notice that for the local step the memory requirement scales linearly with respect to the number $n_m$ of monomers, while for the global step, the memory requirement is independent of $n_m$. Therefore, for large polymers, the memory needed by MDD is controled by the local step. In contrast, for small polymers, the most demanding step in terms of memory is the global step.
Conclusions and remarks
=======================
The domain decomposition algorithm introduced above performs well, in comparison to the two standard methods considered. More importantly, our approach is an effective *preconditionning technique* for DMM iterations. Indeed, MDD provides a rapid and accurate approximation, both in terms of energy and density matrix, regardless of the quality of the initial guess. In contrast, DMM outperforms MDD when the initial guess is good, but only performs poorly, or may even diverge, when this is not the case. The combination of the two methods seems to be optimal. More generally, our MDD algorithm could constitute a good preconditionner to all variational methods, such as the Orbital Minimization method [@Siesta].
Regarding the comparison with DMM, the following comments are in order.
- All our calculations have been performed on a single processor machine. Potentially, both DMM and MDD should exhibit the same speed-up when parallelized. We therefore consider the comparison valid, at least qualitatively, for parallel implementations. The parallelization of the MDD is currently in progress, and hopefully will confirm the efficiency of the approach.
- We recall the Fermi level has to be provided to the DMM method. This is an additional argument in favor of the MDD approach.
- The MDD method, in contrast to the other linear scaling methods, does not perform any truncation in the computations. So, once the profile of $C$ is choosen, the method does not suffer of any instabilities, contrary to DMM (or OM) for which divergences have been observed for the polymer ${\mathcal P}_3$.
- The domain decomposition method makes use of several threshold parameters. For the three polymers we have considered, the optimal values of these parameters, except for the stopping criterion $\epsilon_a$ (equation (\[eq:stopping\])), are the same. We do not know yet if this interesting feature is a general rule.
- Recall our method solves problem (\[eq:infF-blocks\]), which is only an approximation of problem (\[eq:infF2\]). Therefore, the relative error obtained in the limit is only a measure of the difference between (\[eq:infF-blocks\]) and (\[eq:infF2\]). In principle, such a difference could be made arbitrarily small by an appropriate choice of the parameters of problem (\[eq:infF-blocks\]).
- Finally, let us emphasize that there is much room for improvement in both the local and the global steps. We have designed an overall multilevel strategy that performs well, but each subroutine may be significantly improved. Another interesting issue is the interplay between the nonlinear loop in the Hartree-Fock or Kohn-Sham problems (Self-Consistent Field - SCF - convergence [@cances-handbook; @outperform; @EDIIS]) and the linear subproblem considered in the present article. Future efforts will go in these directions.
[Acknowledgments.]{} We would like to thank Guy Bencteux (EDF) for valuable discussions and for his help in the implementation. C.L.B. and E.C. would like to acknowledge many stimulating discussions with Richard Lehoucq (Sandia National Laboratories).
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---
author:
- Flor Álvarez
- Lars Almon
- Patrick Lieser
- Tobias Meuser
- Yannick Dylla
- Björn Richerzhagen
- Matthias Hollick
- Ralf Steinmetz
bibliography:
- 'References.bib'
title: 'Conducting a Large-scale Field Test of a Smartphone-based Communication Network for Emergency Response'
---
<ccs2012> <concept> <concept\_id>10003033.10003079.10003082</concept\_id> <concept\_desc>Networks Network experimentation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003106.10010582.10011668</concept\_id> <concept\_desc>Networks Mobile ad hoc networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003094</concept\_id> <concept\_desc>Networks Network dynamics</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003097</concept\_id> <concept\_desc>Networks Network mobility</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
This work was supported in part by the BMBF within the SMARTER project, in part by the LOEWE initiative (Hessen, Germany) through the NICER project, and in part by the DFG as part of the CRC 1053 MAKI.
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---
abstract: 'Fracton topological phases possess a large number of emergent symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to design highly parallelized decoding algorithms. Here we design and implement decoding algorithms for the three-dimensional X-cube model where decoding is subdivided into a series of two-dimensional matching problems, thus significantly simplifying the most time consuming component of the decoder. Notably, the rigid structure of its point excitations enable us to obtain high threshold error rates. Our decoding algorithms bring to light some key ideas that we expect to be useful in the design of decoders for general topological stabilizer codes. Moreover, the notion of parallelization unifies several concepts in quantum error correction. We conclude by discussing the broad applicability of our methods, and explaining the connection between parallelizable codes and other methods of quantum error correction. In particular we propose that the new concept represents a generalization of single-shot error correction.'
author:
- 'Benjamin J. Brown'
- 'Dominic J. Williamson'
bibliography:
- 'QEC.bib'
title: Parallelized quantum error correction with fracton topological codes
---
The hardware of a fault-tolerant quantum computer [@Barends14; @Corcoles15; @Kelly15; @Takita16] will be supported by a classical decoder that processes syndrome data to determine how best to correct the errors the system suffers. In the absence of a self-correcting quantum memory [@Brown16], this will ideally be achieved with microscopic electronics that are locally integrated among the physical qubits. These systems will promptly deal with the errors as they occur [@Terhal15]. However, studies have shown that cellular automata decoders significantly compromise the error rate a system can tolerate [@Harrington; @Pastawski11; @Herold15; @Herold17; @Breuckmann17; @Dauphinais17]. It is likely that early generation quantum computers will use decoders with high-threshold error rates that rely on long-range classical communication [@Kitaev03; @Dennis02; @Wang03; @Raussendorf07; @Fowler09]. While these decoders can correct a magnitude of errors that is better aligned with the rate at which they occur on codes realized with modern technology [@Kelly15], their runtime increases with the size of the system and, eventually, will not be able to operate at the high clock speed of the quantum hardware as it is scaled up.
Fracton topologically ordered phases [@Vijay15; @Vijay16] are remarkable due to the glassy dynamics [@Chamon05; @Haah11; @Castelnovo11; @Bravyi11b; @Bravyi13a; @PhysRevB.95.155133] of their point-like excitations that are energetically constrained to follow specific trajectories. Notably, fracton models are structured such that they give rise to a significant number of global emergent symmetries [@Kitaev03; @Roberts18]. These are relations among the stabilizer generators where the product of a nontrivial subset gives identity. Emergent symmetries generally occur when a physical symmetry is gauged [@wegner1971duality; @Vijay16; @PhysRevB; @kubica2018ungauging; @you2018symmetric; @shirley2018FoliatedFracton]. A well-studied example is the [X-cube]{} model; a three-dimensional model that has a series of two-dimensional emergent symmetries [@Vijay16].
In this work we consider fracton models from the perspective of active quantum error correction [@Terhal15; @Campbell17]. We design decoding algorithms to deal with Pauli-errors that act on the X-cube model. Remarkably, we find that we can decode the model by performing minimum-weight perfect matching on two-dimensional planes of the three-dimensional model in parallel, thus significantly speeding up the decoding process. We attribute this surprising feature to the emergent symmetries of the model. We expect that our method of decoding will be adaptable to any code whose emergent symmetries are known. Given that decoders based on matching typically have high thresholds, we believe our method will enable us to find high-performance decoding algorithms for more general classes of codes. In particular, we argue that our approach is particularly well suited to topological codes.
In what follows, after describing the X-cube model and the implementation of our decoding algorithms, we go on to discuss the prospect of parallelizing other codes, and how the notion of parallelization relates to other modes of decoding. In particular, we argue that single-shot error correction is a special case of parallelized quantum error correction in the fault-tolerant setting.
**The X-cube model** is defined on a three-dimensional cubic lattice with periodic boundary conditions and linear size $L$ [@Vijay16]. The model has a single qubit on each face of the lattice, and its stabilizer group [@GottesmanThesis] is generated by a vertex stabilizer $A_v$ for each vertex $v$ of the lattice, and cell operators, $B_c^{\mathbf{r}}$ and $B_c^{\mathbf{g}}$, supported on cells $c$ of the lattice. The support of $A_v$, shown in Fig. \[Fig:Stabilizers\](a), includes all the qubits lying on faces that have vertex $v$ on their boundary, i.e., $A_v := \prod_{f,\, v\in \partial f} X_f$ with $X_f$ and $Z_f$ the standard Pauli operators acting on $f$.
![The support of stabilizer generators $A_v$, $B^{\mathbf{r}}_c$ and $B^{\mathbf{g}}_c$ are shown in (a), (b) and (c), respectively. The faces of the lattice can be three-colored as in (d). (e) Lineon excitations which are created by applying Pauli-X errors over dual lines. The color of the excitation is determined by the orientation of the line operator that creates the excitation. (f) Fracton excitations of the vertex stabilizers caused by Pauli-Z operators supported on the red faces. \[Fig:Stabilizers\]](StabilizersAndExcitations.pdf)
It is helpful to three-color the faces of the lattice to define the cell operators. A face $f$ is colored red, green or blue, denoted $\mathbf{c}(f) = \mathbf{r}, \,\mathbf{g},\,\mathbf{b}$, if it is aligned with the $zx$-, $xy$- or $yz$-plane, respectively, see Fig. \[Fig:Stabilizers\](d). A cell operator of color ${\mathbf{c} = \mathbf{r},\,\mathbf{g},\,\mathbf{b}}$ is then defined to have support on all qubits on the boundary of the cell except for the faces of color $\mathbf{c}$, i.e. $B_c^{\mathbf{c}} := \prod_{f \in \partial c,\, \mathbf{c}(f) \not=\mathbf{c}} Z_f $. We remark that the stabilizers of a cell form an overcomplete generating set since $B_c^{\mathbf{r}} B_c^{\mathbf{g}} B_c^{\mathbf{b}} = 1 $. We show the support of $B_c^{\mathbf{r}}$ and $B_c^{\mathbf{g}}$ in Figs. \[Fig:Stabilizers\](b) and (c), respectively.[^1] The codespace is the common $+1$ Eigenspace of the elements of the stabilizer group. The mutually anticommuting pairs of logical operators that generate rotations about the codespace, $\overline{X}_j$ and $\overline{Z}_j$ with $ 1 \le j \le 6L-3$ can be chosen to have string-like support [@Vijay16].
We next look at the structure of the syndrome data of the X-cube model. A syndrome is a list of defects that lie on vertices and cells of the lattice such that $A_v |\psi \rangle = (-1) |\psi \rangle$, or $B_c^\mathbf{c} |\psi \rangle = (-1) |\psi \rangle$ for two of the three color labels. A cell defects is assigned a color $\mathbf{c}$ if $B_c^\mathbf{c} = +1$ and the other two stabilizers at cell $c$ are violated. The reader more familiar with condensed-matter physics may think of a defect as an excitation of the Hamiltonian ${H = -\sum_v A_v - \sum_c (B_c^{\mathbf{r}} + B_c^{\mathbf{g}})}$ [@Brown16]. Vertex defects, commonly referred to as fractons, are created by Pauli-Z errors. Cell defects, or lineons, are created by Pauli-X errors. We decode the vertex and cell defects independently.
A pair of cell defects are created by a Pauli-X error incident to a face of a cell. The defects are colored depending on the face the error occurs on. An error on a red, green, or blue face will create a pair of red, green or blue defects, respectively. If two or more errors occur on the faces of the same cell the color of the defect respects a $\mathbb{Z}_2 \times \mathbb{Z}_2$ fusion rule. In other words, similar to the color code [@Bombin06], the combination of two differently colored defects gives rise to a defect of the remaining distinct color. For instance, the combination of a $\mathbf{r} $ and a $\mathbf{g}$ defect gives a $\mathbf{b}$ defect at the cell where they meet. We can therefore draw strings of errors along edges of the dual lattice, with a color that is determined by their orientation, where the corners must respect the color fusion rules, see Fig. \[Fig:Stabilizers\](e).
Importantly, the X-cube model possesses an emergent $\mathbb{Z}_2$ cell-defect conservation symmetry on planes of the dual lattice that are orthogonal to a coordinate axis. We define the color $\mathbf{c}(\tilde{\Gamma})$ of a dual lattice plane $\tilde{\Gamma}$ in similar fashion to the faces of the lattice. Specifically, the color of any $zx$-, $xy$-, or $yz$-plane is given by $\mathbf{r}, \,\mathbf{g},$ or $\mathbf{b}$, respectively. Hence, for any dual lattice plane $\tilde{\Gamma}$ we have a relation $\prod_{c \in \tilde{\Gamma}} B_c^{\mathbf{c}(\tilde{\Gamma})} = 1$ by taking the product of all the cell operators of color $\mathbf{c}(\tilde{\Gamma})$. For example, the product of all the cell operators of color $\mathbf{r}$ lying on a plane of constant $y$ will return identity.
It follows that the total number of defects of two different colors is conserved modulo 2 on a plane of the third distinct color. For instance, given that both red and green defects require that $B_c^\mathbf{b} = -1$, on blue planes $\tilde{\Gamma}$, where $\prod_{c \in \tilde{\Gamma}} B_c^\mathbf{b} = 1$, we have that the parity of the number of red and green defects must be equal.
Finally, we briefly describe the vertex defects. Vertex defects are created on the vertices at the corners of the faces affected by a Pauli-Z error. Further, vertex defects respect a $\mathbb{Z}_2$ fusion rule. In Fig. \[Fig:Stabilizers\](f) we show a Pauli-Z operator with the vertex defects it creates marked by white vertices. Similar to the cell defects, the vertex defects respect an emergent $\mathbb{Z}_2$ symmetry on each of the $xy$-, $yz$-, and $zx$-lattice planes, $\Gamma$. Specifically, we have the relation $\prod_{v \in \Gamma} A_v = 1$ for any such $\Gamma$.
**A decoding algorithm** takes a syndrome that consists of a collection of defects caused by some Pauli error $E$, and then attempts to return a correction operator $C$ that will restore the code to its initial state. We say that a subset, or cluster, of defects can be neutralized if they can be created by a Pauli error supported on qubits that are nearby the cluster. We look for small clusters of defects that can be neutralized with a low-weight Pauli operator. With this strategy, in the limit that the error rate is small, it is highly likely that $CE$ is a member of the stabilizer group. Our strategy to find neutralizable sets is to pairwise group nearby defects within specifically chosen subregions of the lattice that reflect the emergent symmetries of the model. The intuition behind this approach is that defects are created in clusters such that all emergent symmetries are respected. Conversely, in what follows we find that we we obtain neutralizable clusters by ensuring that every defect of the cluster is paired multiple times, once to another defect within each emergent symmetry it is involved in. The performance of the decoder is determined by its ability to conduct well-informed pairing during each of the pairing subroutines.
We expect that with some imagination most strategies for decoding can be adapted to perform the pairing subroutines [@Harrington; @Duclos-Cianci10; @Wootton12; @Bravyi13a; @Anwar14; @Torlai17]. Of particular interest is the decoder due to Delfosse and Nickerson [@Delfosse17] that is readily adapted to the problem at hand to complete each matching subroutine in almost linear time. Moreover this decoder, and others similar [@Harrington; @Bravyi13a; @Anwar14], should permit a generalization of our work to qudit stabilizer codes. Here we make pairings using the Kolmogorov implementation [@Kolmogorov09] of the minimum-weight perfect matching algorithm due to Edmonds [@Edmonds65]. The algorithm takes a graph with weighted edges and returns a subset of the edges of the original graph such that each vertex has exactly one incident edge and the sum of the weights of the edges that are returned is minimal. We use this algorithm by assigning each defect a vertex of the graph, and adding edges to the graph that are weighted to approximate the logarithm of the likelihood that a Pauli-error caused the pair of defects that are connected by the edge [@Dennis02; @Stace10; @Fowler12b; @Hutter14a; @Nickerson17; @Criger18].
Most interestingly, all the decoders we present use matching subroutines on the defects lying on $L\times L$ planes of the system rather than its full volume. Parallelization here significantly speeds up our algorithm. The subroutines could be performed in almost $L^2$ time [@Delfosse17], thus enabling us to decode the X-cube model in sublinear time.
![A Pauli-X error. On the plane marked by the dotted grid we match all the green and red excitations. We use the blue excitations to help find the correct matching. (a) shows an error where two green defects are caused a string like error with blue defects at its corners. Nearby errors can lead to confusion. At (b) an error moves a blue defect away from the plane, and at (c) an additional error introduces a blue defect that could also mislead the decoder into finding an incorrect matching along the red dashed line. \[Fig:MatchingStrategy\]](LineonMatchingStrategy.pdf)
We first propose a decoder that neutralizes cell defects. Our decoder matches all of the defects of each plane that are constrained by a planar $\mathbb{Z}_2$ symmetry. For instance we match the green and blue defects on each plane of constant $y$. The connected components of the defects, where connections are given by edges returned from all $3L$ different matching calculations, are neutral clusters. We show this explicitly in Appendix \[App:CellDecoder\]. Here we focus on a single plane to better understand the performance of the decoder.
In Fig. \[Fig:MatchingStrategy\] we show a plane upon which we perform matching. Here the red and green defects are paired. The number of blue defects is not conserved on this plane. If a red and a green defect meet at a common cell they are expended to create a blue defect, see Fig. \[Fig:MatchingStrategy\](a) where a local error creates two blue defects. Single blue defects can also be created on or moved from the plane by applying errors to qubits on faces that are parallel to the plane of interest. See Figs. \[Fig:MatchingStrategy\](b) and (c) where a single blue defect is created on the plane, and a blue defect is moved away from the plane, respectively.
Our proposed decoder that neutralizes vertex defects gives a threshold of $\sim 9.4\%$ for an independent and identically distributed noise model. Notably, our decoder makes use of the rigid structure of the defects in our fracton code. Specifically, we use the locations of the blue defects as a ‘breadcrumb trail’ to assign more accurate weights to the edges in the red and green defect matching. This gives the decoder for the fracton model an advantage as the locations of the blue defects reduce the degeneracy of the lowest weight string errors that is typically present in two-dimensional models [@Stace10; @Duclos-Cianci10; @Criger18; @Beverland18].
To illustrate the potential advantages that are to be gleaned from the rigid excitations of the X-cube model we show an error that can be corrected by a decoder that uses information about the blue defects in Fig. \[Fig:MatchingStrategy\]. We first remark that a standard decoder that assigns weights according to the separation of the defects will not be able to deal with this error. This is because the sum of the edge weights along the incorrect path, shown by the red dashed line, is equal to the sum of the edge weights along the path that would yield a correction that successfully restores the system to its initial state.
In contrast, we find the decoder that accounts for the blue defects will succeed on this particular example. The path of the successful correction matches the red and green defects via three blue defects, leaving only one corner without a blue defect. The edges that connect the defects along the incorrect path must include two corners with no blue defect. As such, the minimum-weight matching decoder that accounts for blue defects during the pairing will deal with this error with certainty. To exploit this additional information offered by the blue defects we penalize the pairing of red and green defects around corners than do not support a blue defect. We describe the implementation of this decoder in Appendix \[App:CellDecoder\].
![Pauli-Z errors on a plane of the lattice. By comparison with the toric code we see this error cannot be corrected. \[Fig:LogicalErrors\]](LogicalErrors.pdf)
We decode the vertex defects by pairing each of them with another defect that shares its $x$ coordinate, a defect that shares its $y$ coordinate, and a defect that shares its $z$. We show that a cluster that respects this pairing can be neutralized in Appendix \[App:VertexDecoder\]. Therefore, conducting pairing subroutines on the $xy$-, $yz$- and $zx$- planes connects groups of defects to give neutralizable subsets. Performing matching on these planes, where we suppose the likelihood of an error creating a pair of defects is proportional to their separation, gives a threshold of $\sim 3.8\%$.
We also argue that the threshold should not exceed ${\sim 10\%}$. We consider the qubits on the plane shown in Fig. \[Fig:LogicalErrors\]. If we project these qubits into the page, we effectively have the edges of the square lattice toric code where the syndrome is duplicated twice; once at the front of the image and once at the back. If errors only occur on these qubits, then we expect that the decoder will make incorrect pairings with high probability at the threshold error rate of the toric code, $\sim10\%$ [@Dennis02]. Given that our argument only considers errors on one plane of qubits we should not expect that this bound is tight.
The defects of the X-cube model have a lot of structure due to a subextensive number of global emergent symmetries. We can improve the decoder by communicating outcomes of the previous iterations of the matching subroutine to find more accurate weightings for later rounds of pairing. We detail the implementation of our iterative decoder in Appendix \[App:VertexDecoder\]. Our improved decoder presents a threshold of $\sim 4.4 \%$. We suggest that more refined schemes of belief propagation should be applicable to design very high threshold decoders with fracton codes. However, given the upper bound we have obtained for this particular model, optimization of the X-cube model is unlikely to be of practical value.
It is interesting to compare our results to the threshold for the point defects of the three-dimensional toric code, $\sim 3.3\%$ [@Kubica17]. Unlike the point-defects of the X-cube model, these defects are constrained by only a single global emergent symmetry. As such, its syndrome is considerably less structured. Following similar reasoning, it is unsurprising that our thresholds fall short of the high threshold the three-dimensional toric code presents against bit-flip errors, $\sim 23.5\%$ [@Kubica17]. Indeed, the stabilizer defects that arise due to bit-flips are highly constrained by an extensive number of local emergent symmetries. Nevertheless, we believe that the encouraging thresholds we have obtained here, together with the potential for parallelization, motivates further study of fracton topological codes in the context of active quantum error correction. Additional exploration may lead to the discovery of codes with very high thresholds.
**Parallelized quantum error correction** is a new concept we have introduced that unifies a number of concepts of recent interest in the study of decoding algorithms. Moreover, we have developed several new tools that enable us to decode highly symmetric models with fracton topological order. Below we discuss parallelizable codes in the broader context of quantum error correction, and motivate potential directions of study in this area.
We first compare parallelizable codes with single-shot error correction [@Bombin15a; @Brown16a; @Campbell18; @fawzi2018constant]. Typically, in the fault-tolerant setting where measurements are unreliable, we collect syndrome data over a long period and decode once we have a sufficiently large history [@Dennis02]. Models with single-shot error correction can be decoded within planes of constant time without an extensive syndrome history. In contrast, the X-cube decoder we present can be parallelized over spatially distinct planes. In this sense, we might regard single-shot error correction as a special case of parallelization. This analogy becomes clearer still in the fault-tolerant setting. Indeed, one can conceive of decoding schemes for the four-dimensional spacetime history of the X-cube model where matching is performed on three-dimensional emergent spacetime symmetries. Foliation might be used to compare these error-correction protocols evenhandedly [@Bolt16; @Brown18]. In Appendix \[App:TimeCorrelated\] we show how the resilience of single-shot codes to time-correlated errors [@Bombin16] can be understood from the perspective of parallelized quantum error correction.
Beyond these models, self-correcting memories [@Brown16] are both single-shot [@Bombin15a] and fully parallelizable due to their local emergent symmetries. As such, these memories are readily decoded using cellular automata [@Dennis02; @Pastawski11; @Breuckmann17; @Kubica18]. Like single-shot codes then [@Bombin15a], we might regard parallelizable codes as some intermediate class of codes between self-correcting models and two-dimensional codes that do not permit parallelization.
**In conclusion** a key idea in our work is the use of defect pairing on subsets of defects that respect emergent symmetries of the model. Although a similar method of decoding has been applied to a number of low-dimensional stabilizer codes elsewhere in the literature, to the best of our knowledge this generic method has not been explicitly identified before. As discussed in Appendix \[App:Relations\], we expect our approach is broadly applicable to topological stabilizer codes. Further study may even reveal our method applies to general stabilizer codes where the symmetries of the model are known.
The decoding framework we have presented refines the clustering decoder [@Bravyi13a] where clusters are grown around defects until the cluster respects all of the symmetries of the model. In contrast, we deal with each symmetry separately. Further, our refinement permits the use of minimum-weight perfect matching, which tends to yield high thresholds. As such, we hope our work inspires the discovery of new high-performance parallelized decoders for exotic topological codes. Fracton topological codes offer a rich playground to begin such an exploration. Our decoder should carry over directly to other foliated fracton models [@shirley2017fracton; @shirley2018FoliatedFracton; @shirley2018Fractional; @Slagle2018]. Also of interest are the type-II fracton models which admit no string operators [@Haah11; @Vijay16]. This class includes the cubic code [@Haah11; @Bravyi11b; @Bravyi13a] which, notably, encodes qubits at a greater rate than the surface code. Due to its fractal structure, which leads to partial self-correction, we might expect this model to give rise to impressive thresholds.
**We Acknowledge** S. Bartlett for supportive and inspiring conversations, N. Delfosse for helpful discussions on the complexity of decoding algorithms and to A. Kubica for offering a simplified explanation of the color code decoders by projection. The authors acknowledge the facilities of the Sydney Informatics Hub at the University of Sydney and, in particular, access to the high performance computing facility Artemis. BJB is supported by the University of Sydney Fellowship Programme and the Australian Research Council via the Centre of Excellence in Engineered Quantum Systems(EQUS) project number CE170100009.
The cell defect decoder {#App:CellDecoder}
=======================
Here we describe in detail the decoder that corrects Pauli-X errors which give rise to cell defects. We first argue that collections of defects that are connected via the proposed pairing subroutines will give rise to neutralizable subsets. We then describe our implementation of the decoder that achieves the pairing, and how we use the minimum-weight perfect matching algorithm to account for corner defects.
We perform many minimum-weight matching subroutines. We first establish some terminology where we divide the subroutines into one of three subsets, either the red, green or blue matching, where the color depends on the orientation of the plane on which the matching subroutine is performed. Minimum-weight perfect matching is performed on each plane between all the defects that have a different color to the plane, as these defects respect a parity conservation symmetry. Specifically, each defect of color red and green will be paired with either a red or a green defect with the same $x$ coordinate, and defects of color red and blue (green and blue) will be paired with another defect of color red or blue (green or blue) that has the same $z$ ($y$) coordinate. We say that the collection of edges that connect red and green defects on any plane of constant $x$ are the result of the blue matching. Likewise, the collection of edges that connect red and blue (green and blue) defects on any plane of constant $z$ ($y$) are the result of the green (red) matching.
We now define a correctable cluster of cell defects, and then go on to argue that the correctable cluster is neutralizable.
A cluster of cell defects is correctable if for every defect included in the subset the subset also includes the defect it is paired with on each of the three differently colored matchings; the red, the green, and the blue matching.
We remark that a defect may be paired with the same defect in two different matchings. For instance, a green defect may be displaced from another green defect along the $z$ axis and share a common $x$ and $y$ coordinate. In which case, these two defects may be paired in both the red and blue matching. This gives a correctable cluster.
Given that the union of the edges returned from all of the minimum-weight perfect matching subroutines divides the syndrome into correctable clusters, it remains to show that we can find a Pauli operator that will neutralize a correctable cluster of defects. We therefore prove the following theorem.
![\[Fig:LineonCorrection\] The process of finding a correction operator for a correctable collection of cell defects. (a) An odd parity of each of the differently colored subsets of defects is shown in the bulk of the lattice. (b) Each of the defects are moved freely onto a plane. (c) The defects are then moved onto a one-dimensional line where the planes meet. Finally, (d) the defects are moved along their respective one dimensional lines to neutralize at a common point. ](LineonCorrection.pdf)
\[Thm:CorrectingLineons\] We can efficiently find a Pauli operator that neutralizes a correctable cluster of defects.
The constructive proof of the theorem is sketched in Fig. \[Fig:LineonCorrection\]. The proof relies on the fact that a correctable cluster has an equal parity of each of the three differently colored cell defects. We therefore prove the following lemma before proceeding with the proof of Theorem \[Thm:CorrectingLineons\].
A matched correctable cluster of defects has an equal number of defects of each color modulo 2.
![\[Fig:Edges\] Showing that the parity of the number of defects of the same color is equal. Each red defect has one incident edge from the green matching and one from the blue matching (a), and the green and blue defects have only one incident edge from the blue and green matching, see (b) and (c), respectively. These form one-dimensional chains where an even number of red defects can form a one dimensional cycle (d), two greens or two blues can be connected directly (e) and (f), a string of an odd number of connected red defects terminates at two differently colored defects (g) or an even number of red defects can terminate at two defects of the same color (h). The number of differently colored defects in each of the possible combinations that are shown explicitly between (d) and (h) are of equal parity.](Edges.pdf)
To prove the lemma we consider the edges of the blue and green matching between the defects of a correctable cluster. Each red defect is paired to two other defects; one defect in the green matching and another defect through the blue matching. Similarly, each green (blue) defect is paired to only one other defect through the blue (green) matching. As such, the connectivity of each vertex with respect to their color is shown precisely as in Figs. \[Fig:Edges\](a), (b) and (c), where edges are colored to indicate the color of the matching the edge came from.
We exhaustively consider the connected components of the defects of a correctable cluster that are connected via edges of the green and the blue matching. We first remark that the connected components are necessarily one dimensional, where a string of red defects may form a cycle of an even number of defects, or, possibly, a chain terminates with either green or blue defects. Our exhaustive examination will show that the numbers of defects of each color in any connected component have equal parity.
One eventuality is that a connected component gives rise to a one-dimensional cycle of reds. The alternating edge type guarantees the number of reds in the cycle is even, such that the parity of all the defects in this chain is equal, see Fig. \[Fig:Edges\](d). It is also possible for a pair of greens or blues to be connected directly through their respective pairings, see Figs. \[Fig:Edges\](e) and (f). The number of defects of every color in each of these three cases is even.
We next consider connected components of the subset where the red defects form longer chains that are not cyclic. We find that the edges that are taken from differently oriented planes guarantees that the parity of differently colored vertices still remains equal. For instance, in Fig. \[Fig:Edges\](g) the alternation of edges is such that a one-dimensional string of an odd number of red defects terminates at a green defect at one end and a blue defect at the other. In this case we have that the number of defects of each color in this connected component of the subset will be odd.
Alternatively, as we see in Fig. \[Fig:Edges\](h), an even number of red defects in a terminating chain guarantees the terminal defects of the chain are of the same color, thus leading to an even number of defects of each colors in this connected component of the subset. This completes all of the cases.
Given that the representative connected components shown in Figs. \[Fig:Edges\](d)-(h) have equal parity of each color of defect, it follows that the union of any of these connected components must also have equal parity. This concludes the proof of the lemma.
We remark that the argument above explains why the color code decoder due to Delfosse [@Delfosse14; @KubicaThesis] will give rise to connected components that can be locally corrected.
Using the fact above, we argue that we can find a Pauli operator that will neutralize a correctable cluster of defects. To do so, we continually reduce the dimensionality of the problem. For example, see see Fig. \[Fig:LineonCorrection\] where we show a configuration with an odd parity of red, green and blue defects such that each defect shares a plane with another defect and consequently the defect conservation symmetry is respected on all planes. Important to the proof is ensuring that the parity of the number of cell defects of each color remains equal throughout.
To find a correction, we first specify three planes. We move defects onto these planes to reduce the dimensionality of the problem. We choose one plane of constant $x$, $y$ and $z$ as shown in blue, red and green, respectively, in Fig. \[Fig:LineonCorrection\]. We first move all of the defects onto the plane of their respective color, see Fig. \[Fig:LineonCorrection\](b). This is achieved with a Pauli operator that will not create any additional cell defects. Moreover, the parity of the number of each subset of colored defects is conserved.
It may be that two defects of the same color are moved to a common point of a given plane. These two defects are neutralized. It remains to consider how the other defects connected to the two neutralized defects are paired. We find that we can consistently pair the two defects adjacent to the two neutralized defects.
For concreteness, we name the defects $\delta_j$ with $ 1 \le j \le 4$ such that $\delta_j $ is paired to $\delta_{j+1}$ for $j \le 3$. It may be that $\delta_1$ and $\delta_4$ are paired to other defects that we are not interested in. Defects $\delta_2 $ and $\delta_3$ share the same color, $\mathbf{u}$, and meet at a common point after the first move. Let’s say that $\delta_2$ and $\delta_3$ are paired through the $\mathbf{v}$-colored matching with $\mathbf{v} \not= \mathbf{u}$. Now, $\delta_1$ is necessarily paired to $\delta_2$ through the $\mathbf{w}$-colored matching with $\mathbf{w} \not= \mathbf{u},\mathbf{v}$. Likewise, $\delta_3 $ is paired to $\delta_4$ via the same $\mathbf{w}$-colored matching. Given that $\delta_2$ and $\delta_3$ met at a common point after the initial move, it follows that $\delta_1$ and $\delta_4$ can be consistently paired via the $\mathbf{w}$-colored matching, and the remainder of the proof will continue as follows. Moreover, given that a pair of defects of the same color are neutralized, the parity of the number of defects of that color is not changed.
We also remark that two defects of different colors may meet at a point where two planes intersect. Or three differently colored defects meet at the point where all three defects intersect. All of these eventualities respect the color parity of each of the differently colored defects. In the former case, two defects of different colors $\mathbf{u}$ and $\mathbf{v}$ fuse to give a defect of color $\mathbf{w}$. This maintains the color parity of the defects. In the latter case one defect of each color is neutralized. This also maintains consistency among the differently colored subsets.
After the move what remains on each of the planes is an equal parity of each of the differently colored subsets of defects. The next step is to move all the defects onto the lines where the colored planes intersect. See Fig. \[Fig:LineonCorrection\](c). Pairs of differently colored defects that met at these lines of intersection can skip the following step. Otherwise, we require an ‘L-shaped’ Pauli operator, that divides a defect of color $\mathbf{u} =\mathbf{r},\, \mathbf{g},\, \mathbf{b}$ into two defects, colored $\mathbf{v},\mathbf{w} \not= \mathbf{u}$, and move the two charges towards the plane of their respective color, $\mathbf{v}$, $\mathbf{w}$, respectively. The result of applying the L-shaped correction will leave an equal parity of the differently colored defects on each of the lines where the differently colored planes intersect. From here the remaining defects on these lines can be neutralized at the intersection point of all three of the planes as in Fig. \[Fig:LineonCorrection\](d).
We finally argue that the defects that lie on each of these three lines must be of equal parity. Without loss of generality we consider the line that collects defects of color $\mathbf{u}$ where the planes of color $\mathbf{v}$ and $\mathbf{w}$ intersect. We first remark that the parity of defects of color $\mathbf{v}$ and $\mathbf{w}$ on each of the planes of their respectively color are equal. Moreover, every defect of color $\mathbf{v}$ or $\mathbf{w}$ is paired with one other defect of color $\mathbf{v}$ or $\mathbf{w}$ through the $\mathbf{u}$-colored matching.
We recall that pairs of defects that are matched via the $\mathbf{u}$ matching will meet at the same point along the line of intersection between the $\mathbf{v}$ and $\mathbf{w}$ colored plane. If two defects are paired by the same color, they meet at the line, and neutralize along this line. This action preserves the color symmetry of the subset. Altenatively, a pair of defects of color $\mathbf{v}$ and $\mathbf{w}$ combine to give a single defect of color $\mathbf{u}$. In this case, the number of defects of color $\mathbf{v} $ and $\mathbf{w}$ both decrease by one, but we increase the number of defects of color $\mathbf{u}$, thus maintaining the equal parity of defects of a given color. Given that this operation occurs simultaneously on each line of intersection, this completes the argument for neutrality.
Implementation
--------------
![Logical failure rate using the decoder to correct cell defects where errors are independent and identically distributed bit-flips at rate $p$. We find a threshold $\sim 9.4\% $ at the crossing point where we compare system sizes $L = 72, \,84,\,96$. \[Fig:BitFlipThreshold\]](LineonCornerBoost.pdf){width="\columnwidth"}
The decoder requires that each defect is paired with another defect on each plane such that all the charge conservation symmetry is conserved. We achieve this with minimum-weight perfect matching [@Edmonds65; @Kolmogorov09]. The minimum-weight perfect matching takes as input a list of vertices, that are connected via a list of weighted edges. The algorithm returns a list of edges such that each vertex has one and only one incident edge, and the sum of the weights of the edges that are returned is minimal.
We propose a decoder that accounts for the information provided by the defects that are not conserved on a given plane. We consider the $\mathbf{u}$-colored plane where defects of color $\mathbf{v}$ and $\mathbf{w}$ are conserved. Each of these defects are represented by a vertex in the minimum-weight perfect matching algorithm. Further, we add two vertices for each of the defects of color $\mathbf{u} \not= \mathbf{v},\, \mathbf{w}$ on the plane. We assign weights to edges between all of the vertices proportional to their separation according to their Manhattan distance, except where we increase the weight by one if two defects are separated via a corner. Now, pairs of defects of color either $\mathbf{v}$ or $\mathbf{w} $ can be connected via edges that pass through defects of color $\mathbf{u}$. If two such defects are not aligned parallel to a lattice axis the weight of their matching is penalized by one for each corner of the path that does not pass through a defect of color $\mathbf{u}$. Defects that are aligned parallel to a lattice axis can be paired without the corner penalty.
We find the logical failure rate by comparing the support of the correction operator our decoder returns to one logical operator as shown in Fig. \[Fig:LogicalOperators\]. This decoder shows a threshold of $\sim 9.4\%$.
![The support of the one dimensional logical operators of the X-cube model. The logical Pauli-X (Pauli-Z) operators are supported on the blue (green) qubits, and both logical operators intersect at the pink face. \[Fig:LogicalOperators\]](LogicalOperators.pdf)
Decoding Pauli-Z errors {#App:VertexDecoder}
=======================
We next describe the decoder that finds a low-weight Pauli operator to neutralize vertex defects. We briefly summarize how correctable clusters can be found via parallelized matching subroutines, and we explain how a correction operator can be obtained. We then describe the implementation of our decoder and, in particular, explain how we reiterate the matching subroutines to improve our priors on each matching procedure.
We seek clusters of defects such that we can apply a Pauli operator to each cluster that will neutralize all of its defects. Vertex defects must respect the planar emergent symmetries of the X-cube model, and so too must a neutralizable cluster of vertex defects. To this end, each defect included within a neutralizable cluster must be paired with precisely one other defect that shares its $x$ coordinate, one that shares its $y$ coordinate, and one that shares its $z$ coordinate.
We achieve the pairing by performing minimum-weight perfect matching subroutines over two-dimensional planes of the three-dimensional model. It is helpful to define the matching subroutines. We call the collection of matching subroutines that pair defects with a common $x$ coordinate the $x$-matching and, likewise, the collection of matching subroutines that pair defects with a common $y$ ($z$) coordinate the $y$-matching ($z$-matching). Every defect of a neutralizable cluster is paired with as many as three other defects via the combination of these matching procedures. It may be that a pair of defects are matched twice through two different matching subroutines if one defect is displaced from the other along a direction parallel to a lattice axis. This eventuality is consistent with the following discussion. It also follows that every defect must be paired with at least two distinct other defects.
We next describe how a correction operator is obtained for each neutralizable cluster. Similar to the case with the cell-defect decoder, we can simplify the problem considerably by projecting the defects of a neutralizable cluster onto a two-dimensional plane. Without loss of generality we project all the defects of a neutralizable cluster onto a plane of constant $z$. This projection is achieved by moving pairs of defects with the same $z$ coordinate using a Pauli operator similar to that shown in the left of Fig. \[Fig:PlaneonCorrection\]. Given that every defect in a neutralizable cluster is paired with one other with the same $z$ coordinate, we can necessarily make this projection by the definition of a neutralizable cluster.
![\[Fig:PlaneonCorrection\] Correctable clusters of vertex defects. (left) We can move defects paired on a plane of constant $z$ onto other planes of constant $z$ with the operator shown. (right) The projection of defects that were paired over planes of constant $z$ are shown on a single plane. Given that each defect is paired to another defect on the $xz$ and $yz$ plane, the projected edges mark a boundary with projected defects on its corners. The defects on this plane can be corrected by Pauli-Z operators acting on the faces enclosed by the boundary.](PlaneonCorrection.pdf)
It remains to consider the defects that lie on the plane after the projection, see Fig. \[Fig:PlaneonCorrection\](right). We recall that every defect is paired with one other defect with a common $x$ and a common $y$ coordinate by the definition of a neutralizable cluster. This implies that, after the projection, and supposing that the defects are not neutralized at the plane, that each defect is aligned with two other defects; one horizontally and one vertically. In fact, we can consider the edges that pair these defects as a one-dimensional boundary on the plane. Applying a Pauli-Z operator to each of the qubits that lie on the interior of this boundary will neutralize all of the remaining defects.
Implementation
--------------
We finally describe our use of minimum-weight perfect matching to pair defects such that we can find neutralizable clusters. We first describe a simplified version where the pairings are all achieved naïvely. We then describe a generalization of the decoder where we repeat minimum-weight perfect matching several times such that the outcome of the last matching is used to improve the results of the next.
A neutralizable cluster as defined above requires that every vertex defect is paired with two or three other vertex defects; one must share its $x$ coordinate, one must share its $y$ coordinate, and one must share its $z$ coordinate. We achieve the pairing with minimum-weight perfect matching. We perform a minimum-weight perfect matching subroutine on each plane of constant $x$, $y$ and $z$ where we input a graph such that each vertex defect on the given plane is assigned a vertex, and we weight each edge that connects two vertices of the complete graph according to the Manhattan distance that separates the two vertices it connects. The vertices are paired according to the output of the edges of each matching subroutine. We obtain a threshold of $\sim 3.7\%$ using this decoder where we compare the correction operator our decoder returns to the logical operator shown in Fig. \[Fig:LogicalOperators\]. We collected data using $\sim 10^4$ samples with system sizes $L = 36,\,42,\,48,\,54 $ and $60$ to obtain this result. See Fig. \[Fig:PlaneonIter1\].
![Threshold error rate for the vertex excitations where we perform matching over planes that satisfy a $\mathbb{Z}_2$ symmetry, and we assign weights to edges according to the separation of two vertices by their Manhattan distance. We find a threshold $\sim 3.7\% $ using system sizes $L = 36,\, 42,\, 48,\, 54,\, 60$.\[Fig:PlaneonIter1\]](Iter1plot.pdf){width="\columnwidth"}
It is important to consider how we use minimum-weight perfect matching to find an error that was the most likely to have given rise to a configuration of defects. Ideally, we assign weights to edges such that they are proportional to the logarithm of the probability that an error caused those two defects. For instance, in the case of the surface code, weights are typically assigned according to the separation of the defects, as the weight of the error must increase with their separation if we suppose an independent and identically distributed noise model [@Dennis02]. However, this is a suboptimal solution, as we find the most likely error that caused the syndrome, and an optimal decoder will return a representative correction operator for the class of homologically equivalent errors that most likely caused the syndrome [@Wang03; @Duclos-Cianci10]. To overcome this, some references have considered heuristic modifications to the weighting function of edges to account for degeneracy [@Stace10; @Criger18; @Beverland18] or correlated features [@Fowler12b; @Hutter14a; @Nickerson17] of the error model.
A local Pauli-Z error gives rise to at least four vertex defects. To decode this correctly, it would be favorable to use minimum-weight hypergraph matching, where we minimize the sub of the weights to hyper edges that connect multiple vertices. In the absence of an efficient hypergraph-matching algorithm, we turn to belief propagation methods to supplement minimum-weight graph matching. We perform minimum-weight perfect matching several times, where we use the output of the last matching to improve the next. Specifically, we have a prior matching where each vertex defect is paired to another via the $x$, $y$ and $z$ matching. We use this prior matching to better estimate the weights for the edges in the subsequent matching. The priors are initialized by matching with weights given by the Manhattan distance between pairs of vertices. We use the output of the final matching to determine the correction.
To motivate our choice of weighting function we consider the error shown in Fig. \[Fig:DifficultError\]. In the figure, we mark the faces that have experienced errors by coloring them red. The resulting defects could be connected incorrectly with high probability in either the $y$- or the $z$-matching via the red-dotted edges if we choose weights by their separation only. This will lead to a logical failure. However, we note that one of the errors that gives rise to the extensive line of defects is oriented differently from the others. As such, we expect that assigning weights that also account for the locations of other nearby defects may enable us to correct such an error. We communicate the results of different matching subroutines to produce better estimates for edge weights.
![Threshold error rate using the vertex defect decoder after five iterations of belief propagation. We find a threshold $\sim 4.3\%$ using system sizes $L = 36,\,42,\,48,\,54,\,60$. We use $\sim 10^4$ samples to find each data point. \[Fig:Iter5\]](Iter5plot.pdf){width="\columnwidth"}
Once the priors are initialized, we assign weights to edges that connect two vertices using the separation between the other defects a pair of defects are paired to, as well as their own separation. For instance, without loss of generality, we calculate the weight of an edge separating two defects for the $x$-matching. We label these two defects $\delta_1$ and $\delta_2$. The defects are paired to one or two other defects via the priors from the $y$- and $z$-matching. We label the defects paired with $\delta_j$ as $\delta_j^y$ and $\delta_j^z$ respectively for $j = 1,\,2$, and we denote by $\text{sep}(\delta, \delta')$ the Manhattan distance separating defects $\delta$ and $\delta'$. Supposing $\delta_1^y \not= \delta_2$, which also implies $\delta_2^y \not= \delta_1$ and $\delta_1^z \not= \delta_2$, we choose the weight for the edge $e$ connecting $\delta_1$ and $\delta_2$ such that $$W(e) \sim \text{sep}(\delta_1, \delta_2) + \min [\text{sep}(\delta_1^y, \delta_2^y), \text{sep}(\delta_1^z, \delta_2^z) ].$$ We do not consider the separation between paired defects if they are the same defects as those we are trying to determine the weight between. In the case that, say, $\delta_1^y = \delta_2$, we simply take ${W(e) \sim \text{sep}(\delta_1, \delta_2) + , \text{sep}(\delta_1^z, \delta_2^z) }$. Likewise, if $\delta_1^z = \delta_2$, we simply take ${W(e) \sim \text{sep}(\delta_1, \delta_2) + , \text{sep}(\delta_1^y, \delta_2^y) }$. If $\delta_1^y = \delta_2$, it is necessarily true that $\delta_1^z \not= \delta_2$ and vice versa. As such we always find a defect that is paired that is distinct from the defects of the edge we are assigning a weight to.
![\[Fig:DifficultError\] An error configuration marked by red faces that cannot be decoded by pairing defects using minimum-weight perfect matching where edges are weighted according to their separation. The defects may be paired incorrectly via the red dashed lines. We improve the decoder by choosing weights such that we also account for the separation between defects that have paired with the defects of the edge we are trying to assign a weight to. We look at the defects that have been paired via the $x$-matching that are connected to the defects of interest via the blue edges. Due to the difference in orientation of one of the errors, if we assign weights that account for the separation of the defects of interest, and the separation of their other partners, a high weight will be assigned to the incorrect edges. We mark the large separation between the paired defects by green lines.](DifficultError.pdf)
We find that the decoder rapidly converges to a higher threshold as we increase the repetitions of the belief propagation step. We find a threshold of $\sim 4.3\%$ if we reiterate five times. This value was collected using $\sim 10^4$ Monte Carlo samples for system sizes $L = 36,\,42,\,48,\,54,$ and $60$. The data is shown in Fig. \[Fig:Iter5\].
Decoding time-correlated errors using parallelized quantum error correction {#App:TimeCorrelated}
===========================================================================
![(left) A spacetime diagram of a two-dimensional topological code, for instance the toric code, where time runs up along the page. We imagine an error that persists over a long time that will effectively introduce a puncture that extends through the time dimension. One can imagine a situation where a small error will have a catastrophic effect on the logical information. We show two small errors that introduce a defect into the extended puncture at the top and the bottom of the figure. These defects could easily be mismatched. (middle) We imagine a code where decoding can be parallelized on planes of constant time, i.e., a single-shot code. Again time runs up the page. In this situation the puncture that extends through time appears as a small puncture on each plane. Such a puncture is not disastrous to the decoding procedure. (right) We show the decoding problem on a single plane. The small puncture does not extend across the breadth of the plane so it is easy for a decoder to determine if a defect is absorbed by the small puncture. \[Fig:TimeCorrelatedErrors\]](TimeCorrelatedErrors.pdf)
In Ref. [@Bombin16] it was proven that a decoder that performs single-shot error correction [@Bombin15a] can tolerate time correlated errors. This could be due to some malfunctioning hardware that requires maintenance. Here we give a simplified explanation of this result from the perspective of parallelized quantum error correction. We suggest that the threshold theorem against time correlated errors can be generalized to other types of correlated errors provided a code is chosen appropriately such that it can be decoded along parallelized planes that run orthogonal to the direction of the correlation. We also add that we can find solutions to this issue for codes that are not parallelizable. However, this needs to be conducted at the hardware level [@Stace09; @Stace10; @Barrett10], i.e. the circuit used to measure stabilizer data needs to be adapted such that stabilizer data is collected from gauge degrees of freedom or ‘junk qubits’, see also more recent works [@Auger17; @Nagayama17]. Parallelized quantum error correction permits a solution at the software level.
The left of Fig. \[Fig:TimeCorrelatedErrors\] shows the spacetime history of the syndrome data of a code that does not readily permit parallelization, for instance the toric code. Suppose some components of the lattice are defective such that they experience errors at a very high frequency, or perhaps a stabilizer cannot be measured over many error-correction cycles. Such an error could be regarded as a puncture that extends a long distance over the time direction of spacetime. An extensive puncture will leave the encoded information vulnerable to low-weight errors if we do not take additional measures to identify them. We show a low-weight error in red where defects are separated over a long distance by this puncture. This could easily be mismatched by this decoder. It is important to identify the charge absorbed by the puncture. In the case of the toric code this can be solved by measuring superoperators to identify errors [@Stace09; @Stace10; @Barrett10; @Auger17; @Nagayama17]. This may require altering the stabilizer measurement schedule. Temporally parallelized decoders can deal with such an error at the software level where we simply do not measure the stabilizers local to this time-correlated error.
The middle image in Fig. \[Fig:TimeCorrelatedErrors\] depicts the planes of constant time upon which the decoding subroutines of a single-shot decoder will act. Notably, the puncture is separated across all of these planes such that the puncture is very small in each subroutine. A small puncture can be dealt with easily on each plane. The right of Fig. \[Fig:TimeCorrelatedErrors\] shows how a small error close to the puncture will appear on a single constant time plane. A decoder will simply match the red defect to the small puncture in the centre of the lattice.
It may be interesting to extend the result due to Bombín to models with more symmetry, or symmetries other than the emergent symmetries of single-shot codes. One might expect that it is possible to show that self-correcting stabilizer codes [@Alicki10; @Brown16] demonstrate a threshold against error models that are correlated in an arbitrary spatial or temporal direction.
Relations, emergent symmetries and generalized Gauss’s law {#App:Relations}
==========================================================
We consider stabilizer codes that are specified by a set of generators that are local with respect to some natural lattice metric. For example consider a finite number of qubits on each site of a cubic lattice, and generators supported on cubes.
Any nontrivial product of stabilizer generators that equals the identity is called a relation.
Each relation implies an emergent symmetry of parity conservation for the set of all defects that excite an odd number of stabilizer generators that are involved in the relation.
Emergent symmetries should not be confused with physical symmetries, which are operators that commute with the stabilizer generators and hence the Hamiltonian. However, it should be mentioned that physical symmetries lead to emergent symmetries when they are gauged [@wegner1971duality; @Vijay16; @PhysRevB; @kubica2018ungauging; @you2018symmetric; @shirley2018FoliatedFracton]. Throughout this paper we have exclusively discussed emergent symmetries, sometimes referring to them just as symmetries.
For any subset of stabilizer generators whose elements have support contained inside a ball $R$, nontrivial relations among these stabilizers are called local relations.
Here, $R$ is only required to be topologically equivalent to a ball, and also to be of constant extent as the lattice size is scaled up, or finite extent in an infinite lattice. If we instead consider the larger subset of stabilizer generators that have support intersecting a ball $R$, we can define the group generated by products of these generators that leave no support on $R$. This clearly contains the local relations, and since both groups are abelian we can mod out by the local relations. The quotient group is a set of generalized Gauss’s laws that relate the excitations within a ball to stabilizers that can be measured just outside its boundary.
A product of stabilizer generators, each with support intersecting a ball $R$, that has a nonempty support contained within the complement $R^c$ gives rise to a generalized Gauss’s law on $R$.
We remark that these Gauss’s laws are only defined up to local relations. The above definition generalizes directly to any $R$ that is a connected proper subset.
If the region $R$ is instead taken to be a closed manifold with no boundary, e.g. a sphere or torus, global relations are similarly defined to be nontrivial products of stabilizers that equal identity, modulo local relations.
Every global relation, when restricted to a ball, yields a Gauss’s law. But not every Gauss’s law on a finite ball leads to a global relation, it depends upon the boundary conditions.
For the 2D toric code [@Kitaev03] there are two independent global relations. These are given by the product of all star terms, and the product of all plaquette terms, respectively. These imply an emergent symmetry for the $e$ and $m$ excitations, namely that the total number of each type of excitation is even. Similarly, there are two independent Gauss’s laws on a disc, one given by the product of all star terms overlapping the disc and the other given by the product of all plaquette terms. These relate the parity of $e$ and $m$ excitations within the disc to the eigenvalue of $X$ and $Z$ string operators on its boundary. This generalizes directly to any abelian quantum double [@haah].
In a model that fulfills the topological order condition TQO2 in Ref. [@Haah2013], a cluster of charges that satisfies all boundary stabilizers involved in the Gauss’s law for the smallest ball containing it is a topologically trivial superselection sector and hence can be created via a local operator in some (slightly larger) ball containing the charges. In the 2D toric code and the X-cube model we have considered, all the local Gauss’s laws lead to global relations, and hence matching using all the global relations is sufficient to identify if a charge cluster is neutral.
More generally, not every local Gauss’s law will lead to a global relation and hence matching via only global relations may be insufficient to identify neutral clusters. So while the basic philosophy of our decoding strategy (pairing to satisfy all Gauss’s laws) applies to any topological stabilizer code, the decoder must be adapted to capture the Gauss’s laws locally if it is to succeed in a more general setting. We plan to explore this further in future work. It would also be interesting to determine whether our approach can be further generalized to nontopological LDPC codes.
To decode Wen’s plaquette model [@Wen2003] one seeks to separately pair excitations on the black and white squares of a checkerboard lattice, however this is not globally well defined if the model is placed on a cylinder of odd circumference.
The color code has a number of emergent symmetries. It is commonly decoded using matching subroutines that pair defects of two of its three differently colored types [@Delfosse14; @Kubica18], as these matchings respect a symmetry of the model. The union of the edges from two matching subroutines over the defects of two differently colored pairs of defect types gives neutralizable clusters. Lesser known are the fermionic emergent symmetries of the color code, see Appendix B in Ref. [@Kesselring18]. Indeed the product of its Pauli-X stabililizers on the red plaquettes, Pauli-Y type stabilizers on the green plaquettes and Pauli-Z stabilizers on the blue plaquettes also returns identity. Permutations thereupon can be combined to find neutralizable clusters. One might consider the use of these symmetries to find a color code decoder subject to depolarizing noise. Matching decoders for higher-dimensional generalizations of the color code also rely on symmetries of the model [@Aloshious18; @Kubica18].
Some codes have additional symmetries when we consider restricted error models. For instance the standard surface code that only experiences Pauli-Y errors effectively has one-dimensional emergent symmetries. The symmetries can be extended into two dimensions in the case that we also model measurement errors. One might consider using this observation to design a parallelized decoder to decode highly-biased errors using matching [@BrownWorkInProgress]. Further, correlated error models can also be regarded as symmetry respecting. The ballistic error model proposed in Ref. [@Nickerson17] respects an effective symmetry that enables the model to be decoded by mapping the code onto several decoupled copies of the surface code.
The square lattice Ising model, in two dimensions, is a classical self-correcting memory due to the linear scaling of the energy penalty of an error region with the size of its boundary. It can also be decoded by a local cellular automata. This can be understood in terms of the extensive number of local relations in the model, one for each plaquette. Applying our approach to decode this model would lead to a local, parallelized decoder, capable of identifying neutral clusters. Similar considerations hold for the loop excitations of the 3D toric code, and the 4D toric code with loop excitations.
[^1]: A commuting stabilizer model is obtained this way on any lattice whose cells have three-colorable faces. We get a phase equivalent to two copies of the three-dimensional toric code if we apply the prescription to a lattice that has four colorable cells [@Bombin07a; @Kim11; @Brown16a].
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---
abstract: 'The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In this note we prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. In addition, when $n \neq 0$ the bridge numbers of the knots constructed tend to infinity as the number of twists along the annulus increases.'
address:
- University of Texas at Austin
- University of Dallas
author:
- 'J. Luecke'
- 'J. Osoinach'
title: Infinitely many knots admitting the same integer surgery
---
Dehn surgery on knots is a long-standing technique for the construction of 3-manifolds. While well-known theorems of Lickorish [@lickorish] and Wallace [@wallace] state that every orientable 3-manifold can be obtained by Dehn surgery on some link in $S^3,$ this representation is far from unique. In particular, in the Kirby problem list [@kirby], Clark asks the following question:
Is there a homology 3-sphere (or any 3-manifold) which can be obtained by $n$-surgery on an infinite number of distinct knots?
In [@osoinach], the parenthetical version of this question was answered affirmatively by constructing knots using the method of twisting along an annulus. This method was subsequently developed and refined in [@teragaito] to construct knots yielding a small Seifert-fibered manifold; in [@teragaito10] to describe properties of a toroidal manifold so constructed; and in [@bgl] to analyze the bridge number of the knots constructed. In these examples, however, the surgeries used to construct the manifolds from the knots have all had even integer slopes. In [@osoinach] and [@teragaito10] the surgery slope is $0$, and in [@kouno],[@teragaito] and [@bgl] the surgery slopes are multiples of $4$. An homology sphere results exactly when $|n|=1$.
This note uses the annular twist construction to create, for each integer $n$, an infinite family of distinct knots in $S^3$ such that $n$-surgery on each knot in the collection yields the same manifold. When $|n|=1$, the resulting manifold is an homology sphere thereby answering affirmatively Problem 3.6(D) above. The members of each infinite family are distinguished by their hyperbolic volume. Alternatively, at least when $n \neq 0,$ the knots in a family are shown to be different by proving that the bridge numbers tend to infinity as the number of twists along the annulus increases.
The Dehn surgeries on a knot, $k$, in the $3$-sphere are parameterized by their surgery slopes. These surgery slopes are described by $p/q \in \mathbb{Q} \cup \infty$, meaning that the slope is a curve that runs $p$ times meridianally and $q$ times longitudinally (using the preferred longitude) along the boundary of the exterior of $k$. We write $k(p/q)$ for the $p/q$ Dehn surgery on $k$. In this notation, an $n$-surgery on $k$ refers to the integer surgery $k(n/1)$.
\[defL\] Let ${\ensuremath{{\mathcal L}}}= k \cup l_1 \cup l_2 \cup l_3$ be the link picture in Figure \[Example\]. Let $L(\alpha, \beta, \delta, \gamma)$ be the corresponding Dehn surgery on ${\ensuremath{{\mathcal L}}}$. Here the surgery slopes $\alpha,\beta,\delta, \gamma$ will be either in $\mathbb{Q} \cup \infty$, using the meridian-longitude coordinates on the boundary of a knot in $S^3$ (with a right-handed orientation on $S^3$), or an asterisk, meaning that no surgery is done on that component and the component is seen as a knot in the surgered manifold.
=3truein
The main result of this note is the following theorem.
\[mainthm\] For integers $m,n$, $k_n^m=L(*,-1/m,1/m,-1/n)$ is a knot in $S^3$. Furthermore $k_n^{m_1}(n)$ is homeomorphic to $k_n^{m_2}(n)$ for any integers $m_1,m_2$.
1. For a fixed $n \neq 0$, the bridge number of $k_n^m$ tends to infinity as $m$ tends to infinity.
2. For any integer $n$ there is an $M_n>0$ such that if $m_2 > m_1 > M_n$ then $k_n^{m_1}$ and $k_n^{m_2}$ are hyperbolic knots with the hyperbolic volume of $k_n^{m_2}$ larger than that of $k_n^{m_1}$.
In particular, for each integer $n$ there are infinitely many different knots in the family $\{k_n^m\}$.
[*Proof.*]{}
We first show that for any integer $n$, the $n$-surgery on each $k_n^m$ yields the same manifold for each $m$. Figure \[Q\] shows that the knot $k$ is a non-separating, orientation-preserving curve on a twice-punctured Klein bottle, $Q$, cobounded by $l_1$ and $l_2$ and in the complement of $l_3$.
=3truein
Thus $Q - \operatorname{Nbhd}({\ensuremath{{\mathcal L}}})$ is a $4$-punctured sphere, $P$, properly embedded in the exterior, $X_{{\ensuremath{{\mathcal L}}}}$, of ${\ensuremath{{\mathcal L}}}$ in $S^3$. $\partial P$ has one component on each of $\partial \operatorname{Nbhd}(l_1),\partial \operatorname{Nbhd}(l_2)$ of slope $0/1$ and two components on $\partial \operatorname{Nbhd}(k)$ of slope $0/1$. Let $\widehat{P}$ be the annulus in the exterior of $L(0/1,*,*,*)$ obtained by capping off the two components of $P$ along $\partial \operatorname{Nbhd}(k)$. Dehn twisting this exterior along the properly embedded annulus $\widehat{P}$ in $L(0/1,*,*,*)$ (see Remark \[twisting\]), induces a homeomorphism of $L(0/1,-1/m,1/m,-1/n)$ to $L(0/1,1/0,1/0,-1/n)$ for each $m,n$.
\[twisting\] Let ${\ensuremath{{\widehat{R}}}}$ be an annulus embedded in a $3$-manifold $M$ with $\partial {\ensuremath{{\widehat{R}}}}$ the link $L_1 \cup L_2$ in $M$. Let $R={\ensuremath{{\widehat{R}}}}\cap (M - \operatorname{Nbhd}(L_1 \cup L_2))$. Fix an orientation on $M$ and ${\ensuremath{{\widehat{R}}}}$. This induces an orientation on $L_i$ and its meridian $\mu_i$. Let ${\ensuremath{{\widehat{R}}}}\times [0,1]$ be a product neighborhood of ${\ensuremath{{\widehat{R}}}}$ in $M$ so that the corresponding interval orientation on $R \times [0,1]$ corresponds to the meridian orientation of $L_1$. Pick coordinates ${\ensuremath{{\widehat{R}}}}=e^{2 \pi i \theta} \times [0,1]$, with $\theta \in [0,1]$, so that $e^{2 \pi i \theta} \times \{0\}, \theta \in [0,1],$ is the oriented $L_1$. Define the homeomorphism $f_m\colon {\ensuremath{{\widehat{R}}}}\times [0,1] \to {\ensuremath{{\widehat{R}}}}\times [0,1]$ by $(e^{2 \pi i \theta},s,t) \to
(e^{2 \pi i(\theta + mt)},s,t)$. Note that $f_n$ restricted to ${\ensuremath{{\widehat{R}}}}\times \{0,1\}$ is the identity. Assume that the knot $K$ in $M$ intersects ${\ensuremath{{\widehat{R}}}}\times
[0,1]$ in $[0,1]$ fibers. Let $K^m$ be the knot in $M$ gotten by applying $f_m$ to $K
\cap ({\ensuremath{{\widehat{R}}}}\times [0,1])$ (and the identity on $K$ outside this region). We refer to $K^m$ as [*$K$ twisted $m$ times along ${\ensuremath{{\widehat{R}}}}$*]{}, or we say that [*$K^m$ is obtained from $K$ by twisting along ${\ensuremath{{\widehat{R}}}}$*]{}. Furthermore, note that $f_m$ induces a homeomorphism $h_m \colon M-\operatorname{Nbhd}(L_1 \cup L_2) \to M-\operatorname{Nbhd}(L_1 \cup L_2)$ by applying $f_m$ in $R \times [0,1]$ along with the identity outside this neighborhood. We refer to this homeomorphism $h_m$ of $M-\operatorname{Nbhd}(L_1 \cup L_2)$ as [*$m$ Dehn-twisting along the properly embedded annulus $R$*]{}.
=1.5truein
Figure \[A\] shows an annulus $A$ cobounded by $l_1$ and $l_2$ in the complement of $l_3$ (which can be taken to intersect $k$ algebraically zero and geometrically four times and which induces the framing $0/1$ on each of $l_1$ and $l_2$), which becomes an annulus $A_n$ cobounded by $l_1$ and $l_2$ after $(-1/n)$-surgery on $l_3$. Dehn-twisting the exterior of $l_1 \cup l_2$ in $L(*,*,*, -1/n)$ along $A_n$ (really the restriction of $A_n$ to this exterior, Remark \[twisting\]) induces an orientation-preserving homeomorphism of the manifold $L(1/0,-1/0,1/0,-1/n)=S^3$ to the manifold $L(1/0, -1/m, 1/m, -1/n)$. The inverse of this homeomorphism identifies $k_n^m$ as a knot in $S^3$ obtained by twisting $k_n^0$ along $A_n$ (see Remark \[twisting\]).
The following claim finishes the argument that the $n$-surgeries on $k_n^m$ are the same manifold.
\[clm:slope\] For each $m,n$, $L(0/1,-1/m,1/m,-1/n)=k_n^m(n)$.
[*Proof of Claim:*]{} $L(0/1,-1/m,1/m,-1/n)$ is clearly a surgery on $k_n^m$. Our goal is to identify the slope of this surgery, $\alpha(m,n)$, in terms of the coordinates on $k_n^m$ as a knot in $S^3$. Let $P_n$ be the $4$-punctured sphere $P$ after $-1/n$-surgery on $l_3$. Then $\alpha(0,n)$ is the slope of $P_n$ on $k_n^0$.
Twisting along $A_n$ gives a homeomorphism of the exterior of $l_1 \cup l_2 \cup k_n^0$ in $S^3$ to the exterior of $l_1 \cup l_2 \cup k_n^m$ and consequently takes $P_n$ to a $4$-punctured sphere $P_n^m$ in the exterior of $l_1 \cup
l_2 \cup k_n^m$. The slope $\alpha(m,n)$ is the slope of $P_n^m$ on $k_n^m$. We may use $P_n^m$ to compute the linking number of the slope $\alpha(m,n)$ with $k_n^m$ and consequently the coordinates of the slope. Orient $k_n^m$ and take the orientation on $P_n^m$ that induces an orientation on $\partial P_n^m \cap \operatorname{Nbhd}(k_n^m)$ that agrees with that on $k_n^m$. Then twice the linking number of $\alpha(m,n)$ with the oriented $k_n^m$ in $S^3$ is the negative of the linking number between the oriented $k_n^m$ and $l_1 \cup l_2$, given the orientation induced by $P_n^m$ on $l_1 \cup l_2$. By considering $k_n^m$ as twisting $k_n^0$ along $A_n$ away from $l_1 \cup l_2$, one sees that this latter linking number is $-2n$ (one may verify that in the $1/0$ surgery on $l_3$, this linking number is zero, then observe how the linking number changes under $-1/n$-surgery). Thus $\alpha(m,n)$ is the slope $n/1$ as desired.
\[clm:noannulus\] Let $X_n$ be the exterior of $L(*,*,*,-1/n)$ and $T_1,T_2$ be the components of $\partial X_n$ coming from $\operatorname{Nbhd}(l_1),\operatorname{Nbhd}(l_2)$ respectively. For each integer $n \neq -2$, the interior of $X_n$ is hyperbolic. For every integer $n$ (including $-2$), there is no essential annulus properly embedded in $X_n$ with one boundary component on $T_1$ and the other on $T_2$.
[*Proof of Claim:*]{} SnapPy [@snappy] shows that ${\ensuremath{{\mathcal L}}}$ is hyperbolic. The program HIKMOT [@hikmot] certifies this calculation. The sequence of isotopies Figure \[2surgery\](a)-(c) shows that $l_1$ in $L(*,*,*,1/2)$ is a $(2,-1)$-cable on the knot $l_1'$ pictured in Figure \[2surgery\](d) (the 3-manifold $H$ in Figure \[2surgery\] is a neighborhood of the punctured Klein bottle $Q$ and $l_1$ is pushed off $H$). Because the linking number of $l_1'$ with $k$ is one, the exterior of $k \cup l_1 \cup l_2$ in $L(*,*,*,1/2)$ is toroidal. It follows from [@gordon] and [@gordon-wu] that the interior of $X_n$ is hyperbolic as long as $|n+2|>3$.
For $n \in \{1,0,-1,-3,-4,-5 \}$ SnapPy shows that $X_n$ is hyperbolic and HIKMOT certifies this calculation. Thus the interior of $X_n$ is hyperbolic, and in particular $X_n$ is annular, as long as $n \neq -2$.
We must still show that $X_{-2}$ is anannular. As mentioned above, Figure \[2surgery\](d) shows that $X_{-2}$ is the union, along a torus $T$, of the exterior of a $(2,-1)$-cable of the core of a solid torus and the exterior, $X_{-2}'$, of $l_1' \cup l_2 \cup k$ after $1/2$-surgery on $l_3$. SnapPy shows $X_{-2}'$ is hyperbolic and HIKMOT certifies this. Now assume there were an essential annulus in $X_{-2}$ between $T_1$ and $T_2$, and consider its intersection with the incompressible torus $T$. We may surger away any closed curves of intersection which are trivial on $T$. Then an outermost component of intersection with $X_2'$ will give rise to an essential annulus or disk properly embedded in $X_2'$, contradicting the hyperbolicity of $X_2'$.
=4truein
We first verify $(1)$ of Theorem \[mainthm\]. As before, let $A_n$ be the annulus from Figure \[A\] cobounded by $l_1$ and $l_2$ and after $(-1/n)$-surgery on $l_3$. The knot $k_n^m$ is obtained by twisting $k$ along $A_n$ ($m$ times) in the copy of $S^3$ obtained by $(-1/n)$-surgery on $l_3$. As the linking number of $l_1$ and $l_2$ in this copy of $S^3$ is $n$, $l_1 \cup l_2$ is not the trivial link. Then Claim \[clm:noannulus\] along with Corollary 1.4 of [@bgl] shows that for $n \neq 0$ the (genus $0$) bridge number of the knots $k_n^m$ in $S^3$ goes to infinity as $m$ goes to infinity (as the linking number of $l_1$ and $l_2$ is non-zero, Lemma 2.4 of [@bgl] shows there is a catching surface for the pair $(k, A_n)$). Note that since $A_0$ lies on a Heegaard sphere for $S^3$, the bridge numbers of $\{k_0^m\}$ will be bounded.
We now verify $(2)$ of Theorem \[mainthm\]. By Claim \[clm:noannulus\], the interior of $X_n$ is hyperbolic whenever $n \neq -2$. Thurston’s Dehn Surgery Theorem and Theorem 1A of [@neumann], shows that there is an $M_n>0$ such that for $m >M_n$, $k_n^m$ is hyperbolic and its volume increases monotonically with $m$. When $n = -2$, recall from the proof of Claim \[clm:noannulus\] that Figure \[2surgery\](d) shows that $X_{-2}$ is the union, along a torus $T$, of the exterior of a $(2,-1)$-cable of the core of a solid torus and the exterior, $X_{-2}'$, of $l_1' \cup l_2 \cup k$ after $1/2$-surgery on $l_3$. That is, identify $L(*,*,*,1/2)$ as a link in $S^3$ by putting two full left-handed twists along the linking circle $l_3$. Then $L(*, -1/m,1/m,1/2)$ corresponds to $(-1-2m)/m$ surgery on $l_1$ and $(1-2m)/m$ surgery on $l_2$. The Seifert fiber on $l_1$ as a $(2,-1)$-cabling on $l_1'$ is $-2/1$. As the surgery slope intersects this Seifert fiber slope once, this surgery on $l_2$ corresponds to doing a $(-1-2m)/4m$ surgery on $l_1'$ (see Corollary 7.3 of [@gordon2]). As noted above, HIKMOT verifies $k \cup l_1' \cup l_2$ to be hyperbolic. Thus an application of Theorem 1A of [@neumann] to the exterior $X_{-2}$ of this link, shows there is an $M_{-2}$ such that for $m > M_{-2}$, $k_{-2}^m$ is hyperbolic and its volume increases monotonically with $m$.
Since hyperbolic volume and bridge number are knot invariants, either $(1)$ (when $n \neq 0$) or $(2)$ shows that for an integer $n$ the family $\{k_n^m\}$ is infinite.
SnapPy shows the homology spheres that arise in the above construction ($|n|=1$) to be hyperbolic manifolds with $volume(k_{-1}^0(-1))=3.400436870$ and $volume(k_{1}^0(1))=5.7167678901$. SnapPy shows the manifold corresponding to $n=-2$ to be hyperbolic with $volume(k_{-2}^0(-2))=3.110698158$. These calculations are not verified by HIKMOT.
When $K$ is a knot in the $3$-sphere, let $W(K,n)$ be the $4$-manifold obtained by attaching a 2-handle to the $4$-ball along $S^3$ with framing $n$. The 3-manifold $K(n)$ is the boundary of $W(K,n)$. In [@ajot], the following analog of Problem 1 is considered.
[**Problem 2**]{} [*Let $n$ be an integer. Find infinitely many mutually distinct knots $K_1, K_2, \dots$ such that $W(K_i,n)$ is diffeomorphic to $W(K_j,n)$ for each $i, j \in {\ensuremath{\mathbb{N}}}$.*]{}
In [@ajot], such an infinite family of knots is demonstrated when $n \in
\{-4,0,4\}$. These families of knots are constructed by twisting along an annulus similar to the construction above for the family $\{k_n^m, m \in {\ensuremath{\mathbb{Z}}}\}$. Indeed Theorem \[mainthm\] shows that the boundaries of $W(k_n^{i},n)$ and $W(k_n^{j},n)$ are diffeomorphic. So it is natural to ask
[**Problem 3**]{} [*Let $n$ be an integer. Are $W(k_n^{i},n)$ and $W(k_n^{j},n)$ diffeomorphic?*]{}
[**Acknowledgements.**]{} The authors would like to thank Kyle Larson for very helpful conversations, and Neil Hoffman for his help with HIKMOT.
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---
abstract: 'The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional. In this paper we answer this question in the negative. Projection algorithms that minimize smooth convex functions over a product of convex sets are also discussed.'
author:
- |
J.-B. Baillon,$^1$ P. L. Combettes,$^{2}$ and R. Cominetti$^3$\
$\!^1$Université Paris 1 Panthéon-Sorbonne\
SAMM – EA 4543\
75013 Paris, France (Jean-Bernard.Baillon@univ-paris1.fr)\
$\!^2$UPMC Université Paris 06\
Laboratoire Jacques-Louis Lions – UMR 7598\
75005 Paris, France (plc@math.jussieu.fr)\
$\!^3$Universidad de Chile\
Departamento de Ingeniería Industrial\
Santiago, Chile (rccc@dii.uchile.cl)
date:
title: |
There is no variational characterization of the cycles\
in the method of periodic projections
---
Introduction
============
Throughout this paper ${\ensuremath{{\mathcal H}}}$ is a real Hilbert space with scalar product ${{\left\langle{{\cdot}\mid{\cdot}}\right\rangle}}$ and associated norm $\|\cdot\|$. Let $C_1$ and $C_2$ be closed vector subspaces of ${\ensuremath{{\mathcal H}}}$, and let $P_1$ and $P_2$ be their respective projection operators. The method of alternating projections for finding the projection of a point $x_0\in{\ensuremath{{\mathcal H}}}$ onto $C_1\cap C_2$ is governed by the iterations $$\label{e:pocs1}
(\forall n\in{\ensuremath{\mathbb N}})\quad
\begin{array}{l}
\left\lfloor
\begin{array}{l}
x_{2n+1}=P_2x_{2n}\\
x_{2n+2}= P_1x_{2n+1}.
\end{array}
\right.\\[2mm]
\end{array}$$ This basic process, which can be traced back to Schwarz’ alternating method in partial differential equations [@Schw70], has found many applications in mathematics and in the applied sciences; see [@Deut92] and the references therein. The strong convergence of the sequence $(x_{n})_{n\in{\ensuremath{\mathbb N}}}$ produced by to the projection of $x_0$ onto $C_1\cap C_2$ was established by von Neumann in 1933 [@Vonn49]. The extension of to the case when $C_1$ and $C_2$ are general nonempty closed convex sets was considered in [@Che59a; @Lev66b]. Thus, it was shown in [@Che59a] that, if $C_1$ is compact, the sequences $(x_{2n})_{n\in{\ensuremath{\mathbb N}}}$ and $(x_{2n+1})_{n\in{\ensuremath{\mathbb N}}}$ produced by converge strongly to points $\overline{y}_1$ and $\overline{y}_2$, respectively, that constitute a cycle, i.e., $$\label{e:2010-12-21b}
\overline{y}_1=P_1\overline{y}_2
\quad\text{and}\quad
\overline{y}_2=P_2\overline{y}_1,$$ or, equivalently, that solve the variational problem (see Figure \[fig:1\]) $$\label{e:best2}
{\ensuremath{\underset{\substack{{y_1\in C_1,\,y_2\in C_2}}}{\mathrm{minimize}}\;\;\|y_1-y_2\| }}.$$ Furthermore, it was shown in [@Lev66b] that, if $C_1$ is merely bounded, the same conclusion holds provided strong convergence is replaced by weak convergence. As was proved only recently [@Hund04], strong convergence can fail.
Extending the above results to $m\geq 3$ nonempty closed convex subsets $(C_i)_{1\leq i\leq m}$ of ${\ensuremath{{\mathcal H}}}$ poses interesting challenges. For instance, there are many strategies for scheduling the order in which the sets are projected onto. The simplest one corresponds to a periodic activation of the sets, say $$\label{e:pocs2}
(\forall n\in{\ensuremath{\mathbb N}})\quad
\begin{array}{l}
\left\lfloor
\begin{array}{ll}
x_{mn+1}&=P_mx_{mn}\\
x_{mn+2}&=P_{m-1}x_{mn+1}\\
&\;\vdots\\
x_{mn+m}&= P_1x_{mn+m-1},
\end{array}
\right.\\[2mm]
\end{array}$$ where $(P_i)_{1\leq i\leq m}$ denote the respective projection operators onto the sets $(C_i)_{1\leq i\leq m}$. In the case of closed vector subspaces, it was shown in 1962 that the sequence $(x_n)_{n\in{\ensuremath{\mathbb N}}}$ thus generated converges strongly to the projection of $x_0$ onto $\bigcap_{i=1}^mC_i$ [@Halp62]. This provides a precise extension of the von Neumann result, which corresponds to $m=2$. Interestingly, however, for nonperiodic sweeping strategies with closed vector subspaces, only weak convergence has been established in general [@Amem65] and, since 1965, it has remained an open problem whether strong convergence holds (see [@Bail99] for the state-of-the-art on this conjecture). Another long-standing open problem is the one that we address in this paper and which concerns the asymptotic behavior of the periodic projection algorithm for general closed convex sets. It was shown in 1967 [@Gubi67] (see also [@Opti04 Section 7], [@Erem08], and [@Mart72 Théorème 5.5.2] for extensions of this result to more general operators) that, if one of the sets is bounded, the sequences $(x_{mn})_{n\in{\ensuremath{\mathbb N}}}$, $(x_{mn+1})_{n\in{\ensuremath{\mathbb N}}}$, …, $(x_{mn+m-1})_{n\in{\ensuremath{\mathbb N}}}$ converge weakly to points $\overline{y}_1$, $\overline{y}_m$, …, $\overline{y}_2$, respectively, that constitute a cycle, i.e. (see Figure \[fig:2\]), $$\label{e:2010-12-21c}
\overline{y}_1=P_1\overline{y}_2,\;\ldots,\;
\overline{y}_{m-1}=P_{m-1}\overline{y}_{m},\;
\overline{y}_m=P_m\overline{y}_{1}.$$ However, it remains an open question whether, as in the case of $m=2$ sets, the cycles can be characterized as the solutions to a variational problem. We formally formulate this problem as follows.
\[d:cycle\] Let $m$ be an integer at least equal to $2$ and let $(C_1,\ldots,C_m)$ be an ordered family of nonempty closed convex subsets of ${\ensuremath{{\mathcal H}}}$ with associated projection operators $(P_1,\ldots,P_m)$. The set of cycles associated with $(C_1,\ldots,C_m)$ is $$\label{e:2010-12-22a}
{\ensuremath{\mathsf{cyc}}}(C_1,\ldots,C_m)=
{\big\{{(\overline{y}_1,\ldots,\overline{y}_m)\in{\ensuremath{{\mathcal H}}}^m}~\big |~{
\overline{y}_1=P_1\overline{y}_2,\;\ldots,\;
\overline{y}_{m-1}=P_{m-1}\overline{y}_{m},\;
\overline{y}_m=P_m\overline{y}_{1}}\big\}}.$$
\[prob:1\] Let $m$ be an integer at least equal to $3$. Does there exist a function $\Phi\colon{\ensuremath{{\mathcal H}}}^m\to{\ensuremath{\mathbb{R}}}$ such that, for every ordered family of nonempty closed convex subsets $(C_1,\ldots,C_m)$ of ${\ensuremath{{\mathcal H}}}$, ${\ensuremath{\mathsf{cyc}}}(C_1,\ldots,C_m)$ is the set of solutions to the variational problem $$\label{e:best3}
{\ensuremath{\underset{\substack{{y_1\in C_1,\ldots,\,y_m\in C_m}}}{\mathrm{minimize}}\;\;\Phi(y_1,\ldots,y_m) }}\:?$$
Let us note that the motivations behind Question \[prob:1\] are not purely theoretical but also quite practical. Indeed, the variational properties of the cycles when $m=2$ have led to fruitful applications, e.g., [@Gold85; @Merc80; @Noba95; @Pesq96]. Since the method of periodic projections is used in scenarios involving $m\geq 3$ possibly nonintersecting sets [@Proc93], it is therefore important to understand the properties of its limit cycles and, in particular, whether they are optimal in some sense. Since the seminal work [@Gubi67] in 1967 that first established the existence of cycles, little progress has been made towards this goal beyond the observation that simple candidates such as $\Phi\colon(y_1,\ldots,y_m)\mapsto\|y_1-y_2\|+\cdots
+\|y_{m-1}-y_m\|+\|y_m-y_1\|$ fail [@Baus94; @Baus97; @Sign94; @Kosm87]. The main result of this paper is that the answer to Question \[prob:1\] is actually negative. This result will be established in Section \[sec:2\]. Finally, in Section \[sec:3\], projection algorithms that are pertinent to extensions of to $m\geq 3$ sets will be discussed.
A negative answer to Question \[prob:1\] {#sec:2}
========================================
We denote by $S(x;\rho)$ the sphere of center $x\in{\ensuremath{{\mathcal H}}}$ and radius $\rho\in{\ensuremath{\left[0,+\infty\right[}}$, and by $P_C$ the projection operator onto a nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$; in particular, $P_C 0$ is the element of minimal norm in $C$.
Our main result hinges on the following variational property, which is of interest in its own right.
\[t:03\] Suppose that $\dim{\ensuremath{{\mathcal H}}}\geq 2$ and let $\varphi\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}$ be such that its infimum on every nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$ is attained at $P_C 0$. Then the following hold.
1. \[t:03i\] $\varphi$ is radially increasing, i.e., $$\label{e:mono}
(\forall x\in{\ensuremath{{\mathcal H}}})(\forall y\in{\ensuremath{{\mathcal H}}})\quad
\|x\|<\|y\|\quad\Rightarrow\quad\varphi(x)\leq\varphi(y).$$
2. \[t:03ii\] Suppose that, for every nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$, $P_C 0$ is the unique minimizer of $\varphi$ on $C$. Then $\varphi$ is strictly radially increasing, i.e., $$\label{e:monos}
(\forall x\in{\ensuremath{{\mathcal H}}})(\forall y\in{\ensuremath{{\mathcal H}}})\quad
\|x\|<\|y\|\quad\Rightarrow\quad\varphi(x)<\varphi(y).$$
3. \[t:03iii\] Except for at most countably many values of $\rho\in{\ensuremath{\left[0,+\infty\right[}}$, $\varphi$ is constant on $S(0;\rho)$.
\[t:03i\]: Let us fix $x$ and $y$ in ${\ensuremath{{\mathcal H}}}$ such that $\|x\|<\|y\|$. If $x=0$, property amounts to the fact that $0$ is a global minimizer of $\varphi$, which follows from the assumption with $C={\ensuremath{{\mathcal H}}}$. We now suppose that $x\neq 0$. Let $V$ be a 2-dimensional vector subspace of ${\ensuremath{{\mathcal H}}}$ containing $x$ and $y$, and let $\alpha\in[0,\pi]$ be the angle between $x$ and $y$. For every integer $n\geq 3$, consider a polygonal spiral built as follows: set $x_{n,0}=y$ and for $k=1,\ldots,n$ define $x_{n,k}=P_{R_{n,k}} x_{n,k-1}$, where $(R_{n,k})_{1\leq k\leq n}$ are $n$ angularly equispaced rays in $V$ between the rays ${\ensuremath{\left[0,+\infty\right[}}y$ and ${\ensuremath{\left[0,+\infty\right[}}x=R_{n,n}$ (see Figure \[fig:3\]). Clearly, for the segment $C=[x_{n,k-1},x_{n,k}]$, we have $P_C 0=x_{n,k}$, so that the assumption on $\varphi$ gives $\varphi(x_{n,k})\leq\varphi(x_{n,k-1})$, and therefore $\varphi(x_{n,n})\leq\varphi(x_{n,0})=\varphi(y)$. On the other hand, $x_{n,n}$ and $x$ are collinear with $\|x_{n,n}\|=\|y\|(\cos(\alpha/n))^n$ so that for $n$ large enough we have $\|x_{n,n}\|>\|x\|$ and, therefore, the segment $C=[x,x_{n,n}]$ satisfies $P_C 0=x$, from which we get $\varphi(x)\leq\varphi(x_{n,n})\leq\varphi(y)$ as claimed.
\[t:03ii\]: If the minimizer of $\varphi$ on every nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$ is unique, then all the inequalities above are strict.
\[t:03iii\]: Set $g\colon{\ensuremath{\left[0,+\infty\right[}}\to{\ensuremath{\mathbb{R}}}\colon\rho\mapsto\inf\varphi(S(0;\rho))$ and $h\colon{\ensuremath{\left[0,+\infty\right[}}\to{\ensuremath{\mathbb{R}}}\colon\rho\mapsto\sup\varphi(S(0;\rho))$. It follows from that $$(\forall\rho\in{\ensuremath{\left[0,+\infty\right[}})(\forall\rho'\in{\ensuremath{\left[0,+\infty\right[}})\quad
\rho<\rho'\quad\Rightarrow\quad g(\rho)\leq h(\rho)\leq
g(\rho')\leq h(\rho').$$ Hence, $g$ and $h$ are increasing and therefore, by Froda’s theorem [@Rudi76 Theorem 4.30], the set of points at which they are discontinuous is at most countable. Since $g$ and $h$ coincide at every point of continuity, we conclude that, except for at most countably many $\rho\in{\ensuremath{\left[0,+\infty\right[}}$, $\varphi$ is constant on $S(0;\rho)$.
As a straightforward consequence, we get the following.
Suppose that $\dim{\ensuremath{{\mathcal H}}}\geq 2$ and let $\varphi\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}$ be such that its infimum on every nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$ is attained at $P_C 0$. If $\varphi$ is either lower or upper semicontinuous, then $\varphi=\theta\circ\|\cdot\|$, where $\theta\colon{\ensuremath{\left[0,+\infty\right[}}\to{\ensuremath{\mathbb{R}}}$ is increasing. Furthermore, if $P_C 0$ is the unique minimizer of $\varphi$ on every nonempty closed convex set $C\subset{\ensuremath{{\mathcal H}}}$, then $\theta$ is strictly increasing.
Using Theorem \[t:03\] we can provide the following answer to Question \[prob:1\].
\[t:4\] Suppose that $\dim{\ensuremath{{\mathcal H}}}\geq 2$ and let $m$ be an integer at least equal to $3$. There exists no function $\Phi\colon{\ensuremath{{\mathcal H}}}^m\to{\ensuremath{\mathbb{R}}}$ such that, for every ordered family of nonempty closed convex subsets $(C_1,\ldots,C_m)$ of ${\ensuremath{{\mathcal H}}}$, ${\ensuremath{\mathsf{cyc}}}(C_1,\ldots,C_m)$ is the set of solutions to the variational problem $$\label{e:best4}
{\ensuremath{\underset{\substack{{y_1\in C_1,\ldots,\,y_m\in C_m}}}{\mathrm{minimize}}\;\;\Phi(y_1,\ldots,y_m) }}.$$
Suppose that $\Phi$ exists and set $(\forall i\in\{1,\ldots,m-2\})$ $C_i=\{0\}$. Moreover, take $z\in{\ensuremath{{\mathcal H}}}$ and set $C_{m-1}=\{z\}$. Then, for every nonempty closed convex set $C_m\subset{\ensuremath{{\mathcal H}}}$ we have $$\mathop{\rm Argmin}_{y_1\in C_1,\ldots,y_m\in C_m}
\Phi(y_1,\ldots,y_m)=
{\ensuremath{\mathsf{cyc}}}(C_1,\ldots,C_m)=
\{(0,\ldots,0,z,P_{C_m} 0)\}.$$ Hence, Theorem \[t:03\] implies that, except for at most countably many values of $\rho\in{\ensuremath{\left[0,+\infty\right[}}$, the function $\Phi(0,\ldots,0,z,\cdot)$ is constant on $S(0;\rho)$.
Now suppose that $z\in S(0;1)$ and take $\rho\in\left]1,{\ensuremath{{+\infty}}}\right[$ so that $\Phi(0,\ldots,0,z,\cdot)$ and $\Phi(0,\ldots,0,-z,\cdot)$ are constant on $S(0;\rho)$. Clearly, $${\ensuremath{\mathsf{cyc}}}\big(\{0\},\ldots,\{0\},[-z,z],\{\rho z\}\big)
=\{(0,\ldots,0,z,\rho z)\}$$ and $${\ensuremath{\mathsf{cyc}}}\big(\{0\},\ldots,\{0\},[-z,z],\{-\rho z\}\big)
=\{(0,\ldots,0,-z,-\rho z)\},$$ so that $$\begin{aligned}
\Phi(0,\ldots,0,z,\rho z)
&<\Phi(0,\ldots,0,-z,\rho z)\nonumber\\
&=\Phi(0,\ldots,0,-z,-\rho z)\nonumber\\
&<\Phi(0,\ldots,0,z,-\rho z)\nonumber\\
&=\Phi(0,\ldots,0,z,\rho z)\end{aligned}$$ where the inequalities come from the fact that the minima of $\Phi$ characterize the cycles, while the equalities follow from the constancy of the functions on $S(0;\rho)$. Since these strict inequalities are impossible it follows that $\Phi$ cannot exist.
Related projection algorithms {#sec:3}
=============================
We have shown that the cycles produced by the method of cyclic projections are not characterized as the solutions to a problem of the type , irrespective of the choice of the function $\Phi\colon{\ensuremath{{\mathcal H}}}^m\to{\ensuremath{\mathbb{R}}}$. Nonetheless, alternative projection methods can be devised to solve variational problems over a product of closed convex sets. Here is an example.
\[t:2011-02-04\] For every $i\in I=\{1,\ldots,m\}$, let $({\ensuremath{{\mathcal H}}}_i,\|\cdot\|_i)$ be a real Hilbert space and let $C_i$ be a nonempty closed convex subset of ${\ensuremath{{\mathcal H}}}_i$ with projection operator $P_i$. Let ${\ensuremath{\boldsymbol{\mathcal H}}}$ be the Hilbert space obtained by endowing ${{\ensuremath{\raisebox{-0.5mm}{\mbox{\LARGE{$\times$}}}}\!}}_{i\in I}{\ensuremath{{\mathcal H}}}_i$ with the norm $\boldsymbol{y}=(y_i)_{i\in I}\mapsto
\sqrt{\sum_{i\in I}\|y_i\|_i^2}$, and let $\Phi\colon{\ensuremath{\boldsymbol{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}$ be a differentiable convex function such that $\nabla\Phi\colon{\ensuremath{\boldsymbol{\mathcal H}}}\to{\ensuremath{\boldsymbol{\mathcal H}}}\colon{\boldsymbol y}\mapsto
(G_i{\boldsymbol y})_{i\in I}$ is $1/\beta$-lipschitzian for some $\beta\in{\ensuremath{\left]0,+\infty\right[}}$ and such that the problem $$\label{e:best24}
{\ensuremath{\underset{\substack{{y_1\in C_1,\ldots,\,y_m\in C_m}}}{\mathrm{minimize}}\;\;\Phi(y_1,\ldots,y_m) }}$$ admits at least one solution. Let $\gamma\in\left]0,2\beta\right[$, set $\delta=\min\{1,\beta/\gamma\}+1/2$, let $(\lambda_n)_{n\in{\ensuremath{\mathbb N}}}$ be a sequence in $[0,\delta]$ such that $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(\delta-\lambda_n)={\ensuremath{{+\infty}}}$, and let ${\boldsymbol x}_0=(x_{i,0})_{i\in I}\in{\ensuremath{\boldsymbol{\mathcal H}}}$. Set $$\label{e:2011-02-04a}
(\forall n\in{\ensuremath{\mathbb N}})(\forall i\in I)\quad
x_{i,n+1}=x_{i,n}+\lambda_{n}\big(P_i\big(x_{i,n}-\gamma
G_i{\boldsymbol x}_{n}\big)-x_{i,n}\big).$$ Then, for every $i\in I$, $(x_{i,n})_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point $\overline{y}_i\in C_i$, and $(\overline{y}_i)_{i\in I}$ is a solution to .
Set ${\boldsymbol C}={{\ensuremath{\raisebox{-0.5mm}{\mbox{\LARGE{$\times$}}}}\!}}_{i\in I}C_i$. Then ${\boldsymbol C}$ is a nonempty closed convex subset of ${\ensuremath{\boldsymbol{\mathcal H}}}$ with projection operator $P_{\boldsymbol C}\colon{\boldsymbol x}\mapsto(P_ix_i)_{i\in I}$ [@Livre1 Proposition 28.3]. Accordingly, we can rewrite as $$\label{e:genna07-16}
(\forall n\in{\ensuremath{\mathbb N}})\quad
{\boldsymbol x}_{n+1}={\boldsymbol x}_n+
\lambda_n\big(P_{\boldsymbol C}
\big({\boldsymbol x}_n-\gamma\nabla \Phi({\boldsymbol x}_n)\big)
-{\boldsymbol x}_n\big).$$ It follows from [@Livre1 Corollary 27.10] that $({\boldsymbol x}_n)_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a minimizer $\overline{{\boldsymbol y}}$ of $\Phi$ over ${\boldsymbol C}$, which concludes the proof.
The projection algorithm described in the next result solves an extension of to $m\geq 3$ sets.
\[c:2011-02-04\] Let $m$ be an integer at least equal to 3. For every $i\in I=\{1,\ldots,m\}$, let $C_i$ be a nonempty closed convex subset of ${\ensuremath{{\mathcal H}}}$ with projection operator $P_i$, and let $x_{i,0}\in{\ensuremath{{\mathcal H}}}$. Suppose that one of the sets in $(C_i)_{i\in I}$ is bounded and set $$\label{e:2011-02-04b}
(\forall n\in{\ensuremath{\mathbb N}})(\forall i\in I)\quad x_{i,n+1}=
P_i\left({\displaystyle{\frac{1}{m-1}}}\sum_{j\in I\smallsetminus\{i\}}
x_{j,n}\right).$$ Then for every $i\in I$, $(x_{i,n})_{n\in{\ensuremath{\mathbb N}}}$ converges weakly to a point $\overline{y}_i\in C_i$, and $(\overline{y}_i)_{i\in I}$ is a solution to the variational problem $$\label{e:best8}
{\ensuremath{\underset{\substack{{y_1\in C_1,\ldots,\,y_m\in C_m}}}{\mathrm{minimize}}\;\;\sum_{\substack{(i,j)\in I^2\\ i<j}}\|y_i-y_j\|^2 }}.$$ Moreover, $\overline{y}=(1/m)\sum_{i\in I}\overline{y}_i$ is a minimizer of the function $\varphi\colon{\ensuremath{{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}\colon y\mapsto\sum_{i\in I}\|y-P_iy\|^2$.
We use the notation of Theorem \[t:2011-02-04\], with $(\forall i\in I)$ ${\ensuremath{{\mathcal H}}}_i={\ensuremath{{\mathcal H}}}$. Set $\beta=1-1/m$, $\gamma=1$, $$\Phi\colon{\ensuremath{\boldsymbol{\mathcal H}}}\to{\ensuremath{\mathbb{R}}}\colon(y_i)_{i\in I}\mapsto\frac{1}{2(m-1)}
\sum_{\substack{(i,j)\in I^2\\ i<j}}\|y_i-y_j\|^2,$$ ${\boldsymbol C}={{\ensuremath{\raisebox{-0.5mm}{\mbox{\LARGE{$\times$}}}}\!}}_{i\in I}C_i$, and ${\boldsymbol D}={\big\{{(y,\ldots,y)\in{\ensuremath{\boldsymbol{\mathcal H}}}}~\big |~{y\in{\ensuremath{{\mathcal H}}}}\big\}}$. Then [@Baus93; @Sign94] $$\label{e:2011-02-05a}
{\ensuremath{\operatorname{Fix}}}P_{\boldsymbol C}P_{\boldsymbol D}=
\operatorname{Argmin}\Phi\quad\text{and}\quad
{\ensuremath{\operatorname{Fix}}}P_{\boldsymbol D}P_{\boldsymbol C}=
{\big\{{(y,\ldots,y)}~\big |~{y\in\operatorname{Argmin}\varphi}\big\}}.$$ Since one of the sets in $(C_i)_{i\in I}$ is bounded, $\operatorname{Argmin}\varphi\neq{\ensuremath{{\varnothing}}}$ [@Sign94 Proposition 7]. Now let $y\in\operatorname{Argmin}\varphi$, and set ${\boldsymbol y}=(y,\ldots,y)$ and ${\boldsymbol x}=P_{\boldsymbol C}{\boldsymbol y}$. Then yields ${\boldsymbol y}=P_{\boldsymbol D}P_{\boldsymbol C}{\boldsymbol y}$ and therefore ${\boldsymbol x}=P_{\boldsymbol C}
(P_{\boldsymbol D}P_{\boldsymbol C}{\boldsymbol y})
=P_{\boldsymbol C}P_{\boldsymbol D}{\boldsymbol x}$. Hence ${\boldsymbol x}\in\operatorname{Argmin}\Phi$ and thus $\operatorname{Argmin}\Phi\neq{\ensuremath{{\varnothing}}}$. On the other hand, is a special case of and the gradient of $\Phi$ is the continuous linear operator $$\nabla\Phi\colon{\boldsymbol y}\mapsto\left(y_i-{\displaystyle{\frac{1}{m-1}}}
\sum_{j\in I\smallsetminus\{i\}}y_j\right)_{i\in I}$$ with norm $m/(m-1)=1/\beta$. Note that, since $m>2$, $2\beta>1=\gamma$. Moreover, $\delta=\min\{1,\beta/\gamma\}+1/2>1$. Thus, upon setting, for every $n\in{\ensuremath{\mathbb N}}$, $\lambda_n\equiv 1\in\left]0,\delta\right[$ in , we obtain and observe that $\sum_{n\in{\ensuremath{\mathbb N}}}\lambda_n(\delta-\lambda_n)={\ensuremath{{+\infty}}}$. Altogether, the convergence result follows from Theorem \[t:2011-02-04\]. Finally, set $\overline{\boldsymbol y}=(\overline{y}_1,\ldots,\overline{y}_m)$ and $\overline{\boldsymbol z}=P_{\boldsymbol D}
\overline{\boldsymbol y}$. Then yields $$(\overline{y},\ldots,\overline{y})=
\overline{\boldsymbol z}=P_{\boldsymbol D}
\overline{\boldsymbol y}=P_{\boldsymbol D}(P_{\boldsymbol C}
P_{\boldsymbol D}\overline{\boldsymbol y})=
P_{\boldsymbol D}P_{\boldsymbol C}\overline{\boldsymbol z}$$ and hence $\overline{y}\in\operatorname{Argmin}\varphi$.
Alternative projection schemes can be derived from Theorem \[t:2011-02-04\]. For instance, Corollary \[c:2011-02-04\] remains valid if is replaced by $$\label{e:2011-02-06b}
(\forall n\in{\ensuremath{\mathbb N}})(\forall i\in I)\quad x_{i,n+1}=
P_i\left({\displaystyle{\frac{1}{m}}}\sum_{j\in I}x_{j,n}\right),$$ which amounts to taking $\gamma=\beta$ instead of $\gamma=1$ in the above proof. We then recover a process investigated in [@Baus94; @Sign94; @Pier85].
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abstract: 'We show that every $K_4$-free graph $G$ with $n$ vertices can be made bipartite by deleting at most $n^2/9$ edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size $n/3$. This proves an old conjecture of P. Erdős.'
author:
- 'Benny Sudakov [^1]'
title: 'Making a $K_4$-free graph bipartite'
---
Introduction
============
The well-known Max Cut problem asks for the largest bipartite subgraph of a graph $G$. This problem has been the subject of extensive research, both from the algorithmic perspective in computer science and the extremal perspective in combinatorics. Let $n$ be the number of vertices and $e$ be the number edges of $G$ and let $b(G)$ denote the size of the largest bipartite subgraph of $G$. The extremal part of Max Cut problem asks to estimate $b(G)$ as a function of $n$ and $e$. This question was first raised almost forty years ago by P. Erdős [@E1] and attracted a lot of attention since then (see, e.g., [@AKS; @ABKS; @BS; @A; @Sh; @EGS; @EFPS; @BL; @Ed]).
It is well known that every graph $G$ with $e$ edges can be made bipartite by deleting at most $e/2$ edges, i.e., $b(G) \geq e/2$. To see this just consider a random partition of vertices of $G$ into two parts $V_1, V_2$ and estimate the expected number of edges in the cut $(V_1, V_2)$. A complete graph $K_n$ on $n$ vertices shows that the constant $1/2$ in the above bound is asymptotically tight. Moreover, this constant can not be improved even if we consider restricted families of graphs, e.g., graphs that contain no copy of a fixed [*forbidden*]{} subgraph $H$. We call such graphs $H$-free. Indeed, using sparse random graphs one can easily construct a graph $G$ with $e$ edges such that it has no short cycles but can not be made bipartite by deleting less than $e/2-o(e)$ edges. Such $G$ is clearly $H$-free for every forbidden graph $H$ which is not a forest. It is a natural question to estimate the error term $b(G)-e/2$ as $G$ ranges over all $H$-free graph with $e$ edges. We refer interested reader to [@AKS; @ABKS; @A; @Sh], where such results were obtained for various forbidden subgraphs $H$.
In this paper we restrict our attention to [*dense*]{} ($e=\Omega(n^2)$) $H$-free graphs for which it is possible to prove stronger bounds for Max Cut. According to a long-standing conjecture of Erdős [@E2], every triangle-free graph on $n$ vertices can be made bipartite by deleting at most $n^2/25$ edges. This bound, if true, is best possible (consider an appropriate blow-up of a $5$-cycle). Erdős, Faudree, Pach and Spencer proved that for triangle-free $G$ of order $n$ it is enough to delete $(1/18-\epsilon)n^2$ edges to make it bipartite. They also verify the conjecture for all graphs with at least $n^2/5$ edges. Some extensions of their results were further obtained in [@EGS]. Nevertheless this intriguing problem remains open. Erdős also asked similar question for $K_4$-free graphs. His old conjecture (see e.g., [@EFPS]) asserts that it is enough to delete at most $(1+o(1))n^2/9$ edges to make bipartite any $K_4$-free graph on $n$ vertices. Here we confirm this in the following strong form.
\[main\] Every $K_4$-free graph $G$ with $n$ vertices can be made bipartite by deleting at most $n^2/9$ edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size $n/3$.
This result can be used to prove the following asymptotic generalization.
\[easy\] Let $H$ be a fixed graph with chromatic number $\chi(H)=4$. If $G$ is a graph on $n$ vertices not containing $H$ as a subgraph, then we can delete at most $(1+o(1))n^2/9$ edges from $G$ to make it bipartite.
Another old problem of Erdős, that is similar in spirit, is to determine the best local density in $K_r$-free graphs for $r\geq 3$ (for more information see, e.g., [@KS1; @KS2] and their references). One of Erdős’ favorite conjectures was that any triangle-free graph $G$ on $n$ vertices should contain a set of $n/2$ vertices that spans at most $n^2/50$ edges. The blow-up of a $5$-cycle in which we replace each vertex by an independent set of size $n/5$ and each edge by a complete bipartite graph shows that this estimate can not be improved. On the other hand, for $r>3$ Chung and Graham [@CG] conjectured that Turán graph has the best local density for subsets of size $n/2$. In particular, their conjecture implies that every $K_4$-free graph on $n$ vertices should contain a set of $n/2$ vertices that spans at most $n^2/18$ edges.
Krivelevich [@K] noticed that for regular graphs a bound in the local density problem implies a bound for the problem of making the graph bipartite. Indeed, suppose $n$ is even, $G$ is a $d$-regular $K_4$-free graph on $n$ vertices and $S$ is a set of $n/2$ vertices. Then $dn/2 = \sum_{s \in S} d(s) = 2e(S) + e(S,\bar{S})$ and $dn/2 = \sum_{s \notin S} d(s) = 2e(\bar{S}) +
e(S,\bar{S})$, i.e. $e(S) = e(\bar{S})$. Deleting the $2e(S)$ edges within $S$ or $\bar{S}$ makes the graph bipartite, so if we could find $S$ spanning at most $n^2/18$ edges we would delete at most $n^2/9$ in making $G$ bipartite. Unfortunately, the converse reasoning does not work. Nevertheless, we believe that the result of Theorem \[main\] provides some supporting evidence for conjecture of Chung and Graham.
The rest of this short paper is organized as follows. The proof of our main theorem appears in the beginning of next section. Next we show how to obtain Corollary \[easy\] using this theorem together with well known Szemerédi’s Regularity Lemma [@S] (see also [@KS]). The last section of the paper contains some concluding remarks and open questions.
[**Notation.**]{} We usually write $G=(V,E)$ for a graph $G$ with vertex set $V=V(G)$ and edge set $E=E(G)$, setting $n=|V|$ and $e=e(G)=|E(G)|$. If $X \subset V$ is a subset of the vertex set then $G[X]$ denotes the restriction of $G$ to $X$, i.e. the graph on $X$ whose edges are those edges of $G$ with both endpoints in $X$. We will write $e(X)=e(G[X])$ and similarly, we write $e(X,Y)$ for the number of edges with one endpoint in $X$ and the other in $Y$. $N(v)$ is the set of vertices adjacent to a vertex $v$ and $d(v)=|N(v)|$ is the degree of $v$. For any two vertices $u,v$ we denote by $N(u,v)$ the set of common neighbors of $u$ and $v$, i.e., all the vertices adjacent to both of them. We will also write $d(u,v)=|N(u,v)|$. Finally if three vertices $u,v,$ and $w$ are all adjacent then they form a triangle in $G$ and we denote this by $\t=\{u,v,w\}$.
Proofs
======
Main result
-----------
In this subsection we present the proof of our main theorem. We start with the following well known fact (see, e.g., [@A]), whose short proof we include here for the sake of completeness.
\[4-partite\] Let $G$ be a $4$-partite graph with $e$ edges. Then $G$ contains a bipartite subgraph with at least $2e/3$ edges.
[**Proof.**]{} Let $V_1, \ldots, V_4$ be a partition of vertices of $G$ into four independent sets. Partition these sets randomly into two classes, where each class contains exactly two of the sets $V_i$. Consider a bipartite subgraph $H$ of $G$ with these color classes. For each fixed edge $(u,v)$ of $G$ the probability that $u$ and $v$ will lie in the different classes is precisely $(2\cdot
2)/{4 \choose 2}=2/3$. Therefore, by linearity of expectation, the expected number of edges in $H$ is $2e/3$, completing the proof. $\Box$
Next we need two simple lemmas.
\[codegree\] Let $G$ be a graph with $e$ edges and $m$ triangles. Then it contains a triangle $\{u,v,w\}$ such that $$d(u,v)+d(u,w)+d(v,w) \geq \frac{9m}{e}.$$
[**Proof.**]{} A simple averaging argument, using that $\sum_{(x,y)\in E(G)} d(x,y)=3m$ and Cauchy-Schwartz inequality, shows that there is a triangle $\{u,v,w\}$ in $G$ with $$\begin{aligned}
\hspace{0.4cm}
d(u,v)+d(u,w)+d(v,w) &\geq& \frac{1}{m}
\sum_{\{x,y,z\}=\t}\Big(d(x,y)+d(x,z)+d(y,z)\Big)=
\frac{1}{m}\sum_{(x,y)\in E(G)} d^2(x,y) \\
&\geq&
\frac{e}{m} \left(\frac{\sum_{(x,y)\in E(G)} d(x,y)}{e}\right)^2
=\frac{(3m)^2}{m e}=\frac{9m}{e}. \hspace{3.4cm} \Box\end{aligned}$$
\[maxcut-1\] Let $G$ be a graph on $n$ vertices with $e$ edges and $m$ triangles. Then $G$ contains a bipartite subgraph of size at least $4e^2/n^2-6m/n$.
[**Proof.**]{} Let $v$ be a vertex of $G$ and let $e_v$ denotes the number of edges spanned by the neighborhood $N(v)$. Consider the bipartite subgraph of $G$ whose parts are $N(v)$ and its complement $V(G)\setminus N(v)$. It is easy to see that number of edges in this subgraph is $\sum_{u \in N(v)} d(u)-2e_v$. Thus averaging over all vertices $v$ we have that $$\begin{aligned}
\label{A}
b(G) &\geq& \frac{1}{n}\sum_v\bigg(\sum_{u \in N(v)}d(u)-2e_v\bigg)
= \frac{1}{n}\sum_v d^2(v)-\frac{2}{n}\sum_v e_v\\
&\geq& \left(\frac{\sum_v d(v)}{n}\right)^2-6m/n=4e^2/n^2-6m/n \nonumber.\end{aligned}$$ Here we used Cauchy-Schwartz inequality together with identities $\sum_v e_v=3m$, $\sum_v
d(v)=2e$. $\Box$
Now we can obtain our first estimate on the Max Cut in $K_4$-free graphs. This result can be used to prove the conjecture for graphs with $\leq n^2/4$ edges.
\[maxcut-2\] Let $G$ be a $K_4$-free graph on $n$ vertices with $e$ edges. Then it contains a bipartite subgraph of size at least $2e/7+8e^2/(7n^2)$.
[**Proof.**]{} Let $v$ be a vertex of $G$ and denote by $e_v$ the number of edges spanned by the neighborhood of $v$. Consider a subgraph of $G$ induced by the set $N(v)$. This subgraph $G[N(v)]$ has $d(v)$ vertices, $e_v$ edges and contains no triangles, since $G$ is $K_4$-free. Therefore by previous lemma (with $m=0$) it has a bipartite subgraph $H$ of size at least $4e_v^2/d^2(v)$. Let $(A,B), A \cup B=N(v)$ be the bipartition of $H$. Consider a bipartite subgraph $H'$ of $G$ with parts $(A',B')$, where $A \subset A'$, $B \subset B'$ and we place each vertex $v \in V(G) \setminus N(v)$ in $A'$ or $B'$ randomly and independently with probability $1/2$. All edges of $H$ are edges of $H'$, and each edge incident to a vertex in $V(G) \setminus N(v)$ appears in $H'$ with probability $1/2$. As the number of edges incident to vertices $V(G)\setminus N(v)$ is $e-e_v$, by linearity of expectation, we have $b(G) \geq \mathbb{E}\big[e(H')\big] \geq (e-e_v)/2+4e_v^2/d^2(v)$. By averaging over all vertices $v$ $$\label{B}
b(G) \geq \frac{1}{2}e+\frac{1}{n}\sum_v\Big(4e_v^2/d^2(v)-e_v/2\Big).$$
To finish the proof we take a convex combination of inequalities (\[A\]) and (\[B\]) with coefficients $3/7$ and $4/7$ respectively. This gives
$$\begin{aligned}
b(G) &\geq& \frac{3}{7}\left(\frac{1}{n}\sum_v d^2(v)-\frac{2}{n}\sum_v e_v\right) +
\frac{4}{7} \left(\frac{1}{2}e+\frac{1}{n}\sum_v\Big(4e_v^2/d^2(v)-e_v/2\Big)\right)\\
&=&\frac{2}{7}e+\frac{1}{7n}\sum_v\Big(3d^2(v)-8e_v +16e_v^2/d^2(v)\Big)\\
&=&\frac{2}{7}e+
\frac{1}{7n}\sum_vd^2(v)\Big(3-8\big(e_v/d^2(v)\big)+16\big(e_v/d^2(v)\big)^2\Big)\\
&\geq& \frac{2}{7}e +\frac{2}{7n}\sum_vd^2(v)
\geq \frac{2}{7}e+\frac{2}{7} \left(\frac{\sum_vd(v)}{n}\right)^2=\frac{2}{7}e
+\frac{8}{7}e^2/n^2,\end{aligned}$$
where we used that $3-8t+16t^2=(4t-1)^2+2\geq 2$ for all $t$, $\sum_vd(v)=2e$ and Cauchy-Schwartz inequality. $\Box$
[**Remark.**]{} The above result is enough for our purposes, but one can get a slightly better inequality by taking a convex combination of (\[A\]) and (\[B\]) with coefficients $1/(1+a)$ and $a/(1+a)$ with $a=1.38$.
\[technical\] Let $f(t)=t/18+\frac{2}{9}\big(5/2-t-1/t\big)^2$. Then $f(t)\leq 1/9$ for all $t \in [3/2, 2]$ and equality holds only when $t=2$.
[**Proof.**]{} Note that $f(2)=1/9$ and $$f(t)-1/9=\frac{4t^4-19t^3+31t^2-20t+4}{18t^2}=
\frac{(t-2)(4t^3-11t^2+9t-2)}{18t^2}.$$ Consider $g(t)=4t^3-11t^2+9t-2$ in the interval $[3/2, 2]$. The derivative of this function $g'(t)=12t^2-22t+9$ is zero when $t=\frac{22\pm\sqrt{52}}{24}$, so the largest root of $g'(t)$ is less than $3/2$. Therefore $g(t)$ is strictly increasing function for $t \geq 3/2$ and so $g(t)>g(3/2)=1/4>0$ for all $t\in [3/2, 2]$. Since $18t^2>0$ and $t-2$ is negative for $t<2$ we conclude that $f(t)-1/9<0$ for all $t \in [3/2, 2)$. $\Box$
Having finished all the necessary preparations we are now in a position to complete the proof of our main result.
[**Proof of Theorem \[main\].**]{} It is easy to see that complete 3-partite graph with parts of size $n/3$ has $(n/3)^3=n^3/27$ triangles and that every edge of this graph is contained in exactly $n/3$ of them. To make this graph bipartite we need to destroy all these triangles. Since deletion of one edge can destroy at most $n/3$ of them, altogether we need to delete at least $\frac{n^3/27}{n/3}=n^2/9$ edges. To finish the proof it remains to show that deletion of $\leq n^2/9$ edges is sufficient to make every $K_4$-free graph bipartite.
Let $G$ be a $K_4$-free graph on $n$ vertices with $e$ edges. Turán’s theorem [@T] says that $e \leq n^2/3$, with equality only when $G$ is a complete 3-partite graph with parts of size $n/3$. By Lemma \[maxcut-2\], we need to delete at most $e-b(G) \leq 5e/7-8e^2/(7n^2)=\big(\frac{5}{7}(e/n^2)-\frac{8}{7}(e/n^2)^2\big)n^2$ edges to make $G$ bipartite. The function $g(t)=5t/7-8t^2/7$ is increasing in the interval $t \leq 1/4$ and so $g(t) \leq g(1/4)=3/28$. Therefore if $e\leq n^2/4$ we can delete at most $3n^2/28<n^2/9$ edges to make $G$ bipartite.
Next, consider the case when $n^2/4\leq e \leq n^2/3$ and let $m$ be the number of triangles in $G$. By Lemma \[maxcut-1\], we can delete at most $e-b(G) \leq e-\big(4e^2/n^2-6m/n\big)$ edges to make $G$ bipartite. So we can assume that $e-4e^2/n^2+6m/n \geq n^2/9$ or we are done. Then the number of triangles in $G$ satisfies $m\geq \frac{n}{6}\big(n^2/9+4e^2/n^2-e\big)$ and Lemma \[codegree\] implies that $G$ contains a triangle $\t=\{u,v,w\}$ with $$d(u,v)+d(u,w)+d(v,w) \geq \frac{9m}{e}\geq 6e/n + n^3/(6e) -3n/2.$$ Let $V_1=N(u,v), V_2=N(u,w)$, $V_3=N(v,w)$ and let $X=V(G)\setminus(\cup_{i=1}^3 V_i)$. Since $G$ is $K_4$-free and $(u,v), (u,w),
(v,w)$ are edges of $G$ we have that sets $V_i, 1\leq i\leq 3$ are independent and disjoint. Consider a 4-partite subgraph $G'$ of $G$ with parts $V_1, V_2, V_3$ and $X$. This graph has $e(G')=e-e(X)$ edges where $e(X)$ is the number of edges spanned by $X$. By Turán’s theorem $e(X) \leq |X|^2/3$ and we also know that $$|X|=n-\sum_i |V_i|=n-\big(d(u,v)+d(u,w)+d(v,w)\big) \leq
5n/2-6e/n-n^3/(6e).$$
Since $G'$ is 4-partite we can now use Lemma \[4-partite\] to deduce that $b(G)\geq b(G') \geq 2e(G')/3 =\frac{2}{3}\big(e-e(X)\big).$ Therefore the number of edges we need to delete to make $G$ bipartite is bounded by $$\begin{aligned}
e-b(G) &\leq& e-2\big(e-e(X)\big)/3=e/3+2e(X)/3 \leq e/3+2|X|^2/9\\
&\leq& e/3+\frac{2}{9} \Big(5n/2-6e/n-n^3/(6e)\Big)^2\\
&=&\left(\frac{1}{18}(6e/n^2)+\frac{2}{9}
\Big(5/2-6e/n^2-(6e/n^2)^{-1}\Big)^2\right)n^2\\
&=&f\big(6e/n^2\big) \cdot n^2,\end{aligned}$$ where $f(t)=t/18+\frac{2}{9}\big(5/2-t-1/t\big)^2$. As $n^2/4 \leq e \leq n^2/3$ we have that $3/2 \leq t=6e/n^2\leq 2$. Then, by Lemma \[technical\], $f(6e/n^2) \leq 1/9$ with equality only if $e=n^2/3$. This shows that we can delete at most $n^2/9$ edges to make $G$ bipartite and we need to delete that many edges only when $e(G)=n^2/3$, i.e., $G$ is a complete 3-partite graph with parts of size $n/3$. . $\Box$
Forbidding fixed $4$-chromatic subgraph
---------------------------------------
In this short subsection we show how to use Theorem \[main\] to deduce a similar statement about graphs with any fixed forbidden $4$-chromatic subgraph. The proof is a standard application of Szemerédi’s Regularity Lemma and we refer the interested reader to the excellent survey of Komlós and Simonovits [@KS], which discusses various results proved by this powerful tool.
We start with a few definitions, most of which follow [@KS]. Let $G=(V,E)$ be a graph, and let $A$ and $B$ be two disjoint subsets of $V(G)$. If $A$ and $B$ are non-empty, define the [*density of edges*]{} between $A$ and $B$ by $ d(A,B) = \frac{e(A,B)}{|A||B|}$. For $\epsilon>0$ the pair $(A,B)$ is called [*$\epsilon$-regular*]{} if for every $X \subset A$ and $Y \subset B$ satisfying $|X|>\epsilon |A|$ and $ |Y|>\epsilon |B|$ we have $|d(X,Y)-d(A,B)| < \epsilon$. An [*equitable partition*]{} of a set $V$ is a partition of $V$ into pairwise disjoint classes $V_1,\cdots,V_k$ of almost equal size, i.e., $\big| |V_i|-|V_j| \big| \leq 1$ for all $i,j$. An equitable partition of the set of vertices $V$ of $G$ into the classes $V_1,\cdots,V_k$ is called [*$\epsilon$-regular*]{} if $|V_i| \leq \epsilon |V|$ for every $i$ and all but at most $\epsilon k^2$ of the pairs $(V_i,V_j)$ are $\epsilon$-regular. The above partition is called [*totally $\epsilon$-regular*]{} if all the pairs $(V_i,V_j)$ are $\epsilon$-regular. The following celebrated lemma was proved by Szemerédi in [@S].
\[szemeredi\] For every $\epsilon>0$ there is an integer $M(\epsilon)$ such that every graph of order $n>M(\epsilon)$ has an $\epsilon$-regular partition into $k$ classes, where $k \leq M(\epsilon)$.
In order to apply the Regularity Lemma we need to show the existence of a complete multipartite subgraph in graphs with a totally $\epsilon$-regular partition. This is established in the following lemma which is a special case of a well-known result, see, e.g., [@KS].
\[key\] For every $\delta>0$ and integer $t$ there exist an $0<\epsilon=
\epsilon(\delta,t)$ and $n_0=n_0(\delta,t)$ with the following property. If $G$ is a graph of order $n>n_0$ and $(V_1,\cdots,V_4)$ is a totally $\epsilon$-regular partition of vertices of $G$ such that $d(V_{i},V_{j})\geq \delta$ for all $i<j$, then $G$ contains a complete $4$-partite subgraph $K_4(t)$ with parts of size $t$.
[**Proof of Corollary \[easy\].**]{} Let $H$ be a fixed $4$-chromatic graph of order $t$ and let $G$ be a graph on $n$ vertices not containing $H$ as a subgraph. Suppose $\delta>0$ and let $\epsilon=\min\big(\delta, \epsilon(\delta,t)\big)$, where $\epsilon(\delta,t)$ is defined in the previous statement. Then, by Lemma \[szemeredi\], for sufficiently large $n$ there exists an $\epsilon$-regular partition $(V_1,\cdots,V_k)$ of vertices of $G$.
Consider a new graph $G'$ on the vertices $\{1, \ldots, k\}$ in which $(i,j)$ is an edge iff $(V_i,V_j)$ is an $\epsilon$-regular pair with density at least $\delta$. We claim that $G'$ contains no $K_4$. Indeed, any such clique in $G'$ corresponds to $4$ parts in the partition of $G$ such that any pair of them is $\epsilon$-regular and has density at least $\delta$. This contradicts our assumption on $G$, since by Lemma \[key\], the union of these parts will contain a copy of complete $4$-partite graph $K_4(t)$ which clearly contains $H$.
By applying Theorem \[main\] to graph $G'$, we deduce that there is a set $D$ of at most $k^2/9$ edges of $G'$ whose deletion makes it bipartite. Now delete all the edges of $G$ between the pairs $(V_i,V_j)$ with $(i,j) \in D$. Delete also the edges of $G$ that lie within classes of the partition, or that belong to a non-regular pair, or that join a pair of classes of density less than $\delta$. It is easy to see that the remaining graph is bipartite and the number of edges we deleted is at most $$\hspace{3.2cm}
(k^2/9)(n/k)^2+\epsilon n^2+\delta n^2 \leq \big(1/9+2\delta\big) n^2=(1+o(1))n^2/9.
\hspace{3.2cm}
\Box$$
Concluding remarks
==================
How many edges do we need to delete to make a $K_r$-free graph $G$ of order $n$ bipartite? For $r=3, 4$ this was asked long time ago by P. Erdős. For triangle-free graphs he conjectured that deletion of $n^2/25$ edges is always enough and that extremal example is a blow-up of a $5$-cycle. In this paper we answered the question for $r=4$ and proved that the unique extremal construction in this case is a complete $3$-partite graph with equal parts. Our result suggests that a complete $(r-1)$-partite graph of order $n$ with equal parts is worst example also for all remaining values of $r$. Therefore we believe that it is enough to delete at most $\frac{(r-2)^2}{4(r-1)^2}n^2$ edges for even $r\geq 5$ and at most $\frac{r-3}{4(r-1)}n^2$ edges for odd $r\geq 5$ to make bipartite any $K_r$-free graph $G$ of order $n$. It seems that some of the ideas presented here can be useful to make a progress on this problem for even $r$.
[**Acknowledgment.**]{} I would like to thank József Balogh and Peter Keevash for interesting discussions on the early stages of this project.
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[^1]: Department of Mathematics, Princeton University, Princeton, NJ 08544. E-mail: bsudakov@math.princeton.edu. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.
|
**Gravity with de Sitter and Unitary Tangent Groups**
**Ali H. Chamseddine$^{1,3}$ , Viatcheslav Mukhanov$^{2,4}$ **
[**Abstract**]{}
Einstein Gravity can be formulated as a gauge theory with the tangent space respecting the Lorentz symmetry. In this paper we show that the dimension of the tangent space can be larger than the dimension of the manifold and by requiring the invariance of the theory with respect to 5d Lorentz group (de Sitter group) Einstein theory is reproduced unambiguously. The other possibility is to have unitary symmetry on a complex tangent space of the same dimension as the manifold. In this case the resultant theory is Einstein-Strauss Hermitian gravity. The tangent group is important for matter couplings. We show that in the de Sitter case the 4 dimensional space time vector and scalar are naturally unified by a hidden symmetry being components of a 5d vector in the tangent space. With a de Sitter tangent group spinors can exist only when they are made complex or taken in doublets in a way similar to N=2 supersymmetry.
Introduction
============
The experimental evidence that Lorentz symmetry is preserved for effective four-dimensional theories is overwhelming. In curved space-time this Lorentz symmetry is realized as a local symmetry of the tangent manifold [utiyama]{} [@Kibble]. Moreover, to incorporate spinors in general relativity, we are forced to consider this local symmetry because there are no spinor representations of the diffeomorphism group. Usually the dimension of the tangent space is taken to be equal to the dimension of the curved manifold and then the Lorentz symmetry is simply a manifestation of the equivalence principle, which is valid in torsion-free theories. General relativity could then be formulated as a gauge theory of the Lorentz group where the gauge fields are the spin-connection. In reality one can search for all possible tangent groups in $d$-dimensional space-time [@Wein]. In this paper we will investigate whether it is possible to have a larger group of symmetry in the tangent space and still unambiguously reproduce general relativity. We will show in section 2, that this is indeed possible by taking the tangent space to be real with de Sitter group symmetry. The de Sitter gauge invariant action which is linear in curvature is shown to be identical to Einstein gravity, provided that metricity condition is imposed on the spin and affine connections. In section 3 we consider matter interactions of gravity with the de Sitter group as the tangent group. We then, in section 4, consider a complex tangent space and show that the relevant symmetry in this case is the unitary symmetry. The resultant theory is the Einstein-Strauss theory. Section 5 is the conclusion. An appendix treats the special limit of Poincare symmetry, and examines the relation of our new formalism in three dimensions with Witten’s formulation of Chern-Simons gravity.
Gravity with de Sitter tangent group
====================================
Let us begin with a $d$-dimensional manifold and assume that at every point of this manifold there is a real $N$-dimensional tangent space spanned by linearly independent vectors $\mathbf{v}_{A}$, where $A=1,2...N.$ Assuming that $d\leq N$, the coordinate basis vectors $\mathbf{e}_{\alpha }\equiv
\partial /\partial x^{\alpha },$ where $\alpha =1,2...d,$ span $d$-dimensional space. Next we define the scalar product in the tangent space and take the vectors $\mathbf{v}_{A}$ to be orthonormal[^1] $$\mathbf{v}_{A}\cdot \mathbf{v}_{B}=\eta _{AB}. \label{1}$$where $\eta _{AB}$ is Minkowski matrix. The Lorentz transformations $$\mathbf{\tilde{v}}_{A}=\Lambda _{A}^{\hspace{0.05in}B}\mathbf{v}_{B},\text{
\ \ \ \ \ \ }\Lambda _{A}^{\hspace{0.05in}C}\eta _{CD}\Lambda _{A}^{\hspace{0.05in}D}=\eta _{AB}\text{\ } \label{1a}$$preserve the orthogonality of the vielbein, $\mathbf{\tilde{v}}_{A}\cdot
\mathbf{\tilde{v}}_{B}=\eta _{AB}.$ The scalar product of coordinate basis vectors then induces the metric in $d$-dimensional manifold$$\mathbf{e}_{\alpha }\cdot \mathbf{e}_{\beta }=g_{\alpha \beta }(x^{\gamma }).
\label{2}$$Expanding $\mathbf{e}_{\alpha }$ in $\mathbf{v}_{A}$-basis$$\mathbf{e}_{\alpha }=e_{\alpha }^{B}\mathbf{v}_{B}, \label{3}$$and substituting in (\[2\]) we obtain the following expression for the metric $g_{\alpha \beta }$ $$g_{\alpha \beta }=e_{\alpha }^{A}e_{\beta }^{B}\eta _{AB}, \label{4}$$in terms of components. Tangent space indices are raised and lowered with the Minkowski metric, thus$$e_{A\alpha }=\eta _{AB}e_{\alpha }^{B}=\left( \mathbf{v}_{A}\cdot \mathbf{e}_{\alpha }\right) , \label{4a}$$and $\eta ^{AB}$ is inverse to Minkowski matrix $\eta _{AB}.$ Next we consider parallel transport on the manifold relating vectors in nearby tangent spaces. The affine and spin connections determining the rules for parallel transport of the coordinate basis vectors and vielbein are defined via$$\mathbf{\nabla }_{\mathbf{e}_{\beta }}\mathbf{e}_{\alpha }\equiv \mathbf{\nabla }_{\beta }\mathbf{e}_{\alpha }=\Gamma _{\alpha \beta }^{\nu }\mathbf{e}_{\nu },\ \ \mathbf{\nabla }_{\beta }\mathbf{v}_{A}=-\omega _{\beta
A}^{\quad \hspace{0.03in}B}\mathbf{v}_{B}, \label{5a}$$where $\mathbf{\nabla }_{\beta }$ is the derivative defining the rate of change of vectors along a basis vector $\mathbf{e}_{\beta }$. When applied to a scalar function $f$ this derivative acts as a partial derivative with respect to the appropriate coordinates, that is, $\mathbf{\nabla }_{\beta
}f=\partial f/\partial x^{\beta }$. Notice that $\eta _{AB}$ and $g_{\alpha
\beta }$ as defined in (\[1\]) and (\[2\]) are the sets of scalar functions and, hence, $\mathbf{\nabla }_{\beta }\eta _{AB}=0,$ $\mathbf{\nabla }_{\gamma }g_{\alpha \beta }=\partial g_{\alpha \beta }/\partial
x^{\gamma }\equiv \partial _{\gamma }g_{\alpha \beta }$.
Given $\eta _{AB},$ $g_{\alpha \beta }$ and $e_{\alpha }^{A}$ we derive the consistency (metricity) conditions for the connections by taking derivative of equations (\[1\]), (\[2\]) and (\[4a\]). In particular, we obtain$$\left( \mathbf{\nabla }_{\alpha }\mathbf{v}_{A}\right) \cdot \mathbf{v}_{B}+\mathbf{v}_{A}\cdot \left( \mathbf{\nabla }_{\alpha }\mathbf{v}_{B}\right)
=-\omega _{\alpha AB}^{\quad }-\omega _{\alpha BA}^{\quad }=\mathbf{\nabla }_{\alpha }\eta _{AB}=0, \label{6}$$that is, the spin connection should be antisymmetric with respect to tangent space indices, $\omega _{\alpha AB}^{\quad }=-\omega _{\alpha BA}^{\quad }.$ Applying the derivative $\mathbf{\nabla }_{\gamma }$ to (\[2\]) gives$$\Gamma _{\alpha \gamma }^{\nu }g_{\nu \beta }+\Gamma _{\beta \gamma }^{\nu
}g_{\alpha \nu }=\partial _{\gamma }g_{\alpha \beta } \label{7}$$Assuming that torsion is absent, $\Gamma _{\alpha \beta }^{\nu }=\Gamma
_{\beta \alpha }^{\nu }$, these equations are solved unambiguously, giving the well known result$$\Gamma _{\alpha \beta }^{\gamma }=\frac{1}{2}g^{\gamma \sigma }\left(
g_{\alpha \sigma ,\beta }+g_{\sigma \beta ,\alpha }-g_{\alpha \beta ,\sigma
}\right) , \label{8}$$where $g^{\gamma \sigma }$ is inverse to $g_{\alpha \beta },$ that is, $g^{\alpha \sigma }g_{\sigma \beta }=\delta _{\beta }^{\alpha }.$ We would like to stress that affine connections are determined unambiguously irrespective of the group of tangent space. Finally, from (\[4a\]) we obtain$$\partial _{\beta }e_{A\alpha }=-\omega _{\beta A}^{\quad \hspace{0.03in}B}e_{B\alpha }+\Gamma _{\alpha \beta }^{\nu }e_{A\nu }. \label{9}$$Let us find when these equations can unambiguously be solved for $\omega
_{\beta B}^{\quad \ A}$ in terms of the soldering form $e_{\alpha }^{B}$ and metric $g_{\alpha \beta }.$ The total number of components of $e_{\alpha
}^{B}$ is $Nd$. Given a metric $g_{\alpha \beta },$ whose derivatives determine $\Gamma _{\alpha \beta }^{\nu }$ via (\[8\]), and hence impose $\frac{1}{2}d^{2}\left( d+1\right) $ constraints on $\partial _{\beta
}e_{A\alpha }$, leaves us with $d\left( Nd-\frac{1}{2}d\left( d+1\right)
\right) $ *independent* equations (\[9\]) to determine $\frac{1}{2}dN\left( N-1\right) $ antisymmetric spin connections $\omega _{\beta
AB}^{\quad }.$ Note that for any $N$ and $d$ the number of equations can never exceed the number of independent $\omega _{\beta AB}^{\quad }$ to be determined, and hence for any dimension of tangent space the system of equations is not overdetermined. However, the spin connection is unambiguously determined only if the number of equations is equal to the number of its unknown components:$$d\left( Nd-\frac{1}{2}d\left( d+1\right) \right) =\frac{1}{2}dN\left(
N-1\right) .$$The only solutions of this equation are $N=d$ and $N=d+1.$ The first case is well known and thus we shall concentrate on the second case which corresponds to the larger symmetry group $SO(1,d)$ of the tangent space. In the case of a four-dimensional manifold the tangent space is five dimensional. The metric in 5d tangent space can then be taken either to be $\eta _{AB}=\mathrm{diag}\left( 1,-1,-1,-1,-1\right) $ or $\eta _{AB}=\mathrm{diag}\left( 1,1,-1,-1,-1\right) $. In the first case the gauge group is 5d Lorentz group $SO(1,4)$ which is also the group of symmetry of 4d de Sitter space (de Sitter group), while in the second case the group is $SO(2,3)$ (the group of symmetry of 4d anti de Sitter space). For definiteness and from here on, we consider these cases only. Note that although the consistency equations do not lead to any contradiction for an arbitrary dimension of tangent space the connections are entirely determined by the soldering form only if $N=d$ or $N=d+1.$ Otherwise the spin connection is not unambiguously determined by the fundamental soldering form and the theory is not well defined.
In order to construct gauge invariant Lagrangians we need to define$$e_{A}^{\alpha }=g^{\alpha \gamma }e_{\gamma A}=g^{\alpha \gamma }\eta
_{AB}e_{\gamma }^{B}. \label{10}$$Rewritten in terms of $e_{A}^{\alpha },$ equation (\[9\]) becomes $$\partial _{\beta }e_{A}^{\alpha }=-\omega _{\beta A}^{\quad \hspace{0.03in}B}e_{B}^{\alpha }-\Gamma _{\nu \beta }^{\alpha }e_{A}^{\nu }. \label{11}$$The soldering form $e_{A}^{\alpha }$ is inverse to $e_{\beta }^{B}$ only if the of dimension of the tangent space and the dimension of the manifold match. In case of a de Sitter tangent group contraction over tangent space indices gives$$e_{A}^{\alpha }e_{\beta }^{A}=g^{\alpha \gamma }\eta _{AB}e_{\gamma
}^{B}e_{\beta }^{A}=g^{\alpha \gamma }g_{\gamma \beta }=\delta _{\beta
}^{\alpha }, \label{12}$$however, contraction over space-time indices gives $$e_{A}^{\alpha }e_{\alpha }^{B}\neq \delta _{B}^{A}.$$To prove this, let us introduce the unit vector $\mathbf{n}$ orthogonal to all $\mathbf{e}_{\alpha },$ that is, $\mathbf{n}\cdot \mathbf{e}_{\alpha }=0$ and $\mathbf{n}\cdot \mathbf{n=}\varepsilon ,$ where $\varepsilon =-1$ or $+1 $ for de Sitter and anti de Sitter groups correspondingly. The vectors $\mathbf{n}$ and $\mathbf{e}_{\alpha }$ form a complete basis in tangent space and therefore $$\mathbf{v}_{A}=v_{A}^{\alpha }\mathbf{e}_{\alpha }+n_{A}\mathbf{n.}
\label{13}$$Taking into account (\[4a\]) we have$$v_{A}^{\alpha }=g^{\alpha \gamma }\left( \mathbf{v}_{A}\cdot \mathbf{e}_{\gamma }\right) =g^{\alpha \gamma }\eta _{AB}e_{\gamma }^{B}=e_{A}^{\alpha
}, \label{14}$$that is, the soldering form $e_{A}^{\alpha }$ coincides with the coefficient $v_{A}^{\alpha }$ in expansion (\[13\]). Taking this into account one gets $$\eta _{AB}=\mathbf{v}_{A}\cdot \mathbf{v}_{B}=v_{A}^{\alpha }v_{B}^{\beta
}g_{\alpha \beta }+\varepsilon n_{A}n_{B}=e_{A}^{\alpha }e_{\alpha
B}+\varepsilon n_{A}n_{B}, \label{15}$$or after rasing the tangent space index we obtain$$e_{A}^{\alpha }e_{\alpha }^{B}=\delta _{A}^{B}-\varepsilon n_{A}n^{B}\equiv
P_{B}^{A} \label{16}$$where $P_{B}^{A}$ is a projection operator: $P_{C}^{A}P_{B}^{C}=P_{B}^{A}.$
The components $n_{A}$ satisfy the following relations$$n^{A}e_{A}^{\alpha }=0,\text{ \ }n_{A}n^{A}=\varepsilon . \label{16a}$$To prove this let us note that it follows from (\[13\]) that $\mathbf{v}_{A}\cdot \mathbf{n}=\varepsilon n_{A}.$ Substituting here the expansion $$\mathbf{n}=\tilde{n}^{B}\mathbf{v}_{B}, \label{16b}$$we infer that $\tilde{n}^{B}=\varepsilon n^{B}$ and hence$$\mathbf{n}=\varepsilon n^{B}\mathbf{v}_{B}=\varepsilon \left(
n^{B}e_{B}^{\alpha }\mathbf{e}_{\alpha }+n^{B}n_{B}\mathbf{n}\right) ,
\label{16c}$$from which (\[16a\]) immediately follows.
In vielbein formalism the soldering form $e_{A}^{\alpha }$ is a fundamental quantity and the group of symmetry under which the theory is required to be invariant is the group of *local* Lorentz transformations (\[1a\]), where $\Lambda _{A}^{\quad B}=\Lambda _{A}^{\quad B}\left( x\right) .$ Under Lorentz transformation we have$$\mathbf{\tilde{v}}_{A}=\Lambda _{A}^{\hspace{0.05in}B}\mathbf{v}_{B}=\Lambda
_{A}^{\hspace{0.05in}B}\left( e_{B}^{\alpha }\mathbf{e}_{\alpha }+n_{B}\mathbf{n}\right) =\tilde{e}_{A}^{\alpha }\mathbf{e}_{\alpha }+\tilde{n}_{A}\mathbf{n,} \label{17}$$and hence$$e_{A}^{\alpha }\rightarrow \tilde{e}_{A}^{\alpha }=\Lambda _{A}^{\hspace{0.05in}B}e_{B}^{\alpha } \label{18}$$The transformation law for the spin connection follows from its definition:$$\tilde{\omega}_{\beta A}^{\quad \,B}\mathbf{\tilde{v}}_{B}=-\mathbf{\nabla }_{\beta }\mathbf{\tilde{v}}_{A}$$Substituting $\mathbf{\tilde{v}}_{B}=\Lambda _{A}^{\quad C}\mathbf{v}_{C}$ and taking into account (\[5a\]) we infer that $$\omega _{\mu A}^{\quad B}\rightarrow \tilde{\omega}_{\mu A}^{\quad \hspace{0.03in}B}=\left( \Lambda \omega _{\mu }\Lambda ^{-1}\right)
_{A}^{\,B}+\left( \Lambda \partial _{\mu }\Lambda ^{-1}\right) _{A}^{\,B},
\label{20}$$where $\Lambda $ and $\Lambda ^{-1}$ are the matrices corresponding to Lorentz transformation and its inverse. Up to this point, we have considered only vector representations of the Lorentz group. In general, $$\Lambda =\exp \left( \lambda ^{AB}J_{AB}\right)$$where $J_{AB}$ are corresponding generators of the Lie algebra which satisfy the commutation relations$$\left[ J_{AB},J_{CD}\right] =\frac{1}{2}\left( \eta _{BC}J_{AD}-\eta
_{AC}J_{BD}-\eta _{BD}J_{AC}+\eta _{AD}J_{BC}\right)$$Consider spinors $\psi $ which transforms according to $$\psi \rightarrow \exp \left( \frac{1}{4}\lambda ^{AB}\Gamma _{AB}\right)
\psi , \label{21}$$where$\ \Gamma _{AB}=\frac{1}{2}\left( \Gamma _{A}\Gamma _{B}-\Gamma
_{B}\Gamma _{A}\right) $ are generators of the Lie algebra in the spinor representation and $\Gamma _{A}$ are $d+1$ Dirac matrices satisfying$$\left\{ \Gamma ^{A},\Gamma ^{B}\right\} =2\eta ^{AB},\quad \Gamma ^{\dagger
A}=\Gamma ^{0}\Gamma ^{A}\Gamma ^{0}. \label{22}$$We note that the signature of $\eta ^{AB}$ does not play any significant role in the derivations that follow, and thus our results holds equally well for both de Sitter and anti de Sitter tangent groups. The Dirac action $$\dint d^{4}x\sqrt{g}\,\overline{\psi }i\Gamma ^{C}e_{C}^{\alpha }D_{\alpha
}\psi , \label{23}$$where $$D_{\alpha }\equiv \partial _{\alpha }+\frac{1}{4}\omega _{\alpha }^{\hspace{0.03in}AB}\Gamma _{AB}, \label{24}$$is invariant under gauge transformations (\[18\]), (\[20\]) and (\[21\]). This action is real, thanks to the metricity conditions (\[11\]).
Next one constructs the curvature of the connection $D_{\mu }$ defined by$$\left[ D_{\mu },D_{\nu }\right] =\frac{1}{4}R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\Gamma _{AB}, \label{25}$$where$$R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right)
=\partial _{\mu }\omega _{\nu }^{\,\,\,AB}-\partial _{\nu }\omega _{\mu
}^{\,\,\,AB}+\omega _{\mu }^{\,\,\,AC}\omega _{\nu C}^{\quad B}-\omega _{\nu
}^{\,\,\,AC}\omega _{\mu C}^{\quad B}. \label{26}$$This curvature$\ $transforms as $$\left( R_{\mu \nu }\right) _{A}^{\hspace{0.05in}B}\rightarrow \left( \Lambda
R\Lambda ^{\,-1}\right) _{A}^{\hspace{0.05in}B}, \label{27}$$and hence $$\,R\left( \omega \right) =\,e_{A}^{\mu }R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right) e_{B}^{\nu }, \label{28}$$is invariant under local gauge transformations. The gauge invariant action is then given by $$S=-\frac{1}{2\kappa ^{2}}\dint d^{4}x\sqrt{g}R\left( \omega \right)
\label{29}$$Although this action appears to depend on the non-diagonal $e_{A}^{\mu }$, it is a function of $g_{\mu \nu }$ only.
To prove this we first find how the tangent space covariant derivative acts on the components of a vector $\mathbf{l=}l^{C}\mathbf{v}_{C}.$ Using spinor representation for the vector we have$$D_{\nu }\left( l^{D}\Gamma _{D}\right) =\partial _{\nu }l^{D}\Gamma _{D}+\frac{1}{4}\omega _{\nu }^{\hspace{0.05in}BC}\left[ \Gamma _{BC},\Gamma _{D}\right] l^{D}. \label{30}$$Taking into account the commutation relation $\left[ \Gamma _{BC},\Gamma _{D}\right] =2\left( \eta _{CD}\Gamma _{B}-\eta _{BD}\Gamma _{C}\right) $ one gets$$D_{\nu }\left( l^{D}\Gamma _{D}\right) =\left( \partial _{\nu }l^{D}+\omega
_{\nu \hspace{0.05in}C}^{\hspace{0.05in}D}l^{C}\right) \Gamma _{D},
\label{31}$$and hence we deduce$$D_{\nu }l^{D}=\partial _{\nu }l^{D}+\omega _{\nu \hspace{0.05in}C}^{\hspace{0.05in}D}l^{C}. \label{32}$$In particular, it follows that $$D_{\nu }e^{\rho A}=\partial _{\nu }e^{\rho A}+\omega _{\nu \hspace{0.05in}B}^{\hspace{0.05in}A}e^{\rho B}, \label{33}$$which in turn implies that $$\left[ D_{\mu },D_{\nu }\right] e^{\rho A}=R_{\mu \nu }^{\hspace{0.03in}\hspace{0.05in}AB}\left( \omega \right) e_{B}^{\rho }. \label{34}$$On the other hand, using metricity condition (\[11\]), we have$$D_{\nu }e^{\rho A}=-\Gamma _{\nu \sigma }^{\rho }e^{\sigma A}, \label{35}$$and therefore $$\begin{aligned}
D_{\mu }\left( D_{\nu }e^{\rho A}\right) & =-D_{\mu }\left( \Gamma _{\nu
\sigma }^{\rho }e^{\sigma A}\right) =-\left( \partial _{\mu }\Gamma _{\nu
\sigma }^{\rho }\right) e^{\sigma A}-\Gamma _{\nu \sigma }^{\rho }\left(
D_{\mu }e^{\sigma A}\right) \notag \\
& =-\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }e^{\sigma A}+\Gamma _{\nu
\sigma }^{\rho }\Gamma _{\mu \kappa }^{\sigma }e^{\kappa A}. \label{36}\end{aligned}$$Taking the commutator one gets $$\begin{aligned}
\left[ D_{\mu },D_{\nu }\right] e^{\rho A}& =-\left( \partial _{\mu }\Gamma
_{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma
_{\mu \kappa }^{\rho }\Gamma _{\nu \sigma }^{\kappa }-\Gamma _{\nu \kappa
}^{\rho }\Gamma _{\mu \sigma }^{\kappa }\right) e^{\sigma A} \notag \\
& =-R_{\,\,\,\sigma \mu \nu }^{\rho }\left( \Gamma \right) e^{\sigma A}.
\label{37}\end{aligned}$$Comparing this result with (\[34\]) we arrive at the identity $$R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right)
e_{B}^{\rho }=-R_{\,\,\,\sigma \mu \nu }^{\rho }\left( \Gamma \right)
e^{\sigma A}, \label{38}$$which in turn leads to $$\begin{aligned}
\,R\left( \omega \right) & =e_{A}^{\mu }R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right) e_{B}^{\nu }=-R_{\,\,\,\sigma \mu \nu }^{\nu
}\left( \Gamma \right) e^{\sigma A}e_{A}^{\mu } \notag \\
& =R_{\,\,\,\sigma \nu \mu }^{\nu }\left( \Gamma \right) g^{\sigma \mu
}=R\left( \Gamma \right) . \label{39}\end{aligned}$$This completes the proof that the gauge invariant action (\[29\]) is equivalent to Einstein action and involves only those combinations of $e_{A}^{\mu }$ which reduce to the metric $g_{\mu \nu }$. The remaining $\frac{1}{2}d\left( d+1\right) $ independent combinations of $e_{A}^{\mu }$ components represent the $\frac{1}{2}d\left( d+1\right) $ gauge degrees of freedom associated with $SO(1,d)$. Thus, we conclude that it is possible to formulate Einstein gravity as a gauge invariant theory with the tangent group being de Sitter or anti de Sitter.
We would like to stress that in proving identity (\[39\]) we never (and could not) assume that the soldering form $e_{A}^{\mu }$ has an inverse and, moreover, this result is valid for an arbitrary dimension of tangent space. However, as it was noticed above the theory is well defined only if $N=d$ or $N=d+1.$ We could also consider a gauge invariant action involving higher order curvature invariants. One can show that even in this case the action depends only on the metric $g_{\mu \nu }.$ To give an example consider all possible terms which are of second order in curvature $$R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}R_{\rho \sigma }^{\hspace{0.05in}\hspace{0.05in}CD}\left( c_{1}\ e_{A}^{\mu }e_{B}^{\nu }e_{C}^{\rho
}e_{D}^{\sigma }+c_{2}\ e_{A}^{\mu }e_{C}^{\nu }e_{D}^{\rho }e_{B}^{\sigma
}+c_{3}\ e_{C}^{\mu }e_{D}^{\nu }e_{A}^{\rho }e_{B}^{\sigma }\right) ,$$because other terms are related to these three by symmetry. The first term is identical to $R^{2}\left( \Gamma \right) $, while for the second term we have $$R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right)
e_{A}^{\mu }e_{B}^{\sigma }R_{\rho \sigma }^{\hspace{0.05in}\hspace{0.05in}CD}\left( \omega \right) e_{C}^{\nu }e_{D}^{\rho }=g^{\mu \kappa
}R_{\,\,\,\kappa \mu \nu }^{\sigma }\left( \Gamma \right) g^{\nu \lambda
}R_{\,\,\,\lambda \rho \sigma }^{\rho }\left( \Gamma \right) .$$after using the identity (\[38\]) twice. Similarly, the third term gives $$R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega \right)
e_{A}^{\rho }e_{B}^{\sigma }R_{\rho \sigma }^{\hspace{0.05in}\hspace{0.05in}CD}\left( \omega \right) e_{C}^{\mu }e_{D}^{\nu }=g^{\kappa \rho
}R_{\,\,\,\kappa \mu \nu }^{\sigma }\left( \Gamma \right) g^{\mu \lambda
}R_{\,\,\,\lambda \rho \sigma }^{\nu }\left( \Gamma \right) ,$$which proves that the most general action which is second order in spin-connection curvature is identical to the one that depends on affine-connection curvature.
Matter couplings
================
We have seen that gravity is insensitive to the gauge group of the tangent space. In this section we will show that, to the contrary, matter feels the tangent space group. Let us consider the matter couplings in the case of de Sitter tangent group. In this case the fundamental spinors, vectors and tensors are defined as representations of the 5d Lorentz group of tangent space, and their Lagrangians must be invariant with respect to de Sitter symmetry. In vierbein formulation of gravity, we can exchange space-time tensors with Lorentz tensors. This is no longer valid for de Sitter tangent group because in this case the vielbein $e_{A}^{\mu }$ is not invertible and, for example, a vector in the tangent space is not equivalent to a space-time vector. In fact as we will show now the 5d de Sitter vector is equivalent to 4d space time vector and real space time scalar. Therefore, de Sitter tangent space unifies 4d vectors and scalars.
Let us consider a 5d vector $\mathbf{H,}$ which can be expanded in terms of components as (see (\[13\]), (\[14\])): $$\mathbf{H}=H^{A}\mathbf{v}_{A}=H^{A}e_{A}^{\alpha }\mathbf{e}_{\alpha
}+H^{A}n_{A}\mathbf{n=}H^{\alpha }\mathbf{e}_{\alpha }+\phi \mathbf{n,}
\label{40a}$$where $$H^{\alpha }=H^{A}e_{A}^{\alpha },\text{ \ }\phi =H^{A}n_{A}, \label{41a}$$are the components of a 4d vector and a scalar, respectively. Multiplying the first equation by $e_{\alpha }^{B}$ and taking into account (\[16\]) we derive $$H^{B}=H^{\alpha }e_{\alpha }^{B}+\varepsilon \phi n^{B}; \label{42a}$$since $e_{\alpha }^{B}n_{B}=0$ and $n^{A}n_{A}=\varepsilon $ (see (\[16a\])) it follows from here that $$H^{B}H_{B}=g_{\alpha \beta }H^{\alpha }H^{\beta }+\varepsilon \phi ^{2}.
\label{43aa}$$Let us construct the curvature of $H_{A}$$$F_{AB}=D_{A}H_{B}-D_{B}H_{A}, \label{44a}$$where $D_{A}\equiv e_{A}^{\alpha }D_{\alpha }$ and $D_{\alpha }$ is covariant derivative with respect to tangent space vector indices (see ([32]{})); therefore, the components with only space time indices are scalars with respect to this derivative, for example, $D_{\alpha }H^{\beta
}=\partial _{\alpha }H^{\beta }.$ Taking this into account and using decomposition (\[42a\]) we find$$D_{A}H^{B}=e_{A}^{\beta }e_{\alpha }^{B}\partial _{\beta }H^{\alpha
}+e_{A}^{\beta }H^{\alpha }D_{\beta }e_{\alpha }^{B}+\varepsilon
e_{A}^{\beta }n^{B}\partial _{\beta }\phi +\varepsilon e_{A}^{\beta }\phi
D_{\beta }n^{B}. \label{45a}$$The last term here is equal to zero. In fact, using the definition (\[5a\]) we have $$\partial _{\beta }n_{A}=\varepsilon \mathbf{\nabla }_{\beta }\left( \mathbf{v}_{A}\cdot \mathbf{n}\right) =-\omega _{\beta A}^{\quad \
B}n_{B}+\varepsilon \mathbf{v}_{A}\cdot \mathbf{\nabla }_{\beta }\mathbf{n,}
\label{46aaa}$$and hence $D_{\beta }n_{A}=-\varepsilon \mathbf{v}_{A}\cdot \mathbf{\nabla }_{\beta }\mathbf{n.}$ In turn, one can immediately conclude from $\mathbf{\nabla }_{\beta }\left( \mathbf{e}_{\alpha }\cdot \mathbf{n}\right) =0$ and $\mathbf{\nabla }_{\beta }\left( \mathbf{n}\cdot \mathbf{n}\right) =0$ that $\mathbf{\nabla }_{\beta }\mathbf{n=}0$ and therefore $D_{\beta }n_{A}=0.$ Using metricity condition (\[35\]) to express $D_{\beta }e_{\alpha }^{B}$ in terms of $\Gamma _{\nu \sigma }^{\rho }$ and interchanging indices we then find$$F_{AB}=e_{A}^{\beta }e_{B}^{\alpha }\left( \partial _{\beta }H_{\alpha
}-\partial _{\alpha }H_{\beta }\right) +\varepsilon \left( e_{A}^{\beta
}n_{B}-e_{B}^{\beta }n_{A}\right) \partial _{\beta }\phi . \label{47a}$$Note that $F_{AB}$ is invariant under the $U(1)$ gauge transformation$$H_{A}\rightarrow H_{A}+e_{A}^{\alpha }\partial _{\alpha }\Lambda ,
\label{48a}$$which in terms of the space time components become $H_{\alpha }\rightarrow
H_{\alpha }+\partial _{\alpha }\Lambda ,$ $\phi \rightarrow \phi .$ Squaring (\[47a\]) we will find the gauge invariant Lagrangian density for the massless vector field $$L=-\frac{1}{4}F_{AB}F^{AB}=-\frac{1}{4}F_{\alpha \beta }F^{\alpha \beta }-\frac{1}{2}\varepsilon \partial _{\alpha }\phi \partial ^{\alpha }\phi ,
\label{49a}$$where $$F_{\alpha \beta }=\partial _{\alpha }H_{\beta }-\partial _{\beta }H_{\alpha
}. \label{50a}$$Notice that we get the correct sign for the kinetic energy of the scalar field $\phi $ only in the case of de Sitter group ($\varepsilon =-1$) while for anti de Sitter group $\varepsilon =1$ we get a ghost$.$ We deduce that the formulation of gravity where the tangent group is $SO(1,d)$ instead of $SO(1,d-1)$ unifies spins zero and spin one in one vector field. If we add to the Lagrangian the term (\[43aa\]) both fields acquire the same mass.
We now turn to spinors. Because they should respect 5d tangent Lorentz group it is well known that neither Majorana or Weyl conditions can be imposed on them [@Scherk]. Thus the spinors $\psi $ must be Dirac spinors. The Dirac action in this case is $$\dint \sqrt{g}d^{4}x\left( i\overline{\psi }\Gamma ^{A}D_{A}\psi -i\overline{D_{A}\psi }\Gamma ^{A}\psi \right)$$The spinors do feel the full $SO(1,4)$ local symmetry. This seems to be a very strong constraint as it implies that chiral spinors cannot exist if the tangent group is $SO(1,4).$ This is similar to the situation in case of supersymmetry in five dimensions [@Cremmer], [@CN], or $N=2$ supersymmetry. There, it was shown that it is possible to generalize the Majorana condition by taking a doublet of spinors [@Scherk]. The conclusion we must draw is then that the $SO(1,4)$ tangent group implies that spinors must be treated in the same way as in $N=2$ supersymmetry. To couple the spinors to vectors, some gauge symmetry must be introduced. As an example, let us assume the existence of a $U(1)$ gauge symmetry. In this case the covariant derivative $D_{A}\psi $ becomes $$D_{A}\psi =\left( e_{A}^{\mu }\left( \partial _{\mu }+\frac{1}{4}\omega
_{\mu }^{\hspace{0.03in}AB}\Gamma _{AB}\right) \ +iH_{A}\right) \psi \ ,$$which shows that the spinors exist in a unified interactions with both a scalar and a vector field, as was seen in the decomposition of the vector $H_{A}$ into a vector $H_{\mu }$ and a scalar $\phi .$
Complex gravity and unitary $U(1,d-1)$ tangent group
====================================================
As a tangent space one can also consider a complex vector space with Hermitian scalar product satisfying $$\left( \mathbf{v},\mathbf{u}\right) =\left( \mathbf{u},\mathbf{v}\right)
^{\ast },\text{ \ }\left( \mathbf{v},\alpha \mathbf{u}\right) =\alpha \left(
\mathbf{v},\mathbf{u}\right) , \label{40}$$where $\alpha $ is a complex number. It follows from here that $\left(
\alpha \mathbf{v},\mathbf{u}\right) =\alpha ^{\ast }\left( \mathbf{v},\mathbf{u}\right) .$ As before let us introduce in this space the orthonormal basis $\mathbf{v}_{A}$ $(A=1,...N)$: $$\left( \mathbf{v}_{A},\mathbf{v}_{B}\right) =\eta _{AB}. \label{41}$$The condition of orthogonality is preserved under $U(1,N-1)$ transformations$$\mathbf{\tilde{v}}_{A}=U_{A}^{\hspace{0.05in}C}\mathbf{v}_{C},\ \ \ \ \ \
U_{A}^{\hspace{0.05in}C}\eta _{CD}\left( U_{A}^{\hspace{0.05in}D}\right)
^{\ast }=\eta _{AB}. \label{43}$$For generality let us first consider the complex coordinate basis vectors $\mathbf{e}_{\alpha }$ $(\alpha =1,...d)$ in $d$-dimensional manifold and show that in this case we obtain the Hermitian theory of gravity as formulated by Einstein and Strauss [@Ein], [@ES]. Later on we will show that this theory can be consistently truncated to General Relativity while preserving the unitary structure of the tangent space.
Assuming that $N\geq d$ we can expand the coordinate basis vectors in terms of vielbein vectors, $\mathbf{e}_{\alpha }=e_{\alpha }^{A}\mathbf{v}_{A},$ and then the metric on the manifold can be expressed as$$g_{\alpha \beta }\equiv \left( \mathbf{e}_{\alpha },\mathbf{e}_{\beta
}\right) =e_{\alpha }^{A}e_{\beta }^{B\ast }\eta _{AB}. \label{44}$$This metric is Hermitian$$g_{\alpha \beta }=\left( \mathbf{e}_{\alpha },\mathbf{e}_{\beta }\right)
=\left( \mathbf{e}_{\beta },\mathbf{e}_{\alpha }\right) ^{\ast }=g_{\beta
\alpha }^{\ast }.$$In the case under consideration the affine and spin connections are defined exactly as in (\[5a\]). Taking derivative of (\[44\]) and using definition in (\[5a\]) we obtain$$\partial _{\gamma }g_{\alpha \beta }=\left( \mathbf{\nabla }_{\gamma }\mathbf{e}_{\alpha },\mathbf{e}_{\beta }\right) +\left( \mathbf{e}_{\alpha },\mathbf{\nabla }_{\gamma }\mathbf{e}_{\beta }\right) =\Gamma _{\alpha \gamma
}^{\nu \ast }g_{\nu \beta }+\Gamma _{\beta \gamma }^{\nu }g_{\alpha \nu }.
\label{45}$$These $d^{\,3}$ equations can be solved unambiguously for $\Gamma _{\kappa
\rho }^{\mu }$ in terms of metric $g_{\alpha \beta }$ only if we impose the hermiticity condition$$\Gamma _{\rho \mu }^{\nu \ast }=\Gamma _{\mu \rho }^{\nu }, \label{46}$$which leaves us with $d^{3}$ components to be determined. Unlike the real case equations (\[46\]) can be solved only perturbatively. They were first imposed by Einstein in his formulation of Hermitian gravity which he referred to as the “$+-$” condition [@Ein], [@ES], [@DD]. Similar to (\[6\]) we derive a condition on spin connection$$\omega _{\alpha A}^{\quad C}\eta _{CB}=-\left( \omega _{\alpha B}^{\quad
C}\right) ^{\ast }\eta _{CA}, \label{46a}$$which leaves $N^{2}d$ independent components. Taking derivative of $\left(
\mathbf{v}_{A},\mathbf{e}_{\alpha }\right) =e_{\alpha }^{B}\eta _{AB}$ we derive the following metricity conditions $$\partial _{\gamma }e_{\alpha }^{A}=\omega _{\gamma B}^{\quad \,A}e_{\alpha
}^{B}+\Gamma _{\alpha \gamma }^{\nu }e_{\nu }^{A}. \label{47}$$Taking into account that $d^{3}$ equations (\[45\]) determine $\Gamma
_{\beta \gamma }^{\nu }$ through $\partial _{\gamma }e_{\alpha }^{A}$ we are left with $2Nd^{2}-d^{3}$ equations to find $N^{2}d$ independent components of $\omega _{\alpha A}^{\quad C}.$ The number of equations match the number of unknown components only if $N=d,$ that is, when dimension of complex tangent space coincides with the dimension of the manifold. Hence the gauge group of the tangent space can be only $U(1,d-1)$ [@AHC]. In this case we can define the soldering form $e_{B}^{\beta },$ which is inverse to $e_{\alpha }^{A}:$$$e_{B}^{\alpha }e_{\alpha }^{A}=\delta _{B}^{A},\text{ \ \ }e_{A}^{\alpha
}e_{\beta }^{A}=\delta _{\beta }^{\alpha }.\text{\ } \label{48}$$The metric with upper indices is then given by$$g^{\mu \nu }=e_{A}^{\mu }e_{B}^{\nu \ast }\eta ^{AB}, \label{49}$$and it is inverse to $g_{\alpha \beta }$ $$g_{\alpha \nu }g^{\beta \nu }=\delta _{\alpha }^{\beta }\neq g_{\alpha \nu
}g^{\nu \beta }. \label{50}$$Similar to (\[34\]) the curvature of the connection $\omega _{\mu
A}^{\quad B}$ can be defined as $$\begin{aligned}
\left[ D_{\mu },D_{\nu }\right] e_{A}^{\sigma }& \equiv R_{\mu \nu A}^{\quad
\hspace{0.03in}\hspace{0.03in}B}\left( \omega \right) e_{B}^{\sigma } \notag
\\
& =\left( \partial _{\mu }\omega _{\nu A}^{\quad B}-\partial _{\nu }\omega
_{\mu A}^{\quad B}+\omega _{\mu A}^{\quad C}\omega _{\nu C}^{\quad B}-\omega
_{\nu A}^{\quad C}\omega _{\mu C}^{\quad B}\right) e_{B}^{\sigma }.
\label{52}\end{aligned}$$On the other hand, using the metricity condition, we have $$\begin{aligned}
\left[ D_{\mu },D_{\nu }\right] e_{A}^{\sigma }& =-\left( \partial _{\mu
}\Gamma _{\rho \nu }^{\sigma }-\partial _{\nu }\Gamma _{\rho \mu }^{\sigma
}+\Gamma _{\kappa \mu }^{\sigma }\Gamma _{\rho \nu }^{\kappa }-\Gamma
_{\kappa \nu }^{\sigma }\Gamma _{\rho \mu }^{\kappa }\right) e_{A}^{\sigma }
\notag \\
& \equiv -R_{\hspace{0.03in}\hspace{0.03in}\rho \mu \nu }^{\sigma }\left(
\Gamma \right) e_{A}^{\rho }, \label{53}\end{aligned}$$and it follows from here that$$R_{\hspace{0.03in}\hspace{0.03in}\rho \mu \nu }^{\sigma }\left( \Gamma
\right) =-e_{\rho }^{A}R_{\mu \nu A}^{\quad \,\,\,\,B}\left( \omega \right)
e_{B}^{\sigma }. \label{54}$$In particular, the scalar curvature $$\begin{aligned}
R\left( \omega \right) & =\eta ^{AC}e_{C}^{\mu \ast }R_{\mu \nu A}^{\quad
\,\,\,\,B}\left( \omega \right) e_{B}^{\nu }=-\eta ^{AC}e_{C}^{\mu \ast
}R_{\,\,\,\rho \mu \nu }^{\nu }\left( \Gamma \right) e_{A}^{\rho } \notag \\
& =g^{\rho \mu }R_{\,\,\,\rho \nu \mu }^{\nu }\left( \Gamma \right) =R\left(
\Gamma \right) , \label{55}\end{aligned}$$is $U\left( 1,d-1\right) $ gauge invariant. The scalar curvature is real,$$R^{\ast }\left( \omega \right) =R\left( \omega \right) . \label{56}$$ To prove this we first note the identity$$\left( R_{\mu \nu A}^{\quad \,\,\,B}\left( \omega \right) \right) ^{\ast
}=-R_{\mu \nu C}^{\quad \,\,\,\,D}\left( \omega \right) \eta ^{CB}\eta _{DA},
\label{57}$$which follows from equation (\[52\]) taking into account (\[46a\]). Using this relation together with (\[54\]) we obtain $$\begin{aligned}
\left( R_{\hspace{0.03in}\hspace{0.03in}\rho \mu \nu }^{\sigma }\left(
\Gamma \right) \right) ^{\ast }& =-e_{\rho }^{A\ast }\left( R_{\mu \nu
A}^{\quad \,\,\,B}\left( \omega \right) \right) ^{\ast }e_{B}^{\sigma \ast
}=e_{\rho }^{A\ast }R_{\mu \nu C}^{\quad \,\,\,D}\left( \omega \right) \eta
^{CB}\eta _{DA}e_{B}^{\sigma \ast } \notag \\
& =-\eta ^{CB}e_{C}^{\kappa }e_{B}^{\sigma \ast }R_{\,\,\,\kappa \mu \nu
}^{\lambda }\left( \Gamma \right) \eta _{DA}e_{\lambda }^{D}e_{\rho }^{A\ast
}=-g^{\kappa \sigma }R_{\,\,\,\kappa \mu \nu }^{\lambda }\left( \Gamma
\right) g_{\lambda \rho }. \label{58}\end{aligned}$$It follows from here that the tensor $$R_{\rho \kappa \mu \nu }\left( \Gamma \right) =R_{\,\,\,\kappa \mu \nu
}^{\lambda }\left( \Gamma \right) g_{\lambda \rho }, \label{59}$$is antihermitian with respect to exchange of first two indices $$\left( R_{\kappa \rho \mu \nu }\left( \Gamma \right) \right) ^{\ast
}=-R_{\rho \kappa \mu \nu }\left( \Gamma \right) , \label{60}$$and it is antisymmetric with respect to exchange of the last two indices (see (\[53\]). Taking this into account we have$$R^{\ast }\left( \Gamma \right) =\left( g^{\rho \mu }g^{\nu \sigma }R_{\sigma
\rho \nu \mu }\right) ^{\ast }=g^{\mu \rho }g^{\sigma \nu }R_{\rho \sigma
\mu \nu }=R\left( \Gamma \right) , \label{61}$$and because $R\left( \omega \right) =R\left( \Gamma \right) ,$ this completes the proof of reality of gauge invariant scalar curvature.
The identity (\[60\]) was not noticed by Einstein and this forced him to construct Hermitian combinations of the curvature tensor. As we see this is not necessary because one can use instead the real scalar curvature as Lagrangian density.
If we write the connection as $$\omega _{\mu A}^{\quad B}=\bar{\omega}_{\mu A}^{\quad B}+\frac{1}{d}\hat{\omega}_{\mu }\delta _{A}^{B}, \label{62}$$where $$\bar{\omega}_{\mu A}^{\quad A}=0,\text{ \ }\hat{\omega}_{\mu }=\omega _{\mu
A}^{\quad A}, \label{63}$$the curvature splits into two pieces $$R_{\mu \nu A}^{\quad \,\,\,B}\left( \omega \right) =R_{\mu \nu A}^{\quad
\,\,B}\left( \bar{\omega}\right) +\frac{1}{d}R_{\mu \nu C}^{\quad
\,\,\,C}\left( \hat{\omega}\right) \delta _{A}^{B}, \label{64}$$where $$\begin{aligned}
R_{\mu \nu A}^{\quad \,\,\,B}\left( \bar{\omega}\right) & =\left( \partial
_{\mu }\bar{\omega}_{\nu A}^{\quad B}-\partial _{\nu }\bar{\omega}_{\mu
A}^{\quad B}+\bar{\omega}_{\mu A}^{\quad C}\bar{\omega}_{\nu C}^{\ \ \ \ B}-\bar{\omega}_{\nu A}^{\quad C}\bar{\omega}_{\mu C}^{\ \ \ \ B}\right) ,
\notag \\
R_{\mu \nu C}^{\quad \,\,\,C}\left( \omega \right) & =\partial _{\mu }\hat{\omega}_{\nu }-\partial _{\nu }\hat{\omega}_{\mu }. \label{65}\end{aligned}$$It follows from here that $$R\left( \omega \right) =\eta ^{AC}e_{C}^{\mu \ast }R_{\mu \nu A}^{\quad
\,\,\,B}\left( \bar{\omega}\right) e_{B}^{\nu }+\frac{1}{d}g^{\nu \mu
}R_{\mu \nu C}^{\quad \,\,\,C}\left( \hat{\omega}\right) =R\left( \bar{\omega}\right) +\frac{1}{d}\tilde{R}\left( \hat{\omega}\right) , \label{66}$$where $\tilde{R}=g^{\nu \mu }R_{\mu \nu A}^{\quad \,\,\,A}$ is another scalar curvature invariant. Therefore it can be added to the action with an arbitrary coefficient leading to the following most general gauge invariant first order action$$S=\dint d^{4}x\left\vert \det e_{\mu }^{A}\right\vert \left( \alpha R\left(
\bar{\omega}\right) +\beta \tilde{R}\left( \hat{\omega}\right) \right) .
\label{67}$$It must be stressed that we are using here a second order formalism where the field $\omega _{\mu A}^{\quad B}$ is determined by the metricity condition and not by the field equations. The best strategy to analyze this action is to solve for $\omega _{\mu A}^{\quad B}$ in a perturbative expansion in terms of $e_{\mu }^{A}.$
We can understand the above results by noting that the gauge invariant action allows to use the gauge invariance to reduce the independent components of $e_{\alpha }^{A}$ to those of $g_{\alpha \beta }.$ In other words we expect that because of $U(1,d-1)$ gauge invariance, the action depends only on the metric$$g_{\alpha \beta }=e_{\alpha }^{A}e_{\beta }^{B\ast }\eta _{AB}\equiv
G_{\alpha \beta }+iB_{\alpha \beta }.$$This theory was considered before using a first order formalism where the spin-connection was determined from the equations of motion [@AHC]*.* This is possible only when the action depends quadratically on the spin-connection. However, the $U(1)$ part $\hat{\omega}$ of the $U(1,d-1) $ connection being abelian, appears linearly. This then imposes a constraint on the antisymmetric part of the metric$$\partial _{\alpha }\left( \left\vert \det e_{\mu }^{A}\right\vert B^{\alpha
\beta }\right) =0,$$which thus remains undetermined [@AHC]. This is to be contrasted with the second order formalism where all spin-connections are determined from the metricity condition. * *
We arrive to an interesting case by requiring that the metric $g_{\alpha
\beta }$ to be real. This is equivalent to truncating the $B_{\alpha \beta }$ field. Let $$e_{\alpha }^{A}=e_{\alpha \left( 0\right) }^{A}+ie_{\alpha \left( 1\right)
}^{A},$$so that $$\begin{aligned}
G_{\alpha \beta } &=&\left( e_{\alpha \left( 0\right) }^{A}e_{\beta \left(
0\right) }^{B}+e_{\alpha \left( 1\right) }^{A}e_{\beta \left( 1\right)
}^{B}\right) \ \eta _{AB}, \\
B_{\alpha \beta } &=&\left( e_{\alpha \left( 1\right) }^{A}e_{\beta \left(
0\right) }^{B}-e_{\alpha \left( 0\right) }^{A}e_{\beta \left( 1\right)
}^{B}\right) \ \eta _{AB},\end{aligned}$$Truncating $B_{\alpha \beta }$ gives $\frac{1}{2}d\left( d-1\right) $ constraints on the $2d^{2}$ (real) fields $e_{\alpha \left( 0\right) }^{A}$ and $e_{\alpha \left( 1\right) }^{A}$. In this case the affine connection is also real and its $\frac{1}{2}d^{2}\left( d+1\right) $ components are Christoffel connection for the metric $G_{\alpha \beta }.$ The remaining$$2d^{3}-\frac{1}{2}dd\left( d-1\right) -\frac{1}{2}d^{2}\left( d+1\right)
=d^{3}$$independent equations (\[47\]) are then enough to unambiguously determine $d^{3}$ components of $\omega _{\mu A}^{\quad B}.$ This implies that it is possible to enlarge the tangent group to become $U(1,d-1)$ and still obtain the Einstein gravity without any modification. The coupling to matter will, however, feel the tangent group $U(1,d-1)$.
**Matter coupling.** When the tangent group is $U(1,3)$ then from the previous discussion it should be clear that neither the Majorana nor the Weyl condition could be imposed, except if a doublet of spinors is taken. Thus, as with the $SO(1,4)$ case we must take a Dirac spinor, or a doublet of Majorana or Weyl spinors, again as in the $N=2$ supersymmetric case. We note the isomorphism of the algebras $$U\left( 1,3\right) \sim SO(1,5)\times SO(1,1).$$It is easy to see that $U\left( 1,3\right) $ has ten compact generators and six non-compact generators, while $SO(1,5)$ has ten compact generators and five non-compact generators and $SO(1,1)$ has one non-compact generator. Thus spinors in the case of unitary tangent group will exhibit conformal local symmetry.
Gravity has a universal coupling to matter. One way to classify the fields is according to their behavior under the diffeomorphism group, or equivalently under the tangent Lorentz group. A complex scalar field has the following couplings$$\dint d^{4}x\sqrt{\det g}g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu
}\phi ^{\ast }.$$For a massless vector it can be easily seen that the action can be written in terms of a complex space-time vector $H_{\mu }$ with the action $$\ \dint d^{4}x\sqrt{\det g}g^{\mu \rho }g^{\nu \sigma }F_{\mu \nu }F_{\rho
\sigma }^{\ast }.$$Similarly we can treat the case of fields which are in the vector representations of the gauge group. The fermions have more complicated couplings. First, a Dirac spinor has the $U(1,d-1)$ transformation$$\psi \rightarrow e^{i\lambda _{B}^{\hspace{0.03in}A}\Gamma _{A}\Gamma
^{B}}\psi ,$$where $\Gamma ^{A}$ and $\Gamma _{A}$ satisfy the relations $$\left\{ \Gamma ^{A},\Gamma ^{B}\right\} =0,\quad \left\{ \Gamma _{A},\Gamma
_{B}\right\} =0,\quad \left\{ \Gamma ^{A},\Gamma _{B}\right\} =\delta
_{B}^{A},$$and thus $\Gamma _{A}\Gamma ^{B}$ are the generators of $U(1,d-1).$ We can define the Hermitian Dirac matrices $$\begin{aligned}
\gamma ^{\mu }& =e_{A}^{\mu }\Gamma ^{A}+e^{\mu A}\Gamma _{A}, \\
\left\{ \gamma ^{\mu },\gamma ^{\nu }\right\} & =g^{\mu \nu }+g^{\nu \mu },\end{aligned}$$The covariant derivative is given by $$D_{\mu }\psi =\partial _{\mu }\psi +\omega _{\mu B}^{\quad A}\Gamma
_{A}\Gamma ^{B}\psi ,$$Hermitian Dirac action is then $$\dint d^{4}x\left\vert \det e_{\mu }^{A}\right\vert \overline{\psi }\gamma
^{\mu }D_{\mu }\psi .$$Therefore, Dirac spinors do couple to both the symmetric and antisymmetric components of the Hermitian metric.
Conclusions
===========
We have shown that Einstein gravity exhibits universality when formulated as a gauge theory of tangent space group. Besides of the well known natural case when the tangent space has the same dimension as the manifold, we discovered two other possibilities for General Relativity to be reproduced and the theory still remains unambiguous. Namely, we have shown that in the four dimensional case the tangent space can be five dimensional and possess (anti) de Sitter group of symmetry. This group is important when we incorporate matter couplings to the gravitational field. As an example, we have shown that de Sitter tangent space group allows us to unify 4d vectors and scalars which become components of the same five dimensional vector in tangent space. Even more dramatic are the consequences of the tangent space symmetry group on fermions. They become fundamentally five dimensional and neither Majorana nor Weyl conditions could be imposed on them. This situations is similar to $N=2$ supersymmetry where we are forced to generalize the Majorana condition by taking a doublet of spinors. We also would like to note that if we impose an extra $U\left( 1\right) $ local symmetry in the tangent space then the spinors would exist in a unified interaction with both scalar and vector fields.
Another interesting possibility arise when we consider complex tangent space of the same dimension as the manifold. In this case the group of symmetry is the unitary group. This gives rise generically to the theory of Hermitian gravity, where the basic fields are the symmetric and antisymmetric components of the metric, which coincide with the basic fields appearing in effective open string field theory. It is interesting that this theory can be consistently truncated to Einstein gravity, while still preserving the unitary group of tangent space. In turn, this has interesting and nontrivial consequences for the coupling to matter which should respect this symmetry. In a forthcoming paper [@CM] we shall explore the implications of these new formulations of gravity, especially in regard to the spontaneous breakdown of these larger symmetries down to the $SO(1,d-1)$ symmetry.
Appendix: The Poincare limit and 3d CS gravity
==============================================
In this appendix we examine the special case when the radius of the de Sitter tangent group becomes infinite, which corresponds to Poincare symmetry. Later we shall also investigate the correspondence with Chern-Simons gravity in three dimensions which also have de Sitter or Poincare symmetry [@AT], [@Witten].
The $SO(1,d)$ group generators satisfy the commutation relations$$\left[ J_{AB},J_{CD}\right] =-\frac{1}{2}\left( \eta _{AC}J_{BD}-\eta
_{BC}J_{AD}-\eta _{AD}J_{BC}+\eta _{BD}J_{AC}\right) .$$Splitting the range of the index $A=a,\overline{d},$ where $a=0,1,\cdots
,d-1,$ and similarly for the other indices we get the usual $SO(1,d-1)$ for the $J_{ab},$ while for $J_{a\overline{d}}\equiv RP_{a}$ we have $$\left[ P_{a},P_{b}\right] =-\frac{1}{R^{2}}J_{ab}.$$Thus, in the limit $R\rightarrow \infty $ the de Sitter tangent group becomes the inhomogeneous Lorentz group, i.e. $ISO(1,d-1)$ also known as the Poincare group. The covariant derivative $$D_{\mu }=\partial _{\mu }+\omega _{\mu }^{\,\,AB}J_{AB},$$implies that the field $\omega _{\mu }^{a\overline{d}}$ must be defined as $\omega _{\mu }^{a\overline{d}}\equiv \frac{1}{2R}b_{\mu }^{a}$ so that $$D_{\mu }=\partial _{\mu }+\omega _{\mu }^{\,\,ab}J_{ab}+b_{\mu }^{\,a}P_{a},$$is independent of the radius $R.$ The curvatures in terms of the redefined fields are $$\begin{aligned}
R_{\mu \nu }^{\quad ab}& =\partial _{\mu }\omega _{\nu }^{\,\,\,ab}-\partial
_{\nu }\omega _{\mu }^{\,\,\,ab}+\omega _{\mu }^{\,\,\,ac}\omega _{\nu
c}^{\quad b}-\omega _{\nu }^{\,\,\,ac}\omega _{\mu c}^{\quad b}-\frac{1}{4R^{2}}\left( b_{\mu }^{\,a}b_{\nu }^{\,b}-b_{\mu }^{\,b}b_{\nu
}^{\,a}\right) , \\
R_{\mu \nu }^{\quad a\overline{d}}& =\frac{1}{2R}\left( \partial _{\mu
}b_{\nu }^{\,a}-\partial _{\mu }b_{\nu }^{\,a}+\omega _{\mu
}^{\,\,\,ac}b_{\nu c}-\omega _{\nu }^{\,\,\,ac}b_{\mu c}\right) .\end{aligned}$$The zero torsion condition on $e_{A}^{\mu }$ is consistent in the limit $R\rightarrow \infty $ if we define $$e_{\overline{d}}^{\mu }\equiv \frac{1}{R}c^{\mu },$$so that $$\ \partial _{\mu }c^{\nu }-\frac{1}{2}b_{\mu }^{\,a}e_{a}^{\nu }+\Gamma
_{\rho \mu }^{\nu }c^{\rho }=0,$$which allows us to calculate $b_{\mu }^{a}$ in terms of $c^{\mu }.$ The field $\omega _{\mu }^{\quad ab}$ is solved from the condition $$\ \partial _{\mu }e_{a}^{\nu }+\omega _{\mu a}^{\quad b}e_{b}^{\nu }+\frac{1}{2R^{2}}b_{\mu }^{a}c^{\nu }+\Gamma _{\rho \mu }^{\nu }e_{a}^{\rho }=0.$$Writing the gravitational action in terms of the rescaled fields, we expand $e_{A}^{\mu }R_{\mu \nu }^{\hspace{0.05in}\hspace{0.05in}AB}\left( \omega
\right) e_{B}^{\nu }$ to get $$\begin{aligned}
& e_{a}^{\mu }e_{b}^{\nu }\left( \partial _{\mu }\omega _{\nu }^{\hspace{0.05in}ab}-\partial _{\nu }\omega _{\mu }^{\,\hspace{0.05in}ab}+\omega _{\mu
}^{\,\,\,ac}\omega _{\nu c}^{\quad b}-\omega _{\nu }^{\,\,\,ac}\omega _{\mu
c}^{\quad b}-\frac{1}{4R^{2}}\ \left( b_{\mu }^{\,a}b_{\nu }^{\,b}-b_{\mu
}^{\,b}b_{\nu }^{\,a}\right) \right) \notag \\
& \ +\frac{1}{R^{2}}e_{a}^{\mu }c^{\nu }\left( \partial _{\mu }b_{\nu
}^{\,a}-\partial _{\nu }b_{\mu }^{\,a}+\omega _{\mu }^{\hspace{0.05in}ac}b_{\nu c}-\omega _{\nu }^{\hspace{0.05in}ac}b_{\mu c}\right) .\end{aligned}$$Therefore it is clear that in the limit $R\rightarrow \infty $ the connection $\omega _{\mu }^{\,\,ab}$ coincides with the $SO(1,d-1)$ Lorentz connection and the action becomes identical to the Einstein-Hilbert action. The fields $b_{\mu }^{\,a}$ and $c^{\mu }$ drop out of the action. Thus in the limit of $ISO(1,d-1)$ the action is indistinguishable from the $SO(1,d-1) $ invariant action for gravity.
For matter couplings, especially for the vector $H_{A}$, the gauge transformation is $$\delta H_{A}=\lambda _{AB}H^{B},\quad \lambda _{AB}=-\lambda _{BA}.$$Denoting $H_{\overline{d}}=\phi $ and $\lambda _{a\overline{d}}=\frac{1}{2R}\lambda _{a}$, the gauge transformations of $H_{a}$ and $\phi $ are $$\begin{aligned}
\delta H_{a}& =\lambda _{ab}H^{b}+\frac{1}{2R}\lambda _{a}\phi , \\
\delta \phi & =-\frac{1}{2R}\lambda _{a}H^{a}.\end{aligned}$$Thus, in the limit $R\rightarrow \infty $ the fields $H_{a}$ and $\phi $ remain in the action as spin one and spin zero fields, but they decouple in the transformations and become independent.
When our $SO(1,d)$ gauge invariant gravitational action is taken in three dimensions, it is natural to ask whether the action obtained is identical to the Chern-Simons action which was also shown by Achucarro-Townsend [@AT] and Witten [@Witten] to be equivalent to the Einstein action in three dimensions, but with a cosmological constant. In the Chern-Simons construction one uses only the gauge field $\omega _{\mu }^{AB}$ where the CS action is $$I_{\mathrm{CS}}=\frac{1}{2}\dint d^{3}x\epsilon ^{\mu \nu \rho }\epsilon
_{ABCD}\left( \omega _{\mu }^{\,\,AB}\partial _{\nu }\omega _{\rho
}^{\,\,CD}+\frac{2}{3}\omega _{\mu }^{\,\,AB}\omega _{\nu }^{\,\,CE}\omega
_{\rho E}^{\quad \,D}\right) .$$Using the same decomposition for $\omega _{\mu }^{AB}$ as before, we get $$\frac{1}{R}\dint d^{3}x\epsilon ^{\mu \nu \rho }\epsilon _{abc}b_{\mu
}^{a}\left( \partial _{\nu }\omega _{\rho }^{\,\,bc}+\omega _{\nu
}^{\,\,be}\omega _{\rho e}^{\quad c}-\frac{1}{12R^{2}}b_{\nu }^{\,b}b_{\rho
}^{\,\,c}\right) ,$$which is the the first order formulation of the Einstein action plus a cosmological constant, with the dreibein field $b_{\mu }^{\,a}$. The special case with the $ISO(1,d-1)$ gauge group can be recovered by rescaling the action by $R$ and then taking the limit $R\rightarrow \infty .$ In our treatment, there is also the additional field $e_{A}^{\mu }$ which is not a gauge field. The field $b_{\mu }^{\,a}$ is given by $$b_{\mu }^{\,a}=2e_{\nu }^{a}\nabla _{\mu }c^{\nu },$$where $e_{\nu }^{a}$ is the inverse of $e_{a}^{\nu }.$ Our action can be expressed in terms of $e_{a}^{\nu }$ and a non-propagating field $c^{\mu }.$ Comparing the two formulations, we deduce that the field $b_{\mu }^{\,a}$ must be identified with $e_{\mu }^{a}$. Although $e_{\mu }^{a}$ is not a gauge field, it can be shown, using the torsion constraint, that its diffeomorphism transformation with parameters $\zeta ^{\mu }$ can yield the same gauge transformation as $b_{\mu }^{\,a}$ with the gauge parameter $\lambda ^{a}=e_{\mu }^{a}\zeta ^{\mu }$ [@Witten]. It then clear that although both formulations have the same gauge symmetry, they have different field configurations. Moreover, the usual matter couplings in the CS formulation are not possible because the dreibein $b_{\mu }^{\,a}$ is a gauge field. Any direct coupling to matter breaks gauge invariance, except for coupling to Wilson lines. In our case since $e_{A}^{\mu }$ is not a gauge field, a gauge invariant metric can be easily formed $g^{\mu \nu
}=e_{A}^{\mu }e^{\nu A}$ and coupled to any form of matter desired.
The work of AHC is supported in part by the Alexander von Humboldt Foundation and by the National Science Foundation 0854779. V.M. is supported by TRR 33 The Dark Universe and the Cluster of Excellence EXC 153 Origin and Structure of the Universe.
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[^1]: We use the notation and methods of Misner, Thorne and Wheeler ([@MTW]), in particular Chapters 9 and 10.
|
---
author:
- 'Hiroshi Isono,'
- Toshifumi Noumi
- and Toshiaki Takeuchi
bibliography:
- 'WT.bib'
title: 'Momentum space conformal three-point functions of conserved currents and a general spinning operator'
---
Introduction
============
Recent development in higher dimensional conformal field theory (CFT) (see, e,g., [@Qualls:2015qjb; @Rychkov:2016iqz; @Simmons-Duffin:2016gjk; @Penedones:2016voo] for review) has boosted research activities in rather new applications to cosmology [@Maldacena:2002vr; @Antoniadis:2011ib; @Maldacena:2011nz; @Creminelli:2012ed; @Schalm:2012pi; @Mata:2012bx; @McFadden:2013ria; @Ghosh:2014kba; @Bzowski:2012ih; @Kundu:2014gxa; @Arkani-Hamed:2015bza; @Kundu:2015xta; @Shukla:2016bnu; @Isono:2016yyj; @Arkani-Hamed:2018kmz; @Goon:2018fyu] and condensed matter physics [@Chowdhury:2012km; @Huh:2013vga; @Jacobs:2015fiv; @Myers:2016wsu; @Lucas:2016fju; @Lucas:2017dqa]. For example, inflationary correlation functions are highly constrained by conformal symmetry (realized at the future boundary of de Sitter space) and their analytic properties in momentum space are crucial to identify the particle spectrum at the inflationary scale [@Chen:2009zp; @Baumann:2011nk; @Noumi:2012vr; @Arkani-Hamed:2015bza]. These applications have then pushed forward studies on CFT in momentum space [@Ferrara:1974nf; @Bzowski:2012ih; @Bzowski:2013sza; @Bzowski:2015pba; @Bzowski:2015yxv; @Coriano:2012hd; @Coriano:2013jba; @Bzowski:2017poo; @Kundu:2014gxa; @Arkani-Hamed:2015bza; @Kundu:2015xta; @Shukla:2016bnu; @Coriano:2018bbe; @Bzowski:2018fql; @Gillioz:2018mto; @Coriano:2018bsy; @Farrow:2018yni; @Skvortsov:2018uru]. However, its understanding is still limited compared to the well-developed position space one, hence further studies on conformal correlators in momentum space are encouraged.
The purpose of this paper is to construct three-point functions of conserved currents and a general tensor in momentum space. Recent studies on conformal correlators in momentum space have been triggered by a seminal work [@Bzowski:2013sza] by Bzowski et al, which systematically studied the conformal Ward-Takahasi (WT) identities in momentum space and constructed three-point functions of conserved currents and scalars. For example, three-point functions of primary scalars are given up to an overall normalization factor by [@Ferrara:1974nf; @Bzowski:2012ih; @Bzowski:2013sza; @Bzowski:2015pba; @Bzowski:2015yxv] $$\begin{aligned}
\langle \varphi_1({{\boldsymbol{p}}}_1)\varphi_2({{\boldsymbol{p}}}_2)\varphi_3({{\boldsymbol{p}}}_3)\rangle'=J_{0\{0,0,0\}}(p_1,p_2,p_3)\,,\end{aligned}$$ where we introduced $\langle\,\ldots\,\rangle=(2\pi)^d\delta^d(\sum {{\boldsymbol{k}}}_i)\langle\,\ldots\,\rangle'$ and the so-called triple-$K$ integral (a more precise definition will be given later): $$\begin{aligned}
J_{N\{k_{1},k_{2},k_{3}\}}(p_{1},p_{2},p_{3})
& =\int_{0}^{\infty}\frac{dz}{z}\:z^{2d-\Delta_{t}-k_{t}+N}
\prod_{i=1}^3\left(p_{i}z\right)^{\nu_{i}+k_{i}}K_{\nu_{i}+k_{i}}(p_{i}z)
\,.\end{aligned}$$ Similarly, three-point functions involving conserved currents of spin $1$ and $2$ and scalars can be written in terms of the triple-$K$ integrals $J_{N\{k_{1},k_{2},k_{3}\}}$ with general indices [@Bzowski:2013sza].
So far, momentum space correlators involving a tensor of general spin and dimension have not been studied very much compared to position space ones [@Costa:2011mg; @Sotkov:1976xe; @Sotkov:1980qh]. However, they are necessary especially when we go beyond three-point functions because general tensors appear as an intermediate state whatever external operators we consider. In [@Isono:2018rrb] two of the present authors and a collaborator constructed three-point functions with two scalars and a general tensor. Using the obtained three-point functions, they further constructed a crossing symmetric basis (the Polyakov block) of scalar four-point functions with a general intermediate state, which may be used, e.g., for the Polyakov type bootstrap approach [@Polyakov:1974gs; @Sen:2015doa; @Gopakumar:2016wkt; @Gopakumar:2016cpb; @Gopakumar:2018xqi] complementary to the ordinary bootstrap approach[^1]. In this paper, along the same line, we construct three-point functions involving conserved currents and a general tensor, as a first step toward studies of four-point functions of conserved currents. We provide their expression in terms of triple-$K$ integrals and a differential operator which relates triple-$K$ integrals with different indices. Furthermore, for correlators with no or one conserved current, we can find closed forms without the differential operator. This is based on the fact that correlators can be expanded in triple-$K$ integrals and the differential operators just shift the indices of them.
The rest of the paper is organized as follows. In the next section we summarize the WT identities in momentum space to clarify our notation. In Sec. \[sec:OOs\], to elaborate on our methodology to solve the WT identities, we review and expand the story for three-point functions of two scalars and a general tensor [@Isono:2018rrb]. We then generalize it to correlators with a single conserved current as well as a scalar and a general tensor (Sec. \[sec:JJs\]) and then to those with two conserved currents and a general tensor (Sec. \[sec:TTs\]). Some technical details are collected in appendices.
Ward-Takahashi identities {#sec:Ward-Takahashi-Identities}
=========================
In this section we clarify our notation by briefly summarizing the conformal Ward-Takahashi (WT) identities and conservation laws of currents in momentum space.
#### Notation
In this paper we discuss three-point functions with a general spinning operator. A standard technique to handle symmetric traceless tensors $\mathcal{O}_{\mu_{1}\mu_{2}\ldots\mu_{s}}$ is to contract all the tensor indices with a null vector $\boldsymbol{\epsilon}$ called the polarization vector [@Bargmann:1977gy; @Costa:2011mg]. We also use the shorthand notation, $${\epsilon^{s}.\mathcal{O}}=\epsilon^{\mu_{1}}\epsilon^{\mu_{2}}\cdots\epsilon^{\mu_{s}}\mathcal{O}_{\mu_{1}\mu_{2}\ldots\mu_{s}}\,.\label{eq:polarization_vector}$$ For example, we write three-point functions of three symmetric traceless tensors as $$\begin{aligned}
&\langle{\epsilon_{1}^{s_{1}}.O_1}(\boldsymbol{p_{1}}){\epsilon_{2}^{s_{2}}.O_2}(\boldsymbol{p_{2}}){\epsilon_{3}^{s_{3}}.O_3}(\boldsymbol{p_{3}})\rangle
{{\nonumber}}\\
&\quad
=(2\pi)^d\delta^d({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2+{{\boldsymbol{p}}}_3)
\langle{\epsilon_{1}^{s_{1}}.O_1}(\boldsymbol{p_{1}}){\epsilon_{2}^{s_{2}}.O_2}(\boldsymbol{p_{2}}){\epsilon_{3}^{s_{3}}.O_3}(\boldsymbol{p_{3}})\rangle^{\prime}\,,\end{aligned}$$ where the prime on the r.h.s. implies that the delta function factor for momentum conservation is dropped. We focus on Euclidean correlators in parity invariant $d$-dimensional CFT[^2]. In the following we mostly use the primed correlator. We also use $s_i$ and $\Delta_i$ for the spin and dimension of $O_i$, respectively.
#### Conformal WT identities
We then summarize the conformal WT identities in momentum space. First, the dilatation WT identity is given by $$\begin{aligned}
\left(\sum_{i=1}^3{{\boldsymbol{p}}}_i\cdot{{\boldsymbol{\partial}}}_i+2d-\Delta_t\right)
\langle{\epsilon_{1}^{s_{1}}.O_1}(\boldsymbol{p_{1}}){\epsilon_{2}^{s_{2}}.O_2}(\boldsymbol{p_{2}}){\epsilon_{3}^{s_{3}}.O_3}(\boldsymbol{p_{3}})\rangle^{\prime}=0 \,,\label{eq:dilatation_momentum}\end{aligned}$$ where we introduced ${{\Delta}}_t={{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3$ and ${{\boldsymbol{\partial}}}_i={{\partial}}/{{\partial}}{{\boldsymbol{p}}}_i$. On the other hand, the special conformal WT identity parametrized by a transformation parameter $\boldsymbol{b}$ reads (see, e.g., [@Bzowski:2013sza] for details) $$\begin{aligned}
\boldsymbol{b}\cdot\boldsymbol{K}\,
\langle{\epsilon_{1}^{s_{1}}.O_1}(\boldsymbol{p_{1}}){\epsilon_{2}^{s_{2}}.O_2}(\boldsymbol{p_{2}}){\epsilon_{3}^{s_{3}}.O_3}(\boldsymbol{p_{3}})\rangle^{\prime}=0\,,
\label{eq:SCWT}\end{aligned}$$ where we introduced the differential operator ${{\boldsymbol{K}}}$ by $$\begin{aligned}
\boldsymbol{K}&=\boldsymbol{K}_s+\boldsymbol{K}_{\epsilon_1}+\boldsymbol{K}_{\epsilon_2}+\boldsymbol{K}_{\epsilon_3}\,,
\\*
\boldsymbol{b}\cdot\boldsymbol{K}_s&=\sum_{i=1}^{3}\big[{{\boldsymbol{b}}}\cdot{{\boldsymbol{\partial}}}_{i}\left(-2\left(\Delta_{i}-d+1\right)+2{{\boldsymbol{p}}}_{i}\cdot{{\boldsymbol{\partial}}}_{i}\right)-\left({{\boldsymbol{b}}}\cdot {{\boldsymbol{p}}}_{i}\right){{\boldsymbol{\partial}}}_{i}^{2}\big]\,,
\\*
\boldsymbol{b}\cdot\boldsymbol{K}_{\epsilon_i}&=2({{\boldsymbol{\epsilon}}}_{i}\cdot{{\boldsymbol{\partial}}}_{i}){({{\boldsymbol{b}}}\cdot{{\boldsymbol{\partial}}}_{\epsilon_{i}})}-2({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_{i})({{{\boldsymbol{\partial}}}_{i}\cdot{{\boldsymbol{\partial}}}_{\epsilon_{i}})}\,.\end{aligned}$$ Here the differential operator ${{\boldsymbol{\partial}}}_{{{\epsilon}}_i}={{\partial}}/{{\partial}}{{\boldsymbol{\epsilon}}}_i$ acts on ${{\boldsymbol{\epsilon}}}_i$ as if it were unconstrained. Notice that $\boldsymbol{K}_{\epsilon_i}$ trivially acts on the correlator when $O_i$ is scalar.
#### Transverse and trace WT identities
On top of general symmetric traceless tensors, we consider a spin $1$ conserved current $J_\mu$ and the energy-momentum tensor $T_{\mu\nu}$. For simplicity, we focus on Abelian symmetries, even though extension to non-Abelian symmetries is straightforward. In classical theory, they satisfy the transverse WT identities, $$\begin{aligned}
\label{transverse_WT}
p_\mu J^\mu({{\boldsymbol{p}}})=0\,,
\quad
p_\mu T^{\mu\nu}({{\boldsymbol{p}}})=0\,,\end{aligned}$$ and the trace WT identity, $$\begin{aligned}
\label{trace_WT}
T_\mu{}^\mu({{\boldsymbol{p}}})=0\,,\end{aligned}$$ up to equations of motion. In quantum theory, these relations are corrected by local terms in general. However, there are no such corrections for the correlators we consider in this paper[^3]. Also, the trace WT identity may be modified when there exists a trace anomaly (see [@Bzowski:2015pba; @Bzowski:2018fql; @Bzowski:2017poo] for details). For technical simplicity, we assume that spinning operators other than the conserved currents in three-point functions have generic scaling dimensions, so that there appears no anomalous term in our analysis. Under these assumptions, the transverse and trace WT identities are simply equivalent to the classical ones -.
Two scalars and a general tensor {#sec:OOs}
================================
In the following sections we solve the conformal WT identities for correlators involving conserved currents and a general tensor. To elaborate on our methodology, we first review and expand the story for three-point functions of two scalars and a general tensor [@Isono:2018rrb]. We then generalize it to correlators involving one conserved current in the next section. Correlators involving two identical conserved currents will be given in Sec. \[sec:TTs\].
#### General ansatz
Let us consider three-point functions, $$\begin{aligned}
{{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'\,,\end{aligned}$$ of two scalars ${{\varphi}}_1,{{\varphi}}_2$ and a spin $s$ symmetric traceless tensor $O$. Throughout the paper, we assume that the tensor $O$ has a nonzero spin $s\neq0$ and a generic dimension $\Delta_3$, and thus is not conserved. From now on, we write three-point functions as functions of ${{\boldsymbol{p}}}_1$ and ${{\boldsymbol{p}}}_2$ without explicit dependence on ${{\boldsymbol{p}}}_3$. In other words we use ${{\boldsymbol{p}}}_3=-({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)$ to remove the ${{\boldsymbol{p}}}_3$-dependence. The special conformal WT identity then reads $$\begin{aligned}
\label{ssO-scWT}
{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s {{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' = 0\,,\end{aligned}$$ where notice that ${{\boldsymbol{K}}}_{{\epsilon}}$ does not appear since ${{\varphi}}_1,{{\varphi}}_2$ are scalar and there is no explicit ${{\boldsymbol{p}}}_3$-dependence. To solve the WT identities, we employ the following general ansatz[^4]: $$\begin{aligned}
\label{ssO-ansatz}
{{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'
= \sum_{n=0}^s \frac{1}{n!} \xi^n {{\zeta}}^{s-n} A_n(p_1,p_2,p_3) \,,\end{aligned}$$ where $p_i=|{{\boldsymbol{p}}}_i|$ and we introduced $$\begin{aligned}
\label{xizeta}
\xi={{\boldsymbol{\epsilon}}}_3\cdot{{\boldsymbol{p}}}_2\,, \quad {{\zeta}}={{\boldsymbol{\epsilon}}}_3\cdot({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)\,.\end{aligned}$$ Note also that $p_3$ should be understood as $p_3=|{{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2|$. The dilatation WT identity then implies that the function $A_n(p_1,p_2,p_3)$ must be homogeneous of degree ${{\Delta}}_t-2d-s$, $$\begin{aligned}
\label{ssO-An-homog}
A_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) = {{\lambda}}^{{{\Delta}}_t-2d-s} A_n(p_1,p_2,p_3)\,,\end{aligned}$$ where ${{\Delta}}_t={{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3$.
Reformulating WT identities
---------------------------
Our goal is now to determine the function $A_n$ by solving the WT identity with the ansatz and the condition . Let us first investigate the ${{\boldsymbol{b}}}$-dependence of the left hand side of the WT identity . Since it is Lorentz scalar, it is generally of the form, $$\begin{aligned}
\label{SCWT_schematic}
{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s {{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'
= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) P_1 + ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) P_2 + ({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) R\,.\end{aligned}$$ A concrete form of $P_i$ and $R$ is given shortly. The special conformal WT identity is then equivalent to $P_1=P_2=R=0$. Below we rewrite them in terms of the differential operators with respect to $\xi,{{\zeta}},p_1,p_2,p_3$ utilizing various formulae about ${{\boldsymbol{K}}}_s$ summarized in Appendix \[appsec:Ks\].
Let us start with the equations, $P_1=0$ and $P_2=0$. Using the formulae - with ${{\alpha}}={{\beta}}=0$, we may reduce them into the form, $$\begin{aligned}
0 &= [{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3) - 2p_3^{-2}{{\theta}}_{3}{{\zeta}}{{\partial}}_{{\zeta}}] ~ \sum_{n=0}^s \frac{1}{n!} \xi^n {{\zeta}}^{s-n} A_n(p_1,p_2,p_3) \,,
\label{ssO-P1} \\
0 &= [{{\mathcal{K}}}_2(\nu_2) - {{\mathcal{K}}}_3(\nu_3) - 2p_3^{-2}{{\theta}}_{3}{{\zeta}}({{\partial}}_\xi+{{\partial}}_{{\zeta}})] ~
\sum_{n=0}^s \frac{1}{n!} \xi^n {{\zeta}}^{s-n} A_n(p_1,p_2,p_3) \,, \label{ssO-P2}\end{aligned}$$ where we introduced the Euler operator ${{\theta}}_i=p_i{{\partial}}_{p_i}$ for the momentum $p_i$. The differential operators ${{\mathcal{K}}}_i(\nu_i)$ with respect to $p_i$ are defined as $$\begin{aligned}
{{\mathcal{K}}}_i(\nu_i) = p_i^{-2}{{\theta}}_{i}({{\theta}}_{i}-2\nu_i) \,,\end{aligned}$$ where $\nu_i={{\Delta}}_i-(d/2)$. A remark is that the variables $p_1,p_2,p_3,\xi,{{\zeta}}$ of the differential operators have to be regarded as independent variables.
On the other hand, we may express the equation $R=0$ by using the formula with ${{\alpha}}={{\beta}}=0$ as $$\begin{aligned}
\label{ssO-R}
0 = [{{\theta}}_x({{\theta}}_x+\Xi-1) - x({{\theta}}_x-s)({{\theta}}_x+{{\Delta}}_3-1)] \sum_{n=0}^s \frac{1}{n!}x^nA_n(p_1,p_2,p_3)\,,\end{aligned}$$ where $x=\xi/{{\zeta}}$ and ${{\theta}}_x=x{{\partial}}_x$. The differential operator $\Xi$ is defined by $$\begin{aligned}
\label{Xi_def}
\Xi = \frac{1}{2}\bigg( {{\Delta}}_1-{{\Delta}}_2+{{\Delta}}_3-s-{{\theta}}_{1}+{{\theta}}_{2}-\frac{p_1^2-p_2^2}{p_3^2}{{\theta}}_{3} \bigg)\,.\end{aligned}$$ Notice that $\Xi$ commutes with $x$ and $\theta_x$ in particular.
Solving WT identities
---------------------
We proceed to solving the WT identities. Below we first use the WT identities - to determine $A_s$. Using this $A_s$ as an initial condition for a recursion relation derived from Eq. , we provide an expression for the other $A_n$.
### Initial condition $A_s$ {#initial-condition-a_s .unnumbered}
In general, the WT identities - are rather complicated relations among $A_n$ with different $n$. However, their $\mathcal{O}(\zeta^0)$ terms provide differential equations containing $A_s$ only: $$\begin{aligned}
\label{ssO-As-eqs}
0 &= [{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3)]A_s(p_1,p_2,p_3)
= [{{\mathcal{K}}}_2(\nu_2) - {{\mathcal{K}}}_3(\nu_3)]A_s(p_1,p_2,p_3) \,.\end{aligned}$$ Together with the dilatation WT identity , we can solve these equations in terms of the so-called triple-$K$ integrals as $$\begin{aligned}
\label{ssO-As}
A_s = {{\mathcal{C}}}_A J_{s\{0,0,0\}}(p_1,p_2,p_3) \,,\end{aligned}$$ where ${{\mathcal{C}}}_A$ is an undetermined overall coefficient. We also introduced[^5] $$\begin{aligned}
J_{N\{k_{1},k_{2},k_{3}\}}(p_{1},p_{2},p_{3})
& =\int_{0}^{\infty}\frac{dz}{z}\:z^{2d-\Delta_{t}-k_{t}+N}
\prod_{i=1}^3\left(p_{i}z\right)^{\nu_{i}+k_{i}}K_{\nu_{i}+k_{i}}(p_{i}z)
\,,
\label{eq:triple-K_integral}\end{aligned}$$ where $K_\nu(z)$ is the Bessel function of the second kind and $k_t=k_1+k_2+k_3$. See Appendix \[app:tripleK\] for derivation of . There we also summarize various properties of the triple-$K$ integral .
### Recursion relations for $A_n$ {#recursion-relations-for-a_n .unnumbered}
We would then like to determine the other $A_n$. For this purpose, it is convenient to use the other WT identity , which provides a recursion relation, $$\begin{aligned}
(\Xi+n)A_{n+1} = (-s+n)({{\Delta}}_3-1+n)A_n\,. \label{00s-recursion}\end{aligned}$$ Here we emphasize that the differential operator $\Xi$ is acting only on $A_{n+1}$, hence we can express $A_n$ with a lower $n$ as a derivative of higher $A_n$. Indeed, it is easy to find[^6] $$\begin{aligned}
\label{An_OOs}
A_n = \frac{(\Xi+n)_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}A_s
={{\mathcal{C}}}_A\frac{(\Xi+n)_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}J_{s\{0,0,0\}}
\, ,\end{aligned}$$ where $\displaystyle(x)_n=x(x+1)\cdots (x+n-1)$ is the shifted factorial (also dubbed the Pochhammer symbol). Even though this already provides a compact expression, it is useful to explore an expression without differential operators. As given in Eq. , the differential operator $\Xi$ relates triple-$K$ integrals with different indices. Together with , we can always expand $A_n$ by triple-$K$ integrals with a fixed $k_3$ index as $$\begin{aligned}
\label{tripleK_expansion}
A_n=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s\}}\,.\end{aligned}$$ The coefficients $a_{n\{k_1,k_2\}}$ can then be determined algebraically by using Eq. . The result is that the matrix components with $k_1>0$ are all zero and the non-zero components are given by[^7]
$$\begin{aligned}
a_{s-n\{0,k\}}={{\mathcal{C}}}_A\frac{2^{n-k}(1\!-\!\tfrac{s-{{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3}{2})_{n-k}(1+\tfrac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2-\Delta_3-s)_{n-k}}
\quad
(0\leq k\leq n)\,.
\label{closed_OOS}\end{aligned}$$
### Residual WT identities {#residual-wt-identities .unnumbered}
So far, we have not checked yet if the full three-point function with satisfies the WT identities - (only a part of which was used to determine $A_s$). From the position space results, we know that there is only one free parameter ${{\mathcal{C}}}_A$, hence Eq. should be consistent with all the WT identities. To conclude this section, we explicitly show that it is indeed the case.
First, the WT identities - can be expressed in terms of $A_n$ as $$\begin{aligned}
\label{1-3_expand}
\left[{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3)\right]A_n&=2(s-n)p_3^{-2}\theta_3A_n\,,
\\*
\label{2-3_expand}
\left[{{\mathcal{K}}}_2(\nu_2) - {{\mathcal{K}}}_3(\nu_3)\right]A_n&=2(s-n)p_3^{-2}\theta_3A_n+2p_3^{-2}\theta_3A_{n+1}\,.\end{aligned}$$ As we mentioned earlier, $A_n$ can be expanded by triple-$K$ integrals as Eq. . From the formulae -, we find that the differential operators in Eqs. - act on each triple-$K$ integral as $$\begin{aligned}
\left[{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3)\right]J_{N\{k_1,k_2,k_3\}}&=-2k_1J_{N+1\{k_1-1,k_2,k_3\}}+2k_3J_{N+1\{k_1,k_2,k_3-1\}
}\,,
\\
p_3^{-2}\theta_3J_{N\{k_1,k_2,k_3\}}&=-J_{N+1\{k_1,k_2,k_3-1\}}\,.\end{aligned}$$ Then, we may translate the WT identities - into algebraic relations among the coefficients $a_{n\{k_1,k_2\}}$ as $$\begin{aligned}
k_1a_{n\{k_1,k_2\}}=0\,,
\quad
k_2a_{n\{k_1,k_2\}}=a_{n+1\{k_1,k_2-1\}}\,,\end{aligned}$$ which are indeed satisfied by our solution Eq. . The closed form - without differential operators and the explicit check of the full WT identities are new results of the present paper.
#### Summary of the section
In this way, the WT identities can be reformulated into several sets of differential equations. In the present case, some of them are used to derive recursion relations for functional coefficients in the decomposition , while the others provided their initial conditions. This step specifies three-point functions up to a free parameter in terms of triple-$K$ integrals and a differential operator relating triple-$K$ integrals with different indices. We can also expand it by triple-$K$ integrals without using differential operators, whose coefficients can be calculated algebraically. This expression translates the WT identities into algebraic relations among these coefficients. While we succeeded in providing a closed form of the coefficients in this section, it is not easy for more complicated correlators such as the ones discussed in Sec. \[sec:TTs\] to derive a closed form for general spin $s$ of the tensor. However, our algebraic approach is still tractable enough for concrete problems for a given spin $s$, e.g., with the help of computer software.
One scalar, one conserved current and a general tensor {#sec:JJs}
======================================================
We extend the argument in the previous section to solve the conformal WT identities for correlators with a scalar, a conserved current, and a general tensor.
Spin 1 conserved current $J_\mu$
--------------------------------
Let us begin by three-point functions of a scalar ${{\varphi}}$, a spin $1$ conserved current $J_\mu$, and a spin $s$ symmetric traceless operator $O$: $$\begin{aligned}
{{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,,\end{aligned}$$ where the helicity vector ${{\boldsymbol{\epsilon}}}_3$ is null to respect the tracelessness of $O$. Before solving the conformal WT identities, it is convenient to impose the conservation law $\partial_\mu J^\mu=0$ first, which is achieved by parameterizing three-point functions as $$\begin{aligned}
& {{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' =
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1) A + ({{\boldsymbol{\epsilon}}}_2\cdot\boldsymbol{\pi}_2\cdot{{\boldsymbol{\epsilon}}}_3) B \,.\end{aligned}$$ Here we introduced the transverse projector, $$\begin{aligned}
(\pi_i)_{\mu\nu} = {{\delta}}_{\mu\nu}-\frac{(p_i)_\mu(p_i)_\nu}{p_i^2} \label{pi}\,,\end{aligned}$$ and ${{\boldsymbol{\epsilon}}}_2\cdot\boldsymbol{\pi}_2\cdot{{\boldsymbol{p}}}_1=({{\epsilon}}_2)^\mu(\pi_2)_{\mu\nu}(p_1)^\nu$ for example. We further expand $A$ and $B$ as $$\begin{aligned}
\label{OJS_ansatz}
A = \sum_{n=0}^s \frac{1}{n!} \xi^n{{\zeta}}^{s-n} A_n(p_1,p_2,p_3) \,, \quad
B = \sum_{n=0}^{s-1} \frac{1}{n!} \xi^n{{\zeta}}^{s-1-n} B_n(p_1,p_2,p_3)\,,\end{aligned}$$ where $A_n$ and $B_n$ are scalar functions of $p_i$, and $\xi={{\boldsymbol{\epsilon}}}_3\cdot{{\boldsymbol{p}}}_2$ and ${{\zeta}}={{\boldsymbol{\epsilon}}}_3\cdot({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)$ as before.
### Reformulating WT identities
Just as we did in the previous section, we reformulate the conformal WT identities into differential equations for $A_n$ and $B_n$. First, the dilatation WT identity simply gives the homogeneity conditions on $A_n$ and $B_n$ as $$\begin{aligned}
A_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-1} A_n(p_1,p_2,p_3) \,,\label{01s-homoA}\\*
B_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+1} B_n(p_1,p_2,p_3) \,.\label{01s-homoB}\end{aligned}$$ We next consider the special conformal WT identities: $$\begin{aligned}
0 = ({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s + {{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_2})
{{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) \epsilon_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,.\end{aligned}$$ In the previous section we split the WT identities for the two scalar case into three differential equations based on how the transformation parameter ${{\boldsymbol{b}}}$ is contracted. See Eq. . In the present case we find seven equations as below.
Let us first summarize how the differential operator ${{\boldsymbol{K}}}_s$ acts on $A$ and $B$. Similarly to the calculation in the previous section, we find $$\begin{aligned}
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)A &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(-1,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(-1,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(-1,0)}A\,, \label{X-decomp} \\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)B &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P^{(1,1)}_1B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P^{(1,1)}_2B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(1,1)}B\,, \label{Y-decomp}\end{aligned}$$ where $P^{(\alpha,\beta)}_{1}$, $P^{(\alpha,\beta)}_{2}$, and $R^{(\alpha,\beta)}$ are differential operators defined by Eqs. , , and . In this language, we obtain four differential equations from the four terms proportional to ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_i$ ($i=1,2$) as $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) &: \quad
0 = \left( P_1^{(-1,0)} + 2p_3^{-2}{{\theta}}_{3} \right)A \,, \label{e2p1bp1} \\
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) &: \quad
0 = \left( P_2^{(-1,0)} + 2p_3^{-2}{{\theta}}_{3} \right)A \,. \label{e2p1bp2} \\
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) &: \quad
0 = P^{(1,1)}_1 B+ 2{{\partial}}_\xi A \,, \label{e2e3bp1} \\
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) &: \quad
0 = P^{(1,1)}_2 B\,. \label{e2e3bp2}\end{aligned}$$ There are also two terms proportional to ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$, which lead to $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) &: \quad
0 = \left( R^{(1,1)} + 2{{\partial}}_\xi+2{{\partial}}_{{\zeta}}\right)B \,, \label{e2e3be3} \\*
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1)({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) &: \quad
0 = \left( R^{(-1,0)} + 2{{\partial}}_{{\zeta}}\right)A + 2p_3^{-2}{{\theta}}_{3}B \,. \label{e2p1be3} \end{aligned}$$ Finally, the term proportional to ${{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{b}}}$ gives $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{b}}}) &: \quad 0 = \left[
-({{\Delta}}_1-d)+(\xi-{{\zeta}}){{\partial}}_\xi+{{\theta}}_{1}+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_2^2}(d-2-{{\theta}}_{2})
\right]A {{\nonumber}}\\
&\qquad\qquad +\left[\frac{\xi}{p_2^2}(d-2-{{\theta}}_{2})-\frac{{{\zeta}}}{p_3^2}{{\theta}}_{3}\right]B \,. \label{e2b} \end{aligned}$$ For later convenience, we classify these identities into the following two: First, we call the identities - associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1$, ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2$, and ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$ the primary WT identities, following the terminology of [@Bzowski:2013sza]. They are used to determine a functional form of correlators up to several free parameters. On the other hand, we call the identity associated with ${{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{b}}}$ the secondary WT identity, which provides a constraint on the free parameters.
### Solving WT identities
We now proceed to solving the seven differential equations -. Our strategy for this problem is the following: We start with the primary WT identities. As in the two scalar case, we use the first four equations - to determine $A_s$ and $B_{s-1}$. We then use Eqs. - to find recursion relations for $A_n$ and $B_n$, which specify the form of $A_n$ and $B_n$ up to two free parameters. Finally, we use the secondary WT identity to provide a relation between the two. Afterwards, we are left with a single free parameter, which is consistent with the position space result [@Costa:2011mg].
### Initial conditions from primary WT identities {#initial-conditions-from-primary-wt-identities .unnumbered}
Let us first use the $\mathcal{O}(\zeta^0)$ terms of Eqs. -, $$\begin{aligned}
0=& \left[K_1(\nu_1)-K_3(\nu_3)\right]A_s\,,\label{01s-primary1}\\
0=& \left[K_2(\nu_2)-K_3(\nu_3)\right]A_s\,,\label{01s-primary2}\\
0=& \left[K_1(\nu_1)-K_3(\nu_3)\right]B_{s-1}+2A_s\,,
\label{01s-primary3}\\
0=& \left[K_2(\nu_2)-K_3(\nu_3)\right]B_{s-1}\,,\label{01s-primary4}\end{aligned}$$ to determine the initial conditions, $A_s$ and $B_{s-1}$. Under the homogeneity condition , we solve Eqs. - in terms of the triple-$K$ integral as $$\begin{aligned}
A_s={{\mathcal{C}}}_AJ_{s+1\{0,0,0\}}(p_1,p_2,p_3)\,,\end{aligned}$$ where ${{\mathcal{C}}}_A$ is a free parameter. Next, we solve the other two equations. To find a particular solution for Eqs. -, it is convenient to employ the ansatz, $$\begin{aligned}
B_{s-1}=bJ_{s-1+k_t\{k_1,k_2,k_3\}}\end{aligned}$$ with a constant $b$, where $k_t=k_1+k_2+k_3$. Note that it satisfies the homogeneity condition . Using Eq. , we may reduce Eqs. - to the form, $$\begin{aligned}
0&=-2bk_1J_{s+k_t\{k_1-1,k_2,k_3\}}+2bk_3J_{s+k_t\{k_1,k_2,k_3-1\}}+2{{\mathcal{C}}}_AJ_{s+1\{0,0,0\}}\,,
\\*
0&=-2bk_2J_{s+k_t\{k_1,k_2-1,k_3\}}+2bk_3J_{s+k_t\{k_1,k_2,k_3-1\}}\,,\end{aligned}$$ which can be solved, e.g., by $(b,k_1,k_2,k_3)=({{\mathcal{C}}}_A,1,0,0)$. Adding the homogeneous solution $J_{s-1\{0,0,0\}}$, we find the general solution, $$\begin{aligned}
B_{s-1}={{\mathcal{C}}}_AJ_{s\{1,0,0\}}+{{\mathcal{C}}}_B J_{s-1\{0,0,0\}}\,,\end{aligned}$$ with a free parameter ${{\mathcal{C}}}_B$.
### Recursion relations from primary WT identities {#recursion-relations-from-primary-wt-identities .unnumbered}
Next, Eqs. - can be thought of as recursion relations for $A_n$ and $B_n$: $$\begin{aligned}
0=&\left(\Xi+\frac{3}{2}+n\right)B_{n+1}-(-s+1+n)(\Delta_3+n)B_n\,,\label{01s-Bn}\\*
0=&\left(\Xi-\frac{1}{2}+n\right)A_{n+1}-(-s+n)(\Delta_3-1+n)A_n+p_3^{-2}{{\theta}}_3B_n\,.
\label{01s-An}\end{aligned}$$ The first equation has the same form as , hence its solution is $$\begin{aligned}
\nonumber
B_n&=\frac{(\Xi+n+\frac{3}{2})_{s-1-n}}{(-s+1+n)_{s-1-n}({{\Delta}}_3+n)_{s-1-n}}B_{s-1}
\\
&=\frac{(\Xi+n+\frac{3}{2})_{s-1-n}}{(-s+1+n)_{s-1-n}({{\Delta}}_3+n)_{s-1-n}}\left(
{{\mathcal{C}}}_AJ_{s\{1,0,0\}}+{{\mathcal{C}}}_B J_{s-1\{0,0,0\}}
\right)\,.\end{aligned}$$ Similarly, the solution for Eq. is given by $$\begin{aligned}
\nonumber
A_n&=\frac{(\Xi+n-\frac{1}{2})_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}{{\mathcal{C}}}_AJ_{s+1\{0,0,0\}}\\
&\quad
+\sum_{t=0}^{s-1-n}\frac{(\Xi+n-\frac{1}{2})_t}{(-s+n)_{t+1}({{\Delta}}_3-1+n)_{t+1}}p_3^{-2}{{\theta}}_3B_{n+t}\,.\end{aligned}$$
### A closed form without differential operators {#a-closed-form-without-differential-operators .unnumbered}
Just as the two scalar case, it may be convenient to find expressions without the differential operators. Combining Eq. with Eq. , we can expand $A_n$ and $B_n$ in triple-$K$ integrals with a fixed $k_3$ index as $$\begin{aligned}
A_n&=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2+1\{k_1,k_2,n-s\}}\,,
\\
B_n&=\sum_{k_1,k_2\geq0}b_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s+1\}}\,.\end{aligned}$$ Using Eq. , we can compute the coefficients $a_{n\{k_1,k_2\}},b_{n\{k_1,k_2\}}$ algebraically. The result is summarized as follows: The coefficients $a_{n\{k_1,k_2\}}$ with $k_1>0$ and $b_{n\{k_1,k_2\}}$ with $k_1>1$ all vanish, and the non-zero coefficients are given by $$\begin{aligned}
a_{s-n\{0,k\}}&= b_{s-1-n\{1,k\}} \,,
\\*
b_{s-1-n\{0,k\}}&={{\mathcal{C}}}_B\frac{2^{n-k}(\frac{1}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k}(\frac{3}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2-\Delta_3-s)_{n-k}}\,,
\\*
b_{s-1-n\{1,k\}}&=
{{\mathcal{C}}}_A\frac{2^{n-k}(\frac{3}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k}(\frac{1}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\nonumber
\\*
\label{b1k_OJS}
&\quad
-{{\mathcal{C}}}_B\frac{2^{n-k-1}(\frac{3}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k-1}(\frac{3}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-k-1}}{k!(n-k-1)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\,,\end{aligned}$$ where $0 \leq k \leq n$ (the second line of Eq. is interpreted as zero for $n=k$). These coefficients satisfy the identities - out of the primary WT identities as they satisfy the relations - that are equivalent to the four identities.
### Secondary WT identity {#secondary-wt-identity .unnumbered}
We have determined three-point functions ${{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' $ up to the two free parameters ${{\mathcal{C}}}_A$ and ${{\mathcal{C}}}_B$. Finally, we solve Eq. and reduce the number of parameters to one. We again focus on its $\mathcal{O}(\zeta^0)$ terms. By taking the zero-momentum limit, ${{\boldsymbol{p}}}_3\rightarrow 0$, triple-$K$ integrals reduce to monomials of $p=p_1=p_2$ and thus Eq. is simplified as $$\begin{aligned}
0&={{\mathcal{C}}}_A \Bigr[-j_{s+2\{0,0,0\}}-j_{s+2\{0,-1,0\}}+(-\Delta_1+s+2)j_{s+1\{0,0,0\}} {{\nonumber}}\\*
&\quad
+s(d-2)j_{s\{1,0,0\}}+sj_{s+1\{1,-1,0\}}\Bigr]+s{{\mathcal{C}}}_B\Bigr[(d-2)j_{s-1\{0,0,0\}}+j_{s\{0,-1,0\}}\Bigr]\,,\end{aligned}$$ where $j_{N\{k_1,k_2,k_3\}}$ is a numerical number given in Eq. . The general solution is somewhat complicated, so that we provide two illustrative examples. For example, for a scalar with $\Delta_1=4,d=5$ dual to a $6$D bulk scalar with a conformal mass and a spinning operator with $s=2,\Delta_3=\frac{11}{2}$, we have $$\begin{aligned}
{{\mathcal{C}}}_B=-\frac{3245}{656}{{\mathcal{C}}}_A\,.\end{aligned}$$ Also, for a scalar with $\Delta_1=5, d=5$ dual to a $6$D massless bulk scalar and a spinning operator with $s=2,\Delta_3=\frac{11}{2}$, we find $$\begin{aligned}
{{\mathcal{C}}}_B=-\frac{1073}{656}{{\mathcal{C}}}_A\,.\end{aligned}$$
Energy-momentum tensor $T_{\mu\nu}$
-----------------------------------
We next consider three-point functions of a scalar $\phi$, the energy-momentum tensor $T^{\mu\nu}$, and a general tensor: $$\begin{aligned}
{{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2^{2}.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,.\label{TTS}\end{aligned}$$ The conservation law $\partial_\mu T^{\mu\nu}=0$ and the traceless condition $T^\mu_\mu=0$ result in the ansatz, $$\begin{aligned}
{{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2^{2}.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' =
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1^2) A +({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3) B +({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3^2) C\,,\label{OTS_ansatz} \,\end{aligned}$$ where we introduced the transverse-traceless projector, $$\begin{aligned}
(\Pi_i)_{\mu\nu\rho\sigma}=\frac{1}{2}\left\{(\pi_i)_{\mu\rho} (\pi_i)_{\nu\sigma}+(\pi_i)_{\mu\sigma} (\pi_i)_{\nu\rho}\right\}-\frac{1}{d-1}(\pi_i)
_{\mu\nu}(\pi)_{i\rho\sigma}\,,\end{aligned}$$ with $\pi_i$ being the transverse projector defined in Eq. . Here we used a shorthand notation, e.g., ${{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot{{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3 = ({{\epsilon}}_2)^\mu({{\epsilon}}_2)^\nu(\Pi_2)_{\mu\nu\alpha\beta}(\epsilon_3)^{\alpha}(p_1)^{\beta}$, for the tensorial contraction of the transverse-traceless projector. Note that since the last term in Eq. requires two or more ${{\boldsymbol{\epsilon}}}_3$, we have $C=0$ for $s=1$. We also parameterize $A$, $B$, and $C$ as $$\begin{aligned}
\label{OTS_ansatz1}
&A = \sum_{n=0}^s \frac{1}{n!} \xi^n{{\zeta}}^{s-n} A_n(p_1,p_2,p_3)\,, \\
&B = \sum_{n=0}^{s-1} \frac{1}{n!} \xi^n{{\zeta}}^{s-1-n} B_n(p_1,p_2,p_3)\,, \\
\label{OTS_ansatz3}
& C = \sum_{n=0}^{s-2} \frac{1}{n!} \xi^n{{\zeta}}^{s-2-n} C_n(p_1,p_2,p_3)\,.\end{aligned}$$
### Conformal WT identities
The homogeneity conditions following from the dilatation WT identity are $$\begin{aligned}
&A_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3)={{\lambda}}^{\Delta_t -2d-s-2} A_n(p_1,p_2,p_3)\,,\label{homo1-02s}\\
&B_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3)={{\lambda}}^{\Delta_t -2d -s} B_n(p_1,p_2,p_3)\,,\label{homo2-02s}\\
&C_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3)={{\lambda}}^{\Delta_t -2d-s+2} C_n(p_1,p_2,p_3)\,. \label{homo3-02s}\end{aligned}$$ On the other hand, the special conformal WT identity reads $$\begin{aligned}
0 = ({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s + {{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_2}){{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1) {{\epsilon}}_2^2.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,.\label{OTS_CWT}\end{aligned}$$ Just as we did in the previous subsection, we split into a set of differential equations just as before using the formula and the identities, $$\begin{aligned}
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)A &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(-2,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(-2,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(-2,0)}A
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)B &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(0,1)}B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(0,1)}B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(0,1)}B
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)C &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(2,2)}C+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(2,2)}C+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(2,2)}C
\,.\end{aligned}$$ See Eqs. ,, and for the definitions of the differential operators $P_{1,2}$ and $R$. After straightforward but tedious algebraic calculations, we obtain the following results.
#### Primary WT identities
First, the primary WT identities are $6$ identities associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1$ or ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2$, $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) &: \quad
0=\left( P_1^{(-2,0)}+4p_3^{-2}{{\theta}}_3 \right)A \,, \label{02s-initial1} \\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) &: \quad
0=\left( P_2^{(-2,0)}+4p_3^{-2}{{\theta}}_3 \right)A \,, \label{02s-initial2} \\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) &: \quad
0=\left( P_1^{(0,1)}+2p_3^{-2}{{\theta}}_3 \right)B+4{{\partial}}_\xi A \,, \label{02s-initial3} \\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) &: \quad
0=\left( P_2^{(0,1)}+2p_3^{-2}{{\theta}}_3 \right)B \,, \label{02s-initial4} \\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1) &: \quad
0=P_1^{(2,2)}C +2{{\partial}}_\xi B\,, \label{02s-initial5} \\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2) &: \quad
0=P_2^{(2,2)}C\,,\label{02s-initial6}\end{aligned}$$ and 3 identities associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$, $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) &: \quad
0= \left( R^{(-2,0)}+4{{\partial}}_\zeta \right)A + 2p_3^{-2}{{\theta}}_3 B\, ,\label{02s-recursion1}\\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) &:\quad
0= \left( R^{(0,1)}+2{{\partial}}_\xi + 4{{\partial}}_\zeta \right)B + 4 p_3^{-2}{{\theta}}_3 C\,, \label{02s-recursion2}\\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3) &:\quad
0= \left( R^{(2,2)}+4{{\partial}}_\xi +4{{\partial}}_\zeta \right)C\,. \label{02s-recursion3}\end{aligned}$$
#### Secondary WT identities
On the other hand, the secondary WT identities associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_2$ are the following two: $$\begin{aligned}
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{b}}}\,{{\boldsymbol{\epsilon}}}_3)&: \quad
0=\left[{{\theta}}_1+(\xi-\zeta){{\partial}}_\xi+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_2^2}(d-{{\theta}}_2)+d-\Delta_1\right]B {{\nonumber}}\\*
&\qquad\qquad
+2\left[\frac{\xi}{p_2^2}(d-{{\theta}}_2)-\frac{\zeta}{p_3^2}{{\theta}}_3\right]C\,,\label{02s-second1}\\
({{\boldsymbol{\epsilon}}}_2^2\cdot{{\boldsymbol{\Pi}}}_2\cdot{{\boldsymbol{b}}}\,{{\boldsymbol{p}}}_1)&: \quad
0=\left[{{\theta}}_1+(\xi-\zeta){{\partial}}_\xi+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_2^2}(d-{{\theta}}_2)+d-\Delta_1+1\right] A{{\nonumber}}\\
&\qquad\qquad
+\left[\frac{\xi}{p_2^2}(d-{{\theta}}_2)-\frac{\zeta}{p_3^2}{{\theta}}_3\right]B\,. \label{02s-second2}\end{aligned}$$
### Solving the WT identities
We now solve the reformulated WT identities. We start with the primary WT identities to specify a form of $A_n$, $B_n$, and $C_n$ with three free parameters. Similarly to the previous case, we use the first six primary WT identities - to determine the initial conditions. We then solve the recursion relations following from the other three - to obtain all $A_n$, $B_n$, and $C_n$. Finally, we use the secondary WT identities to derive two constraints on the three free parameters, leaving a single free parameter.
### Initial conditions from primary WT identities {#initial-conditions-from-primary-wt-identities-1 .unnumbered}
The $\mathcal{O}(\zeta^0)$ terms of Eqs. - provide differential equations for $A_s$, $B_{s-1}$, and $C_{s-2}$: $$\begin{aligned}
&0=\left[K_1(\nu_1)-K_3(\nu_3)\right]A_s \,, \label{02s-primary1}\\
&0=\left[K_2(\nu_2)-K_3(\nu_3)\right]A_s \,, \label{02s-primary2}\\
&0=\left[K_1(\nu_1)-K_3(\nu_3)\right]B_{s-1}+4A_s\,,
\label{02s-primary3}\\
&0=\left[K_2(\nu_2)-K_3(\nu_3)\right]B_{s-1}\,, \label{02s-primary4}\\
&0=\left[K_1(\nu_1)-K_3(\nu_3)\right]C_{s-2}+2B_{s-1}\,,
\label{02s-primary5}\\
&0=\left[K_2(\nu_2)-K_3(\nu_3)\right]C_{s-2}\,. \label{02s-primary6}\end{aligned}$$ Following the same strategy as the previous subsection, their general solutions consistent with the homogeneity conditions - are given in terms of triple-$K$ integrals as $$\begin{aligned}
A_s&={{\mathcal{C}}}_A J_{s+2\{0,0,0\} }\,,\\
B_{s-1}&=2{{\mathcal{C}}}_A J_{s+1\{1,0,0\}}+{{\mathcal{C}}}_B J_{s\{0,0,0\}}\,,\\
C_{s-2}&={{\mathcal{C}}}_A J_{s\{2,0,0\}}+{{\mathcal{C}}}_B J_{s-1\{1,0,0\}}+{{\mathcal{C}}}_C J_{s-2\{0,0,0\}}\,,\end{aligned}$$ up to three free parameters ${{\mathcal{C}}}_A$, ${{\mathcal{C}}}_B$, and ${{\mathcal{C}}}_C$.
### Recursion relations from primary WT identities {#recursion-relations-from-primary-wt-identities-1 .unnumbered}
To determine the other $A_n$, $B_n$, and $C_n$, we make use of recursion relations obtained from the identities -: $$\begin{aligned}
0=&\left(\Xi-1+n\right)A_{n+1}-(-s+n)(\Delta_3-1+n)A_n+p_3^{-2}{{\theta}}_3B_n\,.\label{02s-An}\\*
0=&\left(\Xi+1+n\right)B_{n+1}-(-s+1+n)(\Delta_3+n)B_n+2p_3^{-2}{{\theta}}_3C_n\,,\label{02s-Bn}\\*
0=&\left(\Xi+3+n\right)C_{n+1}-(-s+2+n)(\Delta_3+1+n)C_n\,.\label{02s-Cn}\end{aligned}$$ Their solutions are $$\begin{aligned}
\nonumber
A_n &=\frac{(\Xi+n-1)_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}A_{s} \\*
&\quad + \sum_{t=0}^{s-1-n}\frac{(\Xi+n-1)_t}{(-s+n)_{t+1}({{\Delta}}_3-1+n)_{t+1}}p_3^{-2}{{\theta}}_3B_{n+t}\,,\\
\nonumber
B_n &=\frac{(\Xi+n+1)_{s-1-n}}{(-s+1+n)_{s-1-n}({{\Delta}}_3+n)_{s-1-n}}B_{s-1} \\*
&\quad+ 2\sum_{t=0}^{s-2-n}\frac{(\Xi+n+1)_t}{(-s+1+n)_{t+1}({{\Delta}}_3+n)_{t+1}}p_3^{-2}{{\theta}}_3C_{n+t}\,,\\
C_n &=\frac{(\Xi+n+3)_{s-2-n}}{(-s+2+n)_{s-2-n}({{\Delta}}_3+1+n)_{s-2-n}}C_{s-2}\,.\end{aligned}$$
### A closed form without differential operators {#a-closed-form-without-differential-operators-1 .unnumbered}
Just as the two scalar case, it may be convenient to find expressions without the differential operators. Combining Eq. with Eq. , we can expand $A_n,B_n,C_n$ in triple-$K$ integrals with a fixed $k_3$ index as $$\begin{aligned}
A_n&=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2+2\{k_1,k_2,n-s\}}\,,
\\
B_n&=\sum_{k_1,k_2\geq0}b_{n\{k_1,k_2\}}J_{n+k_1+k_2+1\{k_1,k_2,n-s+1\}}\,,
\\
C_n&=\sum_{k_1,k_2\geq0}c_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s+2\}}\,.\end{aligned}$$ Using Eq. , we can compute the coefficients $a_{n\{k_1,k_2\}},b_{n\{k_1,k_2\}},c_{n\{k_1,k_2\}}$ algebraically. The result is summarized as follows: The coefficients $a_{n\{k_1,k_2\}}$ with $k_1>0$, $b_{n\{k_1,k_2\}}$ with $k_1>1$, and $c_{n\{k_1,k_2\}}$ with $k_1>2$ all vanish. Nonzero components of $a_n$ and $b_n$ are given in terms of $c_n$ as $$\begin{aligned}
a_{s-n\{0,k\}}&= c_{s-2-n\{2,k\}} \,,
\quad
b_{s-1-n\{0,k\}}= c_{s-2-n\{1,k\}} \,,
\quad
b_{s-1-n\{1,k\}}= 2c_{s-2-n\{2,k\}} \,.\end{aligned}$$ Finally, nonzero components of $c_n$ are[^8] $$\begin{aligned}
c_{s-2-n\{0,k\}}
&={{\mathcal{C}}}_C\frac{2^{n-k}(\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k}(2+\frac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\,,
\\*
c_{s-2-n\{1,k\}}&={{\mathcal{C}}}_B\frac{2^{n-k}(1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k}(1+\frac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\nonumber
\\
&\quad
-{{\mathcal{C}}}_C\frac{2^{n-k}(1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k-1}(2+\frac{d-s-{{\Delta}}_t}{2})_{n-k-1}}{k!(n-k-1)!(2\!-\!\Delta_3\!-\!s)_{n-k}}\,,
\label{c1k_OTS}
\\
c_{s-2-n\{2,k\}}&={{\mathcal{C}}}_A\frac{2^{n-k}(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k}(\frac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\nonumber
\\
&\quad
-{{\mathcal{C}}}_B
\frac{2^{n-k-1}(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k-1}(1+\frac{d-s-{{\Delta}}_t}{2})_{n-k-1}}{k!(n-k-1)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\nonumber
\\
\label{c2k_OTS}
&\quad
+{{\mathcal{C}}}_C \frac{2^{n-k-2}(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-k-2}(2+\frac{d-s-{{\Delta}}_t}{2})_{n-k-2}}{k!(n-k-2)!(2\!-\!\Delta_3\!-\!s)_{n-k}}
\,,\end{aligned}$$ where $0 \leq k \leq n$. These coefficients satisfy the identities - out of the primary WT identities as they satisfy the relations among the coefficients - that are equivalent to the six identities.
### Secondary WT identities {#secondary-wt-identities-1 .unnumbered}
Finally, we use the secondary WT identities and to provide constraints on ${{\mathcal{C}}}_A$, ${{\mathcal{C}}}_B$, and ${{\mathcal{C}}}_C$. In the zero momentum limit $p_3\rightarrow0$, these two equations are reduced to $$\begin{aligned}
0&=2{{\mathcal{C}}}_A\Bigr[-j_{s+2,\{0,0,0\}}-j_{s+2\{0,-1,0\}}+(s-\Delta_1-1)j_{s+1\{1,0,0\}}{{\nonumber}}\\
&\qquad \qquad
+(s-1)\left(d\,j_{s\{2,0,0\}}+j_{s-1\{2,-1,0\}}\right)\Bigr] {{\nonumber}}\\
&\quad+{{\mathcal{C}}}_B\Bigr[-j_{s+1,\{-1,0,0\}}-j_{s+1\{0,-1,0\}}+(s-\Delta_1-1)j_{s\{0,0,0\}}
{{\nonumber}}\\
&\qquad\qquad
+2(s-1)\left(d\,j_{s-1\{1,0,0\}}+j_{s\{1,-1,0\}}\right)\Bigr]{{\nonumber}}\\
&\quad+(s-1){{\mathcal{C}}}_C\left[d\,j_{s-2\{0,0,2\}}+j_{s-1\{0,-1,2\}}\right]
\label{TTS_secondary1}\end{aligned}$$ and $$\begin{aligned}
0&=2{{\mathcal{C}}}_A\Bigr[-j_{s+1,\{-1,0,0\}}-j_{s+1\{0,-1,0\}}+(s-\Delta_1+1)j_{s+2\{0,0,0\}}{{\nonumber}}\\
&\qquad\qquad
+d\,j_{s+1\{1,0,0\}}+j_{s+2\{1,-1,0\}}\Bigr] {{\nonumber}}\\
&\quad
+{{\mathcal{C}}}_B\left[d\,j_{s\{0,0,0\}}+j_{s+1\{0,-1,0\}}\right]\,.
\label{TTS_secondary2}\end{aligned}$$ For example, for a scalar operator dual to a 6-dimensional massless scalar field $\Delta_1=5$ and a spinning operator $s=2,\Delta_3=\frac{11}{2}$, $$\begin{aligned}
{{\mathcal{C}}}_B=-\frac{27707}{4016}{{\mathcal{C}}}_A\,, \quad {{\mathcal{C}}}_C=-\frac{29585757}{76657408}{{\mathcal{C}}}_A\,.\end{aligned}$$ Also for a scalar operator dual to a 6-dimesional scalar field with conformal mass $\Delta_1=4$ and a spinning operator with $s=2,\Delta_3=\frac{11}{2}$, $$\begin{aligned}
{{\mathcal{C}}}_B=-\frac{2757}{1424}{{\mathcal{C}}}_A\,, \quad {{\mathcal{C}}}_C=-\frac{2053563}{113920 }{{\mathcal{C}}}_A\,.\end{aligned}$$ We are now left with a single free parameter for $s\geq2$, which agrees with the position space result. Note that for $s=1$ we have an additional constraint ${{\mathcal{C}}}_C=0$ as we mentioned, hence three-point functions vanish.
Extension to correlators with two conserved currents {#sec:TTs}
====================================================
In this section we work on correlators with two conserved currents and a general tensor. Since there appear three polarization vectors, the special conformal WT identities become somewhat complicated compared to the previous section. However, we demonstrate that the strategy employed there can be carried over to the present problem without any obstruction: [First, we use the initial conditions and recursion relations following from the primary WT identities to determine correlators up to several free parameters. We then impose constraints on these parameters obtained from the secondary WT identities]{}. For illustration, we focus on three-point functions, $$\begin{aligned}
{{\langle}}{{\epsilon}}_1.J({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,, \label{11s}\end{aligned}$$ of two identical spin $1$ conserved currents and a general tensor in this section. An extension to the energy-momentum tensor is given in Appendix \[app:TTs\].
General ansatz
--------------
Following the strategy used in the single conserved current case, let us first provide the following general ansatz[^9]: $$\begin{aligned}
{{\langle}}{{\epsilon}}_1.J({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'
&=
({{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{p}}}_2)({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1)A {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{p}}}_2)({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3)B
+({{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{p}}}_1)C {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{\epsilon}}}_2\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_3)D
+({{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_2)E \,,
\label{11s-ansatz}\end{aligned}$$ where ${{\boldsymbol{\pi}}}_i$ is the transverse projector for ${{\boldsymbol{p}}}_i$, and ${{\boldsymbol{\epsilon}}}_1\cdot{{\boldsymbol{\pi}}}_1\cdot{{\boldsymbol{\pi}}}_2\cdot{{\boldsymbol{\epsilon}}}_2= (\epsilon_1)^\mu(\pi_1)_{\mu\nu}(\pi_2)^{\nu\rho}(\epsilon_2)_\rho$. Also we used the conservation law $\partial_\mu J^\mu=0$, which requires that ${{\boldsymbol{\epsilon}}}_{1,2}$ have to be contracted with the projector ${{\boldsymbol{\pi}}}_{1,2}$. We then expand each term as $$\begin{aligned}
A &= \sum_{n=0}^s\frac{1}{n!}\xi^n{{\zeta}}^{s-n}A_n(p_1,p_2,p_3) \,,
\label{expansionA}
\\
B &= \sum_{n=0}^{s-1}\frac{1}{n!}\xi^n{{\zeta}}^{s-1-n}B_n(p_1,p_2,p_3) \,, \\
C &= \sum_{n=0}^{s-1}\frac{1}{n!}\xi^n{{\zeta}}^{s-1-n}C_n(p_1,p_2,p_3) \,, \\
D &= \sum_{n=0}^{s-2}\frac{1}{n!}\xi^n{{\zeta}}^{s-2-n}D_n(p_1,p_2,p_3) \,, \\
E &= \sum_{n=0}^s\frac{1}{n!}\xi^n{{\zeta}}^{s-n}E_n(p_1,p_2,p_3) \,.
\label{expansionE}\end{aligned}$$ Since we are considering two identical conserved currents $J_\mu$, three-point functions are symmetric under the $1 \leftrightarrow 2$ exchange: $A$, $B+C$, $D$, and $E$ carry an even parity under the exchange, whereas $B-C$ has an odd parity. In particular, it requires the following relations among the initial conditions, $$\begin{aligned}
A^{}_s(p_1,p_2,p_3)&=(-1)^sA_s(p_2,p_1,p_3) \,, \label{11s-flip1} \\
B_{s-1}(p_1,p_2,p_3)&=(-1)^{s-1}C_{s-1}(p_2,p_1,p_3) \,, \label{11s-flip2} \\
D_{s-2}(p_1,p_2,p_3)&=(-1)^{s-2}D_{s-2}(p_2,p_1,p_3) \,, \label{11s-flip3} \\
E_s(p_1,p_2,p_3)&=(-1)^sE_s(p_2,p_1,p_3) \,. \label{11s-flip4} \end{aligned}$$ Similar relations hold for other terms in the expansion -.
Reformulating WT identities
---------------------------
We next reformulate the conformal WT identities. In terms of Eqs. -, the dilatation WT identity yields the following homogeneity conditions: $$\begin{aligned}
A_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-2} A_n(p_1,p_2,p_3) \,,
\label{JJS_d1}\\
B_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} B_n(p_1,p_2,p_3) \,, \\
C_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} C_n(p_1,p_2,p_3) \,, \\
D_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+2} D_n(p_1,p_2,p_3) \,, \\
E_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} E_n(p_1,p_2,p_3) \,.
\label{JJS_d5}\end{aligned}$$ On the other hand, to rewrite the special conformal WT identity, $$\begin{aligned}
0 = ({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s + {{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_1} + {{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_2}){{\langle}}{{\epsilon}}_1.J({{\boldsymbol{p}}}_1) {{\epsilon}}_2.J({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,,\label{JJS_CWT}\end{aligned}$$ it is convenient to note $$\begin{aligned}
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)A &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(-2,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(-2,0)}A+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(-2,0)}A
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)B &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(0,1)}B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(0,1)}B+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(0,1)}B
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)C &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(0,1)}C+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(0,1)}C+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(0,1)}C
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)D &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(2,2)}D+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(2,2)}D+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(2,2)}D
\,,
\\
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s)E &= ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(0,0)}E+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(0,0)}E+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(0,0)}E
\,.\end{aligned}$$ Using these formulae and Eq. , after a straightforward but lengthy calculation, we arrive at the following primary and secondary WT identities.
### Primary WT identities {#primary-wt-identities-1 .unnumbered}
First, the primary WT identities consist of 15 equations, 10 of which are associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_i$ ($i=1,2$) of the form, $$\begin{aligned}
0&=\left(P_1^{(-2,0)}+4p_3^{-2}{{\theta}}_3\right)A \,, \label{11s-initial1} \\
0&=\left(P_2^{(-2,0)}+4p_3^{-2}{{\theta}}_3\right)A \,, \label{11s-initial2} \\
0&=\left(P_1^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)B+2{{\partial}}_\xi A \,, \label{11s-initial3} \\*
0&=\left(P_2^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)B \,, \label{11s-initial4} \\
0&=\left(P_1^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)C \,, \label{11s-initial5} \\
0&=\left(P_2^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)C-2{{\partial}}_\xi A \,, \label{11s-initial6} \\
0&=P_1^{(2,2)}D+2{{\partial}}_\xi C \,, \label{11s-initial7} \\
0&=P_2^{(2,2)}D-2{{\partial}}_\xi B \,, \label{11s-initial8} \\
0&=P_1^{(0,0)}E+2A \,, \label{11s-initial9} \\
0&=P_2^{(0,0)}E+2A \,. \label{11s-initial10} \end{aligned}$$ We will use them to determine the initial conditions $A_s$, $B_{s-1}$, $C_{s-1}$, $D_{s-2}$, and $E_s$. The other 5 are associated to ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$ and given by $$\begin{aligned}
0&=\left(R^{(-2,0)}+2{{\partial}}_\xi+4{{\partial}}_{{\zeta}}\right)A+2p_3^{-2}{{\theta}}_3(B+C) \,, \label{11s-rr1} \\
0&=\left(R^{(0,1)}+4{{\partial}}_\xi+4{{\partial}}_{{\zeta}}\right)B+2p_3^{-2}{{\theta}}_3D \,, \label{11s-rr2} \\
0&=\left(R^{(0,1)}+4{{\partial}}_{{\zeta}}\right)C+2p_3^{-2}{{\theta}}_3D \,, \label{11s-rr3} \\
0&=\left(R^{(2,2)}+2{{\partial}}_\xi+4{{\partial}}_{{\zeta}}\right)D \,, \label{11s-rr4} \\
0&=R^{(0,0)}E+2(B+C) \,, \label{11s-rr5} \end{aligned}$$ which provide recursion relations among $A_n,...,E_n$. These primary WT identities fix the functional form of three-point functions up to five free parameters.
### Secondary WT identities {#secondary-wt-identities-2 .unnumbered}
On the other hand, the secondary WT identities consist of 4 equations associated with ${{\boldsymbol{b}}}\boldsymbol{\cdot\pi_i\cdot}{{\boldsymbol{\epsilon}}}_i$ ($i=1,2$) of the form, $$\begin{aligned}
0&=\left[ {{\theta}}_2+\xi{{\partial}}_\xi+d-{{\Delta}}_2+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_1^2}(d-2-{{\theta}}_1) \right]A {{\nonumber}}\\
&\quad\quad\quad
+\left[ \frac{{{\zeta}}-\xi}{p_1^2}(d-2-{{\theta}}_1)-\frac{{{\zeta}}}{p_3^2}{{\theta}}_3 \right]C
+\frac{1}{p_1^2}(d-2-{{\theta}}_1)E \,, \label{11s-sec1} \\
0&=\left[ {{\theta}}_1+(\xi-{{\zeta}}){{\partial}}_\xi+d-{{\Delta}}_1+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_2^2}(d-2-{{\theta}}_2) \right]A {{\nonumber}}\\*
&\quad\quad\quad
+\left[ \frac{\xi}{p_2^2}(d-2-{{\theta}}_2)-\frac{{{\zeta}}}{p_3^2}{{\theta}}_3 \right]B
+\frac{1}{p_2^2}(d-2-{{\theta}}_2)E \,, \label{11s-sec2} \\
0&=\left[ {{\theta}}_2+\xi{{\partial}}_\xi+d-{{\Delta}}_2+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_1^2}(d-2-{{\theta}}_1) \right]B
-C {{\nonumber}}\\*
&\quad\quad\quad
+\left[ \frac{{{\zeta}}-\xi}{p_1^2}(d-2-{{\theta}}_1)-\frac{{{\zeta}}}{p_3^2}{{\theta}}_3 \right]D
+{{\partial}}_\xi E \,, \label{11s-sec3} \\
0&=\left[ {{\theta}}_1+(\xi-{{\zeta}}){{\partial}}_\xi+d-{{\Delta}}_1+\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_2^2}(d-2-{{\theta}}_2) \right]C
-B
\nonumber
\\*
&\quad
+\left[ \frac{\xi}{p_2^2}(d-2-{{\theta}}_2)-\frac{{{\zeta}}}{p_3^2}{{\theta}}_3 \right]D
-{{\partial}}_\xi E \,. \label{11s-sec4} \end{aligned}$$ Note that symmetry under the exchange $1\leftrightarrow2$ implies that the first two equations are equivalent, and the third and the fourth also. We therefore have only two independent equations to be considered once the exchange symmetry is taken into account.
Solving WT identities
---------------------
We proceed to solving the reformulated WT identities. Just as in the single conserved current case, we use the primary WT identities to determine the form of three-point functions up to several free parameters and then use the secondary to provide constraints on them.
### Initial conditions from primary WT identities {#initial-conditions-from-primary-wt-identities-2 .unnumbered}
The 10 identities - provide a set of recursion relations for $A_n,...,E_n$, which are somewhat complicated to solve for general $n$. However, their $\mathcal{O}(\zeta^0)$ parts provide equations for $A_s$, $B_{s-1}$, $C_{s-1}$, $D_{s-2}$, and $E_s$ only: $$\begin{aligned}
0&=[K_1(\nu_1)-K_3(\nu_3)]A_s \,, \label{11s-primary1} \\
0&=[K_2(\nu_2)-K_3(\nu_3)]A_s \,, \label{11s-primary2} \\
0&=[K_1(\nu_1)-K_3(\nu_3)]B_{s-1}+2A_s \,,
\label{11s-primary3} \\
0&=[K_2(\nu_2)-K_3(\nu_3)]B_{s-1} \,, \label{11s-primary4} \\
0&=[K_1(\nu_1)-K_3(\nu_3)]C_{s-1} \,, \label{11s-primary5} \\
0&=[K_2(\nu_2)-K_3(\nu_3)]C_{s-1}-2A_s \,,
\label{11s-primary6} \\
0&=[K_1(\nu_1)-K_3(\nu_3)]D_{s-2}+2C_{s-1} \,,
\label{11s-primary7} \\
0&=[K_2(\nu_2)-K_3(\nu_3)]D_{s-2}-2B_{s-1} \,,
\label{11s-primary8} \\
0&=[K_1(\nu_1)-K_3(\nu_3)]E_s+2A_s \,, \label{11s-primary9} \\
0&=[K_2(\nu_2)-K_3(\nu_3)]E_s+2A_s \,. \label{11s-primary10} \end{aligned}$$ Their general solutions consistent with the homogeneity conditions - can be constructed in the same manner as the previous section as $$\begin{aligned}
A_s(p_1,p_2,p_3) &= {{\mathcal{C}}}_A J_{s+2(0,0,0)}(p_1,p_2,p_3) \,.\label{11s-A}
\\
B_{s-1}(p_1,p_2,p_3) &= {{\mathcal{C}}}_B J_{s(0,0,0)}(p_1,p_2,p_3)+{{\mathcal{C}}}_AJ_{s+1(1,0,0)}(p_1,p_2,p_3)\,,
\\
C_{s-1}(p_1,p_2,p_3) &= {{\mathcal{C}}}_C J_{s(0,0,0)}(p_1,p_2,p_3)-{{\mathcal{C}}}_AJ_{s+1(0,1,0)}(p_1,p_2,p_3)\,,
\\
D_{s-2}(p_1,p_2,p_3) &= {{\mathcal{C}}}_D J_{s-2(0,0,0)}(p_1,p_2,p_3)-{{\mathcal{C}}}_AJ_{s(1,1,0)}(p_1,p_2,p_3)
\nonumber\\*
&\quad
-{{\mathcal{C}}}_B J_{s-1(0,1,0)}(p_1,p_2,p_3)+{{\mathcal{C}}}_C J_{s-1(1,0,0)}(p_1,p_2,p_3)
\,,
\\
E_s(p_1,p_2,p_3) &= {{\mathcal{C}}}_E J_{s(0,0,0)}(p_1,p_2,p_3)-{{\mathcal{C}}}_AJ_{s+1(0,0,1)}(p_1,p_2,p_3)\end{aligned}$$ with five free parameters ${{\mathcal{C}}}_A$, ${{\mathcal{C}}}_B$, ${{\mathcal{C}}}_C$, ${{\mathcal{C}}}_D$, and ${{\mathcal{C}}}_E$.
### Recursion relations {#recursion-relations .unnumbered}
Next we solve the other five primary WT identities -, which provide the following recursion relations: $$\begin{aligned}
0=&\left(\Xi+n\right)A_{n+1}-(-s+n)(\Delta_3-1+n)A_n+p_3^{-2}{{\theta}}_3(B_n+C_n)\,,\label{11s-re1}\\
0=&\left(\Xi+2+n\right)B_{n+1}-(-s+1+n)(\Delta_3+n)B_n+p_3^{-2}{{\theta}}_3D_n\,,\label{11s-re2}\\
0=&\left(\Xi+n\right)C_{n+1}-(-s+1+n)(\Delta_3+n)C_n+p_3^{-2}{{\theta}}_3D_n\,,\label{11s-re3}\\
0=&\left(\Xi+2+n\right)D_{n+1}-(-s+2+n)(\Delta_3+1+n)D_n\,,\label{11s-re4}\\
0=&\left(\Xi+n\right)E_{n+1}-(-s+n)(\Delta_3-1+n)E_n+p_3^{-2}{{\theta}}_3(B_n+C_n)\,.\label{11s-re5}\end{aligned}$$ Their solutions are $$\begin{aligned}
A_n &= \frac{(\Xi+n)_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}A_s
\nonumber
\\*
&\quad
+\sum_{t=0}^{s-1-n}\frac{(\Xi+n)_t}{(-s+n)_{t+1}({{\Delta}}_3-1+n)_{t+1}}p_3^{-2}{{\theta}}_3(B_{n+t}+C_{n+t})\,,\\
B_n &= \frac{(\Xi+n+2)_{s-1-n}}{(-s+1+n)_{s-1-n}({{\Delta}}_3+n)_{s-1-n}}B_{s-1}
\nonumber
\\*
&\quad
+\sum_{t=0}^{s-2-n}\frac{(\Xi+n+2)_t}{(-s+1+n)_{t+1}({{\Delta}}_3+n)_{t+1}}p_3^{-2}{{\theta}}_3D_{n+t}\,,\\
C_n &= \frac{(\Xi+n)_{s-1-n}}{(-s+1+n)_{s-1-n}({{\Delta}}_3+n)_{s-1-n}}C_{s-1}
\nonumber
\\*
&\quad
+ \sum_{t=0}^{s-2-n}\frac{(\Xi+n)_t}{(-s+n+1)_{t+1}({{\Delta}}_3+n)_{t+1}}p_3^{-2}{{\theta}}_3D_{n+t}\,,\\
D_n &= \frac{(\Xi+n+2)_{s-2-n}}{(-s+2+n)_{s-2-n}({{\Delta}}_3+1+n)_{s-2-n}}D_{s-2}\,,\\
E_n &= \frac{(\Xi+n)_{s-n}}{(-s+n)_{s-n}({{\Delta}}_3-1+n)_{s-n}}E_s
\nonumber
\\*
&\quad
+ \sum_{t=0}^{s-1-n}\frac{(\Xi+n)_t}{(-s+n)_{t+1}({{\Delta}}_3-1+n)_{t+1}}p_3^{-2}{{\theta}}_3(B_{n+t}+C_{n+t})\,.\end{aligned}$$
### Constraints from $1\leftrightarrow2$ exchange symmetry {#constraints-from-1leftrightarrow2-exchange-symmetry .unnumbered}
Before moving on to the secondary WT identities, let us consider implications from the $1\leftrightarrow2$ exchange symmetry of the two conserved currents. The four conditions - are now translated into the constraints on the five free parameters as $$\begin{aligned}
\text{even $s$:}&\quad {{\mathcal{C}}}_B=-{{\mathcal{C}}}_C\,,
\\*
\text{odd $s$:}&\quad {{\mathcal{C}}}_B={{\mathcal{C}}}_C\,,
\quad{{\mathcal{C}}}_A={{\mathcal{C}}}_D={{\mathcal{C}}}_E=0\,.\end{aligned}$$ Therefore, there remain four free parameters for even spin $s$, whereas a single parameter for odd spin $s$. Note that the full correlator is consistent with the exchange symmetry once we impose its constraint on the initial conditions. It is because the WT identities used to derive the recursion relations are compatible with the exchange symmetry.
### Secondary WT identities {#secondary-wt-identities-3 .unnumbered}
Finally, let us impose the secondary WT identities. As we mentioned, there remain two independent secondary WT identities once we require the $1\leftrightarrow2$ exchange symmetry. An immediate observation is that three-point functions vanish for odd spin $s$ because there exist two constraints on a single parameter[^10]. On the other hand, for even spin $s$, there are four free parameters ${{\mathcal{C}}}_A$, ${{\mathcal{C}}}_B=-{{\mathcal{C}}}_C$, ${{\mathcal{C}}}_D$, and ${{\mathcal{C}}}_E$ after imposing the exchange symmetry. The secondary WT identities then provide two independent constraints on the four parameters, leaving two free parameters afterwards. A general form of the two conditions is somewhat complicated, but it is straightforward to derive these two in the same manner as the previous section. For example, for a spinning operator with $\Delta_3=\frac{11}{2},s=2$ in 5 dimension, we find $$\begin{aligned}
{{\mathcal{C}}}_B=-\frac{1}{2}{{\mathcal{C}}}_A-\frac{1}{2}{{\mathcal{C}}}_E\,,
\quad
{{\mathcal{C}}}_D=\frac{291}{128}{{\mathcal{C}}}_A+\frac{611}{192}{{\mathcal{C}}}_E\,.\end{aligned}$$
To summarize, we have provided an expression for three-point functions in terms of triple-$K$ integrals and differential operators. Just as the two scalar and one scalar cases, we may use Eqs. , , and to rewrite $A_n$, $B_n$, $C_n$, $D_n$, and $E_n$ in terms of triple-$K$ integrals into a form similar to the expansion . The coefficients can be found algebraically, e.g., with the help of computer software, even though we leave derivation of a closed form for general cases for future work. Also, we have not explicitly shown that our expression for the full correlator satisfies Eqs. - and -. Since it has the correct number of free parameters (known in position space [@Costa:2011mg]), it should satisfy them automatically. We have checked that it is indeed the case for several examples, leaving a general proof for future work. The same remark applies to correlators with two energy-momentum tensors studied in Appendix \[app:TTs\].
Summary and outlook
===================
In this paper we constructed conformal three-point functions with a symmetric traceless tensor and conserved currents in momentum space[^11]. Reformulating the conformal WT identities into the primary and secondary ones, we decomposed our problem into the following two steps: We first used the primary WT identities to derive recursion relations among functional coefficients in the tensor decomposition and determine the initial conditions. This step specifies three-point functions up to several free parameters. We then used the secondary WT identities to provide constraints on these parameters. Our expression is based on triple-$K$ integrals and a differential operator which relates triple-$K$ integrals with different indices. For correlators with no or one conserved current, we found explicit forms without the differential operator based on the expansion by triple-$K$ integrals.
We would like to conclude the paper with several future directions. First, our present work will be useful for the study of four and higher point functions with conserved currents. In [@Isono:2018rrb], based on symmetries and analyticity, a crossing symmetric basis of scalar four-point functions with a general intermediate operator was constructed. There, three-point functions of two scalars and a general tensor was useful because general tensors appear as intermediate states. It would be interesting to extend the construction to four-point functions with conserved currents. We expect that such a direction will be useful, e.g., for the Polyakov type bootstrap approach [@Polyakov:1974gs; @Sen:2015doa; @Gopakumar:2016wkt; @Gopakumar:2016cpb; @Gopakumar:2018xqi]. Another promising direction is cosmological applications. In [@Arkani-Hamed:2018kmz], in the same spirit as [@Isono:2018rrb], a basis of de Sitter four-point functions of a massless scalar was constructed and its cosmological implication was discussed. It would be interesting to construct a similar basis for primordial graviton non-Gaussianities extending our present work. We hope to report our progress in these directions elsewhere.
Acknowledgements {#acknowledgements .unnumbered}
================
H.I. is supported in part by the “CUniverse” research promotion project by Chulalongkorn University (grant reference CUAASC). T.N. is supported in part by JSPS KAKENHI Grant Numbers JP17H02894 and JP18K13539, and MEXT KAKENHI Grant Number JP18H04352.
Properties of the differential operators ${{\boldsymbol{K}}}_s$ and ${{\boldsymbol{K}}}_{\epsilon_i}$
=====================================================================================================
\[appsec:Ks\]
We summarize properties of the differential operators ${{\boldsymbol{K}}}_s$ and ${{\boldsymbol{K}}}_{\epsilon_i}$ appearing in the special conformal WT identity .
Rewriting ${{\boldsymbol{K}}}_s$
--------------------------------
Let us first consider ${{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s$ acting on the following function with free parameters ${{\alpha}}$ and ${{\beta}}$: $$\begin{aligned}
F = \sum_{n=0}^{s-{{\beta}}}\frac{1}{n!} \xi^n {{\zeta}}^{s-{{\beta}}-n} f_n(p_1,p_2,p_3)\,,\end{aligned}$$ where $\xi={{\boldsymbol{\epsilon}}}_3\cdot{{\boldsymbol{p}}}_2$ and ${{\zeta}}={{\boldsymbol{\epsilon}}}_3\cdot({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)$ with the null helicity vector ${{\boldsymbol{\epsilon}}}_3$, and the function $f_n$ is homogeneous of degree ${{\Delta}}_t-2d-s+{{\alpha}}$, $$\begin{aligned}
f_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) = {{\lambda}}^{{{\Delta}}_t-2d-s+{{\alpha}}}f_n(p_1,p_2,p_3)\,.\end{aligned}$$ Recall that $p_3$ should be understood as $p_3=|{{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2|$. Let us rewrite $({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s) F$ as a linear combination of ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1$, ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2$, and ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$: $$\begin{aligned}
({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s) F = ({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1)P_1^{(\alpha,\beta)}F+({{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2)P_2^{(\alpha,\beta)}F+({{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3)R^{(\alpha,\beta)}F \, ,\end{aligned}$$ where $P_1^{(\alpha,\beta)}$, $P_2^{(\alpha,\beta)}$, and $R^{(\alpha,\beta)}$ are differential operators given shortly. The equation $0=({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s) F$ is then equivalent to $$\begin{aligned}
P_1^{(\alpha,\beta)}F=P_2^{(\alpha,\beta)}F=R^{(\alpha,\beta)}F=0 \, .\end{aligned}$$ In terms of the differential operators with respect to $p_1$, $p_2$, $p_3$, $\xi$, and ${{\zeta}}$, the coefficients $P_1^{(\alpha,\beta)}F$ and $P_2^{(\alpha,\beta)}F$ are given by $$\begin{aligned}
P_1^{(\alpha,\beta)}F &= [{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3) + 2({{\alpha}}-{{\beta}}-{{\zeta}}{{\partial}}_{{\zeta}})p_3^{-2}{{\theta}}_{3}] ~ F \,, \label{albt-P1} \\
P_2^{(\alpha,\beta)}F &= [{{\mathcal{K}}}_2(\nu_2) - {{\mathcal{K}}}_3(\nu_3) + 2({{\alpha}}-{{\beta}}-{{\zeta}}({{\partial}}_\xi+{{\partial}}_{{\zeta}}))p_3^{-2}{{\theta}}_{3}] ~ F \,, \label{albt-P2}\end{aligned}$$ where the differential operator ${{\mathcal{K}}}_i(\nu_i)$ is defined by $$\begin{aligned}
{{\mathcal{K}}}_i(\nu_i) = p_i^{-2}{{\theta}}_{i}({{\theta}}_{i}-2\nu_i) \label{K_i}\, .\end{aligned}$$ On the other hand, $R^{(\alpha,\beta)}F$ may be expressed with the new variable $x=\xi/{{\zeta}}$ as $$\begin{aligned}
\label{albt-R}
R^{(\alpha,\beta)}F &= 2x^{-1}{{\zeta}}^{s-{{\beta}}-1}\left[{{\theta}}_x\left({{\theta}}_x+\Xi+\frac{{{\alpha}}}{2}-1\right) - x({{\theta}}_x+{{\beta}}-s)({{\theta}}_x+{{\Delta}}_3-1+{{\alpha}}-{{\beta}})\right] {{\nonumber}}\\
&\qquad \times \sum_{n=0}^{s-{{\beta}}} \frac{x^n}{n!} f_n\,,\end{aligned}$$ where the differential operator $\Xi$ is defined as $$\begin{aligned}
\Xi = \frac{1}{2}\bigg( {{\Delta}}_1-{{\Delta}}_2+{{\Delta}}_3-s-{{\theta}}_{1}+{{\theta}}_{2}-\frac{p_1^2-p_2^2}{p_3^2}{{\theta}}_{3} \bigg)\,.\end{aligned}$$ Note that when deriving these formulae, we used the homogeneity condition, $$\begin{aligned}
({{\theta}}_{1}+{{\theta}}_{2}+{{\theta}}_{3})f_n=({{\Delta}}_t-2d-s+{{\alpha}})f_n \,.\end{aligned}$$ It is also convenient to rewrite the expression as $$\begin{aligned}
\label{albt-R-rec}
R^{(\alpha,\beta)}F=2{{\zeta}}^{s-{{\beta}}-1} \sum_{n=0}^{s-{{\beta}}} \frac{x^n}{n!} \left[
(\Xi+\tfrac{{{\alpha}}}{2}+n)f_{n+1} - ({{\beta}}-s+n)({{\Delta}}_3-1+{{\alpha}}-{{\beta}}+n)f_n
\right]\end{aligned}$$ with $f_{s-{{\beta}}+1}=0$. We use it to derive recursion relations among $f_n$.
Formulae for ${{\boldsymbol{K}}}_s+{{\boldsymbol{K}}}_{\epsilon_i}$
-------------------------------------------------------------------
In the special conformal WT identity for correlators with conserved currents, the differential operator ${{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s+{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{\epsilon_i}$ acts on the projectors $(\pi_i)_{\mu\nu}$ and $(\Pi_i)_{\mu\nu\rho{{\sigma}}}$. To massage the complicated identities, it is convenient to find identities about the commutation relation of ${{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s+{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{\epsilon_i}$ and the projectors.
Let us first consider correlators with a spin $1$ conserved current with momentum ${{\boldsymbol{p}}}_i$. The correlator has the tensor structure $({{\boldsymbol{\epsilon}}}_i\cdot{{\boldsymbol{\pi}}}_i)_{\mu}X^{\mu}$, where $X^{\mu}$ is an arbitrary vector function without ${{\boldsymbol{\epsilon}}}_i$ dependence. We also used the shorthand notation $({{\boldsymbol{\epsilon}}}_i\cdot{{\boldsymbol{\pi}}}_i)_{\mu}=({{\epsilon}}_i)^\nu(\pi_i)_{\nu\mu}$. Then, the action of ${{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s+{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{\epsilon_i}$ reads $$\begin{aligned}
&({{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s}+{{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{\epsilon_i}})({{{\boldsymbol{\epsilon}}}_i\cdot{{\boldsymbol{\pi}}}_i})_{\mu}X^\mu\nonumber\\*
&~~ = ({{\boldsymbol{\epsilon}}}_i\cdot{{\boldsymbol{\pi}}}_i)_{\mu}({{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s}) X^{\mu}\nonumber \\*
&~~~~ +2\left({{{\boldsymbol{\epsilon}}}_{i}\cdot{{\boldsymbol{\pi}}}_i\cdot{{{\boldsymbol{\partial}}}_{i}}}\right)({{\boldsymbol{b}}}\cdot {{\boldsymbol{X}}})
-2({{{\boldsymbol{b}}}\cdot{{\boldsymbol{\pi}}}_i\cdot{{\boldsymbol{\epsilon}}}_{i}})({{\boldsymbol{\partial}}}_i\cdot {{\boldsymbol{X}}})+\frac{2(d-2)}{p_i^2}({{{\boldsymbol{b}}}\cdot{{\boldsymbol{\pi}}}_i\cdot{{\boldsymbol{\epsilon}}}_{i}})({{\boldsymbol{p}}}_2 \cdot {{\boldsymbol{X}}}) \,, \label{K_pi}\end{aligned}$$ where ${{\boldsymbol{\partial}}}_i={{\partial}}/{{\partial}}{{\boldsymbol{p}}}_i$.
Next we turn to correlators with the energy-momentum tensor. The correlator has the tensor structure $({{{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i})_{\mu\nu}X^{\mu\nu}$, where $X^{\mu\nu}$ is an arbitrary tensorial function without ${{\boldsymbol{\epsilon}}}_i$ dependence. Also we used the notation $({{{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i})_{\mu\nu}=({{\epsilon}}_i)^\rho({{\epsilon}}_i)^{{\sigma}}(\Pi_i)_{\rho{{\sigma}}\mu\nu}$. We then find $$\begin{aligned}
&({{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s}+{{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{\epsilon_i}})({{{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i})_{\mu\nu}X^{\mu\nu}\nonumber\\
&=({{{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i})_{\mu\nu}({{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s})X^{\mu\nu} \nonumber\\
&\quad
+2\left[({{{\boldsymbol{\epsilon}}}_{i}^2\cdot{{\boldsymbol{\Pi}}}_i\cdot{{{\boldsymbol{\partial}}}_{i}}})_{\mu}b_{\nu} - ({{{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i\cdot {{\boldsymbol{b}}}})_\mu({{\partial}}_i)_\nu
+d({{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i\cdot {{\boldsymbol{b}}})_\mu (p_i)_\nu\right]\label{K-Pi}(X^{\mu\nu}+X^{\nu\mu})\,,\end{aligned}$$ where $({{\partial}}_i)_\nu={{\partial}}/{{\partial}}(p_i)^\nu$ and $({{\boldsymbol{\epsilon}}}_i^2\cdot{{\boldsymbol{\Pi}}}_i\cdot {{\boldsymbol{a}}})_\mu=({{\epsilon}}_i)^\rho({{\epsilon}}_i)^{{\sigma}}(\Pi_i)_{\rho{{\sigma}}\nu\mu}a^\nu$.
Here let us recall that the conservation law is compatible with conformal symmetry only when $\Delta=d-2+s$ without anomalous dimension. Indeed, we have used this relation to derive Eqs. and , where the helicity vector ${{\boldsymbol{\epsilon}}}_i$ is coupled to projectors on both sides consistently.
Triple-$K$ integrals {#app:tripleK}
====================
In this appendix we summarize various properties of triple-$K$ integrals.
Definition
----------
Let us begin with the special conformal WT identities, $$\begin{aligned}
0&= [{{\mathcal{K}}}_1(\nu_1) - {{\mathcal{K}}}_3(\nu_3)] ~ F(p_1,p_2,p_3) \,,
\label{triple-K1} \\
0&= [{{\mathcal{K}}}_2(\nu_2) - {{\mathcal{K}}}_3(\nu_3)] ~ F(p_1,p_2,p_3) \,, \label{triple-K2}\end{aligned}$$ and the dilatation WT identity, $$\begin{aligned}
F(\lambda p_1,\lambda p_2,\lambda p_3)=\lambda^{{{\Delta}}_t-2d}F(p_1,p_2,p_3)\,,
\label{triple-K3}\end{aligned}$$ of scalar three-point functions, where ${{\mathcal{K}}}_i(\nu_i)=p_i^{-2}{{\theta}}_{i}({{\theta}}_{i}-2\nu_i)$. If we require that there is no singularity in the domain $p_i>0$, their solution can uniquely be determined up to an overall constant as [@Bzowski:2013sza] $$\begin{aligned}
\label{triple-K_simple}
F(p_1,p_2,p_3)=\int_{0}^{\infty}\frac{dz}{z}z^{\Delta_t-2d}\prod_{i=1}^3(p_iz)^{\nu_i}K_{\nu_i}(p_iz)\,.\end{aligned}$$ Here $K_{\nu}(x)$ is the modified Bessel function of the second kind, which we call the Bessel $K$ function. It is defined by a hypergeometric series[^12], $$\begin{aligned}
K_\nu(x)=\frac{\pi}{2\sin(\pi\nu)}\left[I_{-\nu}(x)-I_{\nu}(x)\right]
\quad
{\rm with}
\quad
I_\nu(x)=\sum_{j=0}^{\infty}\frac{1}{j!\Gamma (\nu +j+1)}\left(\frac{x}{2}\right)^{\nu +2j}\,,\end{aligned}$$ and satisfies Bessel’s equation, $$\begin{aligned}
\left(\theta_x^2-\nu^2\right)K_\nu(x)=x^2K_\nu(x)\,.\end{aligned}$$ Note that the integral is convergent only when $|{\rm Re}\,\nu_1|+|{\rm Re}\,\nu_2|+|{\rm Re}\,\nu_3|<\frac{d}{2}$. Otherwise, there appears a singularity near $z=0$ and we need to perform analytic continuation [@Bzowski:2015pba; @Bzowski:2015yxv], which may be carried out, e.g., by introducing the Pochhammer contour.
To construct three-point functions with tensors, it is convenient to generalize the integral to the following triple-$K$ integral with indices $N$ and $k_i$ ($i=1,2,3$): $$\begin{aligned}
J_{N\{k_1,k_2,k_3\}}(p_{1},p_{2},p_{3})
& =\int_{0}^{\infty}\frac{dz}{z}\:z^{2d-\Delta_{t}-k_t+N}
\prod_{i=1}^3\left(p_{i}z\right)^{\nu_{i}+k_i}K_{\nu_{i}+k_i}(p_{i}z)
\,,\end{aligned}$$ where $k_t=k_1+k_2+k_3$. It satisfies the differential equations, $$\begin{aligned}
0&= [{{\mathcal{K}}}_1(\nu_1+k_1) - {{\mathcal{K}}}_3(\nu_3+k_3)] ~ J_{N\{k_1,k_2,k_3\}}(p_{1},p_{2},p_{3}) \,,
\\
0&= [{{\mathcal{K}}}_2(\nu_2+k_2) - {{\mathcal{K}}}_3(\nu_3+k_3)] ~ J_{N\{k_1,k_2,k_3\}}(p_{1},p_{2},p_{3}) \,,\end{aligned}$$ and the homogeneity conditions, $$\begin{aligned}
J_{N\{k_1,k_2,k_3\}}(\lambda p_1,\lambda p_2,\lambda p_3)=\lambda^{{{\Delta}}_t+k_t-N-2d}J_{N\{k_1,k_2,k_3\}}(p_{1},p_{2},p_{3})\,.\end{aligned}$$
Differential operators acting on triple-$K$ integrals
-----------------------------------------------------
We then demonstrate how various differential operators act on triple-$K$ integrals, which is useful when we solve the WT identities in the main text. The origin of all the formulae below is the following action of the Euler operator on the Bessel $K$ function: $$\begin{aligned}
\theta_x\big(x^\nu K_\nu(x)\big)=-x^2\big(x^{\nu-1} K_{\nu-1}(x)\big)=
2\nu x^\nu K_\nu(x)-x^{\nu+1}K_{\nu+1}(x)\,.\end{aligned}$$ This can be translated into $$\begin{aligned}
\label{Euler_on_tripleK}
{{\theta}}_1J_{N\{k_1,k_2,k_3\}}=-p_1^2J_{N+1\{k_1-1,k_2,k_3\}}
=2(\nu_1+k_1)J_{N\{k_1,k_2,k_3\}} -J_{N+1\{k_1+1,k_2,k_3\}}
\,.\end{aligned}$$ Here and in what follows we occasionally omit explicit indication of momentum dependence. It then follows that $$\begin{aligned}
\label{K_on_tripleK}
{{\mathcal{K}}}_1(\nu_1)J_{N\{k_1,k_2,k_3\}}=-2k_{1}J_{N+1\{k_{1}-1,k_{2},k_{3}\}}+J_{N+2\{k_{1},k_{2},k_{3}\}}\,.\end{aligned}$$ Similar relations hold for $\theta_{2,3}$ and ${{\mathcal{K}}}_{2,3}$. Combining Eq. with the homogeneity condition of the triple-$K$ integral, $$\begin{aligned}
({{\theta}}_1+{{\theta}}_2+{{\theta}}_3)J_{N\{k_1,k_2,k_3\}}(p_1,p_2,p_3)&=\lambda\frac{d}{d\,\lambda}J_{N\{k_1,k_2,k_3\}}(\lambda p_1,\lambda p_2,\lambda p_3)\Big|_{\lambda=1} {{\nonumber}}\\
&=(\Delta_t+k_t-N-2d)J_{N\{k_1,k_2,k_3\}}(p_1,p_2,p_3)\,,\label{homo-K}\end{aligned}$$ we also find identities among the nearest neighbors, $$\begin{aligned}
\label{k3_down}
&(\Delta_t+k_t+N-d)J_{N\{k_1,k_2,k_3\}} \nonumber\\
&\qquad=J_{N+1\{k_1+1,k_2,k_3\}}+J_{N+1\{k_1,k_2+1,k_3\}}+J_{N+1\{k_1,k_2,k_3+1\}} \end{aligned}$$ and $$\begin{aligned}
\label{k3_up}
&(\Delta_t+k_t-N-2d)J_{N\{k_1,k_2,k_3\}} \nonumber\\
&\qquad=-p_1^2J_{N+1\{k_1-1,k_2,k_3\}}-p_2^2J_{N+1\{k_1,k_2-1,k_3\}}-p_3^2J_{N+1\{k_1,k_2,k_3-1\}}\,.\end{aligned}$$ Finally, we provide the action of the differential operator $\Xi$ given in Eq. on triple-$K$ integrals: $$\begin{aligned}
&(\Xi+a) J_{N\{k_1,k_2,k_3\}}
\nonumber
\\
&=\left(\frac{-\nu_1+\nu_2+{{\Delta}}_3-s}{2}-k_1+k_2+a\right)J_{N\{k_1,k_2,k_3\}}
\nonumber
\\
&\quad
-(\nu_1+k_1+1)J_{N\{k_1+1,k_2,k_3-1\}}+(\nu_2+k_2+1)J_{N\{k_1,k_2+1,k_3-1\}}
\nonumber
\\
&\quad
+\frac{J_{N+1\{k_1+1,k_2,k_3\}}+J_{N+1\{k_1+2,k_2,k_3-1\}}-J_{N+1\{k_1,k_2+1,k_3\}}-J_{N+1\{k_1,k_2+2,k_3-1\}}}{2}
\nonumber
\\
&=\left(\frac{-{{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3-s}{2}-k_1+k_2+a\right)({{\Delta}}_t+k_t+N-d-2)J_{N-1\{k_1,k_2,k_3-1\}}
\nonumber
\\
&\quad
+\frac{k_1-k_2+k_3+N+s-2-2a}{2}J_{N\{k_1+1,k_2,k_3-1\}}
\nonumber
\\
&\quad
+\frac{k_1-k_2-k_3-N+s-2\Delta_3+2-2a}{2}J_{N\{k_1,k_2+1,k_3-1\}}
\label{Xi_on_tripleK}
\,,\end{aligned}$$ where we used Eq. at the second equality.
Zero-momentum limit {#subsec:zero}
-------------------
When solving the secondary WT identities in the main text, we use the zero-momentum limit, ${{\boldsymbol{p}}}_3\rightarrow 0$, of triple-$K$ integrals. In this limit, triple-$K$ integrals reduce to monomials of $p=p_1=p_2$. We write its coefficient as $j_{N\{k_1,k_2,k_3\}}$: $$\begin{aligned}
&j_{N\{k_1,k_2,k_3\}}=p^{-\Delta_t-k_t+2d+N}\lim_{{{\boldsymbol{p}}}_3\rightarrow 0}J_{N\{k_1,k_2,k_3\}} {{\nonumber}}\\*
&=\frac{2^{\frac{d}{2}-1+N}\Gamma\left(\Delta_{3}-\frac{d}{2}+k_{3}\right)}{\Gamma\left(d-\Delta_{3}-k_{3}+N\right)}\prod_{u,v=\pm 1}\Gamma\left(\frac{d-\Delta_{3}-k_{3}+N+u\left(\nu_{1}+k_{1}\right)+v\left(\nu_{2}+k_{2}\right)}{2}\right) \,. \label{zero-p3}\end{aligned}$$ To derive this expression, we have used $$\begin{aligned}
\lim_{x\rightarrow 0}K_\nu (x)=\frac{\Gamma (\nu)2^{\nu -1}}{x^\nu}\quad \text{for}\quad\nu\,\cancel{\in}\,\mathbb{Z} \quad \text{and}\quad\nu > 0 \,,\end{aligned}$$ and the formula [@GR], $$\begin{aligned}
\int_0^{\infty}dx\;x^{-\lambda} K_{\mu}(ax)K_{\nu}(bx)
&=\frac{2^{-2-\lambda}a^{-\nu+\lambda-1}b^\nu}{\Gamma (1-\lambda)}\prod_{u,v=\pm 1}\Gamma\left(\frac{1-\lambda +u\mu+v\nu}{2}\right) \nonumber\\
&\times {}_2F_1\left(\frac{1-\lambda+\mu+\nu}{2},\frac{1-\lambda-\mu+\nu}{2};1-\lambda;1-\frac{b^2}{a^2}\right) \,,\end{aligned}$$ for ${\rm Re}\,(a+b)>0$ and ${\rm Re}\,\lambda<1-|{\rm Re}\,\mu|-|{\rm Re}\,\nu|$, Here ${}_2F_1(a,b;c;x)$ is Gauss’s hypergeometric function, $$\begin{aligned}
{}_2F_1(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{x^n}{n!} \,.\end{aligned}$$
Another derivation of the closed form {#app:another}
=====================================
In this appendix we provide another derivation of the closed form of three-point functions, ${{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$, ${{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1){{\epsilon}}_2.J({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$, and ${{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1){{\epsilon}}_2^2.T({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$.
Two scalars and one tensor
--------------------------
Let us begin with three-point functions ${{\langle}}{{\varphi}}_1({{\boldsymbol{p}}}_1){{\varphi}}_2({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$ of two scalars and one tensor. Our starting point is the triple-$K$ expansion, $$\begin{aligned}
A_n=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s\}}\,,\end{aligned}$$ of the coefficient function $A_n$ ($0\leq n \leq s$) given in Eq. . As we mentioned earlier, the WT identities - can be rephrased in terms of the coefficients $a_{n\{k_1,k_2\}}$ as $$\begin{aligned}
\label{initialWT_OOS}
k_1a_{n\{k_1,k_2\}}=0\,,
\quad
k_2a_{n\{k_1,k_2\}}=a_{n+1\{k_1,k_2-1\}}\,.\end{aligned}$$ In the main text we showed that the closed form obtained algebraically indeed satisfies Eq. . Instead, here we use these two conditions to determine $a_n$: The former requires that the nonzero coefficients appear only at $k_1=0$, whereas the latter implies $$\begin{aligned}
\label{a_n_OOS}
a_{n\{0,k\}}=\frac{a_{n+k\{0,0\}}}{k!}\,.\end{aligned}$$ Therefore, our task is now reduced to determining $a_{n\{0,0\}}$. We then use the recursion relation , taking the form with $\alpha=0$. In Appendix \[subsec:formulae\_rec\] we solve Eq. for general $\alpha$. Applying the general solution , we find $$\begin{aligned}
a_{s-n\{0,0\}}={{\mathcal{C}}}_A\frac{2^{n}(1-\tfrac{s-{{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3}{2})_{n}(1+\tfrac{d-s-{{\Delta}}_t}{2})_{n}}{n!(2-\Delta_3-s)_{n}}\,,\end{aligned}$$ where we used $a_{s\{0,0\}}={{\mathcal{C}}}_A$. Combining with Eq. , we conclude that nonzero coefficients $a_n$ are[^13] $$\begin{aligned}
a_{s-n\{0,k\}}={{\mathcal{C}}}_A\frac{2^{n-k}(1\!-\!\tfrac{s-{{\Delta}}_1+{{\Delta}}_2+{{\Delta}}_3}{2})_{n-k}(1+\tfrac{d-s-{{\Delta}}_t}{2})_{n-k}}{k!(n-k)!(2-\Delta_3-s)_{n-k}}
\quad
(0\leq k\leq n)\,.\end{aligned}$$
Single spin $1$ conserved current
---------------------------------
We next consider three-point functions, ${{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1){{\epsilon}}_2.J({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$, with a spin $1$ conserved current. In this case we expand the coefficient functions in Eq. as $$\begin{aligned}
A_n&=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2+1\{k_1,k_2,n-s\}}\,,
\\
B_n&=\sum_{k_1,k_2\geq0}b_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s+1\}}\,.\end{aligned}$$ First, the primary WT identities - are rephrased as $$\begin{aligned}
&k_1a_{n\{k_1,k_2\}}=0\,,
&&k_2a_{n\{k_1,k_2\}}=a_{n+1\{k_1,k_2-1\}}\,, \label{01s-initial-conseq1}
\\
&k_1b_{n\{k_1,k_2\}}=a_{n+1\{k_1-1,k_2\}}\,,
&&
k_2b_{n\{k_1,k_2\}}=b_{n+1\{k_1,k_2-1\}}\,, \label{01s-initial-conseq2}\end{aligned}$$ which imply that nonzero coefficients are $$\begin{aligned}
\label{ab_nonzero}
a_{n\{0,k\}}=\frac{b_{n-1+k\{1,0\}}}{k!}\,,
\quad
b_{n\{0,k\}}=\frac{b_{n+k\{0,0\}}}{k!}\,,
\quad
b_{n\{1,k\}}=\frac{b_{n+k\{1,0\}}}{k!}\,.\end{aligned}$$ Our task is now to determine $b_{n\{0,0\}}$ and $b_{n\{1,0\}}$. For this purpose, we use the recursion relation , which is of the form with $\alpha=1$. Applying the general solutions -, we find $$\begin{aligned}
b_{s-1-n\{0,0\}}&={{\mathcal{C}}}_B\frac{2^{n}(\frac{1}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n}(\frac{3}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n}}{n!(2-\Delta_3-s)_{n}}\,,
\\
b_{s-1-n\{1,0\}}&
={{\mathcal{C}}}_A\frac{2^{n}(\frac{3}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n}(\frac{1}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n}}{n!(2\!-\!\Delta_3\!-\!s)_{n}}
\nonumber
\\
&\quad
-{{\mathcal{C}}}_B\frac{2^{n-1}(\frac{3}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-1}(\frac{3}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-1}}{(n-1)!(2\!-\!\Delta_3\!-\!s)_{n}}
\,,\end{aligned}$$ where we used $b_{s-1\{0,0\}}={{\mathcal{C}}}_B$ and $b_{s-1\{1,0\}}={{\mathcal{C}}}_A$. All the other coefficients are obtained by using Eq. .
Single energy-momentum tensor
-----------------------------
Finally, let us consider three-point functions, ${{\langle}}{{\varphi}}({{\boldsymbol{p}}}_1){{\epsilon}}_2^2.T({{\boldsymbol{p}}}_2){{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}'$, with an energy-momentum tensor. First, we expand the coefficient functions in Eqs. - as $$\begin{aligned}
A_n&=\sum_{k_1,k_2\geq0}a_{n\{k_1,k_2\}}J_{n+k_1+k_2+2\{k_1,k_2,n-s\}}\,,
\\*
B_n&=\sum_{k_1,k_2\geq0}b_{n\{k_1,k_2\}}J_{n+k_1+k_2+1\{k_1,k_2,n-s+1\}}\,,
\\*
C_n&=\sum_{k_1,k_2\geq0}c_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s+2\}}\,.\end{aligned}$$ Then, the primary WT identities - are rephrased as $$\begin{aligned}
\label{abc_nonzero1}
&k_1a_{n\{k_1,k_2\}}=0\,,
&&k_2a_{n\{k_1,k_2\}}=a_{n+1\{k_1,k_2-1\}}\,,
\\
\label{abc_nonzero2}
&k_1b_{n\{k_1,k_2\}}=2a_{n+1\{k_1-1,k_2\}}\,,
&&k_2b_{n\{k_1,k_2\}}=b_{n+1\{k_1,k_2-1\}}\,,
\\
\label{abc_nonzero3}
&k_1c_{n\{k_1,k_2\}}=b_{n+1\{k_1-1,k_2\}}\,,
&&
k_2c_{n\{k_1,k_2\}}=c_{n+1\{k_1,k_2-1\}}\,,\end{aligned}$$ which imply that nonzero coefficients are $$\begin{aligned}
c_{n\{0,k\}}&=\frac{c_{n+k\{0,0\}}}{k!}\,,
\\
c_{n\{1,k\}}&
=b_{n+1\{0,k\}}
=\frac{c_{n+k\{1,0\}}}{k!}\,,
\\
c_{n\{2,k\}}&
=\frac{1}{2}b_{n+1\{1,k\}}=a_{n+2\{0,k\}}=\frac{c_{n+k\{2,0\}}}{k!}\,.\end{aligned}$$ Our task is now to determine $c_{n\{0,0\}}$, $c_{n\{1,0\}}$, and $c_{n\{2,0\}}$. For this purpose, we use the recursion relation , taking the form with $\alpha=2$. Applying the general solutions -, we find $$\begin{aligned}
c_{s-2-n\{0,0\}}
&={{\mathcal{C}}}_C\frac{2^n(\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(2+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}
\,,
\\
c_{s-2-n\{1,0\}}&
={{\mathcal{C}}}_B\frac{2^n(1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(1+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}
\nonumber
\\
&\quad
-{{\mathcal{C}}}_C\frac{2^{n}(1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-1}(2+\frac{d-s-{{\Delta}}_t}{2})_{n-1}}{(n-1)!(2\!-\!\Delta_3\!-\!s)_n}\,,
\\
c_{s-2-n\{2,0\}}&
={{\mathcal{C}}}_A\frac{2^n(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}
\nonumber
\\
&\quad
-{{\mathcal{C}}}_B
\frac{2^{n-1}(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-1}(1+\frac{d-s-{{\Delta}}_t}{2})_{n-1}}{(n-1)!(2\!-\!\Delta_3\!-\!s)_n}
\nonumber
\\
&\quad
+{{\mathcal{C}}}_C \frac{2^{n-2}(2+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-2}(2+\frac{d-s-{{\Delta}}_t}{2})_{n-2}}{(n-2)!(2\!-\!\Delta_3\!-\!s)_n}
\,,\end{aligned}$$ where we used $c_{s-2\{0,0\}}={{\mathcal{C}}}_C$, $c_{s-2\{1,0\}}={{\mathcal{C}}}_B$, and $c_{s-2\{2,0\}}={{\mathcal{C}}}_A$. All the other coefficients are obtained by using Eqs. -.
Useful formulae {#subsec:formulae_rec}
---------------
In this section we encountered WT identities of the form, $$\begin{aligned}
\label{WT_closed}
\left(\Xi+n+\frac{3}{2}\alpha\right)F_{n+1}=(-s+n+\alpha)(\Delta_3-1+n+\alpha)F_n\,,\end{aligned}$$ with $F_n$ being a sum over triple-$K$ functions given by $$\begin{aligned}
F_n=\sum_{k_1,k_2\geq0}f_{n\{k_1,k_2\}}J_{n+k_1+k_2\{k_1,k_2,n-s+\alpha\}}\,.\end{aligned}$$ In particular, we are interested in its $k_2=0$ sector. Using the formula , we find that Eq. implies $$\begin{aligned}
&f_{s-\alpha-n\{k_1,0\}}
\nonumber
\\
&=2\frac{(1\!-\!\frac{\alpha}{2}+k_1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2}+n\!-\!1)(1+\frac{\alpha}{2}\!-\!k_1+\frac{d-s-{{\Delta}}_t}{2}+n\!-\!1)}{(1+n\!-\!1)(2\!-\!\Delta_3\!-\!s+n\!-\!1)}f_{s-\alpha-n+1\{k_1,0\}}
\nonumber
\\
&\quad
+\frac{(k_1-1-\alpha)}{(1+n\!-\!1)(2\!-\!\Delta_3\!-\!s+n\!-\!1)}f_{s-\alpha-n+1\{k_1-1,0\}}\,.\end{aligned}$$
It is convenient to note that its solution is generally given by $$\begin{aligned}
&f_{s-\alpha-n\{k_1,0\}}
\nonumber
\\
&=\frac{2^n(1\!-\!\frac{\alpha}{2}+k_1+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(1+\frac{\alpha}{2}\!-\!k_1+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{k_1,0\}}
\nonumber
\\
&\quad
+(k_1\!-\!1\!-\!\alpha)\sum_{m=0}^{n-1}\frac{(2\!-\!\frac{\alpha}{2}\!+\!k_1\!+\!\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2}\!+\!m)_{n-m-1}(2\!+\!\frac{\alpha}{2}\!-\!k_1\!+\!\frac{d-s-{{\Delta}}_t}{2}\!+\!m)_{n-m-1}}{(1+m)_{n-m}(2\!-\!\Delta_3\!-\!s+m)_{n-m}}
\nonumber
\\
\label{general_recursive}
&\qquad\qquad\qquad\qquad\quad
\times 2^{n-m-1}f_{s-\alpha-m\{k_1-1,0\}}\,,\end{aligned}$$ where $f_{s-\alpha-m\{k,0\}}=0$ for $k<0$ in the last line. We can then derive concrete expressions for $f_{s-\alpha-n\{k_1,0\}}$ recursively in $k_1$. First, for $k_1=0$ we have $$\begin{aligned}
\label{k1=0_general}
f_{s-\alpha-n\{0,0\}}
&=\frac{2^n(1\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(1+\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{0,0\}}
\,.\end{aligned}$$ Combining this with Eq. , we find $$\begin{aligned}
f_{s-\alpha-n\{1,0\}}
&=\frac{2^n(2\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{1,0\}}
\nonumber
\\*
\label{k1=1_general}
&\quad
-\alpha\frac{2^{n-1}(2\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-1}(1+\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-1}}{(n-1)!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{0,0\}}
\,,\end{aligned}$$ where we used $(a+m)_{n-m}(a)_m=(a)_n$ and $\displaystyle\sum_{m=0}^{n-1}\frac{1}{(a+m)(a+m+1)}=\frac{n}{a(a+n)}$. Similarly, we arrive at $$\begin{aligned}
f_{s-\alpha-n\{2,0\}}
&=\frac{2^n(3\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_n(-1+\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_n}{n!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{2,0\}}
\nonumber
\\*
&\quad
-(\alpha\!-\!1)
\frac{2^{n-1}(3\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-1}(\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-1}}{(n-1)!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{1,0\}}
\nonumber
\\*
\label{k1=2_general}
&\quad
+\alpha(\alpha\!-\!1)\frac{2^{n-3}(3\!-\!\frac{\alpha}{2}+\tfrac{{{\Delta}}_1-{{\Delta}}_2-{{\Delta}}_3-s}{2})_{n-2}(1+\frac{\alpha}{2}+\frac{d-s-{{\Delta}}_t}{2})_{n-2}}{(n-2)!(2\!-\!\Delta_3\!-\!s)_n}f_{s-\alpha\{0,0\}}
\,,\end{aligned}$$ where we used $\displaystyle\sum_{m=0}^{n-1}\frac{m}{(a+m-1)(a+m)(a+m+1)}=\frac{n(n-1)}{2a(a+n)(a+n-1)}$.
Correlators with two energy-momentum tensors {#app:TTs}
============================================
This appendix summarizes the results for correlators with two energy-momentum tensors: $$\begin{aligned}
{{\langle}}{{\epsilon}}_1^{2}.T({{\boldsymbol{p}}}_1) {{\epsilon}}_2^{2}.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,.\end{aligned}$$ The strategy is parallel to the two spin $1$ conserved current case discussed in Sec. \[sec:TTs\]. First, we perform tensor decomposition to write the correlator as a sum of $14$ terms as $$\begin{aligned}
{2}
&{{\langle}}{{\epsilon}}_1^2.T({{\boldsymbol{p}}}_1) {{\epsilon}}_2^2.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' {{\nonumber}}\\
&=
({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1^2)X_A
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3)X_B {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{\epsilon}}}_3^2) X_C
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2)_\mu({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1)^\mu X_D {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{\epsilon}}}_3)_\mu({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{\epsilon}}}_3)^\mu X_E
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1)_{\mu\nu}({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2)^{\mu\nu}X_F {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3) X_G
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1^2) Y_G {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{\epsilon}}}_3^2)X_H
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1^2)Y_H {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2{{\boldsymbol{\epsilon}}}_3)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{\epsilon}}}_3^2)X_I
&&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{\epsilon}}}_3^2)({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1{{\boldsymbol{\epsilon}}}_3)Y_I {{\nonumber}}\\
&\quad
+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{p}}}_2)_\mu({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{\epsilon}}}_3)^\mu X_J
&\;&+({{\boldsymbol{\epsilon}}}_1^2\cdot\boldsymbol{\Pi}_1\cdot {{\boldsymbol{\epsilon}}}_3)_\mu({{\boldsymbol{\epsilon}}}_2^2\cdot\boldsymbol{\Pi}_2\cdot {{\boldsymbol{p}}}_1)^\mu Y_J\,.
\label{22s-ansatz}\end{aligned}$$ Note that $X_C$, $X_I$ and $Y_I$ do not exist when $s=2$ because we have only two ${{\boldsymbol{\epsilon}}}_3$. We then expand each term in $\xi={{\boldsymbol{\epsilon}}}\cdot{{\boldsymbol{p}}}_2$ and ${{\zeta}}={{\boldsymbol{\epsilon}}}\cdot({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)$ as $$\begin{aligned}
&X_A = \sum_{n=0}^{s}\frac{1}{n!}\xi^{n}\zeta^{s-n}A_n \,,
&&X_B = \sum_{n=0}^{s-2}\frac{1}{n!}\xi^{n}\zeta^{s-n}B_n \,, {{\nonumber}}\\
&X_C =\sum_{n=0}^{s-4}\frac{1}{n!}\xi^{n}\zeta^{s-n-4}C_n \,,
&&X_D =\sum_{n=0}^{s}\frac{1}{n!}\xi^{n}\zeta^{s-n}D_n \,, {{\nonumber}}\\
&X_E =\sum_{n=0}^{s-2}\frac{1}{n!}\xi^{n}\zeta^{s-n-2}E_n \,,
&&X_F =\sum_{n=0}^{s}\frac{1}{n!}\xi^{n}\zeta^{s-n}F_n \,, {{\nonumber}}\\
&X_G =\sum_{n=0}^{s-1}\frac{1}{n!}\xi^{n}\zeta^{s-n-1}G_n \,,
&&Y_G =\sum_{n=0}^{s-1}\frac{1}{n!}\xi^{n}\zeta^{s-n-1}G_n^\star \,, {{\nonumber}}\\
&X_H =\sum_{n=0}^{s-2}\frac{1}{n!}\xi^{n}\zeta^{s-n-2}H_n \,,
&&Y_H =\sum_{n=0}^{s-2}\frac{1}{n!}\xi^{n}\zeta^{s-n-2}H_n^\star \,,{{\nonumber}}\\
&X_I =\sum_{n=0}^{s-3}\frac{1}{n!}\xi^{n}\zeta^{s-n-3}I_n \,,
&&Y_I =\sum_{n=0}^{s-3}\frac{1}{n!}\xi^{n}\zeta^{s-n-3}I_n^\star \,,{{\nonumber}}\\
&X_J =\sum_{n=0}^{s-1}\frac{1}{n!}\xi^{n}\zeta^{s-n-1}J_n \,,
&&Y_J =\sum_{n=0}^{s-1}\frac{1}{n!}\xi^{n}\zeta^{s-n-1}J_n^\star \,. \end{aligned}$$ Note that the exchange symmetry ${{\epsilon}}_1,p_1\leftrightarrow{{\epsilon}}_2,p_2$ implies for example $$\begin{aligned}
&A_s(p_1,p_2,p_3)=(-1)^sA_s(p_2,p_1,p_3)\,,
&&B_{s-2}(p_1,p_2,p_3)=(-1)^{s-2}B_{s-2}(p_2,p_1,p_3) \nonumber\,,\\
&C_{s-4}(p_1,p_2,p_3)=(-1)^{s-4}C_{s-4}(p_2,p_1,p_3)\,,
&&D_{s}(p_1,p_2,p_3)=(-1)^{s}D_{s}(p_2,p_1,p_3) \nonumber\,,\\
&E_{s-2}(p_1,p_2,p_3)=(-1)^{s-2}E_{s-2}(p_2,p_1,p_3)\,,
&&F_{s}(p_1,p_2,p_3)=(-1)^{s}F_{s}(p_2,p_1,p_3) \,,\\*
&G_{s-1}(p_1,p_2,p_3)=(-1)^{s-1}G^\star_{s-1}(p_2,p_1,p_3)\,,
&&H_{s-2}(p_1,p_2,p_3)=(-1)^{s-2}H^\star_{s-2}(p_2,p_1,p_3) \nonumber\,,\\
&I_{s-3}(p_1,p_2,p_3)=(-1)^{s-3}I^\star_{s-3}(p_2,p_1,p_3)\,,
&&J_{s-1}(p_1,p_2,p_3)=(-1)^{s-1}J^\star_{s-1}(p_2,p_1,p_3) \nonumber\,.\end{aligned}$$ Based on this ansatz, we solve the special conformal WT identity, $$\begin{aligned}
0 = ({{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_s +{{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_1}+ {{\boldsymbol{b}}}\cdot{{\boldsymbol{K}}}_{{{\epsilon}}_2})
{{\langle}}{{\epsilon}}_1^{2}.T({{\boldsymbol{p}}}_2) {{\epsilon}}_2^{2}.T({{\boldsymbol{p}}}_2) {{\epsilon}}_3^s.O({{\boldsymbol{p}}}_3) {{\rangle}}' \,,\end{aligned}$$ as well as the dilatation WT identity, which yields the homogeneity conditions. $$\begin{aligned}
A_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-4} A_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
B_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} B_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
C_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+4} C_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
D_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-2} D_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
E_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+2} E_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
F_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} F_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
G_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-2} G_n(p_1,p_2,p_3) \,, \label{TTS-hom} \\
G^*_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s-2} G^*_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
H_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} H_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
H^*_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} H^*_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
I_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+2} I_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
I^*_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s+2} I^*_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
J_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} J_n(p_1,p_2,p_3) \,, {{\nonumber}}\\
J^*_n({{\lambda}}p_1,{{\lambda}}p_2,{{\lambda}}p_3) &= {{\lambda}}^{{{\Delta}}_t-2d-s} J^*_n(p_1,p_2,p_3) \,. {{\nonumber}}\end{aligned}$$
List of primary and secondary WT identities
-------------------------------------------
The special conformal WT identities are decomposed into $42$ primary WT identities and $16$ secondary WT identities. First, $28$ of the primary ones are associated to ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_1$ and ${{\boldsymbol{b}}}\cdot{{\boldsymbol{p}}}_2$:
$$\begin{aligned}
&0=\left(P_1^{(-4,0)}+8p_3^{-2}{{\theta}}_3\right)X_A \,, ,
&&0=\left(P_2^{(-4,0)}+8p_3^{-2}{{\theta}}_3\right)X_A \,, \nonumber\\
&0=\left(P_1^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)X_B +4{{\partial}}_\xi Y_G\,,
&&0=\left(P_2^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)X_B -4{{\partial}}_\xi X_G\,, \nonumber\\
&0=P_1^{(4,4)}X_C+2{{\partial}}_\xi Y_I
&&0=P_2^{(4,4)}X_C-2{{\partial}}_\xi X_I \,, \nonumber\\
&0=\left(P_1^{(-2,0)}+4p_3^{-2}{{\theta}}_3\right)X_D +8X_A\,,
&&0=\left(P_2^{(-2,0)}+4p_3^{-2}{{\theta}}_3\right)X_D +8X_A\,, \nonumber\\
&0=P_1^{(2,2)}X_E+2X_B +2{{\partial}}_\xi Y_J\,,
&&0=P_2^{(2,2)}X_E+2X_B -2{{\partial}}_\xi X_J\,,\nonumber\\
&0=P_1^{(0,0)}X_F+2X_D \,,
&&0=P_2^{(0,0)}X_F+2X_D \,, \nonumber\\
&0=\left(P_1^{(-2,1)}+6p_3^{-2}{{\theta}}_3\right)X_G+4{{\partial}}_\xi X_A\,,
&&0=\left(P_2^{(-2,1)}+6p_3^{-2}{{\theta}}_3\right)X_G\,,
\label{TTS_initial}\\
&0=\left(P_1^{(-2,1)}+6p_3^{-2}{{\theta}}_3\right)Y_G\,,
&&0=\left(P_2^{(-2,1)}+6p_3^{-2}{{\theta}}_3\right)Y_G-4{{\partial}}_\xi X_A\,,\nonumber\\
&0=\left(P_1^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)X_H+2{{\partial}}_\xi X_G\,,
&&0=\left(P_2^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)X_H\,,
\nonumber\\
&0=\left(P_1^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)Y_H\,,
&&0=\left(P_2^{(0,2)}+4p_3^{-2}{{\theta}}_3\right)Y_H-2{{\partial}}_\xi Y_G\,,\nonumber\\
&0=\left(P_1^{(2,3)}+2p_3^{-2}{{\theta}}_3\right)X_I+2{{\partial}}_\xi X_B\,,
&&0=\left(P_2^{(2,3)}+2p_3^{-2}{{\theta}}_3\right)X_I-4{{\partial}}_\xi X_H\,,\nonumber\\
&0=\left(P_1^{(2,3)}+2p_3^{-2}{{\theta}}_3\right)Y_I+4{{\partial}}_\xi Y_H\,,
&&0=\left(P_2^{(2,3)}+2p_3^{-2}{{\theta}}_3\right)Y_I-2{{\partial}}_\xi X_B\,,\nonumber\\
&0=\left(P_1^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)X_J+2{{\partial}}_\xi X_D+4X_G\,,
&&0=\left(P_2^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)X_J+4X_G\,,
\nonumber\\
&0=\left(P_1^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)Y_J+4Y_G\,,
&&0=\left(P_2^{(0,1)}+2p_3^{-2}{{\theta}}_3\right)Y_J-2{{\partial}}_\xi X_D+4Y_G\,.
\nonumber\end{aligned}$$
The other $14$ are associated with ${{\boldsymbol{b}}}\cdot{{\boldsymbol{\epsilon}}}_3$: $$\begin{aligned}
0&=\left(R^{(-4,0)}+4{{\partial}}_\xi+8{{\partial}}_{{\zeta}}\right)X_A+2p_3^{-2}{{\theta}}_3(X_G+Y_G) \,, \nonumber\\
0&=\left(R^{(0,2)}+4{{\partial}}_\xi+8{{\partial}}_\zeta\right)X_B+4p_3^{-2}{{\theta}}_3(X_I+Y_I) \,, \nonumber\\
0&=\left(R^{(4,4)}+4{{\partial}}_\xi+8{{\partial}}_\zeta\right)X_C\,, \nonumber\\
0&=\left(R^{(-2,0)}+2{{\partial}}_\xi+4{{\partial}}_\zeta\right)X_D+4(X_G+Y_G)+2p_3^{-2}{{\theta}}_3(X_J+Y_J) \,,\nonumber\\
0&=\left(R^{(2,0)}+2{{\partial}}_\xi+4{{\partial}}_{{\zeta}}\right)X_E+4(X_I+Y_I) \,, \nonumber\\
0&=R^{(0,0)}X_F+2(X_J+Y_J) \,, \nonumber\\
0&=\left(R^{(-2,1)}+6{{\partial}}_\xi+8{{\partial}}_\zeta \right)X_G +2p^{-2}_3{{\theta}}_3 X_B+4p^{-2}_3{{\theta}}_3 X_H\,,
\label{TTS_recursion} \\
0&=\left(R^{(-2,1)}+2{{\partial}}_\xi+8{{\partial}}_\zeta \right)Y_G +2p^{-2}_3{{\theta}}_3 X_B+4p^{-2}_3{{\theta}}_3 Y_H\,,\nonumber\\
0&=\left(R^{(0,2)}+8{{\partial}}_\xi+8{{\partial}}_\zeta\right)X_H+2p_3^{-2}{{\theta}}_3 X_I\,,\nonumber\\
0&=\left(R^{(0,2)}+8{{\partial}}_\zeta\right)Y_H+2p_3^{-2}{{\theta}}_3 Y_I\,,\nonumber\\
0&=\left(R^{(2,3)}+6{{\partial}}_\xi + 8{{\partial}}_\zeta\right)X_I+4p_3^{-2}{{\theta}}_3 X_C\,,\nonumber\\
0&=\left(R^{(2,3)}+2{{\partial}}_\xi + 8{{\partial}}_\zeta\right)X_I+4p_3^{-2}{{\theta}}_3 X_C\,,\nonumber\\
0&=\left(R^{(0,1)}+4{{\partial}}_\xi+4{{\partial}}_\zeta\right)X_J+2X_B +2p_3^{-2}{{\theta}}_3 X_E+8 X_H\,,\nonumber\\
0&=\left(R^{(0,1)}+4{{\partial}}_\zeta\right)Y_J+2X_B +2p_3^{-2}{{\theta}}_3 X_E+8 Y_H\,. {{\nonumber}}\end{aligned}$$ To write down the secondary WT identities, it is convenient to introduce $$\begin{aligned}
{\hat{L}}&=-\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_1^2}{{\theta}}_1+{{\theta}}_2+{{\theta}}_\xi-\Delta_2 +1+d+d\frac{{{\boldsymbol{p}}}_1\cdot{{\boldsymbol{p}}}_2}{p_1^2}\,,\\*
{\hat{S}}&=\frac{\xi-\zeta}{p_1^2}\left(d-{{\theta}}_1\right) +\frac{\zeta}{p_3^2}{{\theta}}_3\,,\\*
{\hat{s}}&=\frac{1}{p_1^2}(d-{{\theta}}_1)\,,\end{aligned}$$ which become simpler in the zero momentum limit ${{\boldsymbol{p}}}_3\rightarrow 0$, that is $p_1=p_2=p$ and $\zeta=0$, $$\begin{aligned}
{\hat{L}}&= \theta_1 \,_{|p_1\rightarrow p}+\theta_2 \,_{|p_2\rightarrow p}+\xi{{\partial}}_\xi-d+1 \,,\nonumber\\*
{\hat{S}}&= \xi\;{\hat{s}}= \frac{\xi}{p^2}(d-\theta_1 \,_{|p_1\rightarrow p}) \,,\end{aligned}$$ where we set $\Delta_2=d$. Half of the $16$ secondary WT identities are associated with $({{\boldsymbol{b}}}\cdot\boldsymbol{\Pi}_1\cdot{{\boldsymbol{\epsilon}}}_1)_{\mu\nu}$ and of the form, $$\begin{aligned}
0&=2{\hat{L}}X_A +{\hat{s}}X_D-{\hat{S}}Y_G\,, \label{22s-second1}\\
0&=-{\hat{S}}X_B+{{\partial}}_\xi X_D +2{\hat{L}}X_G -2Y_G +{\hat{s}}X_J \,, \label{22s-second2}\\
0&=-X_B+2{\hat{L}}X_H-{\hat{S}}X_I+{{\partial}}_\xi X_J\,, \label{22s-second3}\\
0&={\hat{L}}X_B+{\hat{s}}X_E -4Y_H-2{\hat{S}}Y_I +{{\partial}}_\xi Y_J\,, \label{22s-second4}\\
0&=-2{\hat{S}}X_C +{{\partial}}_\xi X_E+{\hat{L}}X_I -2Y_I\,, \label{22s-second5}\\
0&={\hat{L}}Y_G-2{\hat{S}}Y_H+{\hat{s}}Y_J\,,\label{22s-second6}\\
0&={\hat{L}}X_D+2{\hat{s}}X_F-{\hat{S}}Y_J\,, \label{22s-second7}\\
0&=-{\hat{S}}X_E+2{{\partial}}_\xi X_F+{\hat{L}}X_J-Y_J \,.\label{22s-second8}\end{aligned}$$ Note that the condition does not exits for $s=2$ because it requires $3$ or more ${{\boldsymbol{\epsilon}}}_3$. The other $8$ equations are associated with $({{\boldsymbol{b}}}\cdot\boldsymbol{\Pi}_2\cdot{{\boldsymbol{\epsilon}}}_2)_{\mu\nu}$, but they are equivalent to the above 8 identities because of the $1\leftrightarrow2$ exchange symmetry.
Solutions for even $s$
----------------------
Similarly to the spin $1$ current case in Sec. \[sec:TTs\], it is easy to show that three-point functions vanish when the tensor $O$ has an odd spin $s$. We therefore focus on the even spin case.
### Initial conditions {#initial-conditions .unnumbered}
We solve the $\mathcal{O}(\zeta^0)$ terms of Eq. to find the initial conditions: $$\begin{aligned}
A_s&={{\mathcal{C}}}_A J_{s+4\{0,0,0\}}\,,\nonumber\\
B_{s-2}&=-4{{\mathcal{C}}}_A J_{s+2\{1,1,0\}}+2{{\mathcal{C}}}_G J_{s+1\{0,0,1\}}+{{\mathcal{C}}}_B J_{s\{0,0,0\}}\,,\nonumber\\
C_{s-4}&={{\mathcal{C}}}_A J_{s\{2,2,0\}} -{{\mathcal{C}}}_G \left(2J_{s-1\{1,1,1\}}+J_{s-1\{1,2,0\}}+J_{s-1\{2,1,0\}}\right){{\nonumber}}\\
&\quad -\left({{\mathcal{C}}}_B J_{s-2\{1,1,0\}}-{{\mathcal{C}}}_H J_{s-2\{2,0,0\}}-{{\mathcal{C}}}_H J_{s-2\{0,2,0\}}\right) +{{\mathcal{C}}}_I J_{s-3\{0,0,1\}} +{{\mathcal{C}}}_C J_{s-4\{0,0,0\}}\,,\nonumber\\
D_s&=-4{{\mathcal{C}}}_A J_{s+3\{0,0,1\}}+{{\mathcal{C}}}_D J_{s+2\{0,0,0\}}\,,\nonumber\\
E_{s-2}&=4{{\mathcal{C}}}_A J_{s+1\{1,1,1\}}-{{\mathcal{C}}}_D J_{s\{1,1,0\}}-2{{\mathcal{C}}}_G J_{s\{0,0,2\}} +({{\mathcal{C}}}_J-{{\mathcal{C}}}_B ) J_{s-1\{0,0,1\}}+{{\mathcal{C}}}_E J_{s-2\{0,0,0\}}\,,
\nonumber\\
F_s&=2{{\mathcal{C}}}_A J_{s+2\{0,0,2\}}-{{\mathcal{C}}}_D J_{s+1\{0,0,1\}}+{{\mathcal{C}}}_F J_{s\{0,0,0\}}\,,\nonumber\\
G_{s-1}&=-G^{\star}_{s-1}=2{{\mathcal{C}}}_A J_{s+3\{1,0,0\}}+{{\mathcal{C}}}_G J_{s+2\{0,0,0\}}\,,{{\nonumber}}\\
H_{s-2}&=H^{\star}_{s-2}
={{\mathcal{C}}}_A J_{s+2\{2,0,0\}} +{{\mathcal{C}}}_G J_{s+1\{1,0,0\}}+{{\mathcal{C}}}_H J_{s\{0,0,0\}}\,,\nonumber\\
I_{s-3}&=-I^{\star}_{s-3}
=-2{{\mathcal{C}}}_A J_{s+1\{2,1,0\}}+{{\mathcal{C}}}_G\left(2J_{s\{1,0,1\}}+J_{s\{2,0,0\}}\right) \nonumber\\
&\quad\quad\quad\quad\quad
+{{\mathcal{C}}}_B J_{s-1\{1,0,0\}}-2{{\mathcal{C}}}_H J_{s-1\{0,1,0\}}+{{\mathcal{C}}}_I J_{s-2\{0,0,0\}}\,,\nonumber\\
J_{s-1}&=-J^{\star}_{s-1}
=-4{{\mathcal{C}}}_A J_{s+2\{1,0,1\}}+{{\mathcal{C}}}_D J_{s+1\{1,0,0\}}-2{{\mathcal{C}}}_G J_{s+1\{0,0,1\}}+{{\mathcal{C}}}_J J_{s\{0,0,0\}}\,.\end{aligned}$$
### Recursion relations {#recursion-relations-1 .unnumbered}
On the other hand, recursion relations follow from Eq. . For notational simplicity, we introduce differential operators ${{\mathbb{D}}}_{(a,b,c)}$ and ${\widehat}{{\mathbb{D}}}{}_{t(a,b,c)}$ as $$\begin{aligned}
{{\mathbb{D}}}_{(a,b,c)} &= \frac{(\Xi+n+a)_{s-n-b}}{(-s+n+b)_{s-n-b}({{\Delta}}_3-1+n+c)_{s-n-b}},\\
{\widehat}{{\mathbb{D}}}{}_{t(a,b,c)} &= \frac{(\Xi+n+a)_t}{(-s+n+b)_{t+1}({{\Delta}}_3-1+n+c)_{t+1}}\,.\end{aligned}$$ Using this notation, the solutions for recursion relations are given by $$\begin{aligned}
A_n &={{\mathbb{D}}}_{(0,0,0)}+\sum_{t=0}^{s-1-n}{\widehat}{{\mathbb{D}}}{}_{t(0,0,0)}p_3^{-2}{{\theta}}_3(G_{n+t}+G^{\star}_{n+t})\,,\nonumber\\
B_n &={{\mathbb{D}}}_{(2,2,2)}B_{s-2}+\sum_{t=0}^{s-3-n}{\widehat}{{\mathbb{D}}}{}_{t(2,2,2)}p_3^{-2}{{\theta}}_3(I_{n+t}+I^{\star}_{n+t})\,,\nonumber\\
C_n &={{\mathbb{D}}}_{(2,4,4)}C_{s-4}\,,\nonumber\\
D_n &={{\mathbb{D}}}_{(0,0,0)}D_{s} + \sum_{t=0}^{s-1-n}{\widehat}{{\mathbb{D}}}{}_{t(0,0,0)}(2G_{n+t}+2G^{\star}_{n+t}+p_3^{-2}{{\theta}}_3J_{n+t}+p_3^{-2}{{\theta}}_3J^{\star}_{n+t})\,,\nonumber\\
E_n &={{\mathbb{D}}}_{(2,2,2)}E_{s-2} + 2\sum_{t=0}^{s-3-n}{\widehat}{{\mathbb{D}}}{}_{t(2,2,2)}(I_{n+t}+I^{\star}_{n+t})\,,\nonumber\\
F_n &={{\mathbb{D}}}_{(0,0,0)}F_{s} + \sum_{t=0}^{s-1-n}{\widehat}{{\mathbb{D}}}{}_{t(0,0,0)}p_3^{-2}{{\theta}}_3(J_{n+t}+J^{\star}_{n+t})\,,\nonumber\\
G_n &={{\mathbb{D}}}_{(2,1,1)}G_{s-1}+ \sum_{t=0}^{s-2-n}{\widehat}{{\mathbb{D}}}{}_{t(2,1,1)}p_3^{-2}{{\theta}}_3(B_{n+t}+2H_{n+t})\,,\nonumber\\
G^{\star}_n &={{\mathbb{D}}}_{(0,1,1)}G^\star_{s-1}+ \sum_{t=0}^{s-2-n}{\widehat}{{\mathbb{D}}}{}_{t(0,1,1)}p_3^{-2}{{\theta}}_3(B_{n+t}+2H^{\star}_{n+t})\,,\nonumber\\
H_n &={{\mathbb{D}}}_{(4,2,2)}H_{s-2}+\sum_{t=0}^{s-3-n}{\widehat}{{\mathbb{D}}}{}_{t(4,2,2)}p_3^{-2}{{\theta}}_3 I_{n+t}\,,\nonumber\\
H^{\star}_n &={{\mathbb{D}}}_{(0,2,2)}H^{\star}_{s-2}+\sum_{t=0}^{s-3-n}{\widehat}{{\mathbb{D}}}{}_{t(0,2,2)}p_3^{-2}{{\theta}}_3 I^{\star}_{n+t}\,,\nonumber\\
I_n &={{\mathbb{D}}}_{(4,3,3)}I_{s-2}+2\sum_{t=0}^{s-4-n}{\widehat}{{\mathbb{D}}}{}_{t(4,3,3)}p_3^{-2}{{\theta}}_3 C_{n+t}\,,\nonumber\\
I^{\star}_n &={{\mathbb{D}}}_{(2,3,3)}I^{\star}_{s-2}+2\sum_{t=0}^{s-4-n}{\widehat}{{\mathbb{D}}}{}_{t(2,3,3)}p_3^{-2}{{\theta}}_3 C_{n+t}\,,\nonumber\\
J_n &={{\mathbb{D}}}_{(2,1,1)}J_{s-1} + \sum_{t=0}^{s-2-n}{\widehat}{{\mathbb{D}}}{}_{t(2,1,1)}(B_{n+t}+p_3^{-2}{{\theta}}_3E_{n+t}+4H_{n+t})\,,\nonumber\\
J^{\star}_n &={{\mathbb{D}}}_{(0,1,1)}B_{s-1} + \sum_{t=0}^{s-2-n}{\widehat}{{\mathbb{D}}}{}_{t(0,1,1)}(B_{n+t}+p_3^{-2}{{\theta}}_3E_{n+t}+4H^{\star}_{n+t})\,.\end{aligned}$$
### Secondary WT identities {#secondary-wt-identities-4 .unnumbered}
So far we have $10$ free parameters ${{\mathcal{C}}}_A,\ldots,{{\mathcal{C}}}_J$. While there are $8$ secondary WT identities, it turns out that there is one degeneracy among them when applied to the solutions for the primary WT identities. As a result, we are left with $3$ free parameters for $s\geq4$. As we mentioned, the operators $X_C$, $X_I$ and $Y_I$, and the secondary WT identity do not exist for $s=2$, while there still exists one degeneracy. Correspondingly, there remain $2$ free parameters for $s=2$.
[^1]: Indeed, momentum space was employed in the pioneering work [@Polyakov:1974gs] to make analyticity manifest and utilize dispersion relations to accomplish the bootstrap program in the $\text{O}(N)$ model. It was recently revisited and extended in [@Sen:2015doa]. We hope that our momentum space approach to conformal correlators is helpful for proceeding in this direction.
[^2]: In $d=3$ there appears degeneracy in tensor structures, which requires a separate argument (see, e.g., [@Bzowski:2013sza]). We leave the analysis in $d=3$ for future work, focusing on $d\geq4$.
[^3]: For example, let us consider three-point functions of one scalar, one conserved current and a general tensor. The WT identity is then stated as $$\begin{aligned}
{{{\boldsymbol{p}}}_2}_\mu \langle \varphi({{\boldsymbol{p}}}_1) J^\mu({{\boldsymbol{p}}}_2) \epsilon_3^s.O({{\boldsymbol{p}}}_3)\rangle^\prime
=-q_1\langle \varphi({{\boldsymbol{p}}}_1)\epsilon_3^s.O(-{{\boldsymbol{p}}}_1)\rangle^\prime
-q_3\langle \varphi({{\boldsymbol{p}}}_3)\epsilon_3^s.O(-{{\boldsymbol{p}}}_3)\rangle^\prime\,,
\end{aligned}$$ where $q_1\text{ and }q_3$ are the $\text{U}(1)$ charges of the operators $\varphi \text{ and } \mathcal{O}_{\mu_{1}\mu_{2}\ldots\mu_{s}}$. The local terms in the right-hand side vanish since they are proportional to two-point functions of two different operators. The same argument applies to the other correlators studied in this paper. Note that we follow the definition of three-point functions in [@Osborn:1993cr]. On the other hand, if we employ another definition used in [@Bzowski:2013sza], there may appear local terms. See, e.g., Sec. 2.3 of [@Bzowski:2018fql] for details of the two different conventions.
[^4]: The $1\leftrightarrow2$ exchange symmetry is obscured by the ansatz because the correlator is expanded in $\xi={{\boldsymbol{\epsilon}}}_3\cdot{{\boldsymbol{p}}}_2$ and $\zeta={{\boldsymbol{\epsilon}}}_3\cdot({{\boldsymbol{p}}}_1+{{\boldsymbol{p}}}_2)$. To make it manifest, it is more convenient to employ the expansion in ${{\boldsymbol{\epsilon}}}_3\cdot({{\boldsymbol{p}}}_1\pm{{\boldsymbol{p}}}_2)$. See [@Isono:2018rrb] for details. However, it turns out that the ansatz is more useful to derive a closed form such as Eq. .
[^5]: Note that the integral is convergent only when $|{\rm Re}\,\nu_1|+|{\rm Re}\,\nu_2|+|{\rm Re}\,\nu_3|<s+\frac{d}{2}$. Otherwise, there appears a singularity near $z=0$ and we need to perform analytic continuation [@Bzowski:2015pba; @Bzowski:2015yxv], which may be carried out, e.g., by introducing the Pochhammer contour.
[^6]: Eq. is nothing but the hypergeometric differential equation, hence its solution is given by $$\begin{aligned}
\sum_{n=0}^s \frac{1}{n!}x^nA_n(p_1,p_2,p_3) \propto {}_2F_1\left( -s,{{\Delta}}_3-1; \Xi; x \right)\,,\end{aligned}$$ where we chose a polynomial solution in $x$. The proportionality constant is fixed by $A_s={{\mathcal{C}}}_AJ_{s(0,0,0)}$.
[^7]: In Appendix \[app:another\] we provide an alternative derivation of the coefficients , which is useful when working on correlators with at most two spinning operators. However, it turns out to be not straightforward to apply it to correlators of three spinning operators. On the other hand, the algebraic calculation presented in this section provides a general framework applicable to any correlator.
[^8]: Note that the second terms of Eqs. - are interpreted as zero for $k=n$. Similarly, the third term of Eq. is zero for $k=n,n-1$.
[^9]: Note that we have $D=0$ for $s=1$ since the $D$ term requires two or more ${{\boldsymbol{\epsilon}}}_3$. It, however, does not affect our argument very much because three-point functions vanish for odd spin $s$ as we explain shortly.
[^10]: To be precise, we need to check that the two secondary WT identities provide nontrivial conditions on ${{\mathcal{C}}}_B$. However, it is easy to show that it is indeed the case and thus ${{\mathcal{C}}}_A={{\mathcal{C}}}_B={{\mathcal{C}}}_C={{\mathcal{C}}}_D={{\mathcal{C}}}_E=0$ is required for odd spin $s$ by the exchange symmetry and the secondary WT identities.
[^11]: Our strategy will be applicable to correlators with an antisymmetric tensor or a more general tensor with mixed symmetry by introducing Grassmann odd polarization vectors, which we leave for future work.
[^12]: The Bessel $K$ function $K_n(x)$ with an integer index $n$ is defined by the limit $\displaystyle K_n(x)=\lim_{\epsilon\rightarrow 0} K_{n+\epsilon}(x)$.
[^13]: In contrast to the approach presented in the main text, it is not manifest in this derivation if the full correlator satisfies the WT identity , which has to be checked separately.
|
---
abstract: |
This paper presents a numerical method to conceive and design the kinematic model of an anthropomorphic robotic hand used for gesturing and grasping. In literature, there are few numerical methods for the finger placement of human-inspired robotic hands. In particular, there are no numerical methods, for the thumb placement, that aim to improve the hand dexterity and grasping capabilities by keeping the hand design close to the human one. While existing models are usually the result of successive parameter adjustments, the proposed method determines the fingers placements by mean of empirical tests. Moreover, a surgery test and the workspace analysis of the whole hand are used to find the best thumb position and orientation according to the hand kinematics and structure. The result is validated through simulation where it is checked that the hand looks well balanced and that it meets our constraints and needs. The presented method provides a numerical tool which allows the easy computation of finger and thumb geometries and base placements for a human-like dexterous robotic hand.\
*Keywords: robotic hand, gesture, hand dexterity, Kapandji test, thumb opposability, grasping.*
author:
- 'Giulio Cerruti$^{1}$, Damien Chablat$^{2}$, David Gouaillier$^{3}$ and Sophie Sakka$^{4}$[^1][^2][^3] [^4]'
bibliography:
- 'Bibliography.bib'
title: '**Design method for an anthropomorphic hand able to gesture and grasp**'
---
=4
Introduction {#sec:Introduction}
============
The human hand is an astonishingly advanced mechanism which is too complicated to be faithfully replicated. Many techniques are used to model its kinematics, such as direct measurements of the human limbs or optimization algorithms. Currently, the most accurate hand model ever built is the anatomically-correct testbed (ACT) [@ACT]. In this design, the structure was machined with the same shape and mass to that of the human bones, and the joints were designed to preserve the same DoFs and passive stiffness of the human joints. However, the ACT is conceived to deeply investigate the human hand structure, function and control for medical purposes. Its complexity hinders the practical use of the artifact in robotics. In general, simplified joint structures and link shapes are commonly chosen by robotic hand designers. Furthermore, the number of independent DoFs are usually reduced due to technological limits (hardware and software) and mechanical constraints.\
Each human hand has its own peculiarities, and yet, its functional capabilities do not significantly change from the others. This implies that different link lengths, width and joint placements do not greatly alter the hand performance. As a consequence, no true optimal design exists for the construction of a human-like robotic hand. Nevertheless, an appropriate model has to be realized in order to fulfill all functional requirements: the hand is supposed to be able to express emotions and give information through gestures, grasping objects and manipulating them. These abilities require high interaction among fingers, in particular between each finger and the thumb. Therefore, finger geometry and kinematics are less important than the thumb kinematics and its interaction with the opposed fingers.\
Our project aims to design a self-contained human-inspired hand with following constraints: have the size and weight similar to a 6 years old child’s hand (length approximately 120 mm and weight less than 0.6 Kg) in order to be embodied on a humanoid robot of a comparable height, be able to grasp small objects (such as a full soda can) and perform gestural communication (e.g. thumb-up, ok, pointing and counting). This document is the first step in the design process for which we solve geometric problems only. Section \[sec:hi conception\] presents a literature review on human hands and existing robotic hands. It will be shown that no hand that meets our constraints exists. Sections \[sec:hand design\] addresses the problem presenting the kinematic aspects of our hand. Section \[sec:thumb base selection\] proposes a new design method to place the thumb base in order to fulfill the required functional capabilities. Finally, results are discussed in Section \[sec:validation and discussion\] and conclusions are drawn in Section \[sec:Conclusions\].
Literature Review {#sec:hi conception}
=================
The human hand {#subsec:human hand}
--------------
### Hand skeleton
The human hand is composed of 27 bones [@HumanHandModel] which make up its three main parts: wrist, palm and fingers (Figure \[fig:handSkeleton\]). The wrist is formed by 8 small bones called carpals which join the ulna and radius forearm bones to the hand. The palm is composed by 5 bones called metacarpals which connect the fingers and the thumb to the wrist. The joints between the wrist and the finger metacarpals are called the carpometacarpal (CMC) joints while the one linking the metacarpal of the thumb is called the radiocarpal (RC) joint. Each finger is composed by 3 long bones called phalanges whose names are given according to their distance to the palm: proximal, middle and distal. The proximal phalanx (PP) is linked to the metacarpal bone through the metacarpophalangeal joint (MCP) while the remaining phalanges are connected to each other through the interphalangeal joints. The joint between the proximal and the middle phalanx is the proximal interphalangeal joint (PIP) while the joint closest to the end of the finger is the distal interphalangeal joint (DIP). The thumb has one bone less with respect to the fingers (no middle phalanx) and it consequently has one interphalangeal joint plus the MCP joint.
### Hand kinematics
The hand joints are classified into three main types: hinge (1 DoF), condyloid (2 DoFs) and saddle (2 DoFs) joints. The hinge joints on the human hand are the CMC, PIP and the DIP joints. They allow the fingers to flex (move toward the palm) and to extend (move further from palm). The condyloid joints are the finger MCP joints that allow flextion/extension and abduction/adduction of the fingers, the latter being the motion of spreading and gathering them. Hence, each finger can be represented as a kinematic serial chain of 5 DoFs: 1 at the CMC, 2 at the MCP, 1 at the PIP and a last one at the DIP. Finger PIP and DIP joints are orthogonal to the bone axis when the phalanx is fully extended and they progressively bend toward the middle (due to the bone surface) while flexing. As a result, all fingers converge to a common point improving the opposition of the thumb to the ring and little fingers. According to Vitruvian man’s hand and the study conducted by Isobe [@IsobeHandPosition], human fingertips approximatively lie on a common circle when abducted. The circle has a radius equal to the middle finger length and it is centered at the MCP joint of the middle finger. The thumb has the same number of DoFs of fingers but differently distributed: the RC joint is a saddle joint, the MCP is a condyloid joint and the IP joint is a hinge joint [@fiveLinkModelThumb]. The thumb is the only finger able to turn and oppose to the other four fingers. The opposability of the thumb enables humans to grip and hold objects that they would not be able to take otherwise. Figure \[fig:handKinematics\] shows the summarized DoFs of the human hand plus a further DoF in the palm that represents the small motions that occur among the carpal bones while flexing the wrist. However, some joints are relatively immobile due to the interousseous ligaments that stabilize the hand joints. The CMC joints at the bases of the index, middle and ring metacarpals can be neglected and the small intra-carpal motions can be collected into the wrist motion. Note that the CMC joint of the little finger should not be omitted since it forms the hollowed shape of the palm [@jointStructure] when the little finger moves in opposition to the thumb. Not all DoFs in the human hand are independent. Tendons couple some joint like the PIP and DIP joints of the fingers. In the next paragraphs we will refer to the independent DoFs simply as DoFs.
Robotic hands {#subsec:robotis hand kinematics}
-------------
The number of joints and DoFs of human-inspired robotic hands is generally chosen according to the tasks for which they are conceived. For instance, robotic hands oriented to reproduce the human hand manipulation capabilities adopt nearly the same kinematic structure presented in the previous paragraph. Excluding the wrist, the Shadow Dexterous Hand [@ShadowHand] has 18 independent DoFs, 5 of which are dedicated to the thumb, with the PIP and DIP joints coupled. The Awiwi hand, mounted on the DLR Hand Arm System [@DLRhandArmSystem], has even more independent DoFs (19): its PIP and DIP joints of the index and middle are not coupled and it has a simplified thumb with 4 joints and 4 DoFs. The UB Hand IV [@UBhandIV] and DLR/HIT Hand II [@DLRHIThandII] (15 DoFs) do not include the CMC joint of the little finger in their kinematic models and their thumbs have 4 joints with the MCP and the IP coupled.\
Robotic hands designed only for grasping require less complex kinematic models with less joints and DoFs. Generally they have all finger joints coupled, the CMC joint of the little finger coupled with the thumb base motion and fingers with no abduction/adduction motion (MCP with 1 DoF). The Harada Robot Hand [@HaradaHand], for instance, has only 14 joints (3 per finger and 2 for the thumb) and 5 DoFs: 1 DoF per finger and 1 DoF for the thumb. Another example is the Prensilia IH2 Azzurra hand [@AzzurraHand] that has 11 joints, 5 of which actuated: 2 DoFs for the thumb, 1 DoF for index and middle and the last DoF for both ring and little fingers. Robotic hands that aim to both manipulate and grasp find a compromise between hand motion capabilities and control complexity. Among these we find hands such as the Robonaut II Hand [@RobonautIIhand], Elu2-Hand [@EluHand] (Shunk hand) and the iCub hand [@iCubHand]. Table \[tab:RoboticHandsComparison\] lists the DoFs and the weight of the aforementioned hands and it specifies if hands fully embed their actuators and electronic components. As it can be seen, none of them fulfill our requirements: half of them are not self-contained (i.e. they have hardware placed externally from the hand), the DLR/HIT Hand II is too heavy, the Harada Robot Hand, Prensilia IH2 Azzurra hand and TUAT/Karlsruhe Hand [@TUATKarlsruheHand] do not have enough DoFs to perform the required gestures. Only the Elu2-Hand could fulfill our functional needs, however, its size is similar to that of an adult’s hand and it cannot be scaled to the demanded size.
In general, anthropomorphic robotic hands kinematics are not designed on the base of the finger-thumb interaction, even though hand performance strictly depends on the thumb dexterity and opposability. Thumb capabilities are determined by its kinematics and base placement. The problem of thumb placement has already been addressed in literature. Grebenstein *et al.* [@7BillionPerfectHands] developed cardboard prototypes to incrementally refine the hand kinematics in order to obtain thumb functionalities and hand aesthetics similar to the human one. Wang *et al.* [@designGuidelineForTheThumb] approached the problem using a numerical method based on the Euler rotation theorem. The thumb placement was computed passing from a lateral posture to an opposing one in order to ensure basic grasping modes. This paper solves the problem using a numerical method based on a surgery test and a workspace analysis of the whole hand. The best thumb position and orientation is found according to the hand kinematics and structure.
Our hand design {#sec:hand design}
===============
The design of the kinematic model of our hand started from mechanical and functional observations which lead to the following objectives:
- be aesthetically anthropomorphic in order to be easily accepted by human beings;
- be proportioned to a 6 years old child’s hand;
- weigh less than 0.6 Kg - to be embodied on the robot;
- be compliant - to have a safe human-robot interaction;
- be silent.
while from a functional point of view it has to:
- 70$\%$ communicate information and emotions using gestures: pointing, counting, thumb-up, ok, etc …;
- 30$\%$ grasp light objects, like a smart-phone or a full soda can.
Kinematics design {#subsec:Our kinematics design}
-----------------
To have a light and small hand the number of actuators has to be as small as possible. A robotic hand that is able to grasp various objects does not need numerous actuators. Thanks to differential mechanisms and under-actuated fingers capable to adapt to different surfaces [@Berglin] only one actuator is sufficient [@TUATKarlsruheHand]. Also finger geometry and couplings based on postural synergies [@BicchiSynergies] allow to use a reduced number of actuators. However, in our case, non-verbal communication should be added to grasping capabilities. For this reason, the design of the kinematic model is firstly oriented to satisfy the required gestures. To achieve this, the whole human hand kinematic model is firstly considered with all its joints coupled. Then, for each gesture, joint couplings are incrementally relaxed to minimally increase the number of actuators. After that, the number of joints is reduced to meet the weight and size limits. Obviously, this last step imposes a trade off between well-mimicked gestures and a light and small structure. To identify the less relevant joints, gestures are classified according to the feelings they trigger in the recipient: low priority is given to gestures inducing negative sensations, while, middle and high priorities are associated to the ones inspiring neutral and positive feelings respectively. Hence, to meet weight and size constraints, joints purely involved in performing low priority gesture are removed. The thumb is defined with four joints, abduction/adduction finger motions are removed as well as the CMC joint at the base of the little finger. Figure \[fig:firstImplemendtedModelPaper\] shows our final hand kinematic model: 1 DoF per finger (they can only flex/extend) and 3 DoFs for the thumb. All joints can rotate by 90$^{\circ}$ about their axes. Their range of motion normally goes from 0$^{\circ}$ to 90$^{\circ}$, where 0$^{\circ}$ refers to the hand configuration in which all fingers are fully extended on the same plane.
Finger and thumb proportions {#subsec:finger and thumb proportions}
----------------------------
No best practice to define finger lengths, widths and placements exists. Modular designs of some or all fingers are often used to help mechanical modeling and construction. Each finger has the same link length, width, joint RoM and DoFs and their placement on the palm determines the similarity to the human counterpart. Nevertheless, a better affinity with the human hand is achieved respecting human finger diversities. Link lengths and widths are derived from body proportions under the assumption that the structure and bones proportions of hands are maintained despite their physical size [@basicResearchHandBiomechanics]. Link lengths are commonly disclosed as percentage of the hand length [@Davidoff], therefore, they can be easily computed knowing the distance between the wrist and the middle fingertip. Link widths measurements are determined as a function of the hand width using two linear equations as lower and upper bounds. This approach took inspiration from the comparison done by [@fingerWidths] between the Grainer model [@greinerHand] and Buchholz and Armstrong model [@BuchholzHand]. Table \[tab:FingerProportions\] gives the link lengths and widths for a humanoid robot of similar height to a 6 year old child.
-- -- -- --
-- -- -- --
: Link lengths and widths for a humanoid robot of similar height to a 6 year old child.
\[tab:FingerProportions\]
Finger placement {#subsec:finger placement}
----------------
Finger bases are computed using a circle centered at the MCP joint of the middle finger [@IsobeHandPosition]. Since link lengths are fixed (Table \[tab:FingerProportions\]), the angle between each finger base determines the linking point to the palm. Small angles return closer finger bases, with similar heights from the wrist, while big angles locate finger bases at further distances and with various heights from the wrist. Obviously, too close finger bases are not mechanically feasible, while far finger bases result in an unaesthetic design. A compromise between the two is necessary in order to have a well-balanced hand (Figure \[fig:15DegFingerPos\]). Along the $z$ axis, fingers are placed so that the fingertips lie on a common $xy$ plane when they are completely flexed. This allows the palm to have the same arched shape as the human hand.\
Designed fingers can only flex and extend (1 DoF) via three parallel hinge joints. Hence, MCP joints are inclined to preserve the finger convergence of the human hand towards the center of the palm. To determine maximum angles of inclination of the finger bases, intra-finger collisions are checked while flexing the joints. For each finger, finger distances are computed increasing the inclination angle from $0^\circ$ until a collision is detected (Figure \[fig:firstImplemendtedModelPaper\]). Table \[tab:Baseplacement\] shows the resulting finger base positions and orientations. The distance along y between the little and the ring bases is smaller than the others, slightly complicating the mechanical design. However, with these results the aesthetics of the human hand is globally met.
-- -- --
-- -- --
: Finger base positions and orientations with respect to the hand frame (Figure \[fig:firstImplemendtedModelPaper\]).
\[tab:Baseplacement\]
Thumb placement {#subsec:thumb placement}
---------------
To approximately attain human hand performances the thumb geometry and kinematics need to be carefully designed. To gesture and grasp, 4 joints with 3 DoFs are chosen for the thumb kinematics (Figure \[fig:firstImplemendtedModelPaper\]). Its proportions are calculated as described in the previous Section \[subsec:finger and thumb proportions\]. The thumb base configuration is computed following three criteria: dexterity, opposability and aesthetics. In order to imitate human thumb motions, the robotic thumb has to be able to reach a certain number of desired positions on the opposite fingers. These positions are defined by a well-known surgery test, known as Kapandji test [@Kapandji], used after pollicization, which consists in reconstructing the thumb by displacing the index finger from its MCP joint to the MCP joint of the thumb. The test contains all motion directions and an interesting set of positions which are useful to check the thumb range of motion and its dexterity. However, it cannot provide information about thumb grasping abilities. Reaching all required positions does not necessarily imply that the thumb is effectively able to interact with the opposed finger. Hence, an additional test is carried out to explore the physical interaction between the thumb fingertip and the opposed fingers. This test returns the intersection volume between the thumb and fingers as a thumb opposability index [@GifuIII], which denotes the hand grasping abilities from a kinematics point of view. Note that no forces are considered in this context; an additional evaluation metric should be taken into account to determine finger forces and grasping stability. Finally, the global appearance of the hand has to be equilibrated and easily accepted. Indeed, the hand will be mounted on a social robot and its aesthetics is fundamental to enhance and encourage human-robot interaction.\
In summary, the thumb base placement is the most delicate design step. In order to obtain the needed functionalities (gesturing and grasping) the thumb has to be dexterous and opposable. These two properties are respectively achieved through:
- The Kapandji test;
- The thumb opposability index.
In addition, an aesthetic check is done to evaluate the overall hand appearance. This last test implies a cyclic design in which the intervals of search among the thumb base parameters are adapted at each iteration.
Thumb base selection {#sec:thumb base selection}
====================
In this section the tests used to design a dexterous and opposable thumb are presented. According to the thumb geometry and kinematics, the thumb base configuration is determined by calculating the thumb base position ($x$, $y$, $z$) and orientation about the $z$ ($\theta_z$) axis with respect to the hand frame (Figure \[fig:firstImplemendtedModelPaper\]). The orientations about $x$ and $y$ axes are not included because of the 2 DoFs of the base.
The Kapandji test {#subsec:kapandji test}
-----------------
The Kapandji test assigns a score for each reached position of the thumb on the opposite fingers. As shown in [@Kapandji] the thumb passes from the index base to its fingertip and proceeds to the little fingertip touching the end of the fingers in between. Finally, it reaches the little base to complete the test. Based on this idea, the thumb is asked to reach all finger joints and fingertips for a total of 16 desired positions. If the thumb attains all 4 positions defined on each finger, the thumb base is collected as a candidate solution (Figure \[fig:problemDefinitionForOneFinger\]).
### Problem formulation {#subsubsec: problem formulation}
The thumb and the finger in opposition are treated as two independent serial chains attached to a common base. On each chain an end-effector is defined according to the desired position that has to be reached. The objective is to determine if both kinematic chains can converge to a common position.\
*Definitions:*
- $j = t$ or $f_i$ respectively referring to the thumb fingertip and the $i^{th}$ end-effector ($i =$ MCP, PIP, DIP and fingertip) on the $f^{th}$ finger;
- $\mathbf{q}_j$ vector of the independent joint variables of the $j^{th}$ chain;
- $T_j(\mathbf{q}_j)$ direct geometric model (DGM) of the $j^{th}$ serial kinematic chain. It computes the end-effector configuration passing from the joint space to the Cartesian space;
- $J_j(\mathbf{q}_j)$ Jacobian matrix. It provides the $j^{th}$ end-effector differential displacement (linear and angular) in terms of the differential variation of the joint variables. In this context, only its upper part is used in order to uniquely consider the linear displacement of the end-effector: $\mathbf{dP} = J_j(\mathbf{q}_j)\mathbf{dq}_j$.
The problem can be solved using the inverse geometric model where the desired frame changes at each iteration:
1. Initialization of current joint variables $\mathbf{q}^c_t$ and $\mathbf{q}^c_{f_i}$ to a random value or desired initial value within the joint domain;
2. Computation of the current end-effector transforms $T_t(\mathbf{q}^c_t)$ and $T_{f_i}(\mathbf{q}^c_{f_i})$ of each chain (thumb and opposing finger);
3. Computation of the position error defined as:\
$\mathbf{dP} = P_t(\mathbf{q}^c_t) - P_{f_i}(\mathbf{q}^c_{f_i})$, where $P_t(\mathbf{q}^c_t)$ and $P_{f_i}(\mathbf{q}^c_{f_i})$ are the position vectors of $T_t(\mathbf{q}^c_t)$ and $T_{f_i}(\mathbf{q}^c_{f_i})$ respectively;
4. If ($||\mathbf{dP}||$ is as small as required): the algorithm stops;\
else: $\mathbf{dP}$ is scaled if it is too big to be used in the differential model $\mathbf{dP} = \mathbf{dP}/|| \mathbf{dP} || dx$, where $dx$ is a properly defined small displacement;
5. Computation of thumb and finger current Jacobians $J_t(\mathbf{q}^c_t)$ and $J_{f_i}(\mathbf{q}^c_{f_i})$;
6. Calculation of joint variations $\mathbf{dq}_t = J_t(\mathbf{q}^c_t)^+ \mathbf{dP}/2$ and $\mathbf{dq}_{f_i} = J_{f_i}(\mathbf{q}^c_{f_i})^+ (-\mathbf{dP}/2)$;
7. Updating joint configurations $\mathbf{q}^c_t = \mathbf{q}^c_t + \mathbf{dq}_t$ and $\mathbf{q}^c_{f_i} = \mathbf{q}^c_{f_i} + \mathbf{dq}_{f_i}$ within the joint domain;
8. Reiteration from step 2.
In our context, $\mathbf{q}_{f_i}$ is always a scalar for each finger $f$, independently from the chosen end-effector since all finger joints are coupled. Finger joint couplings influence the thumb base selection since end-effector trajectories change. In this context, transmission ratios are set to 1:1. Note that, if more than 3 DoFs are available on a single chain, this algorithm can be extended when computing the joint displacements. Indeed, a correction in orientation, if desired, could be attained as a secondary objective, working in the null space of the Jacobian matrix of the serial chain. This iterative approach can return some undesired solutions: the thumb end-effector reaches the desired position passing through the finger in opposition. Consequently, a non-linear optimization algorithm with collision avoidance constraints is used: $$\begin{aligned}
&\min_{\mathbf{q}_t,\mathbf{q}_{f_i}} ||\mathbf{dP(\mathbf{q}_t,\mathbf{q}_{f_i})}||\\
\text{s.t.}&\\
& q_{t_l \min} \leq q_{t_l} \leq q_{t_l \max}, & l = 1, \text{ \ldots, DoFs}(\mathbf{q}_t)\\
& q_{{f_i}_l \min} \leq q_{{f_i}_l} \leq q_{{f_i}_l \max}, & l = 1, \text{ \ldots, DoFs}(\mathbf{q}_{f_i})\\
& d_{sph} \geq r_t + r_{f_i} - \epsilon\\
\label{eq:optProblem}
\end{aligned}$$ where $r_t$ is the thumb radius at the fingertip (the half of the thumb width defined in Table \[tab:FingerProportions\]), $r_{f_i}$ is the radius of the opposed finger at the desired point of interest (half of the joint or fingertip width defined in Table \[tab:FingerProportions\]), $\epsilon$ is an arbitrary fixed scalar (e.g. 0.5 mm) which defines the intersection acceptance between the two chains, considering that finger surfaces can be made of soft materials, and $d_{sph}$ is the distance between two sphere centers, one located within the thumb and the other one within the opposed finger. Spheres are placed so that their surfaces overlap with the finger ones. Obviously, contour and constraining inequalities have to be adapted according to the finger shapes.
Volume of intersection {#subsec:volume of intersection}
----------------------
Volumes of intersection among fingers discerns their kinematic degree of interaction. Collision points are identified by exploring the finger workspaces. To evaluate the thumb opposability a modified version of the performance index [@GifuIII] is used:
$$I = \frac{1}{d_t^3} \sum_{i = 1}^{k} \sum_{j = 1}^{e} w_{ij} v_{ij}
\label{eq:indexThumbOpposability}$$
where $d_t$ is the thumb length, $k$ is the number of fingers (thumb excluded), $e$ is the number of end-effectors considered on each opposed finger, $w_{ij}$ is a weighing coefficient, $v_{ij}$ is the volume of intersection between the thumb and the finger end-effector. This performance index includes and weights the interaction between the thumb fingertip and each finger portions involved in the Kapandji test. Intersection volumes ($v_{ij}$) are computed sampling the Cartesian space in $x$, $y$ and $z$. The number of cells which compose the grid strictly depends on the hand workspace boundaries and the size of the sampling interval (set to 2 mm). Each contact is checked by visiting the thumb and finger joint spaces and it is stored in the corresponding cell. Consequently, the number of cells holding collisions reveals the intersection volume per finger:
$$v_{ij} = n_{ij} \Delta V
\label{eq:intersectionVolue}$$
where $n_{ij}$ is the number of cells in which the thumb fingertip and the $j^{th}$ end-effector of the $i^{th}$ finger intersect and $\Delta V$ is the volume of a cell.\
Differently from [@designGuidelineForTheThumb], the end-effector is not considered as a single point. Indeed, a single point does not determine the potential grasping capabilities of the hand. For this reason, a cloud of points within the finger and thumb mechanical structures is taken into account. In this case, $v_{ij}$ approximatively discloses the amount of effective interaction between the thumb and the fingers close to the Kapandji positions (Figure \[fig:thumbFingerEeIntersection\]).
Solution selection {#subsec:solution selection}
------------------
To select the best position and orientation of the first joint of the thumb the following algorithm is implemented:
1. Thumb bases are generated using a brute force exploration. Intervals are defined according to aesthetics and mechanical constraints;
2. For each base the Kapandji test is performed;
3. If the candidate passes the Kapandji test: the thumb opposability index is computed beside the relative standard deviation ($\sigma_r$) of the intersection volume:
$$\sigma_r = \frac{\sigma^2}{\bar{v}} 100
\label{eq:relStdDev}$$
where $\sigma^2 = \sum_{i = 1}^{k}(v_i - \bar{v})^2/ k$, $v_i = \sum_{j = 1}^{e} v_{ij}$ and $\bar{v}$ is the mean value of $v_i$.
4. If the relative standard deviation is lower than a desired threshold (in our case set to 20$\%$) the candidate is stored (the thumb opposability is enough equilibrated).
5. The solution is selected picking the stored candidate with the highest thumb opposability index.
Table \[tab:ThumbBaseplacement\] lists some stored candidates. The one with the highest TOI is chosen.
-- -- -- --
-- -- -- --
: Candidate thumb bases. TOI is the Thumb Opposability Index
\[tab:ThumbBaseplacement\]
Validation and discussion {#sec:validation and discussion}
=========================
The hand model is designed and simulated using Python and *NAOqi*[@NAOqi] APIs. Geometric and kinematic parameters are inspected by direct visualization of the final robotic hand model. The Kapandji test is verified by moving the fingers and thumb in the collected configurations obtained during the algorithm execution and it is checked that the hand is able to reach the desired list of positions within the demanded precision. Figure \[fig:LittleThumbFingertip\] shows the furthest (a) and the closest (b) positions involved in the test. Hand grasping capabilities have been examined on various objects (e.g. soda can and a smartphone) using CAD software. However, no conclusions can be drawn on the grasping success since only a kinematic analysis has been conducted at this stage. Beside common gestures, like “pointing” (Figure \[fig:hangGestures\]-a), “thumb-up” or “ok” gestures, the hand is able to count from one to ten in the Chinese way and from one to five in the European one. For example, Figure \[fig:hangGestures\]-b shows the number three in the European counting. Note that, in order to give more emphasis to the finger kinematics a simplified palm is represented.
(a)
(b)
(a)
(b)
Conclusions {#sec:Conclusions}
===========
A simple and effective numerical method for designing dexterous and highly opposable anthropomorphic robotic hands has been presented. The method is conceived to realize a fully embedded, small sized and light hand for a social humanoid robot of similar height to a 6 year old child. It consists in a design tool that generates the hand kinematics which are able to express emotions (gestures) and grasp small objects. Hand geometry has been conceived using anthropomorphic data and heuristics methods which allow a kinematic solution close to that of a human hand. The main focus has been given to the thumb base position and orientation, since its interaction with the opposed fingers strongly effects the hand performance. Three tests drove its selection: aesthetic, surgical and interaction tests. The first played a relevant role to obtain a fairly equilibrated hand, the second provided a human-like dexterous thumb, while the last one defined its opposability. Hand designers can follow this method to immediately obtain the hand kinematic model and structure. Future work will consist in improving the candidate selection among thumb bases using genetic algorithms rather than a brute force approach.
[^1]: $^{1}$Giulio Cerruti is with Institut de Recherche en Communications et Cybernétique de Nantes (IRCCyN), 1 rue de la Noë BP 92101, 44321 Nantes Cedex 3, and Aldebaran in the Advanced Mechatronic Lab (AM-Lab Nantes), 6 rue Saint Domingue, 44200 Nantes, France
[^2]: $^{2}$D. Chablat is with the LUNAM University, IRCCyN UMR CNRS 6597, France [Damien.Chablat@irccyn.ec-nantes.fr]{}
[^3]: $^{3}$D. Gouaillier is head of Aldebaran AM-Lab Nantes, 44200 Nantes, France [dgouaillier@aldebaran.com]{}
[^4]: $^{4}$S. Sakka is with the LUNAM University, IRCCyN UMR CNRS 6597, and University of Poitiers, 93 Avenue du Recteur Pineau, 86000 Poitiers, France [Sophie.Sakka@irccyn.ec-nantes.fr]{}
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abstract: |
We compute exact asymptotic of the statistical density of random matrices belonging to invariant random matrices ensemble (RMT) orthogonal, unitary and symplectic ensembles, where all its eigenvalues lie within the interval $[\sigma, +\infty[$ or $]-\infty,\tau]$ or $[\sigma,\tau]$. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied in [@D], [@S] and [@boo].\
[**Math Subject classification:**]{} 15B52, 15B57, 60B10.\
[**Key-words:**]{} Random matrices, Probability measures, Logarithmic potential.
author:
- Mohamed BOUALI
title: Constrained Eigenvalues Density of Invariant Random Matrices Ensembles
---
Introduction
============
Random matrix theory has been successfully applied in various branches of physics and mathematics, including in subjects ranging from nuclear physics, quantum chaos, disordered systems, and number theory. Of particular importance are invariant random matrices with density $e^{-\tr(Q(x))}$ with respect the Lebesgue measure where $Q$ is some regular potential. There are three interesting classes of matrices distributed with such density: $(n \times n)$ real symmetric (Gaussian Orthogonal Ensemble (GOE)), $(n \times n)$ complex Hermitian (Gaussian Unitary Ensemble (GUE)) and $(2n \times 2n)$ self-dual Hermitian matrices ( Gaussian Symplectic Ensemble (GSE)). In these models the probability distribution for a matrix $X$ in the ensemble is given by $$p_n(X)\propto \exp\Big(-n\frac\beta 2\tr Q(X)\Big).$$ where $\tr$ is the trace function and $\beta$ is the Dyson index and $n$ is a rescaling factor.
A crucial result in the theory of random matrices is the celebrated Wigner semi-circle law. It states that for regular potential $Q$ and for large $n$ and on an average, the $n$ eigenvalues of a matrix distributed as the density $p_n(X)$, lie within a finite interval $[-\sqrt 2, \sqrt 2]$, often referred to as the Wigner ’sea’. Within this sea, the statistical density of eigenvalues has a semi-circular form that vanishes at the two edges $-\sqrt 2$, $\sqrt 2$. $$\rho(\lambda)=\frac1\pi\sqrt{2-\lambda^2}.$$ See for example [@LE], [@dei]. The above result means that, if one looks at the statistical density of eigenvalues of a typical system described by one of the three ensembles above, for a large enough $n$, it will resemble closely to the Wigner semi-circle law. From the semi-circle law, we know that on an average half the eigenvalues are positive and half of them are negative.
An important question will be studied in this work, namely what is the statistical density of eigenvalues which lie within the interval $[\sigma, +\infty[$ or $]-\infty,\tau]$ or $[\sigma,\tau]$.
In this paper we calculate the asymptotic density of eigenvalues in this conditioned ensemble, we will see that it is quite different from the Wigner semi-circle law. We prove the following result. The density of eigenvalues in the interval $[\sigma,\tau]$ is given in this way $$\int_{\Bbb R}f(x)\mu^{\sigma,\tau}(dx)=\frac1{\pi}\int_{\sigma}^{\tau} f(x)p_{\tau,\sigma}(x)\frac1{\sqrt{(\tau-x)(x-\sigma)}}dx,$$ where $p_{\tau,\sigma}(x)$ is some positive function which will be given later.
Also We prove that as $\tau\rightarrow +\infty$ the density present an inverse square root at $\sigma$ and $p_{\tau,\sigma}(x)\rightarrow p_{\sigma}(x)$, $\mu^{\tau,\sigma}\rightarrow\mu^\sigma$ with $$\int_{\Bbb R}f(x)\mu^{\sigma}(dx)=\frac1{\pi}\int_{\sigma}^{L(\sigma)} f(x)p_{\sigma}(x)\sqrt{\frac{L(\sigma)-x}{x-\sigma}}dx,$$ where $L(\sigma)$ is some constant which depend on $\sigma$.
The same hold if one keep $\tau$ fixed and let $\sigma\rightarrow -\infty$ with $\sigma$ replaces by $\tau$. Moreover if $\sigma\to-\infty$ and $\tau\to +\infty$ the case of unconstraint eigenvalues, one recovers the Wigner semi-circular law.
The paper is organized as follows. In the second section we begin by recalling some result about potential theory and equilibrium measure and we enunciate our fundamental equilibrium measure which is the key of the work (theorem \[t1\]). In section 3 we prove the existence of such measure and we give its density explicitly. Section 4 is dedicated to prove that the measure of theorem \[t1\] is the limit of the global density of eigenvalues such that the intervals $]-\infty,\sigma]$ or $[\tau,+\infty[$ or $\Bbb R\setminus[\sigma,\tau]$ are devoid of eigenvalues. At the end we argue our study with some examples and numerical simulations.
Preliminary
===========
Let $\Sigma$ be a closed interval, and $Q$ be a lower semi-continuous function on $\Sigma$. If $\Sigma$ is unbounded we assume that $$\lim_{|x|\to +\infty}\Big(Q(x)-\log(1+x^2)\Big)=\infty.$$ For given $Q$ and $\Sigma$, we wish to compute the equilibrium measure. We start by some general results.
For any probability measure $\mu$ on $\Sigma$, we define the potential of $\mu$ by: for $x\in\Sigma\setminus{\rm supp}(\mu)$ $$U^\mu(x)=\int_\Sigma\log\frac{1}{|x-y|}\mu(dy),$$ and the energy by $$E_{Q, \Sigma}(\mu)=\int_\Sigma U^\mu(x)\mu(dx)+\int_\Sigma Q(x)\mu(dx).$$ From the inequality $$|x-y|\leq\sqrt{1+x^2}\sqrt{1+y^2},$$ it can be seen that $E_{Q, \Sigma}(\mu)$ is bounded below. Let $$E^*_{Q, \Sigma}=\inf_{\mu\in\mathfrak{M}(\Sigma)}E_{Q, \Sigma}(\mu),$$ where $\mathfrak{M}(\Sigma)$ is the set of probability measures on the closed set $\Sigma$.
If $\mu(dx)=f(x)dx$, where $f$ is continuous function with compactly support $\subset\Sigma$, the potential is a continuous function, and $E_{Q, \Sigma}(\mu)<\infty$.
—\[p2\] There is a unique measure $\mu^*\in\mathfrak{M}(\Sigma)$ such that $$E^*_{Q, \Sigma}=E_{Q, \Sigma}(\mu^*),$$ moreover the support of $\mu^*$ is compact.
This measure $\mu^*$ is called the [*equilibrium measure.*]{}
See theorem II.2.3 [@F]. The next proposition is a method to find the equilibrium measure
—\[p1\] Let $\mu\in\mathfrak{M}(\Sigma)$ with compact support. Assume the potential $U^\mu$ of $\mu$ is continuous and, there is a constant $C$ such that\
[(i)]{} $U^\mu(x)+\frac12Q(x)\geq C$ on $\Sigma$.\
[(ii)]{} $U^\mu(x)+\frac12Q(x)= C$ on [supp]{}$(\mu)$. Then $\mu$ is the equilibrium measure $\mu=\mu^*$.
The constant C is called the (modified) Robin constant. Observe that $$E^*_{Q, \Sigma}=C+\frac12\int_\Sigma Q(x)\mu^*(dx).$$
— Let $\Sigma'\subset\Sigma$ be a closed interval of $\Sigma$, if we consider the restriction of the function $Q$ initially defined on $\Sigma$ to the closed interval $\Sigma'$ and if the equilibrium measure $\mu$ associate to $(\Sigma, Q)$ satisfies supp$(\mu)\subset\Sigma'$. Then the equilibrium measure for the couple $(\Sigma', Q)$ is $\mu$.
We saw that if $\Sigma=\Bbb R$, and $Q$ a polynomials of degree $2m$ then the equilibrium measure $\mu_0$ for $(Q,\Sigma)$ is given by Pastur formula $$\int_{\Bbb R}f(x)\mu_0(dx)=\frac1{\pi}\int_{a_0}^{b_0} f(x)q(x)\sqrt{(b_0-x)(x-a_0)}dx,$$ where $q$ is a polynomial of degree $2m-2$ given by $$q(x)=\frac1{2\pi}\int_{a_0}^{b_0}\frac{Q'(x)-Q'(t)}{x-t}\frac{dt}{\sqrt{(b_0-t)(t-a_0)}},$$ The numbers $a_0$ and $b_0$ are given by $$\int_{a_0}^{b_0}\frac{Q'(t)dt}{\sqrt{(b_0-t)(t-a_0)}}=0,\;\;\int_{a_0}^{b_0}\frac{tQ'(t)dt}{\sqrt{(b_0-t)(t-a_0)}}=2\pi.$$
We come to our first result. Let $Q$ be a convex polynomial of degree $2m$ with $m\geq 1$ and $\Sigma=[\sigma,\tau]$ with $-\infty\leq\sigma<\tau\leq+\infty$.
[**A.**]{} The equilibrium measure of the pair $(Q,\Sigma)$ depend on the parameters $\sigma$ and $\tau$. We denoted it by $$\mu(\sigma,\tau,dx).$$ Its support is a compact interval $[a,b]$, $a=a(\sigma,\tau)$ and $b=b(\sigma,\tau)$. Its density $h(x):=h(\sigma,\tau; x)$ has singularity at $a$ and $b$. The following cases are present
\[t1\]
(1)
: [ Soft edges in $a$ and $b$]{}: $$h(x)=\frac1\pi q(x)\sqrt{(x-a)(b-x)}\chi(x),$$ with $\chi$ the indicator of the interval $[a,b]$. $q$ is a polynomial of degree $2m-2$, given by the Pastur formula $$q(x)=\frac1{2\pi}\int_{a}^{b}\frac{Q'(x)-Q'(t)}{x-t}\frac{dt}{\sqrt{(b-t)(t-a)}}.$$
(2)
: [ Hard edge in $a$ and soft edge in $b$]{}: In that case $a=\sigma$: $$h(x)=\frac1\pi p(x)\sqrt{\frac{b-x}{x-a}}\chi(x),$$ $p$ is a polynomial of degree $2m-1$ and is given in this way: $$p(x)=\frac1{2\pi} \int_{a}^{b}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{t-a}{b-t}}dt+\frac12Q'(x).$$
(3)
: [ Soft edge in $a$ and hard edge in $b$]{}: In that case $b=\tau$: $$h(x)=\frac1\pi s(x)\sqrt{\frac{x-a}{b-x}}\chi(x),$$ $s$ is a polynomial of degree $2m-1$ and is given in this way: $$s(x)=\frac1{2\pi} \int_{a}^{b}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{b-t}{t-a}}dt-\frac12Q'(x).$$
(4)
: [ Hard edges in $a$ and $b$]{}: then $a=\sigma$, $b=\tau$: $$h(x)=\frac1\pi r(x)\frac1{\sqrt{(x-a)(b-x)}}\chi(x),$$ where $r$ is a polynomial with degree $2m$, $$r(x)=\frac1{2\pi}\int_{\sigma}^{\tau}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{(t-\sigma)(\tau-t)}dt+\frac12
\big(\frac{\sigma+\tau}{2}-x\big)Q'(x)+1.$$
[**B.**]{} Given $\sigma$ and $\tau$. How to determine which case is present and how to find the parameters $a$ and $b$?
\[t22\]
(i)
: For $\sigma=-\infty$ and $\tau=\infty$, let $a=a(-\infty,\infty)=a_0$ and $b=b(-\infty,\infty)=b_0$, and denote $\mu_0$ the equilibrium measure. One can see that we are in the first case (1). If $\sigma\leq a_0$ and $\tau\geq b_0$, then $a=a_0$ and $b=b_0$ and the equilibrium measure is given by $\mu_0$.
(ii)
: Fix $\tau=\infty$ and keep the barrier at $\sigma$ moving
(ii.a)
: If $\sigma\leq a_0$ then $\mu(\sigma,\infty,dx)=\mu_0(dx).$
(ii.b)
: If $\sigma> a_0$ then $a(\sigma,\infty)=\sigma$, with hard edge in $a(\sigma,\infty)=\sigma$ and soft edge in $b=b(\sigma,\infty)$.
(iii)
: We move the barrier $\tau$ in decreasing order from $\infty$.\
Assume (ii.a) is true
(iii.a)
: If $\tau\geq b_0$ then $\mu(\sigma,\tau,dx)=\mu_0(dx).$
(iii.b)
: If $\tau< b_0$ then $b(\sigma,\tau)=\tau$, with soft edge in $a=a(-\infty,\tau)$ and hard edge in $b(\sigma,\tau)=\tau$.
Assume (ii.b) is true
(iii.c)
: If $\tau\geq b(\sigma,\infty)$ then $\mu(\sigma,\tau,dx)=\mu(\sigma,\infty;dx).$ Hard edge in $a(\sigma,\infty)=\sigma$ and soft edge in $b=b(\sigma,\infty)$.
(iii.d)
: If $\tau< b(\sigma,\infty)$, then $a(\sigma,\tau)=\sigma$ and $b(\sigma,\tau)=\tau$. Hard edges in $\sigma$ and $\tau$.
[**Proof of theorem \[t1\].**]{} We give just the proof of (2) and (4). The others steps follow in the same way.\
(2). Let $\mu^\sigma$ be the equilibrium measure of $\Sigma=[\sigma,+\infty[$, $G_\sigma$ be the Cauchy-Stieljes transform of $\mu^\sigma$, $\displaystyle G_\sigma(z)=\int_a^{b}\frac1{z-t}\mu^\sigma(dt).$
$G_\sigma$ is an holomorphic function on $\Bbb C\setminus[a,b]$ and $$[G_\sigma]=-2i\pi\mu^\sigma.$$ Moreover, for $x\in[a,b]$ $${\rm Re}G_\sigma(x)=\frac12Q'(x).$$ Let $$\label{ww}\widetilde G_\sigma(z)=G_\sigma(z)\sqrt{\frac{z-a}{z-b}}.$$ The function $\widetilde G _\sigma$ is holomorphic in $\Bbb C\setminus[a,b]$ and satisfies $$[\widetilde G_\sigma]=-i\sqrt{\frac{x-a}{b-x}}Q'(x)\chi(x).$$ Since $\widetilde G_\sigma$ goes to $0$ at infinity, using Liouville theorem one gets $$\label{w}\begin{aligned}
\widetilde G_\sigma(z)&=\frac1{2\pi}\int_a^{b}\frac{1}{z-t}Q'(t)\sqrt{\frac{t-a}{b-t}}dt\\
&=-\frac1{2\pi}\int_a^{b}\frac{Q'(z)-Q'(t)}{z-t}\sqrt{\frac{t-a}{b-t}}dt+Q'(z)\frac1{2\pi}\int_a^{b}
\frac{1}{z-t}\sqrt{\frac{t-a}{b-t}}dt.
\end{aligned}$$ Using equations (\[w\]) and (\[ww\]) and lemma \[li\] in appendix, it follows $$[G_\sigma]=-2i\sqrt{\frac{b(\sigma)-x}{x-\sigma}} p(x)\chi(x).$$ Hence one deduces the formula as announced, since $[G_\sigma]=-2i\pi\mu^\sigma$.
(4). Let $\mu^{\sigma,\tau}$ be the equilibrium measure of $\Sigma=[\sigma,\tau]$, and $\displaystyle G_{\sigma,\tau}=\int_a^b\frac1{z-t}\mu^{\sigma,\tau}(dt)$. As in the previous $$[G_\sigma]=-2i\pi\mu^{\sigma,\tau}.$$ Moreover, for $x\in[a,b]$ $${\rm Re}G_\sigma(x)=\frac12Q'(x).$$ Let $$\label{ww}\widetilde G_\sigma(z)=G_\sigma(z)\sqrt{(z-a)(z-b)}.$$ The function $\widetilde G _\sigma$ is holomorphic in $\Bbb C\setminus[a,b]$ and satisfies $$[\widetilde G_\sigma]=i\sqrt{(x-a)(b-x)}Q'(x)\chi(x).$$ Since $\widetilde G_\sigma$ goes to $1$ at infinity, using Liouville theorem one gets $$\label{w}\begin{aligned}
\widetilde G_\sigma(z)&=-\frac1{2\pi}\int_a^{b}\frac{1}{z-t}Q'(t)\sqrt{(t-a)(b-t)}dt+1\\
&=\frac1{2\pi}\int_a^{b}\frac{Q'(z)-Q'(t)}{z-t}\sqrt{(t-a)(b-t)}dt-Q'(z)\frac1{2\pi}\int_a^{b}
\frac{1}{z-t}\sqrt{(t-a)(b-t)}dt+1.
\end{aligned}$$ using equations (\[w\]) and (\[ww\]) and lemma \[li\] in appendix, it follows $$[G_\sigma]=-2i\frac1{\sqrt{(b-x)(x-a)}} r(x)\chi(x).$$ Hence one deduces the formula as announced.
[**Proof of theorem \[t22\].**]{}\
[*Proof of (ii).*]{}\
(ii.b). Let $\Sigma_\sigma=[\sigma,+\infty[$ with $\sigma\geq a_0$ and $Q$ any function satisfying the hypotheses as above. Let $\mu^\sigma(dx):=\mu(\sigma,\infty,dx)$ be the measure defined as in (2) of theorem \[t1\], with $b=b(\sigma,\infty)=b(\sigma)$ given in the proposition \[pp1\] in appendix, $$p(x)=\frac1{2\pi} \int_{\sigma}^{b(\sigma)}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{t-\sigma}{b(\sigma)-t}}dt+\frac12Q'(x).$$ Remark that from lemma \[li\] and proposition \[pp1\] in appendix, one gets $$\displaystyle p(\sigma)=\int_\sigma^{b(\sigma)}\frac{Q'(t)}{\sqrt{(t-\sigma)(b(\sigma)-t)}}=\varphi(\sigma,b(\sigma))\geq 0.$$ Moreover a simple computation give $$p(x)=p(\sigma)+(x-\sigma)q(x),$$ where $$q(x)=\frac1{2\pi}\int_{\sigma}^{b(\sigma)}\frac{Q'(x)-Q'(t)}{x-t}\frac{dt}{\sqrt{(b(\sigma)-t)(t-\sigma)}}.$$ By convexity of the polynomial $Q$ and the fact that $p(\sigma)\geq 0$, one gets for all $x\geq\sigma$, $p(x)\geq 0$.
Expanding the function $\widetilde G_\sigma$ near infinity, it follows $$\widetilde G_\sigma(z)=\sum_{n=0}^{+\infty}\frac{a_n}{z^{n+1}},$$ where $\displaystyle a_n=\frac1{2\pi}\int_{\sigma}^{b(\sigma)}t^nQ'(t)\sqrt{\frac{t-\sigma}{b(\sigma)-t}}dt,$ Since $$\frac1{2\pi}\int_{\sigma}^{b(\sigma)}Q'(t)\sqrt{\frac{t-\sigma}{b(\sigma)-t}}dt=-\frac1{2\pi}\psi(\sigma,b(\sigma))+1=1,$$ which follow from proposition \[pp1\] in appendix. Thus $$\widetilde G_\sigma(z)=\frac1z+o(\frac1z),$$ moreover $$G(z)=\sqrt{\frac{z-\sigma}{z-b(\sigma)}}\widetilde G_\sigma(z),$$ hence, $$G(z)=\frac1z+o(\frac1z).$$
Which prove that $\mu(\sigma,\infty;dx)$ is a probability.
It remains to prove that $\mu(\sigma,\infty;dx)$ is the equilibrium measure. Since for $a=\sigma$ and $b=b(\sigma)$, $$G_\sigma(z)=-\sqrt{\frac{z-b(\sigma)}{z-\sigma}}p(z)+\frac12Q'(z).$$ Moreover $Q$ is holomorphic, hence $p$ is holomorphic too. It follows that
$$\lim_{y\to 0,\,y>0}{\rm Re}G_\sigma(x+iy)=\left\{\begin{aligned}&-\sqrt{\frac{b(\sigma)-x}{\sigma-x}}p(x)+\frac12Q'(x)\;\; if\;\;x\geq b(\sigma),\\
&\quad\frac12Q'(x)\;\qquad\qquad\qquad\quad{if}\,\sigma\leq x\leq b(\sigma).
\end{aligned}\right.$$ An easy computation shows that $$\frac{d}{dx}U^{\mu^\sigma}(x)=-\lim_{y\to 0,\,y>0}{\rm Re}G_\sigma(x+iy).$$ Hence
$$2\frac{d}{dx}U^{\mu^\sigma}(x)+Q'(x)=\left\{\begin{aligned}&2\sqrt{\frac{b(\sigma)-x}{\sigma-x}}p(x)\quad if\;\;x\geq b(\sigma),\\
&0\qquad\qquad\qquad\;{if}\,\sigma\leq x\leq b(\sigma).
\end{aligned}\right.$$ We have proved in the beginning that $p(x)\geq0$ for all $x\geq\sigma\geq a_0$, hence $$2U^{\mu^\sigma}(x)+Q(x)\left\{\begin{aligned}&\geq C\qquad\qquad if\;x\geq b(\sigma),\\
&=C\qquad\qquad\;{if}\,\sigma\leq x\leq b(\sigma).
\end{aligned}\right.$$ (ii.a). For $\sigma\leq a_0$, The equilibrium measure is $\mu_0(dx)$. In fact, if there is some $a\in[\sigma,a_0[$ such that $p(x)\geq 0$ for all $x\in[a,a_0[$, where $$p(x)=\frac1{2\pi} \int_{a}^{b(a)}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{t-a}{b(a)-t}}dt+\frac12Q'(x).$$ Then by the equation $$p(x)=p(a)+(x-a)q(x),$$ one gets for $x\in[a,a_0[$ $$p(a)+(x-a)q(x)\geq 0.$$ As $x\to a$, we obtain $$p(a)=\varphi(a,b(a))\geq 0.$$ This contradicts the minimality of $a_0$ see proposition \[pp1\] in appendix. This complete the proof.\
[*Prove of (iii).*]{} We prove just (iii.c) and (iii.d), the rest hold with the same lines as in (ii) with little modification.\
$1)$ $a_0<\sigma$ and $\Sigma=[\sigma,\tau]$.\
(iii.c) First case $\tau<b(\sigma)$. The density of the measure is $$\mu^{\sigma,\tau}(dx)=h(x,\sigma,\tau)=\frac1\pi\frac{r(x)}{\sqrt{(\tau-x)(x-\sigma)}}\chi_{[\sigma,\tau]}(x)dx,$$ with $$r(x)=\frac1{2\pi}\int_{\sigma}^{\tau}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{(t-\sigma)(\tau-t)}dt-\frac12\Big(x-\frac{\sigma+\tau}{2})\Big)Q'(x)+1.$$ Since by some algebra one gets $$r(x)=r(\tau)+(\tau-x)p(x),$$ where $$p(x)=\frac1{2\pi} \int_{\sigma}^{\tau}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{t-\sigma}{\tau-t}}dt+\frac12Q'(x),$$ moreover $p(x)\geq 0$ for all $x\geq\sigma$. It remains to prove that $r(\tau)\geq 0$.
A simple computation gives $$r(\tau)=-\frac1{2\pi}\psi(\sigma,\tau),$$ hence $$r(\tau)=-\frac1{2\pi}h(\sigma,\tau)(\tau-\sigma).$$ We saw that the function $x\mapsto h(\sigma,x)$ increases for all $x>\sigma$, and $b(\sigma)\geq\tau>\sigma$, it follows that $$r(\tau)=-\frac1{2\pi}h(\sigma,\tau)(\tau-\sigma)\geq-\frac1{2\pi}h(\sigma,b(\sigma)(\tau-\sigma)=0.$$
If $$G_{\sigma,\tau}(z)=\int_\sigma^{\tau}\frac1{z-t}\mu^{\sigma,\tau}(dt),$$ then as in the previous, one has $$G_{\sigma,\tau}(z)=\frac1{\sqrt{(z-\sigma)(z-\tau)}}S(z),$$ where $$S(z)=\frac1{2\pi}\int_{\sigma}^{\tau}\frac{Q'(z)-Q'(t)}{z-t}\sqrt{(t-\sigma)(\tau-t)}dt-\frac12\Big(z-\frac{\sigma+\tau}{2}
-\sqrt{(z-\sigma)(z-\tau)}\Big)Q'(z)+1.$$ Since for $z\in\Bbb C\setminus[\sigma,\tau]$, $$S(z)=\frac1{2\pi}\int_{\sigma}^{\tau}\frac{-Q'(t)}{z-t}\sqrt{(t-\sigma)(\tau-t)}dt+1,$$ thus $$\lim_{|z|\to +\infty}S(z)=1,$$ and $$\lim_{|z|\to+\infty}|z|G(z)=\lim_{|z|\to +\infty}S(z)=1.$$ It follows that $\mu^{\sigma,\tau}$ is a probability measure. Moreover $$\lim_{y\to 0,\,y>0}{\rm Re}G_\sigma(x+iy)=\frac12Q'(x),\quad\forall x\in[\sigma,\tau].$$ Hence $$2\frac{d}{dx}U^{\mu^{\sigma,\tau}}(x)+Q'(x)=0\quad\forall\;x\in [\sigma,\tau].$$ This shows that $\mu^{\sigma,\tau}$ is the equilibrium measure.
(iii.d) Second case $\tau> b(\sigma)$. In this case we saw that $\mu(\sigma,\infty,dx)$ is a probability measure on $\Sigma=[\sigma,+\infty[$ with support $[\sigma,b(\sigma)]$, hence it is a probability on $[\sigma,\tau]$. Moreover for all $\tau\geq b(\sigma)$
$$\lim_{y\to 0,\,y>0}{\rm Re}G_\sigma(x+iy)=\left\{\begin{aligned}&-\sqrt{\frac{x-b(\sigma)}{x-\sigma}}p(x)+\frac12Q'(x)\;\; if\;\tau\geq x\geq b(\sigma),\\
&\quad\frac12Q'(x)\;\qquad\qquad\qquad\quad{if}\,\sigma\leq x\leq b(\sigma).
\end{aligned}\right.$$ Hence $\mu(\sigma,\infty,dx)$ is an equilibrium measure on $[\sigma,\tau]$. By unicity one gets $\mu^{\sigma,\tau}(dx)=\mu(\sigma,\infty,dx)$.
One can prove (iii.d) in this way. Assume there is some $b\in]b(\sigma),\tau]$ such that the measure $\displaystyle\mu(\sigma,b,dx)=\frac1{\pi}\frac{r(x)}{\sqrt{(x-\sigma)(b-x)}}dx$ define a probability measure on $[\sigma,b]$. Then as in the previous, for all $x\in[\sigma,b]$ $$r(x)=r(b)+(b-x)p(x),$$ with $$p(x)=\frac1{2\pi} \int_{\sigma}^{b}\frac{Q'(x)-Q'(t)}{x-t}\sqrt{\frac{t-\sigma}{b-t}}dt+\frac12Q'(x),$$ We saw that $\displaystyle r(b)=-\frac1{2\pi}\psi(\sigma,b)$. In other hand, the measure $\mu(\sigma,b,dx)$ is a probability, hence $\displaystyle-\frac1{2\pi}\psi(\sigma,b)\geq 0$, and $\displaystyle h(\sigma,b)=\frac{\psi(\sigma,b)}{b-\sigma}\leq 0$ which gives $h(\sigma,b)\leq 0$. Furthermore, the function $x\mapsto h(\sigma,x)$ increases strictly, hence $$0=h(\sigma,b(\sigma))<h(\sigma,b)\leq 0.$$ Which gives a contradiction. This prove that the support of the measure is $[\sigma,b(\sigma)]$ and $$r(x)=-\frac1{2\pi}\psi(\sigma,b(\sigma))+(b(\sigma)-x)p(x)=(b(\sigma)-x)p(x),$$ hence $$\mu(\sigma,\tau,dx)=\mu(\sigma,\infty,dx).$$
To see how to pass from the hard edge in $\tau$ to the soft edge in $b(\sigma)$, for $\tau< b(\sigma)$, one can write $r(x)$ in the following sense $$r(x)=r(\tau)+(\tau-x)p(x),$$ since $r(\tau)=-\frac1{2\pi}\psi(\sigma,\tau)$. As $\tau\to b(\sigma)$, one gets $r(b(\sigma))=0$ see proposition \[pp1\] in appendix, hence $$r(x)=(b(\sigma)-x)p(x),$$ and the density $$\frac1{\pi}\frac{r(x)}{\sqrt{(\tau-x)(x-\sigma)}},$$ becomes $$\frac1{\pi}p(x)\sqrt{\frac{b(\sigma)-x}{x-\sigma}}.$$ The same hold in all the others cases.
One can change the conditions on the potential $Q$, to be a sufficiently regular and convex function with $\displaystyle\lim_{|x|\to +\infty}Q(x)-\log(1+x^2)=+\infty$ and $\displaystyle\lim_{x\to \pm\infty} Q'(x)=\pm\infty$.
Constrained eigenvalues density
===============================
We consider the invariant random matrices ensemble with Dyson index $\beta = 1,2,4$, corresponding to real, complex, and quaternion entries, respectively. The probability distribution of the entries is given by $$\Bbb{P}_{n}(dX)=\frac1{C_n}\exp\Big(-n\frac{\beta}{2}\tr Q(X)\Big)dX.$$ where $Q$ is a convex polynomial with even degree $2m$, $m\geq 1$, $C_n$ is a normalizing constant and $dX$ is the Lebesgue measure on the space $H_n=Herm(n,\Bbb F)$ of hermitian matrices with respectively real, complex or quaternion coefficients $\Bbb F=\Bbb R$, $\Bbb C$, or $\Bbb H$. Consequently the joint probability density of eigenvalues is given by $$\Bbb{P}_{n}(d\lambda_1,...,d\lambda_n)=\frac1{C_n}e^{-n\frac{\beta}{2}
\sum\limits_{i=1}^nQ(\lambda_i)}|\Delta(\lambda)|^{\beta}d\lambda_1...d\lambda_n,$$ where $\Delta(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)$ is the Vandermonde determinant, and $n$ is a normalizing factor.
More generally one can consider this probability with an arbitrary real number $\beta>0$.
For $\sigma,\tau\in[-\infty,+\infty]$, $\sigma<\tau$, consider $$\Omega_{n, \sigma,\tau}=\Big\{X\in H_n\mid\,\lambda_{\rm min}(X)\geq \sigma\; {\rm and}\;\lambda_{\rm \max}(X)\leq\tau\Big\},$$ the subset of Hermitian matrices for which all its eigenvalues are in $[\sigma,\tau]$.
We wish to study $\Bbb{P}_{n}(\Omega_{n, \sigma,\tau})$, the probability for a hermitian matrix $X\in H_n$, to have all its eigenvalues in $\Omega_{n, \sigma,\tau}$. It is the probability that all the eigenvalues lies in the interval $[\sigma, \tau]$ that is $$\Bbb{P}_{n}(\Omega_{n, \sigma,\tau})=\frac1{C_n}\int_{[\sigma,\tau]^n}e^{-n\frac{\beta}{2}
\sum\limits_{i=1}^nQ(\lambda_i)}|\Delta(\lambda)|^{\beta}d\lambda_1...d\lambda_n.$$
Let $\nu^{\sigma,\tau}_{n}$ be the probability measure defined on $[\sigma,\tau]$ by : for all continuous functions $f$ $$\int_{[\sigma,\tau]}f(x)\nu^{\sigma,\tau}_{n}(dx)=\int_{[\sigma,\tau]^n}\frac1n\sum_{i=1}^nf(\lambda_i)\Bbb{P}_{n}(d\lambda_1,d\lambda_2,...,d\lambda_n),$$ which means that $$\nu^{\sigma,\tau}_{n}=\Bbb{E}_{n}\big(\frac1n\sum_{i=1}^n\delta_{\lambda_i}\big),$$ where $\Bbb{E}_{n}$ is the expectation with respect to the measure $\Bbb{P}_{ n}$.
As $n$ goes to infinity we prove, the measure $\nu^{\sigma,\tau}_{n}$ converges to some probability measure, which is the statistical density of the eigenvalues in the interval $[\sigma,\tau]$. In other word one has the following theorem
—\[t2\] There exist a unique $a=a(\sigma,\tau)$, and $b=b(\sigma, \tau)$, such that, the measure $\nu^{\sigma,\tau}_{n}$ converges for the tight topology to the probability measure $\mu(\sigma,\tau,dx)$ of theorem \[t1\], with support $[a,b]$, and density $h(\sigma, \tau,x)$. This means for all continuous functions $\varphi$ on $[\sigma,\tau]$, $$\lim_{n\to\infty}\int_\sigma^\tau\varphi(x)\nu_{n}^{\sigma,\tau}(dx)=\int_\sigma^\tau\varphi(x)\mu(\sigma,\tau,dx).$$
[*Examples.*]{}\
1) The density of eigenvalues such that all eigenvalues lie in the interval $[\sigma,+\infty[$ is given by the measure $\mu^\sigma(dx):=\mu(\sigma,\infty,dx)$, see (2) theorem \[t1\].\
2) The density of eigenvalues in $\Bbb R$ (unconstrained condition) is given by Pastur formula $\mu(-\infty,\infty,dx)$, see (1) theorem \[t1\].
Proof of theorem \[t2\]
-----------------------
In the rest of the section we prove theorem \[t2\], for this purpose we need some preliminary results.
Let $K_n$ be the function on $\Sigma^n=[\sigma,\tau]^n$, defined by $$K_n(x)=\sum_{i\neq j}k_n(x_i,x_j),$$ where $$k_n(x,y)=\log\frac{1}{|x-y|}+\frac12Q(x)+\frac12Q(y).$$ The function $K_n$ is bounded below, moreover if $\sigma=-\infty$ or $\tau=+\infty$, $\lim_{|x|\to+\infty}K_n(x)=+\infty$, it follows that $K_n$ attaint it minimum at a point say, $x^{(n, \sigma,\tau)}=(x_1^{(n, \sigma,\tau)},...,x_n^{(n, \sigma,\tau)} )$. Let $$\delta^{\sigma,\tau}_n=\frac1{n(n-1)}\inf_{\Sigma^n} K_n(x),$$ and $$\rho^{\sigma,\tau}_n=\frac1n\sum_{i=1}^n\delta_{x^{(n, \sigma,\tau)}_i}.$$
For a probability measure $\mu$ on $\Sigma$, consider the energy $$E_{\delta, \sigma}(\mu)=\int_{\Sigma^2}\log\frac{1}{|s-t|}\mu(ds)\mu(dt)+\int_{\Sigma}Q(s)\mu(ds).$$ and $$E^*_{\delta, \sigma}=\inf_{\mu}E_{\delta, \sigma}(\mu),$$ where the minimum is taken over all compactly support measures with support in $\Sigma$. Moreover, defined the scaled density $$\Bbb{P}_{ n}(dx)=\frac1{Z_n}e^{-n\frac{\beta}{2}
\sum\limits_{i=1}^nQ(x_i)}|\Delta(x)|^{\beta}dx_1...dx_n.$$ where $Z_n$ is a normalizing constant.
—\[p3\]
(1)
: $\displaystyle\lim_{n\to\infty}\delta^{\sigma,\tau}_n=E^*_{\sigma,\tau}=E_{\sigma}(\mu^{\sigma,\tau}).$
(2)
: The measure $\rho^\sigma_n$ converge for the tight topology to the equilibrium measure $\mu^{\sigma,\tau}$.
(3)
: $\displaystyle\lim_{n\to\infty}-\frac1{n^2}\log Z_n=\frac\beta2E^*_{ \sigma,\tau}$.
[**Proof.**]{}—\
[**Step 1 and 2:**]{} For a probability measure $\mu$, $$\int_{\Bbb R^n_+}K^\sigma_n(x)\mu(dx_1)...\mu(dx_n)=n(n-1)\int_{\Sigma^2}\log\frac{1}{|x-y|}\mu(dx)\mu(dx)+n(n-1)\int_{\Sigma}Q(x)\mu(dx),$$ hence $$\delta_n^{\sigma,\tau}\leq E_{\sigma,\tau}(\mu).$$ For $\mu=\nu^{\sigma,\tau}$, $$\label{34}\delta_n^{\sigma,\tau}\leq E^*_{\sigma,\tau}.$$ Moreover $$K_n(x^{(n, \sigma,\tau)})=\sum_{i\neq j}k_n\Big(x_i^{(n, \sigma,\tau)},x_j^{(n, \sigma,\tau)}\Big)\geq(n-1)\frac12\Big(\sum_{i=1}^nh\Big(x_i^{(n, \sigma,\tau)}\Big)+
\sum_{i=1}^nh\Big(x_i^{(n,\sigma,\tau)}\Big)\Big),$$ where $\displaystyle h(x)=Q(x)-\log(1+x^2),$ Since $$\int_{\Sigma}h(t)\rho^{\sigma,\tau}_n(dt)=\frac1n\sum_{i=1}^nh\Big(x_i^{(n, \sigma,\tau)}\Big),$$ it follows that, $$\int_{\Sigma}h(t)\rho^{\sigma,\tau}_n(dt)\leq\delta^{\sigma,\tau}_n\leq E^*_{\sigma,\tau}.$$ Using the fact that $\lim_{x\to \infty}h(x)=+\infty,$ then by the Prokhorov criterium there is some subsequence $\rho_{n_k}$, which convergent to $\rho$ for the tight topology.
For $\ell\geq 0$, let $k_n^{ \ell}(x,y)=\inf(k_n(x,y),\ell)$, defined $$E^\ell_{\sigma,\tau}(\mu)=\int_{\Sigma^2}k^{ \ell}_n(x,y)\mu(dx)\mu(dy),$$
$$\label{35}E^\ell_{ \sigma,\tau}(\rho^{\sigma,\tau}_{n_k})\leq\delta_n^{\sigma,\tau}+\frac{\ell}{n}\leq E^*_{\sigma,\tau}+\frac{\ell}{n}.$$
The cut kernel $k^{\sigma, \ell}_{\alpha\pm\e}(x,y)$ is bounded and continuous, and the probability measure $\rho^{\sigma,\tau}_{n_k}$ converge tightly to $\rho^{\sigma,\tau}$, hence $\displaystyle\lim_{k\to+\infty}E_{\sigma,\tau}(\rho^{\sigma,\tau}_{n_k})=E_{\sigma,\tau}(\rho^{\sigma,\tau})$. As $\ell$ goes to $+\infty$, by the monotone convergence theorem one obtains $$E_{ \sigma,\tau}(\rho^{\sigma,\tau})\leq E^*_{\sigma,\tau}.$$
By the definition of the equilibrium measure we obtains $E^*_{\sigma,\tau}=E_{\sigma,\tau}(\rho^{\sigma,\tau})=E_{\sigma,\tau}(\mu^{\sigma,\tau})$, it follows by unicity of the equilibrium measure that $\rho^{\sigma,\tau}=\mu^{\sigma,\tau}$. Which means the only possible limit for a subsequence of $\rho_n^{\sigma,\tau}$ is $\mu^{\sigma,\tau}$, hence the sequence $\rho_n^{\sigma,\tau}$ it self converge to $\mu^{\sigma,\tau}$. Moreover from equation (\[35\]) one gets $$\lim\limits_n\delta^{\sigma,\tau}_n=E^*_{\sigma,\tau}.$$ [**Step 3:**]{} We saw for every $x\geq 0$, $K^\sigma_n(x)\geq n(n-1)\delta^{\sigma,\tau}_n,$ hence $${Z}_n\leq e^{-\frac\beta 2n(n-1)\delta^{\sigma,\tau}_n}c^n ,$$ where $\displaystyle c=\int_0^{+\infty}e^{-\frac\beta 2Q(x)}dx$, hence $$\frac1{n^2}\log{Z}_n\leq -\frac\beta2\frac{n-1}{n}\delta^{\sigma,\tau}_n+\frac1n\log(c).$$ Then $$\limsup_n\frac1{n^2}\log{Z}_n\leq -\frac\beta 2E^*_{\sigma, \tau}.$$ Furthermore $${Z}_n\geq \int_{\Bbb R^n}e^{-\frac\beta 2K_n(x)-\frac\beta 2Q(x)-\sum\limits_{i=1}^n\log h(\sigma,\tau,x_i)}\prod_{i=1}^n\mu^{\sigma,\tau}(dx_i),$$ Applying Jensen’s inequality we obtain
$${Z}_n\geq\exp\int_{\Bbb R^n}\Big(-\frac\beta 2K_n(x)-\frac\beta 2Q(x)-\sum\limits_{i=1}^n\log h(\sigma,\tau,x_i)\Big)\prod_{i=1}^n\mu^{\sigma,\tau}(dx_i),$$ hence $${Z}_n\geq e^{-\frac\beta 2\big(n(n-1)E^*_{\sigma,\tau}\big)}\exp\Big(-\frac\beta 2n\int_a^bQ(x)h(\sigma,\tau, x)dx\Big)\exp\Big(-n\int_a^bh(\sigma,\tau, x)\log h(\sigma,\tau, x)\Big)dx.$$ The function $x\mapsto h(\sigma,\tau, x)\log h(\sigma, \tau, x)$ is continuous on $[a,b]$ $(a:=a(\sigma,\tau); b:=b(\sigma,\tau))$. Hence $$\liminf_n\frac1{n^2}\log{Z}_n\geq -\frac\beta 2E^*_{\sigma,\tau},$$ and the conclusion hold $$-\frac\beta 2E^*_{ \sigma,\tau}\leq\liminf_n\frac1{n^2}\log{Z}_n\leq\limsup_n\frac1{n^2}\log{Z}_n\leq -\frac\beta 2E^*_{\sigma,\tau}.$$ [**Proof of theorem \[t2\]**]{}—The proof of the theorem follows the proof in ([@F], theorem IV.5.1).
Examples.
=========
[*Example 1.*]{}
If one considers the Gaussian invariant ensemble, for such ensemble $Q(x)=x^2$. The solutions of the two equations $\varphi(b_0,a_0)=\psi(b_0,a_0)=0$ are $(a(-\infty,\infty),b(-\infty,\infty))$ see proposition \[pp1\] in appendix, one obtains $a(-\infty,\infty)=a_0=-\sqrt 2$, and $b(-\infty,\infty)=b_0=\sqrt 2$, moreover $$r(x)=\frac1{\pi}\int_{\sigma}^{\tau}\sqrt{(t-\sigma)(\tau-t)}dt+x(-x+\frac{\sigma+\tau}{2})+1.$$ [*Two Hard edges.*]{} By a simple computation it yields the density of the measure $\mu^{\sigma,\tau}$: for $-\sqrt 2\leq\sigma<\tau\leq\sqrt 2$, $$\label{eq001} h(\sigma,\tau,x)=\frac1{\pi}\frac1{\sqrt{(\tau-x)(x-\sigma)}}\Big(\frac{(\sigma-\tau)^2}{8}+1+x\frac{\sigma+\tau}{2}-x^2\Big),$$
[*Two Soft edges.*]{} For $\sigma\leq -\sqrt 2$ and $\tau\geq\sqrt 2$ one gets two soft edges in $-\sqrt 2,\sqrt 2$ and $\mu^{\sigma ,\tau}=\mu^{-\sqrt 2 ,\sqrt 2}$ is the semicircle law with density $$h(\sigma,\tau,x)=h(-\infty,\infty,x)=\frac1\pi\sqrt{2-x^2},$$ $h(-\infty,\infty,x)$ is the density of unconstrained eigenvalues.
[*Hard edge in $\sigma$ and Soft edge in $b(\sigma)$.*]{} For $-\sqrt 2<\sigma<\sqrt 2$ and $\tau>\sqrt 2$, one gets a hard edge in $\sigma$ and a soft edge in $b(\sigma)<\tau$ and from proposition \[pp1\] appendix we obtain $\displaystyle b(\sigma)=\frac23\Big(\frac\sigma2+\sqrt{\sigma^2+6}\Big)$ and the density with support $[\sigma,b(\sigma)]$ is given in this way $$h(\sigma,\tau,x)=h(\sigma,\infty,x)=\frac1{2\pi}\sqrt{\frac{b(\sigma)-x}{x-\sigma}}\Big(2x+b(\sigma)-\sigma\Big)\chi(x),$$ The density $h(\sigma,\infty,x)$ represents the density of eigenvalues of Gaussian hermitian random matrices to have all its eigenvalues in $[\sigma,+\infty[$. This cases agree with results of Dean-Majumdar see for instance [@D] and [@S].
[*Soft edge in $a(\tau)$ and Hard edge in $\tau$.*]{} For $\sigma<-\sqrt 2$ and $\tau<\sqrt 2$, we obtain a hard edge in $\tau$ and a soft edge in $\displaystyle a(\tau)=\frac23\big(\frac \tau2-\sqrt{\tau^2+6}\big)$ and the density with support $[a(\tau),\tau]$ is $$h(\sigma,\tau,x)=h(-\infty,\tau,x)=\frac1{2\pi}\sqrt{\frac{x-a(\tau)}{\tau-x}}\Big(a(\tau)-\tau-2x\Big)\chi(x),$$ $h(-\infty,\tau,x)$ is the density such that all eigenvalues of Gaussian hermitian matrices lie in $]-\infty,\tau]$.
[*Example 2.*]{} Let $Q(x)=x^2-\mu_n\log|x|$ with $\mu_n$ a nonnegative sequence of real numbers. It has been proved in [@boo] that, if $\displaystyle\lim_{n\to\infty}\frac{\mu_n}n=\alpha$, the density of eigenvalues to be all in the half line $[\sigma,+\infty[$ is $$f_{\alpha, a}(x)=\frac{1}{2\pi}\sqrt{\frac{b-x}{x-a}}\Big(2x+b-a-2\alpha\sqrt{\frac{a}{b}}\frac1x\Big)\chi(x),$$ where $a\geq\sigma$ and $b>a$ are the unique solutions of the following equations $$\label{01}b+a-\frac{2\alpha}{\sqrt {ab}}=0,\;\;\frac{3}{4}(b-a)^2+a(b-a)+2\alpha\frac{\sqrt a}{\sqrt b}-2\alpha-2=0,$$ and $\chi$ is the characteristic function of the interval $[a,b]$. One can see that the two previous equations are just the following $\varphi(a,b)=0$ and $\psi(a,b)=0$ as in proposition\[pp1\] in appendix.
If $\alpha>0$, for all $\sigma>0$ the previous equations admit a unique solutions $a(\sigma,\alpha)\geq\sigma$ and $b(\sigma,\alpha)>a(\sigma,\alpha)$. It has been proved that there exist some critical value $a_c(\alpha)$ and $b_c(\alpha)$ as in theorem \[t22\] $(a_0,b_0)$, which is a transition point from a hard edge in $a$ to a soft edge in $a_c$.
For $\alpha>0$ then $a_c>0$ and for $ 0\leq\sigma\leq a_c$, the density of eigenvalues to be all in the interval $[\sigma,+\infty[$ is $$\label{e1}f_\alpha(x)=\sqrt{(b_c-x)(x-a_c)}\Big(1+\frac\alpha{\sqrt {a_c b_c}}\frac1x\Big)\chi(x).$$
As $\alpha\to 0^+$ we find from equation (\[01\]) the case of Gaussian random unitary ensemble. As $\alpha\to 0^+$ then $a_c(\alpha)\to-\sqrt 2$ and $b_c(\alpha)\to\sqrt 2$ and one recovers the Wigner semicircle law.
Also in that case one can find the density of eigenvalues to lie within in the interval $[0,+\infty[$ which is given by $$f(x)=\frac{1}{2\pi}\sqrt{\frac{b-x}{x}}\Big(2x+b\Big)\chi(x),$$ with $b=\frac23\sqrt 6$ and $\chi$ the indicator function of the interval $[0,\frac23\sqrt 6]$.
[**Approximation Density.**]{} Here we give a numerical simulation of the density of eigenvalues for the case where $Q(x)=x^2-\mu_n\log|x|$.
Recall that for $\lim_{n\to+\infty}\frac{\mu_n}{n}=\alpha$, the density of positive eigenvalues is $\displaystyle f_{\alpha}(x)$ of equation (\[e1\]).
Let $$f_n(x)=\frac1{\sqrt n}\sum_{k=0}^{n-1}\varphi^{\mu_n}_k(\sqrt nx)^2,$$ where $\displaystyle\varphi^{\mu_n }_k(x)=\frac1{\sqrt {d_{k,n}}}H^{\mu_n}_k(x)x^{\mu_n}e^{-\frac{x^2}{2}},$ and $H_k^{\mu_n}$ is the truncated orthogonal Hermite polynomial on the positive real axis, which satisfies $$\int_0^{+\infty}H_k^{\mu_n}(x)H_m^{\mu_n}(x)x^{2\mu_n}e^{-x^2}dx=0,\quad {\rm for}\; m \neq k,$$ and $$\int_0^{+\infty}(H_k^{\mu_n}(x))^2x^{2\mu_n}e^{-x^2}dx=d_{k, n}.$$ It has been proved in theorem 5.1 in [@boo], that as n go to $+\infty$, the density $f_n$ converge tightly to the density $f_\alpha$ where $\displaystyle\alpha=\lim_{n\to\infty}\frac{\mu_n}n$.
[**First case:**]{} $n=7, \mu_7=0$, hence $\alpha=0$. $$\displaystyle f_0(x)=\frac1{2\pi}\sqrt{\frac{\frac23\sqrt 6-x}{x}}\big(2x+\frac23\sqrt 6\big), {\rm with\;\; support}\;\; [0,\frac23\sqrt 6].$$
Exact density $f_0$ of positive eigenvalues .\
Approximative density $f_7$ of positive eigenvalues.\
[**Second case:**]{} $n=5, \mu_5=\frac52$, hence $\alpha=\frac12$. $$\displaystyle f_{\frac12}(x)=\frac1{\pi}\sqrt{(1.9-x)(x-0.1)}\big(1+\frac1{2\sqrt{0.19}x}\big), {\rm with\;\; support}\;\; [0.1,1.9].$$
Exact density $f_{\frac12}$ of positive eigenvalues .\
Approximative density $f_5$ of positive eigenvalues.\
Appendix
\[li\] For $z\in\Bbb C\setminus[a,b]$ $$\frac1{\pi}\int_a^b\frac1{z-t}\sqrt{\frac{t-a}{b-t}}dt=\sqrt{\frac{z-a}{z-b}}-1.$$ $$\frac1{\pi}\int_a^b\frac1{z-t}\sqrt{\frac{b-t}{t-a}}dt=-\sqrt{\frac{z-b}{z-a}}+1.$$ $$\frac1\pi\int_a^b\frac1{z-x}\sqrt{(b-x)(x-a)}dx=z-\frac{a+b}{2}-\sqrt{(z-a)(z-b)}.$$
[*Consequence.*]{} $$\frac1\pi\int_a^b\sqrt{(b-x)(x-a)}dx=\frac{(a-b)^2}{8}.$$ $$\frac1{\pi}\int_a^b\sqrt{\frac{t-a}{b-t}}=\frac1{\pi}\int_a^b\sqrt{\frac{b-t}{t-a}}=\frac{b-a}{2}.$$ [**Proof.**]{} The Cauchy-Stieljes transform $$G(z)=\int_a^b\frac1{z-t}\sqrt{\frac{t-a}{b-t}}dt,$$ satisfies $$[G]=-2i\pi\sqrt{\frac{t-a}{b-t}}\chi(t),$$ where for $x\in[a,b]$,$[G]=\displaystyle\lim_{\varepsilon\to 0}G(x+i\varepsilon)-G(x-i\varepsilon)$ is the bounded values distribution, and $\chi$ is the indicator function of $[a,b]$. Moreover $$[\sqrt{\frac{z-a}{z-b}}]=-2i\sqrt{\frac{t-a}{b-t}}\chi(t).$$ The function $$\frac1{\pi}G(z)-\sqrt{\frac{z-a}{z-b}},$$ is holomorphic in $\Bbb C$ with limit $-1$ at infinity. By Liouville theorem one gets $$\frac1{\pi}G(z)-\sqrt{\frac{z-a}{z-b}}=-1.$$ The proof of the second and third relation can be performed by the same lines with few modification.
For $a<b$ $$\int_a^bQ'(t)\frac{dt}{\sqrt{(t-a)(b-t)}}=\int_0^1Q'\big((1-t)b+ta\big)\frac{dt}{\sqrt {t(1-t)}}.$$ $$\int_a^bQ'(t)\sqrt{\frac {t-a}{b-t}}dt=(b-a)\int_0^1Q'\big((1-t)b+ta\big)\sqrt{\frac {1-t}{t}}dt.$$
[**Proof.**]{} We use the change of variable $t=(1-u)b+au$.
For $a\in\Bbb R$, $b>a$, let $$\varphi(a,b)=\int_0^1Q'((1-t)b+ta)\frac{dt}{\sqrt{t(1-t)}},$$ $$\psi(a,b)=(b-a)\int_0^1Q'((1-t)b+ta)\sqrt\frac {1-t}tdt-2\pi.$$ And $$h(a,b)=\frac{\psi(a,b)}{b-a}.$$ $Q$ a convex polynomial with degree $2m$, $m\geq 1$, and strictly positive leading coefficient.
\[pp1\]
(1)
: For all $a\in\Bbb R$ there is a unique $b(a)>a$, such that $\psi\big(a,b(a)\big)=0$.
(2)
: Let $\kappa=\inf\big\{a\in\Bbb R\mid \varphi(a,b(a))\geq 0\big\}$ which is an element of $[-\infty,+\infty[$. The map $a\mapsto b(a)$ define a increasing function on $]\kappa,+\infty[$, and for all $a> \kappa$, $\varphi\big(a,b(a)\big)\geq 0$.
(3)
: There is a unique $a_0\in\Bbb R$, and a unique $b_0>a_0$, such that $\varphi(a_0,b_0)=0\;and\;\psi(a_0,b_0)=0,$ $(b_0=b(a_0))$.
[**Proof.—**]{}
\(1) For $a\in\Bbb R$, by the derivative theorem under the integral sign one gets $$\frac{\partial h(a,b)}{\partial b}=\int_0^1Q''((1-t)b+ta)(1-t)\sqrt\frac {1-t}tdt+\frac{2\pi}{(b-a)^2}.$$ Since the polynomial $Q$ is convex, hence the function $b\mapsto h(a,b)$ increases strictly. Moreover, using Fatou’s lemma and a simple computations give $\displaystyle\lim_{b\to +\infty}h(a,b)=+\infty.$ From the dominate convergence theorem we obtain $\displaystyle\lim_{b\to a^+}h(a,b)=+\infty.$ Hence, the Rolle’s theorem and the monotony of $h$ ensure the existence and uniqueness of $b(a)>a$ such that $h(a,b(a))=0$ which means that $\psi(a,b(a))=0$.
\(2) It is enough to prove that $\big\{a\in\Bbb R\mid \varphi(a,b(a))\geq 0\big\}\neq\emptyset.$ If for all $a\in\Bbb R$, $\varphi(a,b(a))<0$, then for all $a\in\Bbb R$, $$\int_0^1Q'((1-t)b(a)+ta)\frac{dt}{\sqrt{t(1-t)}}<0,$$ since $b(a)>a$, hence $$\pi Q'(a)\leq \int_0^1Q'((1-t)b(a)+ta)\frac{dt}{\sqrt{t(1-t)}}<0.$$ As $a\to +\infty$ one gets a contradiction. Thus $$\kappa=\inf\big\{a\in\Bbb R\mid \varphi(a,b(a))\geq 0\big\}\in[-\infty,+\infty[.$$ The map $a\mapsto b(a)$ is correctly defined from the unicity of the solution.
Furthermore for all $a> \kappa$, $$\varphi(a,b(a))\geq0.$$ follows from the definition of $\kappa$.
[*Growth of $a\mapsto b(a)$.*]{} In the first hand by derivative theorem under the integral sign we have $$\frac{\partial\psi(a,b)}{\partial a}=-\int_0^1Q'((1-t)b+ta)\sqrt{\frac {1-t}t}dt+(b-a)\int_0^1Q''((1-t)b+ta)\sqrt{t {(1-t)}}dt.$$ Using a integration by part in the second member, we obtain $$\frac{\partial\psi(a,b)}{\partial a}=-\frac12\varphi(a,b).$$ Moreover $$\frac{\partial\varphi(a,b)}{\partial a}=\int_0^1Q''((1-t)b+ta)\sqrt{\frac t{1-t}}dt,$$ which is strictly positive by convexity of $Q$. Hence the function $\displaystyle a\mapsto\frac{\partial\psi(a,b)}{\partial a}$ decreases strictly on $]-\infty,b[$ for all $b$. Then for all $(a_1,a)$ with $\kappa<a_1<a$ and $a_1,a\in]-\infty,b(a_1)[$, one gets $$\frac{\partial\psi(a,b(a_1))}{\partial a}<\frac{\partial\psi(a_1,b(a_1))}{\partial a}=-\frac12\varphi(a_1,b(a_1))\leq 0.$$ Hence the function $\displaystyle a\mapsto\psi(a,b(a_1))$ decreases strictly on $]-\infty,b(a_1)[$ .
Let $a_1<a_2$ and assume $b(a_2)<b(a_1)$, hence $a_1<a_2<b(a_2)< b(a_1)$. Moreover, we saw that for $b>a$, the function $\displaystyle b\mapsto h(a,b)=\frac{\psi(a,b)}{b-a}$ is strictly increasing. Then $$h(a_2,b(a_2))< h(a_2,b(a_1))=\frac{\psi(a_2,b(a_1))}{b(a_1)-a_2}\leq\frac{\psi(a_1,b(a_1))}{b(a_1)-a_2}=0,$$ thus $\psi(a_2,b(a_2)<0$, this give a contradiction. Which means for all $a_1< a_2$ then $b(a_1)< b(a_2)$ (here we used $b(a_1)\neq b(a_2)$ for $a_1\neq a_2$).
\(3) Let $$E=\Big\{a\in\Bbb R\mid \varphi(a,b(a))\geq 0\;and\;\psi(a,b(a))=0\Big\}.$$ Where $b(a)$ is the unique solution in $]a,+\infty[$ of the equation $\psi(a,b)=0$. We want to show that $E$ admit a minimum. Indeed from the property (2), $E\neq\emptyset$.
Assume $E$ is unbounded below.
From the monotony of $a\mapsto b(a)$, we have for all $a\leq 0$, $b(a)\leq b(0)$. Hence
Then $\displaystyle\lim_{a\to-\infty}Q'((1-t)b(a)+t a)=-\infty$ for all $t\in[0,1]$, where we use the positivity of the leading coefficient of the polynomial $Q$ and the degree of $Q'$ is $2m-1$.
Letting $a\to -\infty$ in the equation below $$-\varphi(a,b(a))=\int_0^1-Q'((1-t)b(a)+ta)\frac1{\sqrt{t(1-t)}}dt\leq 0,$$ and using Fatou’s lemma we get a contradiction. which prove that $E$ is bounded below. The closeness of $E$ is an immediate consequence of the continuity of the functions $(u,v)\mapsto\varphi(u,v)$ and $(u,v)\mapsto\psi(u,v)$ and the increasing of the function $a\mapsto b(a)$.
We denote in the sequel $a_0=\min E$. From the previous we saw, there exist a unique $b_0:=b(a_0)>a_0$ such that $\varphi(a_0,b_0)\geq 0$ and $\psi(a_0,b_0)=0$.
For $n\in\Bbb N$, let $a_n=a_0-\frac 1n$, Since $a_n\notin E$, hence for all $b\geq a_n$, $\varphi(a_n, b)<0$ or $\psi(a_n, b)\neq 0$. Moreover for all $n\in\Bbb N$, from property (1) of the proposition there exist a unique $b_n>a_n$, such that $\psi(a_n, b_n)=0$, thus $\varphi(a_n,b_n)<0$. Since the sequence $a_n$ converges to $a_0$ and $b_n>a_n$, then $b_n$ is bounded from below. If $b_n$ is unbounded from above, then $\displaystyle\lim_{n\to +\infty}b_n=+\infty$. By continuity of the function $\varphi$ and Fatou’s lemma we have $\displaystyle\lim_{n\to +\infty}\varphi(a_n,b_n)=+\infty\leq 0$ this give a contradiction. Hence the sequence $b_n$ is bounded.
Let $y_n$ be a convergent subsequence of $b_n$ with limit $y\geq a_0$. We have $$\psi(a_n, y_n)=0\;and\;\varphi(a_n,y_n)\leq 0.$$ By continuity of $\varphi(u,v)$ and $\psi(u,v)$, as $n$ goes to $+\infty$ we obtain $$\psi(a_0, y)=0\;and\;\varphi(a_0,y)\leq 0.$$ Since the equation $\psi(a_0,y)=0$ has a unique solution $b_0$ in $]a_0,+\infty[$ and $\psi(a_0,a_0)=-2\pi$ hence $y=b_0$. Thus, $$\varphi(a_0,b_0)\leq 0.$$ Together with the reverse inequality one gets the desired result.
[**Acknowledgments.**]{}
My sincere thanks go to Jacques Faraut for his comments on this manuscript and his important remarks.
[9]{}
A. Aazami and R. Easther, J. Cosmol. Astropart. Phys. JCAP03 013 (2006). M. Bouali, *Generalized Gaussian Random Unitary Matrices Ensemble*. JP Journal of Geometry and Topology ,Vol 16, N1,(2014). ——–, *Generalized $\beta$-Gaussian Ensemble Equilibrium measure method.* ArXiv:1409.0126v1 \[math.PR\]. (Submitted). ——–, *Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble.* ArXiv:1409.0103 \[math.PR\] D. S. Dean & S.N. Majumdar. *Extreme value statistics of eigenvalues of Gaussian random matrices.* Phys. Rev. E 77, 041108, Published 10 April (2008). J-P. Dedieu and G. Malajovich. *On the number of minima of a random polynomial*. Journal of Complexity 24, pp 89–108 (2008). P. Deift & D Gioev. *Random matrix theory: invariant ensembles and universality*. Bulletin (New Series) of the American Mathematical Society. Vol 48, N1, 147-152, (2011). L. Erdos. *Universality of Wigner random matrices: a survey of recent results.* Russ. Math. Surv. 66 507 (2011).
J. Faraut. *Logarithmic potential theory, orthogonal polynomials, and random matrices*, in Modern methods in multivariate statistics, Lecture Notes of CIMPA-FECYT-UNESCO-ANR, Hermann, 2014. S. N. Majumdar, C. Nadal, A. Scardicchio. & P. Vivo. *How many eigenvalues of a Gaussian random matrix are positive?* PHYSICAL REVIEW E 83, 041105 (2011).
Address: College of Applied Sciences Umm Al-Qura University P.O Box (715), Makkah, Saudi Arabia.\
Faculté des Scinces de Tunis, Campus Universitaire El-Manar, 2092 El Manar Tunis.\
E-mail: bouali25@laposte.net & mabouali@uqu.edu.sa
|
---
address: 'LAMFA, Université de Picardie-Jules Verne 33, Rue Saint-Leu 80039 Amiens Cedex France'
author:
- 'F. Digne'
title: 'Présentations duales des groupes de tresses de type affine $\tilde A$'
---
Abstract {#abstract .unnumbered}
========
Artin-Tits groups of spherical type have two well-known Garside structures, coming respectively from the divisibility properties of the classical Artin monoid and of the dual monoid. For general Artin-Tits groups, the classical monoids have no such Garside property. In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type $\tilde A_n$ we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group.
More precisely, if $W$ is a Coxeter group, one can consider the length $l_R$ on $W$ with respect to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter element in $W$ and let $P_c$ be the set of elements $p\in
W$ such that $c$ can be written $c=pp'$ with $l_R(c)=l_R(p)+l_R(p')$. We define the monoid $M(P_c)$ to be the monoid generated by a set $\underline
P_c$ in one-to-one correspondence, $p\mapsto \underline p$, with $P_c$ with only relations $\underline{pp'}=\underline p.\underline p'$ whenever $p$, $p'$ and $pp'$ are in $P_c$ and $l_R(pp')=l_R(p)+l_R(p')$. We conjecture that the group of quotients of $M(P_c)$ is the Artin-Tits group associated to $W$ and that it has a simple presentation (see \[conjecture\] (ii)). These conjectures are known to be true for spherical type Artin-Tits groups. Here we prove them for Artin-Tits groups of type $\tilde A$. Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type $A$.
Introduction
============
Soit $(W,S)$ un système de Coxeter quelconque (avec $S$ fini). Soit $R$ l’ensemble des réflexions de $W$ (c’est-à-dire des conjugués des éléments de $S$). Appelons “longueur de réflexion” $l_R(w)$ d’un élément $w\in W$ le nombre minimum de termes dans une décomposition de $w$ en produits de réflexions. Nous dirons que $v\in W$ divise $w\in W$, noté $v\preccurlyeq w$, si $w=vv'$ avec $l_R(w)=l_R(v)+l_R(v')$. Comme $l_R$ est invariant par conjugaison ceci est équivalent à $w=v''v$ avec $l_R(w)=l_R(v'')+l_R(v)$, autrement dit il n’y a pas lieu de distinguer entre diviseurs à gauche et à droite. La relation $\preccurlyeq$ est clairement une relation d’ordre. Fixons un élément $c$ de Coxeter, c’est-à-dire un produit de tous les éléments de $S$ dans un certain ordre; considérons l’ensemble $P_c$ des diviseurs de $c$ et $M(P_c)$ le mono[ï]{}de engendré par un ensemble $\underline P_c=\{\underline
w\,\mid\,w\in P_c\}$ en bijection avec $P_c$ avec comme relations $\underline w.\underline w'=\underline{ww'}$ si $w$ et $w'$ sont des éléments de $P_c$ tels que $ww'\in P_c$ et $l_R(ww')=l_R(w)+l_R(w')$ ([[*cf.*]{}]{} [@BDM section 2] et [@B 0.2 et 0.4]). La longueur $l_R$ s’étend à $M(P_c)$ en une longueur additive. Les notions de divisibilité à gauche et à droite dans le mono[ï]{}de étendent la relation de divisibilité de $P_c$. Nous conjecturons
[\[conjecture\]]{}
1. Le groupe des fractions de $M(P_c)$ est isomorphe au groupe des tresses d’Artin-Tits associé à $W$.
2. Le mono[ï]{}de $M(P_c)$ (resp. son groupe de fractions) a la présentation suivante comme mono[ï]{}de (resp. comme groupe): l’ensemble des générateurs est $\{\underline r\mid r\in R\cap P_c\}$ et pour chaque couple $(r,t)\in\ R^2$ tel que $rt$ divise $c$ on a la relation ${{\underline r}}.{{\underline t}}=\underline{rtr}.{{\underline r}}$.
Cette conjecture a été prouvée pour les groupes de Coxeter de type $A_n$ dans [@BiKoLee] et [@BDM] et pour les autres types sphériques dans [@B]. Dans cet article nous allons prouver cette conjecture dans le cas où $W$ est de type $\tilde A_n$. Nous montrons de plus que pour exactement un choix $c_0$ de l’élément de Coxeter (à automorphisme du diagramme près) le mono[ï]{}de obtenu est un treillis pour la divisibilité (structure quasi-Garside), ce qui implique l’existence de formes normales dans son groupe de fractions et donne ainsi entre autres conséquences une nouvelle solution au problème des mots dans les groupes d’Artin-Tits de type $\tilde A_n$. Les preuves utilisent en particulier l’introduction d’objets “sans croisements” dans une couronne, et leur interprétation comme tresses dans un cylindre.
Les symétries des présentations obtenues définissent des automorphismes du groupe d’Artin-Tits dont on peut calculer les points fixes grâce à la théorie des structures de Garside. Cela permet par exemple d’obtenir le centralisateur d’une puissance de l’élément de Coxeter $c_0$, qui s’avère être un groupe d’Artin-Tits de type $B$.
Dans la section 2 nous étudions la longueur $l_R$ et déterminons $P_c$ pour un groupe de Coxeter de type $\tilde A$. Nous en donnons une interprétation topologique par des chemins sans croisement dans une couronne. Dans la section 3 nous donnons une présentation du mono[ï]{}de dual $M(P_c)$. Dans la section 4 nous prouvons la conjecture \[conjecture\] (i) dans le cas $\tilde A$. Dans la section 5 nous prouvons qu’on a une structure de Garside et en donnons un certain nombre de conséquences.
Longueur de réflexion; diviseurs d’un élément de Coxeter
========================================================
On sait par Dyer [@Dyer] que la longueur de réflexion d’un élément $w$ d’un groupe de Coxeter quelconque est égale au nombre minimum de termes qu’il faut effacer dans une suite minimale d’éléments de $S$ de produit égal à $w$ pour que le produit des termes restants soit égal à 1. C’est aussi la longueur d’un plus court chemin (croissant) de 1 à $w$ dans le graphe de Bruhat du groupe de Coxeter. Ceci montre immédiatement par exemple que la longueur de réflexion d’un élément de Coxeter est égal à sa longueur de Coxeter. Cela montre aussi que la restriction à un sous-groupe parabolique de $l_R$ est égale à la longueur de réflexion dans ce sous-groupe. Soit $W$ un groupe de Coxeter de type $\tilde A_{n-1}$. Pour étudier $M(P_c)$ nous avons besoin d’une formule explicite pour $l_R$ qui nous permettra en particulier de déterminer quels sont les diviseurs d’un élément de Coxeter. Pour cela nous utilisons une représentation de $W$ comme sous-groupe du groupe des permutations de ${{\mathbb Z}}$. Pour expliquer cette interprétation nous introduisons les définitions suivantes:
- Nous dirons qu’une permutation de ${{\mathbb Z}}$ est $n$-périodique si elle vérifie $w(x+n)=w(x)+n$ pour tout $x\in {{\mathbb Z}}$.
- Si $w$ est une permutation $n$-périodique de ${{\mathbb Z}}$, nous appelons décalage de $w$ l’entier $\dfrac1n\sum_{x=1}^{x=n}(w(x)-x)$.
On sait ([[*cf.*]{}]{} par exemple [@shi]) que $W$ s’identifie aux permutations $n$-périodiques de ${{\mathbb Z}}$ de décalage nul. On peut le voir de la façon suivante: le groupe de Coxeter affine de type $\tilde A_{n-1}$ est le produit semi-direct du réseau des racines de type $A_{n-1}$ par le groupe de Coxeter de ce type. Autrement dit $W={{\mathfrak S}}_n\ltimes\{(a_i)\in{{\mathbb Z}}^n\mid\sum a_i=0\}$. Notons $\overline w$ l’image dans ${{\mathfrak S}}_n$ de $w\in {{\mathfrak S}}_n\ltimes {{\mathbb Z}}^n$. On peut identifier le groupe ${{\mathfrak S}}_n\ltimes{{\mathbb Z}}^n$ au groupe des permutations $n$-périodiques de ${{\mathbb Z}}$, l’image de $k\in\{1,\ldots n\}$ par $w=\overline w.(a_1,\ldots,a_n)$ étant $\overline w(k)+na_k$. Alors le décalage d’une permutation $n$-périodique est donné par le morphisme $\overline w.(a_1,\ldots,a_n)\mapsto a_1+\cdots+a_n$ et $W$ s’identifie au noyau de ce morphisme.
Si $w$ est une permutation $n$-périodique de ${{\mathbb Z}}$, l’ensemble des orbites de $w$ est invariant par translation de $n$. Rappelons qu’on appelle support d’une permutation $w$ le complémentaire dans ${{\mathbb Z}}$ de l’ensemble des points fixes de $w$.
\[decalage\] Soit $w$ une permutation $n$-périodique de ${{\mathbb Z}}$; nous appelons décalage d’une orbite ${{\mathcal O}}$ de $w$ l’entier $\dfrac1n\sum_{x\in{{\mathcal O}}\cap\{1,\ldots,n\}}(w(x)-x)$.
Le décalage de ${{\mathcal O}}$ est donc l’entier $h$ tel que pour tout $a\in{{\mathcal O}}$ on ait $w^k(a)=a+hn$ où $k$ est le cardinal de l’image de l’orbite modulo $n$. Une orbite est finie si et seulement si elle est de décalage nul. Le décalage d’une permutation $n$-périodique est la somme des décalages de ses orbites, donc une permutation $n$-périodique de ${{\mathbb Z}}$ est dans $W$ si et seulement si la somme des décalages des orbites infinies est nulle.
On peut décomposer un élément de $W$ en produit d’éléments de $W$ ayant deux à deux des supports disjoints. On s’intéresse aux décompositions maximales de cette forme. Ceci nous amène à poser la définition suivante:
\[cycle\]
- Nous dirons qu’une permutation $n$-périodique est un cycle si elle n’a qu’une orbite non triviale à translation de $n$ près.
- Nous dirons qu’un élément de $W$ différent de l’identité est un pseudo-cycle si la restriction de $w$ à toute partie stricte stable de son support n’est pas dans $W$.
Toute permutation périodique est produit de cycles disjoints et cette décomposition est unique à l’ordre près. Mais un cycle n’est dans $W$ que si son décalage est nul, c’est-à-dire si les orbites de ce cycle sont finies. Tout élément de $W$ est produit de pseudo-cycles de supports disjoints, mais il n’y a pas unicité des pseudo-cycles ayant des orbites infinies (voir par exemple \[non-additivite\] ci-dessous). Un pseudo-cycle ou bien est un cycle, ou bien a toutes ses orbites non triviales infinies, de décalage total nul, et dans ce cas toute sous-famille de ses orbites a un décalage total non nul.
\[(a,b,c)\]
- Un cycle sera représenté sous la forme $(a,b,c,\ldots,l)_{[h]}$ où $h$ est le décalage du cycle. Cette notation signifiant que $a$, $b$, $c$, …, $l$ sont tous distincts modulo $n$ et que $w$ envoie $a$ sur $b$, $b$ sur $c$, … et $l$ sur $a+hn$. Pour simplifier les notations, si le décalage est nul, nous omettrons l’indice $[0]$.
- Nous représenterons toute permutation $n$-périodique de ${{\mathbb Z}}$ comme un produit de cycles de supports disjoints non vides.
- On pose $s_i=(i,i+1)$ pour $i=1,\ldots, n$. Les $s_i$ sont les générateurs de Coxeter de $W$.
Avec ces conventions les réflexions sont les éléments $(a,b)$ avec $a$ et $b$ quelconques distincts modulo $n$. On obtient exactement une fois chaque réflexion si on impose de plus $a<b$ et $a\in\{1,\ldots,n\}$.
Nous utiliserons à plusieurs reprises la formule suivante qui résulte d’un calcul immédiat.
\[calcul\] On désigne par $a_1,\ldots,a_l$ des entiers distincts modulo $n$ et par $h$ et $k$ des entiers quelconques. On a $$\begin{gathered}
(a_1,\ldots,a_i,\ldots,a_l)_{[h]}(a_1,a_i+kn)=\\
(a_{i+1},\ldots,a_l,a_1+hn)_{[h+k]}(a_2,a_3,\ldots,a_i)_{[-k]}\end{gathered}$$
Pour toute partie $T\subset{{\mathbb Z}}$ invariante par translation de $n$, notons $\tilde W_T$ le fixateur dans $W$ du complémentaire de $T$ et $W_T$ le sous-groupe des éléments de $\tilde W_T$ n’ayant pas d’orbite infinie. Un sous-groupe du type $W_T$ ou $\tilde W_T$ sera dit “[quasi-parabolique]{} ”.
Remarquons que $\tilde W_T$ et $W_T$ sont des groupes de Coxeter de types respectifs $\tilde A_{|\overline T|-1}$ et $A_{|\overline T|-1}$ si $\overline T$ est l’image de $T$ modulo $n$.
\[nu et kappa\] Pour $w\in W$ notons $\nu(w)$ le nombre de classes modulo $n$ d’orbites de $w$ et notons $\kappa(w)$ le plus grand entier $k$ tel que $w=w_1w_2\ldots w_k$ où les $w_i\in W$ sont (des pseudo-cycles) de supports disjoints.
Remarquons que $\kappa(w)$ est le plus grand entier $k$ tel que $w$ soit dans le produit direct de $k$ sous-groupes [quasi-parabolique]{}s. On a $\kappa(w)\leq\nu(w)$.
La proposition suivante donne une formule explicite pour la longueur de réflexion dans $W$.
\[l\_R\] Pour $w\in W$ on a $l_R(w)=n+\nu(w)-2\kappa(w)$.
On pose $f(w)=n+\nu(w)-2\kappa(w)$. On montre les trois propriétés suivantes:
- Si $f(w)\geq 1$, il existe $r\in R$ tel que $f(wr)=f(w)-1$.
- Si $f(w)=0$ alors $w=1$.
- $f(w)\leq l_R(w)$.
Les propriétés (1) et (2) donnent l’inégalité $l_R(w)\leq f(w)$, d’où la proposition.
Prouvons (1). Si $w$ a un cycle de la forme $(a_1,a_2,\ldots,a_l)_{[h]}$ avec $l\neq 1$, la multiplication par $r=(a_1,a_2)$, augmente $\nu(w)$ et $\kappa(w)$ de 1 ([[*cf.*]{}]{} \[calcul\]). Si tous les cycles de $w$ sont de la forme $(a_1)_{[h]}$, soit $w_i$ un des facteurs différents de l’identité dans la décomposition $w=w_1\ldots w_{\kappa(w)}$ comme dans \[nu et kappa\]; comme la somme des décalages des orbites de $w_i$ est nulle, il est produit d’au moins deux cycles $(a_1)_{[h]}$ et $(a_2)_{[k]}$. Posons $r=(a_1,a_2)$; on a $(a_1)_{[h]}(a_2)_{[k]}(a_1,a_2)=(a_1,a_2+kn)_{[h+k]}$. Donc $\nu(wr)=\nu(w)-1$ et $\kappa(wr)=\kappa(w)$, donc $f(wr)=f(w)-1$.
Montrons la propriété (2) : Comme $\kappa(w)\leq\nu(w)\leq n$, si $f(w)$ est nul on doit avoir $\nu(w)=n=\kappa(w)$. La première égalité prouve que chaque orbite est un singleton modulo $n$ et la deuxième prouve alors que chaque orbite est de décalage nul, donc $w=1$.
Pour montrer la propriété (3) nous montrons que pour $r\in R$ et $w\in W$ on a $f(wr)\leq f(w)+1$. Ceci implique par récurrence que si $w$ est produit de $k$ réflexions on a $f(w)\leq k$.
Soit $r=(a,b)$ une réflexion. Si $a$ et $b$ apparaissent modulo $n$ dans la même orbite de $w$ le lemme \[calcul\] montre que que $\nu(wr)=\nu(w)+1$. D’autre part $\kappa(wr)\geq \kappa(w)$, d’où le résultat dans ce cas. Si $a$ et $b$ sont modulo $n$ dans deux orbites différentes du même pseudo-cycle $w_i$ de la décomposition $w=w_1\ldots w_{\kappa(w)}$, le même lemme montre que $\nu(wr)=\nu(w)-1$ et on a $\kappa(wr)=\kappa(w)$. Si $a$ et $b$ apparaissent dans deux pseudo-cycles distincts, alors de même que précédemment $\nu(wr)=\nu(w)-1$ et de plus $wr$ a une décomposition en produit de $\kappa(w)-1$ éléments, donc $\kappa(wr)\geq \kappa(w)-1$, ce qui donne bien $f(wr)\leq
f(w)+1$.
Si $w$ est dans un [quasi-parabolique]{} sa longueur de réflexion dans ce sous-groupe est égale à sa longueur de réflexion dans $W$.
On a déjà vu que la restriction de $l_R$ à un sous-groupe parabolique est la longueur de réflexion dans ce sous-groupe, grâce au résultat de Dyer. On peut aussi voir que la formule de la proposition \[l\_R\] donne la longueur de réflexion dans un parabolique standard. Ceci donne le résultat dans le cas d’un [quasi-parabolique]{} de type $A$. D’autre part, si un élément $w$ est dans le [quasi-parabolique]{} de type $\tilde A_{n-m-1}$ fixant $j+kn$ pour $j$ dans une partie de cardinal $m$ de $[1,n]$ et pour tout $k$, les valeurs de $\nu(w)$ et $\kappa(w)$ diminuent de $m$ dans le [quasi-parabolique]{} et $n$ est remplacé par $n-m$ dans la formule donnant la longueur. Ceci prouve que $l_R$ se restreint bien aussi dans ce cas, d’où le résultat.
Nous allons maintenant chercher quelles réflexions divisent un élément de Coxeter de $W$. La proposition suivante exprime un élément de Coxeter comme permutation.
\[coxeter\] Soit $c$ un élément de Coxeter de $W$, c’est-à-dire le produit des générateurs de Coxeter dans un ordre arbitraire fixé; alors il existe une partition en deux parties non vides $$\{1,\ldots,n\}=\{a,b,\ldots,l\}\coprod\{\alpha,\beta,\ldots,\lambda\}$$ telle que $a<b<\ldots<l$ et $\alpha<\beta<\ldots<\lambda$ et que $$c=(a,b,\ldots,l)_{[1]}(\lambda,\ldots,\beta,\alpha)_{[-1]}.$$
L’ensemble des générateurs de Coxeter est $\{s_1,s_2,\ldots,s_n\}$ avec les notations de \[(a,b,c)\]. On a $c=s_{i_1}s_{i_2}\ldots s_{i_n}$, où les indices $i_j$ sont tous distincts. Quitte à changer $c$ en $c{^{-1}}$, on peut supposer que dans la suite $(s_{i_1},s_{i_2},\ldots
,s_{i_n})$ l’élément $s_1$ est à droite de $s_n$. Alors $s_1$ commute avec tous les $s_i$ qui sont à sa droite sauf éventuellement $s_2$ qui à son tour commute avec tous les éléments qui sont à sa droite sauf éventuellement $s_3$ On peut alors réécrire $c$ comme un produit qui se termine par $s_1s_2s_3\ldots s_i$ pour un certain $i$. On itère le procédé en commençant avec $s_{i+1}$. Finalement on écrit $c$ sous la forme $$(s_{k_h}s_{k_h+1}\ldots s_n)(s_{k_{h-1}}s_{k_{h-1}+1}\ldots
s_{k_h-1})\ldots(s_{k_1}s_{k_1+1}\ldots s_{k_2-1})(s_1s_2\ldots s_{k_1-1}),$$ avec $1<
k_1<k_2<\ldots<k_h\leq n$ qui est une permutation de la forme voulue: elle s’écrit $(k_h,k_{h-1},\ldots,k_1)_{[-1]}
(1,2,\ldots,\widehat{k_1}\ldots,\widehat{k_2},\ldots,\widehat{k_h},\ldots,n)_{[1]}$, où $\widehat{k_i}$ signifie que $k_i$ ne figure pas.
\[X et Xi\] Dans la suite nous fixons un élément de Coxeter $c$ et posons $X=\{a,b,\ldots,l\}+n{{\mathbb Z}}$ et $\Xi=\{\alpha,\beta,\ldots,\lambda\}+n{{\mathbb Z}}$ comme dans la proposition précédente.
Nous allons démontrer:
\[atomes\] Avec les notations ci-dessus, si $x$ et $y$ sont deux entiers distincts, la réflexion $(x,y)$ divise $c$ si et seulement si $x$ et $y$ vérifient l’une des propriétés suivantes:
1. $x\in X$ et $y\in \Xi$ ou $x\in \Xi$ et $y\in X$.
2. $x, y \in X$, et $|y-x|<n$.
3. $x,y\in \Xi$ et $|y-x|<n$.
On calcule $c(x,y)$ dans tous les cas possibles.
Dans le cas (i), quitte à échanger les rôles de $x$ et $y$ et à changer de représentants des cycles de $c$, on peut supposer $x=l$ et $y=\alpha+n$. Le produit $c(x,y)$ vaut $(a,b,\ldots,l,\lambda,\ldots,\beta,\alpha)$ dont la longueur de réflexion vaut bien $n-1$.
Si $x$ et $y$ sont dans $X$, on peut supposer $x=a$ et on écrit $c=(a,b,\ldots,t,y+kn,z,\ldots,l)_{[1]}(\lambda,\ldots,\beta,\alpha)_{[-1]}$ pour un certain $k$. Le produit $c(x,y)$ vaut $(y,b,\ldots,t)_{[k]}(z,\ldots,l,a+n)_{[1-k]}(\lambda,\ldots,\alpha)_{[-1]}$. Comme $\nu(c(x,y))=3$, on a $l_R(c(x,y))=n-1$ si et seulement si $\kappa(c(x,y))$ est égal à $2$, c’est à dire si on peut regrouper les trois orbites en deux parties de décalages nuls (sinon $\kappa(c(x,y))$ est égal à 1). On a donc $\kappa(c(x,y))=2$ si et seulement si $k=0$ ou $k=1$, ce qui donne le cas (ii).
On fait un raisonnement analogue si $x$ et $y$ sont dans $\Xi$ et on obtient le cas (iii).
Les calculs faits dans la preuve précédente montrent aussi que
\[quotients\] Soit $s$ et $t=n-s$ les cardinaux respectifs des images de $X$ et $\Xi$ modulo $n$; un élément de longueur $l_R(c)-1$ divise $c$ si et seulement s’il est d’une des trois formes suivantes:
1. $(a_1,a_2,\ldots,a_s,\alpha_t,\alpha_{t-1},\ldots,\alpha_1)$ où $(a_i)$ et $(\alpha_i)$ sont des sous-suites croissantes formées d’éléments consécutifs respectivement de $X$ et de $\Xi$,
2. $(a_1,a_2,\ldots,a_r)(a_{r+1},\ldots,a_s)_{[1]}
(\lambda,\ldots,\beta,\alpha)_{[-1]}$ avec $(a_i)$ suite croissante d’éléments consécutifs de $X$,
3. $(a,b,\ldots,l)_{[1]}(\alpha_t,\alpha_{t-1},\ldots,\alpha_{r+1})
(\alpha_r,\alpha_{r-1},\ldots,
\alpha_1)_{[-1]}$ avec $(\alpha_i)$ suite croissante d’éléments consécutifs de $\Xi$.
Pour donner la liste des diviseurs de $c$ nous avons besoin de la définition suivante.
On appelle diviseur élémentaire un pseudo-cycle de $W$ de la forme $(a_1,\ldots,a_h,\alpha_k,\ldots,\alpha_1)$ avec $h\geq 0$, et $k\geq 0$ et $h+k\geq 2$, ou de la forme $(a_1,a_2,\ldots,a_h)_{[1]}(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}$, avec $h\geq 1$ et $k\geq 1$, où les $a_i$ sont dans $X$ et les $\alpha_i$ dans $\Xi$, et où $a_1< a_2< a_3< \ldots< a_h< a_1+n$ et $\alpha_1<\alpha_2<\ldots<\alpha_k<\alpha_1+n$.
Remarquons qu’un diviseur élémentaire est un élément de Coxeter d’un [quasi-parabolique]{} de $W$: dans le premier cas il s’agit d’un [quasi-parabolique]{} de type $A_{h+k-1}$, dans le deuxième d’un [quasi-parabolique]{} de type $\tilde A_{h+k-1}$.
Nous allons donner une interprétation topologique des diviseurs de $c$ en termes de chemins dans une couronne.
(0,0)![Deux réflexions sans croisement \[figure0\]](figure0.ps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
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On considère une couronne dans le plan orienté. On fixe des points étiquetés $a$, $b$, …, $l$ sur le cercle extérieur, dans l’ordre cyclique et on fait de même sur le cercle intérieur avec des points étiquetés $\alpha$, …, $\lambda$. Cette couronne peut être vue comme le quotient par la translation de $n$ d’une bande infinie orientée où les points de $X$ et de $\Xi$ respectivement sont marqués dans l’ordre croissant sur chacun des deux bords. Nous allons associer à toute réflexion une classe de chemins continus dans la couronne.
\[representation graphique\] À la réflexion $(x,y)$ on associe la classe d’homotopie à extrémités fixes de l’image dans la couronne d’un chemin continu joignant $x$ à $y$ dans la bande (voir figure \[figure0\], la représentation n’est bien définie qu’une fois fixée l’identification du quotient de la bande par les translations avec la couronne).
Avec cette notation on a:
\[sans autocroisement\] Une réflexion divise l’élément de Coxeter $c$ si et seulement si elle peut être représentée par un chemin sans auto-intersection dans la couronne.
Ce corollaire est une constatation immédiate à partir de la liste donnée dans \[atomes\].
Pour pouvoir décrire tous les diviseurs d’un élément de Coxeter nous utilisons la même interprétation topologique. À un diviseur élémentaire de la forme $(a_1,a_2,\ldots,a_h,\alpha_k,\ldots,\alpha_2,\alpha_1)$ nous associons la classe d’homotopie à extrémités fixes du lacet composé des chemins associés aux réflexions $(a_1,a_2)$, …,$(a_{h-1},a_h)$, $(a_h,\alpha_k)$, $(\alpha_k,\alpha_{k-1})$, …, $(\alpha_2,\alpha_1)$, $(\alpha_1,a_1)$ et à un diviseur élémentaire de la forme $(a_1,a_2,\ldots,a_h)_{[1]}(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}$, nous associons la classe d’homotopie à extrémités fixes de l’union de deux lacets ne se coupant pas et composés respectivement de chemins associés aux réflexions $(a_1,a_2)$, …, $(a_{h-1},a_h)$, $(a_h,a_1+n)$ et $(\alpha_k,\alpha_{k-1})$, …, $(\alpha_2,\alpha_1)$, $(\alpha_1,\alpha_k-n)$ (voir figure \[figure01\]).
(0,0)![La représentation graphique de ${{\underline c}}$ \[figure01\]](figure01.ps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
(3137,3304)(3839,-6122) (6436,-3346)[(0,0)\[lb\]]{} (6931,-4066)[(0,0)\[lb\]]{} (6976,-4876)[(0,0)\[lb\]]{} (6661,-5641)[(0,0)\[lb\]]{} (4276,-3301)[(0,0)\[lb\]]{} (5681,-4261)[(0,0)\[lb\]]{} (5316,-4056)[(0,0)\[lb\]]{} (5341,-2951)[(0,0)\[lb\]]{} (4976,-4211)[(0,0)\[lb\]]{} (5776,-4711)[(0,0)\[lb\]]{}
\[sans autointersection\] On peut définir de la même façon une interprétation topoloique de tout pseudo-cycle. Les diviseurs élémentaires correspondent exactement aux pseudo-cycles qui ont une représentation (lacet ou union de deux lacets) orientée positivement et sans autointersection.
\[sans croisement\] On dit que deux diviseurs élémentaires sont sans croisement s’ils admettent des représentations sans intersection.
En particulier si deux diviseurs élémentaires sont sans croisement, un seul des deux au plus a des orbites infinies car deux lacets d’image non triviale dans le groupe fondamental de la couronne et ayant tous deux leur origine sur la même composante du bord ont nécessairement une intersection.
Avec cette définition on peut caractériser les diviseurs d’un élément de Coxeter:
\[diviseurs de c\] Les diviseurs de l’élément de Coxeter $c$ sont exactement les produits de diviseurs élémentaires deux à deux sans croisement. La longueur d’un tel produit est la somme des longueurs des facteurs et ces facteurs commutent deux à deux.
La remarque qui précède la proposition implique alors qu’un diviseur de $c$ a zéro ou deux orbites infinies (et celles-ci, si elles existent, sont l’une de décalage 1, l’autre de décalage $-1$).
Montrons d’abord l’additivité des longueurs. Dans la formule donnant la longueur, seul le terme $\kappa$ n’est pas toujours additif. Mais pour un élément qui a 0 ou 2 orbites infinies la décomposition en pseudo-cycles est unique, un des pseudo-cycles étant formé de la permutation induite sur l’union des deux orbites infinies. Dans ce cas $\kappa$ est bien additif.
Pour prouver l’assertion sur la forme des diviseurs de $c$ nous utilisons le lemme suivant:
\[reflexions divisant un produit\] Soient $x$ et $y$ deux entiers distincts et $v$ un élément de $W$ ayant 0 ou 2 orbites infinies.
- La réflexion $(x,y)$ divise $v$ si et seulement si elle divise un des pseudo-cycles de $v$.
- La réflexion $(x,y)$ divise un pseudo-cycle si et seulement si $x$ et $y$ sont dans la même orbite de ce pseudo-cycle ou si chacun d’eux est dans une des orbites infinies de ce pseudo-cycle.
Si $x$ est dans une orbite finie de $v$ et $y$ dans une orbite finie ou infinie distincte de celle de $x$, alors le calcul de \[calcul\], appliqué avec $a_1=y$, $a_i=x$ et $k=0$, et en faisant passer $(x,y)$ dans le membre de droite, montre que $\nu(v(x,y))=\nu(v)-1$ et que la suite des décalages non nuls des orbites est la même pour $v$ et $v(x,y)$. Donc $\kappa(v(x,y))=\kappa(v)-1$ et $l_R(v(x,y))=l_R(v)+1$, donc $(x,y)$ ne divise pas $v$. Réciproquement, si $x$ et $y$ sont dans le support d’un même pseudo-cycle de $v$, \[calcul\] appliqué avec $a_1=x$, $a_2=y$ et $h=0$ et avec ou bien $k=1$ ou bien $k=0$ montre que $(x,y)$ divise ce pseudo-cycle. Comme les longueurs des pseudo-cycles de $v$ s’ajoutent d’après le début de la démonstration de la proposition, on en déduit que $(x,y)$ divise $v$.
Nous prouvons maintenant, par récurrence descendante sur la longueur du diviseur, l’assertion de la proposition \[diviseurs de c\] sur les diviseurs de $c$. Il y a exactement un diviseur de longueur $n$ qui est $c$ lui-même. Un diviseur de longueur $k$ est le produit d’un diviseur $v$ de longueur $k+1$ par une réflexion $(x,y)$ qui divise $v$. Par hypothèse de récurrence appliquée à $v$, les pseudo-cycles de $v$ sont des diviseurs élémentaires deux à deux sans croisement. D’après le lemme précédent la réflexion $(x,y)$ divise $v$ si et seulement si elle divise un des pseudo-cycles $v_1$ de $v$. On est ramené à trouver tous les diviseurs de longueur $l_R(v_1)-1$ d’un diviseur élémentaire $v_1$. Si $v_1$ a deux orbites infinies, c’est un élément analogue à $c$ dans le sous-groupe [quasi-parabolique]{} correspondant au support de $v_1$. Les diviseurs cherchés sont donnés par \[quotients\]. Ce sont bien des produits de diviseurs élémentaires deux à deux sans croisement, de supports inclus dans le support de $v_1$, donc aussi sans croisement avec les autres pseudo-cycles de $v$. Si $v_1$ n’a pas d’orbite infinie il s’écrit $(a_1,\ldots,a_h)$ où $l_R(v_1)=h-1$ et ses diviseurs de longueur $h-2$ sont exactement les éléments $(a_1,a_2,a_{i-1},a_{j+1},\ldots,a_h)(a_i,a_{i+1},a_{i+2},\ldots,a_j)$, ce qui est aussi de la forme annoncée.
Réciproquement, si $v$ est un produit de diviseurs élémentaires deux à deux sans croisement et si $l_R(v)<n$ on va montrer qu’on peut trouver une réflexion $r$ telle que $r v$ soit de longueur $l_R(v)+1$ et soit un produit de diviseurs élémentaires deux à deux sans croisement. L’hypothèse de récurrence montre alors que $r v$ divise $c$, donc que $v$ divise $c$. Distinguons plusieurs cas pour $v$.
1. Si $v$ a une orbite infinie, il en a alors exactement deux, d’après la remarque qui suit \[sans croisement\] et ces orbites sont de décalages $1$ et $-1$, donc les cycles correspondants sont de la forme $(a_1,a_2,\ldots,a_h)_{[1]}$ et $(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}$ avec $a_1< a_2< a_3< \ldots< a_h< a_1+n$ et $\alpha_1<\alpha_2<\ldots<\alpha_k<\alpha_1+n$, les $a_i$ étant dans $X$ et les $\alpha_i$ dans $\Xi$; d’autre part $v$ doit avoir aussi une orbite finie car $v\neq c$. Tous les points d’une telle orbite finie sont compris entre deux éléments consécutifs d’une des deux orbites infinies, puisque les diviseurs élémentaires sont sans croisement. On peut supposer que cette orbite est comprise dans $[a_1,a_2]$. Notons-la $\{b_1,\ldots,b_l\}$ avec $a_1<b_1<b_2<\cdots<b_l<a_2$. On a alors $$\begin{gathered}
(a_2,b_1)
(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}
(a_1,a_2,\ldots,a_h)_{[1]}(b_1,\ldots,b_l)=\\
(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}
(a_1,b_1,\ldots,b_l,a_2,\ldots,a_h)_{[1]}.\end{gathered}$$
Si $v$ n’a pas d’orbite infinie, il a au moins une orbite finie de la forme $(a_1,a_2\ldots,a_h,\alpha_k,\ldots,\alpha_2,\alpha_1)$ avec les mêmes conventions que précédemment (cette orbite peut être un singleton). Quitte à échanger les rôles de $X$ et de $\Xi$ on peut supposer que $h\neq 0$. Soit $b_1$ le successeur de $a_h$ dans $X$. Il y a trois cas:
1. Si $b_1=a_1+n$ alors $$\begin{gathered}
(b_1,\alpha_k)
(a_1,a_2\ldots,a_h,\alpha_k,\ldots,\alpha_2,\alpha_1)=\\
(a_1,a_2,\ldots,a_h)_{[1]}
(\alpha_k,\ldots,\alpha_2,\alpha_1)_{[-1]}.\end{gathered}$$
2. Si $k\neq 0$ et si le cycle de $v$ dont le support contient $b_1$ est de la forme $(b_1,b_2\ldots,b_l,\beta_m,\ldots,\beta_2,\beta_1)$ avec $b_1< b_2<\ldots<b_l< b_1+n$ et $\beta_1<\beta_2<\ldots<\beta_m<\beta_1+n$, les $b_i$ étant dans $X$ et les $\beta_i$ dans $\Xi$; alors $$\begin{gathered}
(b_1,\alpha_k)(a_1,a_2,\ldots,a_h,\alpha_k,\ldots,\alpha_2,\alpha_1)
(b_1,b_2,\ldots,b_l,\beta_m,\ldots,\beta_2,\beta_1)=\\
(a_1,a_2,\ldots,a_h,b_1,b_2,\ldots,b_l,\beta_m,\ldots,\beta_2,\beta_1,
\alpha_k,\ldots,\alpha_2,\alpha_1).\end{gathered}$$
3. Si $k=0$ et si le cycle de $v$ dont le support contient de $b_1$ est de la forme $$(b_1,b_2\ldots,b_r,\beta_m,\ldots,\beta_2,\beta_1,b_{r+1},\ldots,b_l)$$ avec $b_{r+1}<b_{r+2}<\cdots<b_l<a_1<a_h<b_1< b_2<\ldots<b_r< b_{r+1}+n$ et $\beta_1<\beta_2<\ldots<\beta_m<\beta_1+n$, les $b_i$ étant dans $X$ et les $\beta_i$ dans $\Xi$; alors $$\begin{gathered}
(a_1,b_1)(a_1,a_2,\ldots,a_h)
(b_1,b_2\ldots,b_r,\beta_m,\ldots,\beta_2,\beta_1,b_{r+1},\ldots,b_l)=\\
(a_1,a_2,\ldots,a_h,b_1,b_2,\ldots,b_r,\beta_m,\ldots,\beta_2,\beta_1,
b_{r+1},\ldots,b_l).\end{gathered}$$
Dans tous les cas le produit est bien comme annoncé.
Enfin puisque les longueurs de diviseurs élémentaires deux à deux sans croisements s’ajoutent et que de tels diviseurs commutent dans $W$, ils commutent aussi dans le mono[ï]{}de.
\[non-additivite\] Il est faux que la longueur d’un produit de pseudo-cycles de supports disjoints est la somme des longueurs des pseudo-cycles, comme le montre l’exemple suivant dans $\tilde A_n$ avec $n\geq 5$: $$w=[(1)_{[-1]}(2)_{[-1]}(3)_{[2]}][(4)_{[1]}(5)_{[1]}(6)_{[-2]}].$$ La longueur de chacun des facteurs est égale à 4 ($\nu=3$, $\kappa=1$, dans un [quasi-parabolique]{} de type $\tilde
A_2$) et la longueur du produit vaut 6 car le même élément s’écrit $$w=[(1)_{[-1]}(4)_{[1]}][(2)_{[-1]}(5)_{[1]}][(3)_{[2]}(6)_{[-2]}],$$ ce qui prouve que $\kappa$ vaut $3$ (et on a $\nu=6$) dans un [quasi-parabolique]{} de type $\tilde A_5$.
Par contre s’il y a au plus deux orbites infinies on a bien additivité des longueurs.
À tout diviseur $w$ de $c$ on peut associer la partition périodique de ${{\mathbb Z}}$ dont les parties sont les supports des pseudo-cycles de $w$.
Nous dirons que deux parties $A$ et $B$ de ${{\mathbb Z}}$ sont sans croisement si pour tous $x$ et $y$ dans $A$ et tous $z$ et $t$ dans $B$ il existe deux chemins sans intersection dans la bande joignant respectivement $x$ à $y$ et $z$ à $t$. On peut alors réexprimer \[diviseurs de c\] et \[sans autointersection\] par:
\[partition\] Une partition périodique de ${{\mathbb Z}}$ dont toute partie infinie rencontre à la fois $X$ et $\Xi$ est associée comme ci-dessus à un diviseur (unique) de $c$ si et seulement si ses parties sont deux à deux sans croisement.
Les mono[ï]{}des duaux {#monoide dual}
======================
L’objectif de cette partie est de prouver la conjecture \[conjecture\] (ii) pour le type $\tilde A$. Nous suivons une démarche analogue à celle de [@B] (ou de [@BDM]). En particulier nous nous plaçons dans le cadre des groupes positivement engendrés telle qu’elle est exposée dans [@B 0.4 et 0.5]. Rappelons-en les résultats principaux dans un cadre plus général car nous ne supposons pas que le nombre de générateurs est fini. Soit $G$ un groupe engendré comme mono[ï]{}de par un ensemble $R$. On dit que $(G,R)$ est un groupe positivement engendré. On définit la longueur dans $l_R(g)$ par rapport à $R$ de $g\in G$ comme le nombre minimum de facteurs dans une décomposition de $g$ en produit de générateurs. On dit que $h\in G$ divise $g\in G$ à gauche et que $k\in G$ divise $g$ à droite si $g=hk$ avec $l_R(g)=l_R(h)+l_R(k)$. On dit qu’un élément est équilibré si ses diviseurs à droite et à gauche sont les mêmes.
Soit $(G,R)$ un groupe positivement engendré, soit $c\in G$ un élément équilibré et soit $P_c$ l’ensemble des diviseurs (à gauche ou à droite) de $c$. On considère un ensemble ${{\underline P}}_c=\{\,\underline w\,\mid\,w\in P_c\}$ en bijection avec $P_c$ et on définit un mono[ï]{}de noté $M(P_c)$ par la présentation suivante: l’ensemble des générateurs est ${{\underline P}}_c$ et les relations sont $\underline w.\underline w'=\underline{ww'}$ pour tous les couples $(w,w')$ tels que $w$, $w'$ et $ww'$ sont des diviseurs de $c$ et que $l_R(ww')=l_R(w)+l_R(w')$.
Les notions standard de divisibilité à gauche ou à droite dans le monoïde $M(P_c)$ étendent les notions correspondantes de divisibilité définies dans $P_c$. Notons que les atomes, [[*i.e.*]{}]{}, les éléments différents de 1 qui ne sont pas produit de deux facteurs différents de 1 sont des éléments de $P_c$.
Remarquons qu’on a un morphisme de mono[ï]{}des $M(P_c)\rightarrow G$ donné par ${{\underline w}}\mapsto
w$ pour $w\in P_c$. L’existence de ce morphisme permet facilement de prouver que $M(P_c)$ a une propriété de simplifiabilité partielle à gauche et à droite ([[*cf.*]{}]{} [@B 0.4.4]): si $am=bm$ ou si $ma=mb$ avec $a$ et $b$ dans $P_c$ et $m\in M(P_c)$ alors $a=b$. On a de plus:
\[conjugaison par c\] Pour tout $w\in P_c$ l’élément $w'=cwc{^{-1}}$ est dans $P_c$ et est l’unique élément de $P_c$ tel que ${{\underline w}}'.{{\underline c}}={{\underline c}}.{{\underline w}}$. L’application ${{\underline w}}\mapsto{{\underline w}}'$ définit un automorphisme du mono[ï]{}de $M(P_c)$.
On appellera “conjugaison par ${{\underline c}}$” l’automorphisme ainsi défini.
On a $c=xw$ avec $l_R(c)=l_R(w)+l_R(x)$. Comme $c$ est équilibré, l’élément $x$ est aussi un diviseur de $c$ à droite, donc on peut écrire $c=w'x$ avec $l_R(w')+l_R(x)=l_R(c)$. On a $w'=cwc{^{-1}}$ et ${{\underline c}}={{\underline w}}'.{{\underline x}}={{\underline x}}.{{\underline w}}$, d’où ${{\underline c}}.{{\underline w}}={{\underline w}}'.{{\underline x}}.{{\underline w}}={{\underline w}}'.{{\underline c}}$. L’application ${{\underline w}}\mapsto{{\underline w}}'$ définit un morphisme de mono[ï]{}des. On a un morphisme en sens inverse en faisant un raisonnement analogue en partant de $w'$. D’où le résultat.
Notons encore la propriété générale suivante de $M(P_c)$:
Tout élément de $M(P_c)$ divise une puissance suffisament grande de $c$.
Tout élément de $M(P_c)$ s’écrit ${{\underline w}}_1\ldots {{\underline w}}_k$ pour un certain $k$, où les $w_i$ sont des éléments de $P_c$. On montre par récurrence sur $k$ que ${{\underline w}}_1\ldots {{\underline w}}_k$ divise ${{\underline c}}^k$: par définition les éléments de ${{\underline P}}_c$ divisent ${{\underline c}}$, donc il existe $x\in P_c$ tel que ${{\underline w}}_k{{\underline x}}={{\underline c}}$. Par \[conjugaison par c\] on a ${{\underline w}}_1\ldots{{\underline w}}_k{{\underline x}}={{\underline c}}{{\underline w}}'_1\ldots{{\underline w}}'_{k-1}$ où les $w'_i$ sont des éléments de $P_c$. Par hypothèse de récurrence ${{\underline w}}'_1\ldots{{\underline w}}'_{k-1}$ divise ${{\underline c}}^{k-1}$, d’où la proposition.
Remarquons que par définition la longueur $l_R$ s’étend en une fonction additive sur $M(P_c)$. Remarquons aussi que les diviseurs de ${{\underline c}}$ dans $M(P_c)$ sont exactement les éléments de ${{\underline P}}_c$. Les éléments ${{\underline r}}\in{{\underline P}}_c$ tels que $r\in R$ sont les atomes du mono[ï]{}de $M(P_c)$.
Revenons à la situation du groupe de Coxeter $W$ de type $\tilde A_{n-1}$ et appliquons les constructions précédentes à un élément de Coxeter $c$ fixé de $W$ comme dans la section précédente, dont on garde les notations.
Nous prouvons maintenant dans ce cas la conjecture \[conjecture\] (ii):
\[presentation\] Le mono[ï]{}de $M(P_c)$ est engendré par les ${{\underline r}}$ où $r$ est une réflexion qui divise $c$ avec comme relations $${{\underline r}}.{{\underline t}}=\underline{rtr}.{{\underline r}}\label{*}$$ si $r$ et $t$ sont deux réflexions distinctes telles que $rt$ divise $c$.
Remarquons qu’un cas particulier de ces relations est que ${{\underline r}}.{{\underline t}}={{\underline t}}.{{\underline r}}$ si $rt$ divise $c$ et que $r$ et $t$ commutent.
La preuve suit les mêmes grandes lignes que celle de [@B 2.1.4]. Le mono[ï]{}de $M(P_c)$ est engendré par les ${{\underline r}}$ où $r$ est une réflexion qui divise $c$ et les relations \[\*\] sont vraies dans $M(P_c)$. Il suffit de voir que ces relations impliquent les autres, c’est-à-dire que pour tout $w\in P_c$ on peut passer d’une écriture de $w$ de longueur $l_R(w)$ à une autre uniquement par les relations $$r.s=(rsr).r\label{**}$$ si $r$ et $s$ sont deux réflexions distinctes telles que $rs$ divise $c$. Prouvons ceci par récurrence sur $l_R(w)$. Si $l_R(w)=1$ il n’y a qu’une écriture de longueur minimale de $w$. Dans le cas général il suffit de prouver que si $t\in R$ et $t\preccurlyeq w\in P_c$ alors à partir d’une écriture minimale de $w$ fixée et par application des relations \[\*\*\] on peut obtenir une écriture minimale de $w$ commençant par $t$. L’hypothèse de récurrence permet alors de conclure. L’élément $w$, divisant $c$, est un produit de diviseurs élémentaires comme dans \[diviseurs de c\]. On fixe une écriture minimale de $w$ obtenue par concaténation d’une écriture minimale de chaque diviseur élémentaire. Par \[reflexions divisant un produit\] une réflexion $t$ divise $w$ si et seulement si elle divise un des diviseurs élémentaires de $w$. Comme deux réflexions qui interviennent dans les écritures de deux diviseurs élémentaires de $w$ distincts commutent entre elles, $t$ commute avec les réflexions qui interviennent dans les écritures minimales des autres diviseurs élémentaires. On est donc ramené à montrer le résultat pour un seul diviseur élémentaire. Si ce diviseur est un élément de Coxeter d’un groupe de type $A$ (c’est-à-dire a toutes ses orbites finies) le résultat est connu ([[*cf.*]{}]{} [@B] et [@BDM]). On est ramené au cas d’un élément de Coxeter d’un groupe de type $\tilde A$. Il suffit donc de prouver le résultat pour $c$ lui-même. Quitte à faire une permutation circulaire des réflexions élémentaires, ce qui revient à une conjugaison donc laisse invariantes les relations \[\*\*\], on peut ramener $c$ à être de la forme ([[*cf.*]{}]{} preuve de \[coxeter\]) $$c=(s_{k_h}s_{k_h+1}\ldots s_{k_{h+1}-1})(s_{k_{h-1}}s_{k_{h-1}+1}\ldots
s_{k_h-1})\ldots(s_1s_2\ldots s_{k_1-1}),$$ avec $1=k_0<k_1<k_2<\ldots<k_h<k_{h+1}=n+1$ et $s_i=(i,i+1)$. On part de cette écriture et on veut faire appara[î]{}tre $t$ à gauche de $c$ par application des relations \[\*\*\]. En fait il suffit de faire appara[î]{}tre $t$ dans une écriture de $c$; on peut ensuite le ramener à gauche par application des relations \[\*\*\]. Chaque $(s_{k_i}s_{k_i+1}\ldots s_{k_{i+1}-1})$ est un élément de Coxeter d’un groupe de type $A$, donc on peut faire appara[î]{}tre à gauche ou à droite, par application des relations \[\*\*\] n’importe quelle réflexion de support inclus dans $[k_i,k_{i+1}]$. On en déduit que si $k_{i-1}\leq a\leq k_i\leq k_j\leq
b\leq k_{j+1}$, on peut faire appara[î]{}tre dans l’écriture de $c$ par application de \[\*\*\] le produit $(k_j,b)(k_{j-1},k_j)\ldots(k_{i+1},k_i)(k_i,a)$ qui est l’écriture d’un élément de Coxeter d’un groupe de type $A$, donc on peut faire appara[î]{}tre $(a,b)$ dans l’écriture de cet élément. Le même type d’argument montre qu’on peut faire appara[î]{}tre $(b,a+n)$ sous les mêmes hypothèses, en faisant appara[î]{}tre le produit $(b,1+n)$ à droite du produit $(s_{k_h}\ldots s_{k_{h+1}})\ldots(s_{k_j}\ldots s_{k_{j+1}})$ et $(1+n,a+n)=(1,a)$ à gauche du produit $(s_{k_i}\ldots s_{k_{i+1}-1})\ldots(s_{k_0}\ldots s_{k_1-1})$.
Il reste à voir qu’on peut faire appara[î]{}tre toute réflexion $t=(a,\alpha)$ où $a\in X$ et $\alpha\in \Xi$. Comme $c$ est la translation d’une position dans le sens croissant de $X$ et d’une position dans le sens décroissant de $\Xi$, il conjugue $(a,\alpha)$ sur $(a',\alpha')$ où $a'$ est translaté de $a$ dans $X$ d’une position dans le sens croissant et $\alpha'$ est translaté de $\alpha$ dans $\Xi$ d’une position dans le sens décroissant. Donc $(a,\alpha)$ peut être ramené par conjugaison par une puissance de $c$ sur une réflexion d’une des formes $(a,b)$ ou $(a,b-n)$, avec $1\leq a,b\leq n$, et par la première partie de cette démonstration on sait qu’on peut faire appara[î]{}tre une telle réflexion dans une écriture de $c$ par application des relations \[\*\*\]. La conjugaison par $c$ est aussi une suite d’applications de \[\*\*\], d’où le résultat.
Présentations duales pour les groupes d’Artin-Tits affines de type $\tilde A$
=============================================================================
Nous gardons les notations des deux sections précédentes, en particulier $c$ est un élément de Coxeter du groupe de Coxeter $W$ de type $\tilde A_{n-1}$. Le premier but de cette section est de prouver le théorème suivant ([[*cf.*]{}]{} conjecture \[conjecture\] (i)):
\[G isom B\] Le groupe de fractions $G(P_c)$ de $M(P_c)$ est isomorphe au groupe d’Artin-Tits de type $\tilde
A_{n-1}$.
Nous noterons ${B(\tilde A_{n-1})}$ le groupe d’Artin-Tits de type $\tilde A_{n-1}$. Il est engendré par ${{\bf s}}_1,{{\bf s}}_2,\ldots {{\bf s}}_n$, avec comme relations ${{\bf s}}_i{{\bf s}}_{i+1}{{\bf s}}_i={{\bf s}}_{i+1}{{\bf s}}_i{{\bf s}}_{i+1}$ pour $i=1,\ldots,n$, si on pose ${{\bf s}}_{n+1}={{\bf s}}_1$, et ${{\bf s}}_i{{\bf s}}_j={{\bf s}}_j{{\bf s}}_i$ si $i\neq j\pm 1\pmod n$. Pour prouver le théorème \[G isom B\] on montre d’abord que dans $G(P_c)$ les éléments ${{\underline s}}_1,\ldots,{{\underline s}}_n$ vérifient les mêmes relations de tresses. Ceci définit un morphisme ${B(\tilde A_{n-1})}\to G(P_c)$. On trouve ensuite dans ${B(\tilde A_{n-1})}$ des éléments dont les images par ce morphisme sont les générateurs ${{\underline r}}$ de $G(P_c)$ et qui vérifient les relations \[\*\], ce qui prouve la bijectivité du morphisme. Pour cette deuxième étape on utilisera l’interprétation de ${B(\tilde A_{n-1})}$ comme groupe fondamental.
\[B to G\] L’application ${{\bf s}}_i\mapsto{{\underline s}}_i $ pour $i=1,\ldots,n$ se prolonge en un homomorphisme ${B(\tilde A_{n-1})}\rightarrow G(P_c)$.
Cela revient à montrer que dans $G(P_c)$ on a ${{\underline s}}_i{{\underline s}}_{i+1}{{\underline s}}_i={{\underline s}}_{i+1}{{\underline s}}_i{{\underline s}}_{i+1}$ pour $i=1,\ldots,n$ et ${{\underline s}}_i{{\underline s}}_j={{\underline s}}_j{{\underline s}}_i$ si $i,j\in[1,n]$ et $|i-j|\geq 2$. Par \[presentation\] on a ${{\underline s}}_i{{\underline s}}_{i+1}={{\underline s}}'{{\underline s}}_i={{\underline s}}_{i+1}{{\underline s}}'$, où $s'=(i,i+2)$. On en déduit ${{\underline s}}_{i+1}{{\underline s}}_i{{\underline s}}_{i+1}={{\underline s}}_{i+1}{{\underline s}}'{{\underline s}}_i={{\underline s}}_i{{\underline s}}_{i+1}{{\underline s}}_i$. On a aussi par \[presentation\] ${{\underline s}}_i{{\underline s}}_j={{\underline s}}_j{{\underline s}}_i$ si $|i-j|\geq 2$.
Avant de montrer que ce morphisme est un isomorphisme, nous rappelons l’interprétation de ${B(\tilde A_{n-1})}$ comme sous-groupe du groupe de tresses à $n$ brins dans ${{\mathbb C}}^*$ ([[*cf.*]{}]{} [@Graham-Lehrer] et [@Allcock]). On considère un $n$-uplet de points de ${{\mathbb C}}^*$. Le groupe des tresses dans ${{\mathbb C}}^*$ de base ce $n$-uplet (“tresses à $n$ brins” dans ${{\mathbb C}}^*$), est isomorphe au groupe d’Artin-Tits $B(B_n)$ de type $B_n$. L’application qui associe à une telle tresse le nombre de tours total des brins autour de 0 est un morphisme à valeur dans ${{\mathbb Z}}$. Le groupe des tresses de type $\tilde A_{n-1}$ est le noyau de ce morphisme. Notons aussi qu’on peut considérer le groupe des tresses à $n$ brins dans ${{\mathbb C}}^*$ de base le $n$-uplet $(x_1,\ldots,x_n)$ comme le sous-groupe du groupe des tresses à $n+1$ brins dans ${{\mathbb C}}$ de base le $n+1$-uplet $(0,x_1,\ldots,x_n)$ tel que le brin issu de l’origine soit trivial (tresses pures relativement à un brin fixé).
Nous nous plaçons dans le cadre de \[representation graphique\] et nous choisissons comme $n$-uplet de base $(a,b,\ldots,l,\alpha,\ldots,\lambda)$ comme dans \[representation graphique\]. Nous associons à chaque $\underline r$ où $r$ est une réflexion de $P_c$ une tresse de ${B(\tilde A_{n-1})}$ la façon suivante: $r$ est représentée par un chemin $\gamma$ dans la couronne, sans auto-intersection, reliant $i$ à $j$ où $i$ et $j$ sont les images dans la couronne de deux points de $X\cup\Xi$. On associe à ce chemin la tresse où tous les points sont fixes sauf les deux points partant respectivement de $i$ et $j$ qui suivent $\gamma$ en sens inverse et s’“évitent par la droite” si l’orientation du plan est choisie dans le sens horaire ce que nous supposerons dans les figures qui suivent. Plus précisément, on peut supposer que $\gamma$ est une application différentiable de $[0,1]$ dans la couronne telle que les tangentes en 0 et 1 soient orthogonales au bord de la couronne. Soit $\vec n(t)$ un vecteur normal dans le sens direct à $\gamma$ en $\gamma(t)$. On considère la tresse où tous les brins sont fixes sauf un brin partant de $i$ donné par $t\mapsto\gamma(t)+\varepsilon\sin(\pi t)\vec n(t)$ et un brin partant de $j$ donné par $t\mapsto\gamma(1-t)-\varepsilon\sin(\pi (1-t))\vec n(1-t)$ où $\varepsilon$ est assez petit pour que la tresse soit dans la couronne.
\[G to B\] L’application que nous venons de définir se prolonge en un isomorphisme de $G(P_c)$ dans ${B(\tilde A_{n-1})}$ inverse de l’homomorphisme défini par \[B to G\].
Il faut voir que les relations \[\*\] sont vérifiées par les images des éléments ${{\underline r}}$. On déduit de \[diviseurs de c\] qu’il y a trois types de couples $(r,t)\in R^2$ tels que $rt$ divise $c$.
1. Si $r$ et $t$ correspondent à des chemins sans intersection ${{\underline r}}$ et ${{\underline t}}$ commutent et il est clair que leurs images dans ${B(\tilde A_{n-1})}$ commutent aussi.
2. Si $r=(i,j)$ et $t=(j,k)$ où $i$, $j$ et $k$ sont deux à deux distincts modulo $n$ et si $i,j,k$ sont les sommets d’un triangle curviligne direct dans la bande, la relation est $\underline{(i,j)}.\underline{(j,k)}=\underline{(i,k)}.\underline{(i,j)}$. Cette relation est vérifiée par les tresses images: si nous notons encore $i$, $j$ et $k$ les images repectives dans la couronne des points $i$, $j$ et $k$, les éléments $\underline{(i,j)}$, $\underline{(j,k)}$ et $\underline{(i,k)}$ correspondent à des chemins respectivement de $i$ à $j$ de $j$ à $k$ et de $k$ à $i$ formant le bord d’un triangle curviligne direct inclus dans la couronnne et la relation pour les tresses correspondantes n’est autre que la relation classique pour les tresses à trois brins (voir figure \[relation standard\]) dans le groupe de tresses de ${{\mathbb C}}^*$ (ou de ${{\mathbb C}}$).
(0,0)![ $\underline{(i,j)}.\underline{(j,k)}=\underline{(j,k)}.\underline{(i,k)}$ \[relation standard\]](figure1.ps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
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3. Si $r=(x,\xi)$ et $t=(x-n,\xi)$ avec $x\in X$ et $\xi\in \Xi$, la relation est $\underline{(x,\xi)}.\underline{(x-n,\xi)}=
\underline{(x+n,\xi)}.\underline{(x,\xi)}$. La relation correspondante est vraie dans ${B(\tilde A_{n-1})}$: on considère l’automorphisme de $M(P_c)$ induit par l’identité sur $X$ et la translation de $n$ sur $\Xi$. La relation revient à dire que $\underline{(x,\xi)}.\underline{(x-n,\xi)}$ est invariant par cet automorphisme. Dans ${B(\tilde A_{n-1})}$ on considère l’automorphisme induit par une isotopie qui est l’identité sur le bord extérieur de la couronne et fait tourner l’autre bord de la couronne d’un tour dans le sens positif. L’application de l’énoncé est compatible avec ces automorphismes. On peut supposer $x\in [1,n]$; en appliquant une puissance convenable de ces deux automorphismes on peut ramener $\xi$ dans $[1,n]$; la relation résulte alors de la figure \[relation speciale\] qui permet de voir l’invariance cherchée.
(0,0)![ $\underline{(x,\xi)}.\underline{(x,\xi+n)}$\[relation speciale\]](figure2.ps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
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Les propositions \[B to G\] et \[G to B\] prouvent le théorème \[G isom B\]
Nous avons ainsi défini pour chaque choix d’un élément de Coxeter $c$ un sous-mono[ï]{}de $M(P_c)$ de ${B(\tilde A_{n-1})}$. Deux tels mono[ï]{}des correspondant à des ensembles $\{a,\ldots,l\}$ et $\{\alpha,\ldots,\lambda\}$ de mêmes cardinaux respectifs sont isomorphes. L’échange de $X$ et de $\Xi$ est aussi un isomorphisme (qui se traduit dans la représentation géométrique par une rotation de $\pi$ de la bande). On en déduit que la classe d’isomorphisme de $M(P_c)$ ne dépend que de la partition de $n$ en $|X\cap [1,n]|+|\Xi\cap[1,n]|$. Réciproquement:
L’application qui à $c$ associe l’ensemble $\{|X\cap [1,n]|,|\Xi\cap[1,n]|\}$ induit une bijection des classes d’isomorphisme des mono[ï]{}des $M(P_c)$ de la section \[monoide dual\] sur les partitions de $n$ en deux parties.
Ce théorème résulte des considérations qui précèdent et de la proposition suivante qui montre que $|X\cap[1,n]|$ et $|\Xi\cap [1,n]|$ sont déterminés par l’action de la conjugaison par ${{\underline c}}$ sur les atomes.
\[orbites de c\] L’orbite par la conjugaison par ${{\underline c}}$ d’un atome $\underline{(i,j)}$ est finie de cardinal $|X\cap[1,n]|$ (resp. $|\Xi\cap[1,n]|$) si $i$ et $j$ sont tous deux dans $X$ (resp. tous deux dans $\Xi$) et infinie si $i\in X$ et $j\in
\Xi$.
Par \[conjugaison par c\], si $r$ et $r'$ sont des réflexions de $P_c$ on a ${{\underline r}}{{\underline c}}={{\underline c}}{{\underline r}}'$ si et seulement si $rc=cr'$. On a vu dans la démonstration de \[presentation\] que la conjugaison par $c$ revient à décaler $X$ d’une position dans le sens croissant et $\Xi$ d’une position dans le sens décroissant. On en déduit le résultat.
Une structure à la Garside pour les groupes d’Artin-Tits affines de type $\tilde A$ {#garside}
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Gardons les notations des sections précédentes. Nous allons étudier les propriétés de la divisibilité dans $P_c$ dans le cas où $c=s_1s_2\ldots s_n$. Dans ce cas on a $c=(2,3,\ldots,n)_{[1]}(1)_{[-1]}$ et les ensembles $\Xi$ et $X$ sont respectivement $\Xi=\{z\in{{\mathbb Z}}\mid z\equiv1\pmod n\}$ et $X=\{z\in{{\mathbb Z}}\mid z\not\equiv1\pmod n\}$.
La propriété fondamentale du mono[ï]{}de $M(P_c)$ dans ce cas (théorème \[M(P\_c) quasi-Garside\]) est une conséquence de la proposition suivante:
\[ppcm d’atomes\] Si $c=s_1\ldots s_n$, deux atomes quelconques de $P_c$ ont un ppcm dans $P_c$.
Considérons deux atomes $r=(x,y)$ et $r'=(x',y')$. On sait que $r$ (resp. $r'$) divise un élément $p\in P_c$ si et seulement si $x$ et $y$ (resp. $x'$ et $y'$) sont dans la même partie de la partition associée à $p$ comme dans \[partition\]. Il y a deux cas.
- Si $r$ et $r'$ sont sans croisement, ils commutent et $rr'$ est dans $P_c$ par \[diviseurs de c\]. Dans la décomposition d’un multiple commun $m$ de $r$ et $r'$ en produit de diviseurs élémentaires, ou bien $r$ et $r'$ divisent deux diviseurs élémentaires de $m$ différents et $rr'$ divise alors $m$, ou bien $r$ et $r'$ divisent le même diviseur élémentaire $c'$ de $m$ qui est un élément de Coxeter d’un sous-groupe [quasi-parabolique]{} $W'$ et on conclut que $rr'$ divise $c'$ soit par \[diviseurs de c\] si $W'$ est de type $\tilde A$ soit par [@BDM 1.8] si $W'$ est de type $A$. Donc le ppcm de $r$ et $r'$ existe et vaut $rr'$.
- Si $r$ et $r'$ se croisent ([[*i.e.*]{}]{} ne sont pas sans croisement), supposons $x<y$, $x'<y'$ et $x$ et $x'$ dans $[1,n]$; si $m$ est un multiple commun de $r$ et $r'$, alors $\{x,y\}$ et $\{x',y'\}$ ne peuvent pas être dans deux parties distinctes de la partition associée à $m$ car ces deux parties ne seraient pas sans croisement. Donc il existe un diviseur élémentaire $m'$ qui divise $m$ et qui est multiple de $r$ et $r'$. Distinguons trois cas.
Si l’ensemble $\{x,y,x',y'\}\cap X$ est inclus dans un intervalle de longueur strictement inférieure à $n$, la partition dont les seules parties non triviales sont $\{x,y,x',y'\}+kn$ avec $k\in{{\mathbb Z}}$ définit un élément de Coxeter $c'$ d’un [quasi-parabolique]{} de type $A$.
Si l’ensemble $\{x,y,x',y'\}\cap X$ n’est pas inclus dans un intervalle de longueur strictement inférieure à $n$ et si $\{x,y,x',y'\}\cap \Xi$ est non vide, la partition dont la seule partie non triviale est $\{x,y,x',y'\}+n{{\mathbb Z}}$ définit un élément de Coxeter $c'$ d’un [quasi-parabolique]{} de type $\tilde A$.
Si l’ensemble $\{x,y,x',y'\}\cap X$ n’est pas inclus dans un intervalle de longueur strictement inférieure à $n$ et si $\{x,y,x',y'\}\cap \Xi$ est vide, notons $c'$ l’élément de Coxeter d’un [quasi-parabolique]{} de type $\tilde A$ défini par la partition dont la seule partie non triviale est $\{1,x,y,x',y'\}+n{{\mathbb Z}}$ et remarquons que la partie associée à $m'$ étant infinie doit contenir $1$.
Dans les trois cas l’élément $c'$ divise $m'$ par \[diviseurs de c\] appliqué à $m'$ et il est multiple de $r$ et $r'$. C’est donc le ppcm de $r$ et $r'$ dans $P_c$.
On peut alors appliquer la généralisation immédiate suivante de [@B 0.5.2]:
\[treillis\] Soit $(G,R)$ un groupe positivement engendré, soit $c$ un élément équilibré et soit $M(P_c)$ comme précédemment. Supposons que deux éléments quelconques de $R$ ont un ppcm dans $P_c$; alors la divisibilité à gauche et la divisibilité à droite donnent à $M(P_c)$ deux structures de treillis.
Dans ce contexte on peut vérifier que les résultats de [@BDM section 2] et [@B 0.5] s’appliquent.
Nous appellerons structure quasi-Garside une structure de mono[ï]{}de vérifiant tous les axiomes de [@B 0.5.1] sauf la finitude du nombre d’atomes. Nos axiomes seront donc:
\[quasi-garside\] Un mono[ï]{}de $M$ est dit quasi-Garside si
1. Pour tout $m\in M$ le nombre de facteurs dans un produit égal à $m$ est borné.
2. $M$ est simplifiable à gauche et à droite.
3. La divisibilité à gauche et la divisibilité à droite donnent à $M$ deux structures de treillis.
4. Il existe un élément $\Delta$ (élément de Garside) dont l’ensemble des diviseurs à gauche est égal à l’ensemble des diviseurs à droite et engendre $M$.
L’axiome (i) signifie que le mono[ï]{}de est atomique au sens par exemple de [@B 0.2.2]; il est équivalent à l’existence d’une longueur $l$ sur le mono[ï]{}de telle que $l(ab)\geq l(a)+l(b)$ pour tout couple $(a,b)$.
Les axiomes (i), (ii) et (iv) étant vérifiés par un mono[ï]{}de défini comme plus haut à partir d’un groupe engendré et d’un élément équilibré, l’énoncé \[treillis\] devient:
\[M(P\_c) quasi-Garside\] Sous les hypothèses de \[treillis\], le mono[ï]{}de $M(P_c)$ est un monode quasi-Garside avec ${{\underline c}}$ comme élément de Garside.
Ce théorème s’applique en particulier à un élément $c$ comme dans \[ppcm d’atomes\]. Un tel choix définit donc une structure quasi-Garside sur le groupe d’Artin-Tits de type $\tilde A_{n-1}$.
Le résultat suivant montre que le choix de $c$ fait dans \[ppcm d’atomes\] est à isomorphisme près le seul pour lequel la divisibilité a une structure de treillis.
Si $c$ est un élément de Coxeter d’un groupe de type $\tilde A_{n-1}$, le mono[ï]{}de $M(P_c)$ muni de l’ordre de la divisibilité a une structure de treillis (et est donc un mono[ï]{}de quasi-Garside) si et seulement si l’un des deux ensembles $\Xi$ ou $X$ est réduit à un seul élément modulo $n$.
L’échange de $\Xi$ et $X$ définit un isomorphisme des mono[ï]{}des correspondants. D’autre part deux éléments de Coxeter tels que les ensembles $X$ correspondants ont même nombre d’éléments modulo $n$ sont conjugués, donc dans ce cas aussi les mono[ï]{}des sont isomorphes. On en déduit par \[ppcm d’atomes\] que si $\Xi$ ou $X$ a un seul élément modulo $n$, on a bien une structure de treillis. Inversement, supposons que modulo $n$, à la fois $\Xi$ et $X$ ont au moins deux éléments. Soient $a<b$ dans $X\cap[1,n]$ (resp. $\alpha\neq \beta$ dans $\Xi\cap[1,n]$). Les éléments $(a,b)$ et $(b,a+n)$ divisent $(a,b)_{[1]}(\alpha)_{[-1]}$ et $(a,b)_{[1]}(\beta)_{[-1]}$ qui sont de longueur 3 et n’ont aucun diviseur commun de longueur 2 d’après \[diviseurs de c\]. Donc $(a,b)$ et $(b,a+n)$ n’ont pas de ppcm.
Remarquons que dans l’exemple précédent c’est le dernier cas de la preuve de \[ppcm d’atomes\], dans lequel on a dû introduire le [quasi-parabolique]{} de type $\tilde A$ défini par $\{1,x,y,x',y'\}+n{{\mathbb Z}}$, qui est en défaut. En fait les raisonnements des autres cas s’appliquent pour tout élément de Coxeter $c$ mais ce dernier cas utilise le fait que $\Xi$ est un singleton modulo $n$.
Donnons quelques conséquences de l’existence d’une structure quasi-Garside sur ${B(\tilde A_{n-1})}$. Ces conséquences sont de simples applications des propriétés générales des mono[ï]{}des de Garside dont on vérifie qu’elles sont encore valables dans le cadre quasi-Garside. Le premier est l’existence de formes normales telles que dues à Garside (voir par exemple [@michel] ou [@BDM section 2], voir aussi [@Charney]).
1. Tout élément de ${B(\tilde A_{n-1})}$ s’écrit de façon unique $a{^{-1}}b$ où $a$ et $b$ sont des éléments de $M(P_c)$ premiers entre eux.
2. Tout élément de $M(P_c)$ s’écrit de façon unique $a_1a_2\ldots a_k$ où pour $i=1,\ldots,k$ l’élément $a_i\in{{\underline P}}_c$ est un (le) diviseur maximal dans ${{\underline P}}_c$ du produit $a_ia_{i+1}\ldots a_k$ et $a_k\neq 1$.
Nous prouvons maintenant:
Le centre de ${B(\tilde A_{n-1})}$ est trivial.
La démonstration suit les mêmes idées que la démonstration classique pour les mono[ï]{}des de tresses ou que la démonstration de [@picantin 4.1]; ces démonstrations ne s’appliquent pas telles quelles car elles supposent qu’il y a un nombre fini d’atomes. On utilise le lemme suivant:
Soit $M$ un mono[ï]{}de quasi-Garside et soit $b$ un élément quasi-central de $M$, c’est-à-dire tel qu’il existe un automorphisme $\tau$ de $M$ vérifiant $xb=b\tau(x)$ pour tout $x\in M$; soit $x$ un diviseur à gauche de $b$ et $y\in M$; posons $\operatorname{ppcm}(x,y)=yz$ avec $z\in M$: alors $z$ divise $b$ à gauche.
Comme $x$ divise $b$ il divise $b\tau(y)=yb$, donc $yz$ divise $yb$ et par simplifiabilité $z$ divise $b$.
On en déduit la proposition: Soit $g\in{B(\tilde A_{n-1})}$ central. On peut écrire $g={{\underline c}}^nb$ avec $b\in M(P_c)$ non divisible par ${{\underline c}}$ et $n\in{{\mathbb Z}}$ convenable. On a alors $b$ quasi-central. Montrons par l’absurde que $b=1$. Sinon, soit $r$ une réflexion de $P_c$ telle que ${{\underline r}}$ divise $b$. Pour tout triplet de réflexions $(r,s,t)$ correspondant à un triangle direct comme dans la preuve de \[G to B\] on a ${{\underline r}}.{{\underline s}}={{\underline s}}.{{\underline t}}={{\underline t}}.{{\underline r}}=\operatorname{ppcm}({{\underline r}},{{\underline s}})=\operatorname{ppcm}({{\underline s}},{{\underline t}})$. On en déduit par le lemme que ${{\underline r}}$ et ${{\underline t}}$ divisent $b$. Comme à partir de $r$, de proche en proche on peut faire appara[î]{}tre n’importe quelle réflexion $r'$ de $P_c$ dans un triangle direct, on en déduit que $b$ est multiple de tous les atomes de $M(P_c)$ donc est multiple de ${{\underline c}}$, ce qui est contradictoire. Tout élément central est donc une puissance de ${{\underline c}}$. Or la conjugaison par ${{\underline c}}$ est un automorphisme d’ordre infini ([[*cf.*]{}]{} \[presentation\] ou \[orbites de c\]), donc aucune puissance de ${{\underline c}}$ autre que ${{\underline c}}^0$ n’est centrale.
Donnons une dernière conséquence de l’existence de la structure quasi-Garside.
\[centralisateur\] Soient $c=s_1\ldots s_n$ et ${{\underline c}}={{\underline s}}_1\ldots {{\underline s}}_n$ comme précédemment. Le centralisateur de ${{\underline c}}^h$ dans le groupe d’Artin-Tits de type $\tilde A_{n-1}$ engendré par ${{\underline s}}_1,\ldots,{{\underline s}}_n$ est un groupe d’Artin-Tits de type $B_{\operatorname{pgcd}(h,n-1)}$.
La théorie générale des mono[ï]{}des de (quasi-)Garside énonce que les points fixes d’un automorphisme $\sigma$ dans un mono[ï]{}de $M$ de (quasi-)Garside forment un mono[ï]{}de $M^\sigma$ de (quasi-)Garside, avec pour atomes certains des ppcm des orbites des atomes et même élément de Garside; le groupe des fractions de ce monoïde est le groupe des points fixes de $\sigma$ dans le groupe des fractions de $M$ ([[*cf.*]{}]{} [@BDM 2.26] dont la démonstration s’étend à la situation quasi-Garside). Nous appliquons ceci à la conjugaison par ${{\underline c}}^h$. Le centralisateur de ${{\underline c}}^h$ dans $G(P_c)$ est donc engendré par les ppcm des orbites des atomes sous la conjugaison par ${{\underline c}}^h$. Comme le montre le calcul fait dans \[presentation\] la conjugaison par ${{\underline c}}$ envoie $\underline{(x,y)}$ sur $\underline{(x',y')}$ où $x'$ et $y'$ s’obtiennent à partir de $x$ et de $y$ en translatant $X$ d’une position dans le sens croissant et $\Xi$ d’une position dans le sens décroissant. Pour simplifier au lieu d’indexer comme précédemment les éléments de $X$ par les entiers non congrus à 1 modulo $n$, nous renumérotons consécutivement les éléments de $X$, en les notant $x_i$ avec $i\in {{\mathbb Z}}$, et nous faisons de même pour les éléments de $\Xi$ qui seront notés $\xi_i$ avec $i\in{{\mathbb Z}}$. La translation de $n$ devient alors $x_i\mapsto x_{i+n-1}$ et $\xi_i\mapsto \xi_{i+1}$. La conjugaison par ${{\underline c}}^h$ envoie $\underline{(x_i,x_j)}$ sur $\underline{(x_{i+h},x_{j+h})}$. L’orbite de $\underline{(x_i,x_j)}$ ne dépend donc que du pgcd de $h$ et de $n-1$. La conjugaison par ${{\underline c}}^h$ envoie $\underline{(\xi_0,x_j)}$ sur $\underline{(\xi_0,x_{j+nh})}$. Le ppcm d’une telle orbite est $\underline{(\xi_0)_{[-1]}(x_j,x_{j+k},x_{j+2k},\ldots,x_{j+n-1-k})_{[1]}}$ où $k$ est le pgcd de $h$ et $n-1$. On voit que les ppcm des orbites d’atomes ne dépendent que du pgcd de $n-1$ et $h$. Donc ${{\underline c}}^h$ et ${{\underline c}}^{\operatorname{pgcd}(n-1,h)}$ ont même centralisateur. On est donc ramené au cas où $h$ divise $n-1$. Le lemme suivant est le cas particulier de la proposition quand $h=n-1$.
Le centralisateur de ${{\underline c}}^{n-1}$ est un groupe d’Artin-Tits de type $B_{n-1}$ et l’image de ${{\underline c}}$ dans ce groupe est un élément de Coxeter de ce groupe.
On a appelé élément de Coxeter d’un groupe d’Artin-Tits le relevé canonique d’un élément de Coxeter du groupe de Coxeter.
Les éléments $\underline{(x_i,x_j)}$ sont centralisés par ${{\underline c}}^{n-1}$. L’orbite de $\underline{(\xi_0,x_j)}$ se compose des éléments $\underline{(\xi_0,x_{j+kn(n-1)})}$ avec $k\in{{\mathbb Z}}$. Le ppcm d’une telle orbite est $\underline{(\xi_0)_{[-1]}(x_j)_{[1]}}$. Le centralisateur $C({{\underline c}}^{n-1})$ de ${{\underline c}}^{n-1}$ est donc engendré par ces éléments et toutes les relations s’obtiennent en égalant les décompositions de ${{\underline c}}$ comme produits de ces générateurs. On a $${{\underline c}}=\underline{(x_1,x_2)}.\underline{(x_2,x_3)}\ldots\underline{(x_{n-2},x_{n-1})}.
\underline{(\xi_0)_{[-1]}(x_{n-1})_{[1]}}.$$
Pour construire un isomorphisme entre $C({{\underline c}}^{n-1})$ et le groupe d’Artin-Tits $B(B_{n-1})$ de type $B_{n-1}$ nous revenons à l’interprétation de $B(\tilde A_{n-1})$ comme tresses dans une couronne. Remarquons que les générateurs ci-dessus de $C({{\underline c}}^{n-1})$ sont des tresses telles que le brin d’origine $\xi_0$ a pour extrémité $\xi_0$ (voir figure \[figure3\]). Or on peut interpréter le groupe de tresses de type $B_{n-1}$ comme le groupe de tresses à $n-1$ brins dans la couronne, les points de base étant les points $x_1,x_2,\ldots,x_{n-1}$ ([[*cf.*]{}]{}, [@Lambropoulou]). On a donc un morphisme de $C({{\underline c}}^{n-1})$ dans $B(B_{n-1})$ par oubli du brin issu de $\xi_0$. Les générateurs de $C({{\underline c}}^{n-1})$ s’envoient sur les générateurs de la présentation duale du groupe $B(B_{n-1})$ et l’image de ${{\underline c}}$ est l’élément de Coxeter de $B(B_{n-1})$. Donc les relations entre les générateurs de ces deux groupes se correspondent. Le morphisme est donc un isomorphisme.
(0,0)![ $\underline{(\xi_0)_{[-1]}(x_{n-1})_{[1]}}$ \[figure3\]](figure3.ps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
(3124,3338)(3839,-6122) (5317,-4107)[(0,0)\[lb\]]{} (5213,-2941)[(0,0)\[lb\]]{} (6890,-3964)[(0,0)\[lb\]]{} (6405,-3299)[(0,0)\[lb\]]{} (6959,-4850)[(0,0)\[lb\]]{} (6681,-5595)[(0,0)\[lb\]]{} (4029,-3333)[(0,0)\[lb\]]{}
Prouvons alors la proposition. Si $h$ divise $n-1$, le centralisateur de ${{\underline c}}^h$ est égal au centralisateur de ${{\underline c}}^h$ dans le centralisateur de ${{\underline c}}^{n-1}$. Le lemme permet donc de terminer la démonstration de la proposition car le centralisateur d’une puissance d’un élément de Coxeter dans un groupe de tresses de type $B$ est connu par les résultats de [@BDM].
On obtient des générateurs standards ([[*i.e.*]{}]{}vérifiant les relations de tresses de type $B$) du centralisateur de ${{\underline c}}^{n-1}$ en prenant $\underline{(x_1,x_2)}$, $\underline{(x_2,x_3)}$, …, $\underline{(x_{n-2},x_{n-1})}$, et $\underline{(\xi_0)_{[-1]}(x_{n-1})_{[1]}}$. D’autre part il est facile de voir que le centralisateur de $(s_1s_2\ldots s_n)^{n-1}$ dans le groupe de Coxeter $W(\tilde
A_{n-1})$ est le groupe le groupe engendré par $(x_1,x_2)$, $(x_2,x_3)$, …, $(x_{n-2},x_{n-1})$, et $(\xi_0)_{[-1]}(x_{n-1})_{[1]}$ donc est l’image du centralisateur de ${{\underline c}}^{n-1}$ dans le groupe d’Artin-Tits. La présentation de cette image s’obtient en ajoutant aux relations de tresses de type $B$ le fait que les $n-2$ premiers générateurs sont d’ordre 2 (le dernier est d’ordre infini).
Remerciements {#remerciements .unnumbered}
=============
Je remercie tout particulièrement Eddy Godelle pour ses nombreuses remarques pertinentes sur une version antérieure de cet article. Je remercie également Jean Michel avec qui j’ai eu plusieurs discussions sur ces résultats.
[BKL]{} D. Allcock, Braid pictures for Artin groups, [*Trans. AMS **354***]{} 3455–3474 (2002). D. Bessis, The dual braid monoid, [*Ann. ENS **36***]{} 647–683 (2003). D. Bessis, F. Digne et J. Michel, Springer theory in braid groups and the Birman-Ko-Lee monoid, [*Pacific J. Math. **205, n$^\circ$ 2***]{} 287–309 (2003). J. Birman, K. H. Ko, S. J. Lee, A new approach to the word and conjugacy problem in the braid groups, [*Adv. Math. **139 n$^\circ$ 2***]{} 322–353 (1998). R. Charney, Geodesic automation and growth functions for Artin groups of finite type, [*Math. Ann. **301 n$^\circ$ 2***]{} 307–324 (1995). M. J. Dyer, On minimal lengths of expressions of Coxeter group elements as product of reflections, [*Proc. Amer. Math. Soc. **129 n$^\circ$ 9***]{} 2591–2595 (2001). J.J. Graham et G.I. Lehrer, Algèbre de diagrammes, algèbres de Hecke et nombres de décomposition aux racines de l’unité, [*Annales scientifiques de l’École Normale Supérieure **36/4***]{} 479-524 (2003). S. Lambropoulou, Solid torus links and Hecke algebras of $B$-type, [*Proceedings of the Conference on Quantum Topology, Manhattan, World Sci. Publishing*]{} 225–245 (1994). J. Michel, A note on words in braid monoids, [*J. of Algebra **215***]{} 366–377 (1999). M. Picantin, The center of thin Gaussian groups, [*Journal of Algebra **245***]{} 92–122 (2001). J. Y. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, [*Lecture Notes in Math. **1179***]{}, Springer (1986).
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---
abstract: 'In this paper we present solutions to three short comings of Smoothed Particles Hydrodynamics (SPH) encountered in previous work when applying it to Giant Impacts. First we introduce a novel method to obtain accurate SPH representations of a planet’s equilibrium initial conditions based on equal area tessellations of the sphere. This allows one to imprint an arbitrary density and internal energy profile with very low noise which substantially reduces computation because these models require no relaxation prior to use. As a consequence one can significantly increase the resolution and more flexibly change the initial bodies to explore larger parts of the impact parameter space in simulations. The second issue addressed is the proper treatment of the matter/vacuum boundary at a planet’s surface with a modified SPH density estimator that properly calculates the density stabilizing the models and avoiding an artificially low density atmosphere prior to impact. Further we present a novel SPH scheme that simultaneously conserves both energy and entropy for an arbitrary equation of state. This prevents loss of entropy during the simulation and further assures that the material does not evolve into unphysical states. Application of these modifications to impact simulations for different resolutions up to $6.4 \cdot 10^6$ particles show a general agreement with prior result. However, we observe resolution dependent differences in the evolution and composition of post collision ejecta. This strongly suggests that the use of more sophisticated equations of state also demands a large number of particles in such simulations.'
author:
- |
Christian Reinhardt,$^1$[^1] Joachim Stadel$^2$\
Institute for Computational Science, University of Zurich, 8057 Zurich, CH
bibliography:
- 'literature.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Numerical aspects of Giant Impact simulations
---
\[firstpage\]
hydrodynamics – planets and satellites: formation – planets and satellites: terrestrial planets
Introduction
============
Collisions are one of the most fundamental processes in planet formation as they are a key mechanism to explain growth of initially micron sized dust grains to massive planets like the ones we observe today in our solar system. The last stage of terrestrial planet formation is dominated by very energetic collisions between roughly Mars-sized planetary embryos called Giant Impacts (GI) [@Agnor1999]. Such GI play a key role in the process of planet formation as they determine and influence many important properties of the final planetary system like the final number, mass and spin state [@Agnor1999], chemical and physical composition [@Benz2007; @Wiechert2001; @Genda2005; @Tonks1992; @Jutzi2013] and the presence of satellites (e.g. @Benz1986, @Canup2004, @Citron2015).
Starting with Benz and Cameron’s [@Benz1986] pioneering work in the mid 80’s simulations of GI between planetary embryos have became an increasingly important tool to study the outcomes of such violent collisions. While most of the earlier work (e.g. @Benz1987, @Cameron1991, @Canup2001, @Canup2004, @Wada2006) concentrated on the giant impact hypothesis of the origin of the moon [@Cameron1976] more recent simulations explore a variety of possible scenarios. Some recent examples include satellite formation in a more general sense (e.g. @Canup2011, @Citron2015), mantle stripping events [@Benz2007] or deriving merging criteria to improve N-Body simulations [@Genda2012; @Leinhardt2012].
Most prior work uses the Smoothed Particles Hydrodynamics (SPH) method to solve the equations of motion of the material. Since SPH interpolates the physical quantities using a discrete set of particles that evolve with the flow, it can simulate very deformed geometries and track the ejected material over a large dynamic range making it an ideal tool for such simulations. For the research presented in this paper we modified the cosmological hydrodynamics code GASOLINE [@Wadsley2004] that has been extensively used in many astrophysical applications and applied it to such GI simulations. The code uses a modern SPH implementation, a tree to calculate gravity and has a modular design that allows one to implement new physics without touching the complicated parallel layer. We have implemented the Tillotson equation of state (EOS) [@Tillotson1962] within GASOLINE to model the condensed materials that planets are made of. To facilitate the use of the Tillotson EOS we developed a library[^2] that provides functions to calculate the pressure, sound speed and temperature allowing its use in any simulation software. Besides the SPH modifications described in this paper we also added a new class of smoothing kernels [@Dehnen2012] that do not suffer from particle clumping and allow increasing the number of neighbours thereby reducing the noise of the method.
In this paper we present solutions to three short comings of SPH that previous work encountered. First of all it is very difficult to get an accurate SPH representation of a planet’s equilibrium initial condition (IC) so one has to evolve (“relax”) the system carefully to reach a true equilibrium state before doing any impacts. This is very time consuming and can, in the worst case, use more computational resources than the actual simulations. In Section \[section:ic\] we present a new method to generate extremely low noise planetary models that do not need relaxation prior to a collision which drastically reduces the total simulation time.
Another issue is the proper treatment of the material–vacuum boundary at a planet’s surface, where standard SPH tends to severely underestimate the material’s density causing it to jump from the condensed to an expanded state. This creates an “atmosphere” of low density material around a planet prior to the impact. A solution to this problem is discussed in Section \[section:newdens\] where we present a modification of the standard SPH density estimator that properly calculates the density for material–vacuum boundaries encountered at the surface of the planets. Another possible remedy would be to apply DISPH [@Saitoh2013] to GI simulations as done by @Hosono2016. Because this SPH scheme does not use the density but pressure as a smoothed (and thus differentiable) quantity, it performs well at contact discontinuities and free surfaces, e.g., encountered at the core-mantle boundary and the planet’s surface. In the present work we focus on undifferentiated bodies. However, the proper treatment of the core–mantle boundary is not a problem and the improvements described here have already seen application in realistic models with an iron core and basalt mantle.
Finally, SPH does not necessarily conserve both energy and entropy for an arbitrary equation of state. In the case of an ideal gas one can define an entropy function [@Springel2002] which allows one to conserve entropy for purely adiabatic flows. For a more complex, maybe even tabular EOS, this is in general not possible. In Section \[section:isph\] we present a new, entropy conserving SPH scheme that can be applied to any equation of state. This prevents loss of entropy during the simulation and assures that the material does not evolve to unphysical states thus also eliminating code crashes resulting from particles reaching negative internal energies. Code tests and applications of the methods to GI simulations are presented in Section \[section:tests\]. Finally a summary of the present work and conclusions are discussed in Section \[section:conclusions\].
Creating quiet particle realizations of planets {#section:ic}
===============================================
In Smoothed Particles Hydrodynamics (SPH) the density of each particle (as well as pressure forces and changes in internal energy) is derived from the positions of its neighbours. Thus it is crucial to have an accurate particle representation of the initial conditions. An inhomogeneous distribution will cause large spurious radial and angular density fluctuations that cause oscillations and requires careful relaxation. Having Poison noise in the IC not only means that we waste computational time on relaxing the models but is in general problematic for SPH as it affects the method’s convergence [@Monaghan1985]. For larger simulations with several million particles, relaxation can take a substantial part of the total simulation time and doing such simulations becomes prohibitive if the IC is too noisy. This makes the traditional approach laborious for parameter studies where the target and impactor properties are varied.
Previous work placed the particles on a uniform 3D lattice either with constant internal energy [@Genda2012] or e.g., an isentropic thermal profile [@Canup2012], inside a sphere to have a very uniform and thus low noise particle distribution. To obtain a density gradient that is consistent with the thermal profile and get rid of the Poison noise at the boundary, these models are then evolved with the hydrodynamics code (usually with a velocity damper that reduces the particles velocities by a given fraction after each step) until an equilibrium representation is reached. This method has two short comings: 1) it does not allow one to imprint a (radial) density gradient and 2) also has severe noise at the planet’s surface where the grid is not adapted to the spherical symmetry of the profile. Some authors have tried to model a density gradient by distorting the uniform grid radially [@Woolfson2007] but the resulting models were out of equilibrium and still needed relaxation.
In the present paper we suggest a different approach that respects the spherical symmetry of the problem and provides a very uniform radial particle distribution by using an equal area tessellation of the sphere. This method not only produces very low radial and angular density fluctuations but it also allows one to imprint a density gradient that closely follows the equilibrium solution thus making relaxation of the models obsolete.
Prior to generating the SPH representation of the planetary bodies one has to (numerically) solve the usual internal structure equations (e.g., @Alibert2014). For our boundary conditions (e.g., $M \left( r = R \right) = M_{tot}$ and $\rho \left( r = R \right) = \rho_0$) it is convenient to solve these for $\rho \left( r \right)$, $M \left( r \right)$ and $u \left( r \right)$. Besides the equation for hydrostatic equilibrium $$\frac{\nabla P \left( \rho, u \right)}{\rho} = - G \frac{M \left( r \right)}{r^2}
\label{eqn:hydrostaticequilibrium}$$ one needs to specify an equation of state and an internal energy profile to have a closed set of equations. For the present work we used the Tillotson equation of state (EOS) [@Tillotson1962] that was originally developed to model hyper-velocity impacts and has been used in many prior simulations (e.g. @Benz1987, @Canup2001, @Marinova2011, @Genda2012 and @Jutzi2013). Despite the simple analytic form the results are in good agreement with measurements [@Benz1986; @Brundage2013] and its ability to properly reproduce the materials Hugoniot curve, thus accurately modelling shocks, is excellent [@Brundage2013]. More details on the EOS and the material parameters used in this study can be found in Appendix \[appendix:eos\].
The simplest internal energy profile is uniform and each particle has the same internal energy. This works well for bodies smaller than a few hundred kilometres where the density is roughly constant because self–gravity is too weak to significantly compress the material (e.g. @Genda2015). More massive planets have increasingly steep density gradients, where the use of a uniform thermal profile would be unphysical and furthermore some particles tend to end up below the minimal allowed energy for the material (the so called cold curve)[^3] when starting from a such initial conditions. Such a profile is also not isentropic meaning that the model is expected to exhibit convection until an internal energy profile with constant entropy is reached.
To create the particle realization of an equilibrium model we divided the sphere into concentric spherical shells and use an equal area tessellation of the surface of each shell. We experimented with two tessellation methods, the Icosahedron package [@Tegmark1999] and HEALPix [@Gorski2005] that was originally developed for analysing data from cosmic microwave background measurements. Both methods in principle suit our purpose. Icosahedral tessellation produces a nearly perfect tangentially homogeneous grid but suffers from larger radial deviations. HEALPix on the other hand produces a better radial distribution but shows a small amount of tangential density fluctuations due to artefacts resulting from the grid. Since, in our case, radial fluctuations are more problematic and the HEALPix grid adapts more flexibly in the inner part of the sphere (see below), it has been used exclusively in all of the work presented here.
We begin particle generation by first solving the above structure equations (Equation \[eqn:hydrostaticequilibrium\]). Our method then iteratively minimizes the axis ratios of the finite volumes so that the tangential to radial size almost identical by adjusting the number of particles for each shell and the spacing between shells keeping the particle mass constant. This procedure results in somewhat more or somewhat less particles than specified, meaning that the total mass is deviating from the desired value. We then recalculate the particle mass given the obtained number of particles and repeat the above until we converge. Our final model has the exact total mass and equilibrium density profile (Figure \[fig:densityprofile\]) for the planet as well as having equal mass for all particles. There are somewhat larger density fluctuations in the inner–most shells because the number of particles, and thus the radial size of the cells, can not be adapted as flexibly due to constraints from the HEALPix grid. We also notice that the smoothed density deviates significantly for the imprinted profile in the outer most shells. There the resulting density as determined by the SPH code is lower than that originally imposed, which causes the bodies to be out of equilibrium. For a condensed material EOS this is even more problematic as the estimated surface density is lower than the material’s reference density ($\rho_0$, the density of the material at zero temperature and pressure) which causes the material to fall into an unphysical regime. For a colder model with a low surface temperature this is less problematic as the Tillotson EOS in this region has the same analytic form as in the condensed states. For hotter models, however, the particles end up in the intermediate expanded states where the EOS is different and the models become very unstable. It is therefore crucial to correct the surface density as calculated by the SPH code and is made possible by the method we present in the following section.
![A slice (2.2 x 2.2 x 0.2 $R_{\oplus}$) through the SPH representation of an equilibrium model ($0.997 M_{\oplus}$ and $10^{5}$ particles) where the colours represent the internal energy of the material. The small plot shows a zoom at the surface of the model, where we marked the smoothing kernel (in red) of a particle in the outer most shell. Obviously such outer most particles have only one side of their kernel sampled by neighbouring particles while the rest is vacuum.[]{data-label="fig:slice"}](plots/slice.png)
![[]{data-label="fig:relaxedmodels"}](plots/relaxedmodels.png)
A modified density estimator for free surfaces {#section:newdens}
==============================================
In Smoothed Particle Hydrodynamics the density of a particle is derived from the masses and positions of its neighbouring particles [@Monaghan1992; @Springel2010; @Price2012] $$\rho_i = \sum_{j=1}^{N}{ m_j W \left( \left| {\boldsymbol{r}}_i - {\boldsymbol{r}}_j \right| \right) }
\label{eqn:sphdensym}$$ where the kernel function $W \left(r \right)$ weights each particle’s contribution according to its distance from the central particle. Modern SPH implementations use a compact kernel and enforce a fixed number of nearest neighbour particles [@Monaghan1992] or enclosed mass [@Springel2002] to automatically adapt the resolution of the method. When the density is estimated in this way a particle at the planet’s surface has about one half of its kernel sampled with neighbours while the rest of the volume is empty, or rather filled with vacuum. This means that the mass to volume ratio, and thus the resulting density, is underestimated because the standard SPH density estimator assumes that the whole kernel is (more or less homogeneously) filled with particles.
One possible treatment of vacuum/material interfaces is thus to correct the density using a correction factor $$V_{\rm fac} = \frac{V}{V_{\rm eff}}$$ that only accounts for the effectively sampled volume $V_{\rm eff}$. One way to estimate $V_{\rm eff}$ is to consider the imbalance of the particle distribution in the kernel $$f_{{\rm imb},i} = \frac{\left| {\boldsymbol{f_i}} \right|}{2 h_i \sum_j{ m_j W_{ij} \left( h_i \right)}}
\label{eqn:fimb}$$ where we used $${\boldsymbol{f_i}} = \sum_j{ \left( {\boldsymbol{r_i}} - {\boldsymbol{r_j}} \right) m_j W_{ij} \left( h_i \right)}
\label{eqn:fi}$$ to measure the asymmetry of the particle distribution inside the kernel. We also tried to estimate the imbalance as suggested by Woolfson [@Woolfson2007] but found that the lower order of this estimator results in density fluctuations larger than the required correction. By using a kernel average (we use the Wendtland $C_2$, a fourth order kernel, with 80 nearest neighbours) as shown in the above equations our resulting density at the surface is much less noisy. The modified density is then obtained from $$\rho_{\rm eff} = \frac{\rho_{\rm SPH}}{V_{\rm fac}}$$ which obviously gives the standard SPH density if the particle distribution inside of the kernel is symmetric ($V_{\rm fac} = 1$).
Since the result was too sensitive to noise for very small and large imbalances we used a linear approximation (shown in green in Figure \[fig:vfac\]) and kept $V_{\rm fac}$ constant in these cases.
![[]{data-label="fig:vfac"}](plots/kernel_centroid.png)
Since the proposed density correction only depends on a given particle’s kernel volume, it is not symmetric (e.g., Equation \[eqn:fimb\] and \[eqn:fi\]) and we have to calculate the standard SPH density from $$\rho_i = \sum_{j=1}^{N}{ m_j W_{ij} \left( h_i \right)}$$ However, we do still use a symmetric formulation to evaluate the equation of motion such that both momentum and energy are conserved (e.g. @Monaghan1992, @Wadsley2004, @Springel2010, @Price2012).
Since the usually used B-spline kernels (see e.g. @Wadsley2004 or @Monaghan1992) can trigger an instability that causes particles to clump when they are getting close we implemented the Wendland kernels proposed by @Dehnen2012 that are both stable and allow a larger effective number of particles thus decreasing the noise for a given resolution. Another cause for clumping, independent of the chosen kernel function, is a negative pressure between particles. This is problematic as the Tillotson EOS calculates a negative pressure in the low density, cold region (Appendix \[appendix:eos\]) to mimic a tensional force between particles that are separated more than their equilibrium distance [@Melosh2007]. Since a fluid at these low densities (and temperatures) is expected to fragment and, for example, form droplets rather than behaving like a continuum we follow Melosh [@Melosh1989] and set the pressure to zero if it would become negative.
![A density profile of the particle representation of an $M=0.997 M_{\oplus}$ protoplanet using $N=10^{5}$ particles. The solid red line shows $\rho(r)$ for the calculated equilibrium. The dots are the smoothed density for the classic SPH (light blue) and our new density estimator (light green). While the results agree for both methods in the inner part of the sphere, the particles in the last shell match the imposed surface density much better for our improved estimator. At this material/vacuum interface only a fraction of the kernel is sampled with SPH particles so we correct the standard SPH density accordingly. Besides avoiding an artificial “atmosphere” of low density material this also provides increased stability to the equilibrium models, so no relaxation is needed prior to use in a simulation. We also observe that our proposed modification tends to slightly overestimate the density compared to the desired values because imbalance of the particle distribution associated with the density gradient triggers an over–correction.[]{data-label="fig:densityprofile"}](plots/target100k-14000-gamma37-density.png)
The resulting particle representations using the modified density estimator are in excellent agreement with the model (Figure \[fig:densityprofile\]) and the particles in the outer most shell remain in the condensed state. The smoothed density profile exhibits a general trend to over estimate the density compared to classic SPH, which is most pronounced in the outer most shell. This occurs because the proposed estimator artificially triggers an imbalance due to a density gradient on the kernel scale. For less steep density gradients, e.g., for lower mass models, the corrected density lies even closer to the desired values at the surface. In order to properly represent very massive planets, like super–earths, one would have to modify the procedure that relates $V_{\rm fac}$ to $f_{{\rm imb},i}$ to account for a density gradient. At this point we leave further improvements of the estimator to later work.
Entropy conserving Smoothed Particles Hydrodynamics {#section:isph}
===================================================
The results with the modified density were improved but some simulations would crash, e.g., due to unphysical entropy loss, even if the original ICs were setup in a physically consistent way as described above. For example, ejected material that re–impacts with the target would all of a sudden see a high density region when its kernel starts to overlap with the target, causing a sudden and unphysical increase in the particle’s density while the internal energy stays constant (Figure \[fig:crash\]). This would not happen, if the method would exactly conserve the particle’s entropy since its internal energy would increase accordingly.
Clearly evolving the entropy rather than the internal energy as a fundamental thermodynamical variable would be desirable. A particle will follow an isentrope for a purely adiabatic flow thus making it impossible to ever fall below the cold curve (if we start from physical initial conditions) because the only possibility is to move to a higher isentrope as a result of shock entropy generation.
In its standard version, GASOLINE evolves the internal energy as [@Wadsley2004] $$\frac{du_i}{dt} = \frac{du_{ad}}{dt} + \frac{du_{\Pi}}{dt}$$ where $$\frac{du_{ad}}{dt} = \frac{P_i}{\rho_i^2} \sum_{j=1}^{n}{m_j {\boldsymbol{v}}_{ij} \cdot \nabla_i W_{ij}}$$ is the work from adiabatic compression or expansion (PdV work) and $$\frac{du_{\Pi}}{dt} = \frac{1}{2} \sum_{j=1}^{n}{m_j \Pi_{ij} {\boldsymbol{v}}_{ij} \nabla_i W_{ij}}
\label{eqn:duavdt}$$ is the contribution due to irreversible ($ds > 0$) shock heating. The artificial viscosity term $\Pi_{ij}$ (AV) is non zero only for converging flows and captures shocks (e.g. @Monaghan1992, @Wadsley2004, @Springel2010, @Price2012). Following previous work (e.g. @Canup2004) we used $\alpha = 1.5$ and $\beta=2\alpha$ for the viscosity parameters.
In the case of an ideal gas it is possible to define an entropy function $A(s) = (\gamma - 1) u \rho^{1-\gamma}$ [@Springel2002] so that one can alternatively evolve the entropy rather than the internal energy given by $$\frac{dA_i}{dt} = -\frac{\gamma-1}{\rho^{\gamma}} \mathcal{L} \left( \rho_i, u_i \right) + \frac{\gamma-1}{\rho_i^{\gamma-1}} \frac{du_{\Pi}}{dt},
\label{eqn:Adot}$$ where $\mathcal{L}$ is a source term. This has the advantage that in the absence of dissipative processes the entropy is exactly conserved ($\mathcal{L}=0$). This is not the case if the internal energy is evolved [@Springel2002]. The main difficulty of this entropy formulation is, however, that it requires the EOS to be both analytic and thermodynamically complete in order to define an entropy function. This is often not the case for a more complex EOS like the Tillotson EOS that we use. It is possible however to calculate the isentropes for an arbitrary EOS from the first law of thermodynamics $$du = \frac{P}{\rho^2} d\rho + T ds$$ where $ds = 0$ for constant entropy. So one can solve $$\frac{du}{d \rho} = \frac{P}{\rho^2}
\label{dudrho}$$ numerically and store the results for different initial values (taken at $u \left( \rho = \rho_0 \right)$) in a lookup table. Since this does not require an analytic expression of the pressure but just a numerical value it can be done for any EOS, even if it is tabular. In the absence of entropy changing processes, like shocks, the evolution of the internal energy for a given change in density can then be simply done by interpolating between these values.
To setup the lookup table we integrated Equation \[dudrho\] for different starting values $v_i = u_i \left( \rho = \rho_0 \right)$ in the direction of both increasing and decreasing densities. The variable $v_i$ thus can be used to label an isentrope and $v_0 = 0$ corresponds to the cold curve. For a given isentrope a particle’s internal energy thus depends only on it’s density $u = u \left( \rho, v \right)$. So in order to evolve a particle’s energy from $\left( \rho_1, u_1 \right)$ to $\left( \rho_2, u_2 \right)$ we first have to determine on which isentrope it lies. For this we calculate the root of $$F\left( v \right) = u_1 - u\left( \rho_1, v \right)$$ which is bracketed by $v_0$ and $v_{max}$ using the Brent algorithm. To find $u \left( \rho, v \right)$ we do a two dimensional bi-cubic interpolation in the look-up table as described in Figure \[fig:splint\]. Since the derivatives of u with respect to $\rho$ are known analytically we only have to fit cubic splines along the direction of v (at constant density). Once a particle’s isentrope is found we do the final interpolation of $u_2 = u_2 \left( \rho_2, v \right)$ for $\rho_2$.
To do realistic (impact) simulations one has to account for the change of entropy due to shocks thus we still need to account for the $d u_{\pi}/dt$ term. In SPH it is possible to obtain the internal energy contribution due to shocks from Equation \[eqn:duavdt\]. This allows one to split the (temporal) evolution of the internal energy into an adiabatic and a shock heating part, enforcing the conservation of entropy for an adiabatic flow and including the contribution of shock heating to the internal energy using the standard SPH formalism.
One time step from $t_i$ to $t_{i+1}$ in the leapfrog algorithm used in GASOLINE[^4] is given by, where ${\boldsymbol{a}}_{i} = {\boldsymbol{a}} \left ( {\boldsymbol{x}}_{i} \right)$ and ${\boldsymbol{a}}_{i+1} = {\boldsymbol{a}} \left ( {\boldsymbol{x}}_{i+1} \right)$ are the accelerations due to the forces acting between the particles.
Implementing such an integrator for a fluid is slightly more complicated because the forces not only depend on the position but also the velocity and internal energy of the particles [@Quinn1997]. These quantities are known at the start but not at the end of each time step so in order to calculate the fluid forces one has to use approximate predicted values to update the velocity from $v_{i+1/2}$ to $v_{i+1}$.
We now describe a single ISPH time–step. 1) The gravitational and SPH forces on all particles ${\boldsymbol{a}} \left( t_i \right)$ and contributions from shock heating (Equation \[eqn:duavdt\]) are calculated at the beginning of the time interval. 2) Then the velocities and internal energies are updated for half a time step (“kick”). 3) Now all particle positions are “drifted” from $x \left( t_i \right)$ to $x \left( t_i + \Delta t \right)$. Before calculating the updated density $\rho \left( t_i + \Delta t \right)$ at the end of the interval its old value $\rho \left( t_i \right)$ is stored. 4) Using this value and the new density, calculate the change in internal energy due to adiabatic expansion or compression using the lookup table as described above. 5) In order to do the closing “kick” the *predicted velocity* and *predicted internal energy* at the end of the interval are determined. 6) The forces ${\boldsymbol{a}} \left( t_i + \Delta t\right)$ and shocking heating are calculated. 7) Finally both the velocity and internal energy are updated using the newly calculated force and shock heating term (closing “kick”). This way of evolving the energy equation matches very well with our time integration scheme and can be integrated rather easily into the code. The actual algorithm is more complex than this because the particles have their own unique time steps based on a block time stepping scheme. While particles are all drifted simultaneously on the smallest time step, such that the positions, and hence densities, are always time synchronized, those on larger time steps will not have their thermal energy advanced (see step 4 above) until they reach the end of their time step.
![[]{data-label="fig:crash"}](plots/c-b0-v10-100k-Particle90069.png)
Tests {#section:tests}
=====
As a first test we simulated the equilibrium models with our code to check if they truly stayed in equilibrium or would expand or collapse after a certain time. For classic SPH the models would remain stable but oscillate, as there was some intrinsic noise in our model and the density at the surface of the body was significantly underestimated as discussed in Section \[section:newdens\]. After several oscillations the particles would settle into an equilibrium configuration because the artificial viscosity in SPH would dampen their motion since it does not distinguish between a converging flow and a shock. As in previous studies (e.g. @Canup2001, @Genda2012) we used a velocity damper that reduced the particle’s velocity each step to speed things up but this has to be done carefully to not over–damp the system as the models will become unstable again as soon as the damping term is removed. For simulations that used our proposed density correction the models remain stable right from the start and do not change at all. Using ISPH does, as expected, not affect the result in any way.
We then checked the code’s ability to properly model an adiabatic flow which is not only a key requirement for any hydrodynamics code but also most important to test how accurately ISPH conserves entropy. For this we cut a sphere from a cubic uniform density and internal energy particle grid and let it evolve only due to pressure forces (neglecting gravity). We ran two simulations, one with classic SPH and one with ISPH both using our proposed modified density estimator. The differences are striking (Figure \[fig:adiabatic\]). While classic SPH suffers from huge scatter in the internal energy the particles in the ISPH simulation closely follow their initial isentrope over more than 17 hours in simulation time showing excellent entropy conservation.
{width="\textwidth"}
Next we verified that we can recover the Rankine-Hugoniot jump conditions (e.g. @Melosh1989) relating the fluid quantities in front and behind the passing shock to test the code’s ability to correctly capture shocks. For this we let two uniform granite slabs ($15 \times 15 \times 8 R_{\oplus}$) that are initially in contact collide with opposite velocities. The initial discontinuity in the velocity field then sends a shock wave through the material causing the fluid variables to jump to the post shock state at the shock front. These values can then be obtained from the simulations and compared to the theoretical results using the Rankine-Hugoniot equations. For both classic SPH and ISPH (with and without the modified density estimator) we obtain the correct values and thus conclude that shocks are adequately resolved in our code.
{width="500pt"}
As a last test we simulated collisions between granite spheres for different impact parameters and compared our results to prior work done in the field. Since there are not many publications on GI between undifferentiated bodies and the early work [@Benz1986] suffers from (for today’s standards) low resolution and short simulation time, a direct comparison of the results was complicated. However we expect the basic features of such a collision to be similar for differentiated and undifferentiated bodies. First we run a simulation of an oblique impact ($b = 0.71$, $v_{imp} = 10km/s$, $M_t = 0.997 M_{\oplus}$ and $M_i = 0.099 M_{\oplus}$) like the one that might have created our moon [@Canup2001; @Canup2004] using a total number of $10^5$ particles for both classic SPH without any of the modifications presented in this paper and ISPH (including the density correction discussed in Section \[section:newdens\]). Prior to the impact both bodies are placed in the xy-plane at a given separation so that they collide with the desired impact angle and velocity. As in previous simulations the impactor is deformed due to tidal forces while approaching the target. The impact sends a strong shock through both bodies destroying most of the impactor and the ejected material forms an arm like structure for both methods which is consistent with prior results [@Canup2001]. Since our planetary models require no relaxation prior to the impact simulation we run the same simulation for an increasing number of particles for $N=10^5$,$8 \cdot 10^5$ up to $6.4 \cdot 10^6$ to check the resolution dependence of the results. While the general morphology (massive target, arm like structure and impactor remnant) is the same the density and internal energy of the material and thus the physical state of the material clearly depends on the resolution. This is most pronounced for the arm like structure as shown in Figure \[fig:grazingimpact\] and Figure \[fig:ejecta\]. This makes the use of a highly sophisticated EOS in low density, poorly resolved, regions questionable as the values of the fluid variables there are highly dependant on the resolution and the treatment of the hydrodynamics itself.
The second simulation we performed was a more head–on impact ($b=0.5$) of an $0.2 M_{\oplus}$ impactor onto a $0.9 M_{\oplus}$ target at $10.62$ km/s with $10^5$ SPH particles using classic SPH and ISPH. We let the simulation evolve for one day and again find that both methods initially produce very similar results. They do however differ in how the ejecta evolves on a longer time scale, as the impactor remnant leaves the system in the classic SPH simulation but remains bound when ISPH is used and re–impacts with the target after 4.4 days. In both cases the impact produces a disk of hot, partially vaporized material extending to about $3 R_{\oplus}$ from the target. Already after one day it assumes a flattened, rotationally supported profile which is more pronounced in the case of ISPH. We also observe a steep density gradient at the interface between the planet and the disk. To analyse disk properties we define it as the bound orbiting material that has a density lower that the reference density $\rho_0$ at the surface of the planet. Despite visible differences, for ISPH we obtain a slightly more massive planet and less massive disk than for classic SPH, the two methods produce very similar disk profiles up to 24 hours in simulation time. To study the long term evolution of the disk and the fate of the impactor remnant in the ISPH collision we continued both simulations for additional seven days beyond impact. As in previous simulations the Balsara switch [@Balsara1995] was used to reduce unwanted dissipation in the disk. After 2.5 days the differences in the disk’s structure due to the method are significant. The disk obtained from ISPH is rotating significantly faster (Figure \[fig:disk\_plot\]) and is less extended than the one resulting in the classic SPH simulation. We also observe that for ISPH the disk remains flatter and is slightly hotter (Figure \[fig:disk\_sph\]). These differences remain until the simulations ends eight days after the impact. At this point we leave a detailed investigation to later work.
![Results from a $b=0.71$, $v=10 km/s$ impact of a Mars-sized ($M=0.099 M_{\oplus}$) impactor onto an Earth-mass target ($M=0.99 M_{\oplus}$) for different resolutions of $10^5$ (green), $8 \cdot 10^5$ (blue) and $6.4 \cdot 10^6$ (red) particles. The top plot shows the internal energy vs density of the particles in the “arm” like structure that forms between the target and the surviving part of the impactor. For increasing resolution it becomes denser and more particles have an internal energy larger than $u_{IV}$ meaning that the material is partially evaporated. So the physical composition (and thus the pressure) in this low density region clearly depends on the resolution. The bottom plot shows the same for the impactor remnant that forms the end of the “arm” and consist mostly of impactor material that was not dispersed during the impact. The light orange region marks where we did the pressure cut–off because it would be negative for the Tillotson EOS. In this region the material is supposed to form small droplets and would ideally be treated by a multi–phase fluid prescription within the hydrodynamics.[]{data-label="fig:ejecta"}](plots/ejecta.png)
![The circum–planetary disk resulting from a $b=0.5$, $v=10.61 km/s$ impact between a $M=0.9 M_{\oplus}$ target and an $M=0.2 M_{\oplus}$ impactor about three days after the collision. The colour shows the materials temperature from 4500K (blue) to 40’000K (red). The top frame (a) shows the result for classic SPH without any of the modifications presented in this paper and the bottom frame (b) shows the outcome when ISPH is used. For classic SPH the disk cooler, slightly more expanded and less flat. []{data-label="fig:disk_sph"}](plots/disk.png){width="46.00000%"}
![The specific angular momentum as a function of radius for the disk resulting from simulation 2 ($b=0.5$, $v=10.65 km/s$, $M_t=0.9 M_{\oplus}$ and $M_i=0.2 M_{\oplus}$) about three days after the impact as shown in Figure \[fig:disk\_sph\]. The green line is result for classic SPH with none of the modifications discussed in this paper and the blue line is obtained using ISPH. In both cases we used the Balsara switch [@Balsara1995] to avoid artificial dissipation due to shear flows. While the methods agree within the first 24h after the collision the evolution is quite different at later times where the disk obtained using ISPH clearly has a larger angular momentum.[]{data-label="fig:disk_plot"}](plots/run2-110k-disk-BS-final.pdf)
Summary and conclusion {#section:conclusions}
======================
In this paper we present three solutions to long standing problems of GI simulations with the SPH method. First we introduce a method to generate low noise particle representation of planets. Comparing the particle’s (smoothed) density to the numerical solution of the equilibrium model shows excellent agreement and a low spread around the desired value. We then propose an improved treatment of a material/vacuum interface as it occurs at a planet’s surface. There the classic SPH density estimator fails, calculating a significantly lower value. As a result the models obtained with `ballic` deviate from equilibrium and start to oscillate. This is even more problematic when the surface temperature is high as in this case the material is partially vaporized. As a solution we propose a modification of the standard SPH density estimator that accounts for only the part of the kernel volume that is sampled with particles, which gives a much better result and introduces less noise compared to previous work. Applying this density correction to our equilibrium models results in stable initial conditions that do not have to be relaxed prior to the simulation, significantly reducing the total simulation time allowing us to reach so far unprecedented number of particles. Finally we present an entropy conserving formulation of SPH that does not rely on an analytic or thermodynamically complete EOS. The evolution of the internal energy is split into a contribution from shock heating using the standard SPH artificial viscosity and an exact adiabatic evolution that is calculated by interpolating between isentropes. For standard SPH the particles significantly deviate from their original isentrope. Our new method conserves entropy and the particles never fall below their adiabatic evolution curves. At the same time, the Rankine-Hugoniot jump conditions for the fluid variables at the discontinuity are successfully recovered, so our scheme properly captures shocks as expected.
A comparison between the new method and classic SPH when applied to GI simulations shows that the methods mostly agree on the general outcome within the first 24 hours after the impact. In one simulation (Simulation 2 in Section \[section:tests\]) the impact produces a flattened, rotationally supported disk of hot, low density material around the target. While the total bound mass is very similar, ISPH produces a slightly more massive central body and less massive disk compared to classic SPH. The other disk properties are, however, almost identical. To study the long term evolution of the ejecta and the disk we continued the simulation until eight days after the impact and observe that the differences between the methods become more pronounced. We also re–ran one simulation (Simulation 1 in Section \[section:tests\]) for $10^5$, $8 \cdot 10^5$ and $6.4 \cdot 10^6$ particles to check if the results within the first few hours after the impact agree and found a clear resolution dependence of the physical state of the material of the ejecta (Figure \[fig:ejecta\]). This suggests that the use of a more sophisticated EOS should be considered with care as these low density regions are poorly resolved and the results are dominated by the numerical resolution and not physics.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Prasenjit Saha for helping with the implementation of an early version of the ISPH lookup algorithm. CR also wishes to express his gratitude to Hidekazu Tanaka and Hidenori Genda for inviting him to the University of Hokkaido and the Tokyo Institute of Technology in Japan in 2014, and we want to thank them for the valuable discussions that we had during their visits in Zurich. All the simulations were performed on the zBox4 CPU cluster at the Institute for Computational Science (University of Zurich). This work was supported by the SNF Grant in “Computational Astrophysics” (200020\_162930/1).
The equation of state {#appendix:eos}
=====================
An equation of state (EOS) is usually defined as a relationship between the pressure, density and internal energy/temperature of a substance. For GI simulations the proper choice of the EOS plays a key role as it not only has to be able to correctly model the behaviour of the material over a wide range of pressures, densities and internal energies but also must capture the strong shocks occurring in both bodies due to the impact. For our work we used the Tillotson EOS [@Tillotson1962] that has been used in many prior simulations (e.g., @Benz1986, @Canup2001, @Marinova2011, @Genda2012, @Jutzi2013) and was especially developed to model hyper-velocity impacts. Depending on the density and internal energy of the material it can be divided into four different regions where the pressure is given by a different analytic expression (Figure \[fig:crash\]). Each material is defined by 10 material constants (Table \[tab:materials\]). Despite its simplicity (compared to purely tabular EOS, maybe with analytic fits in different parts) the results are in good agreement with measurements [@Benz1986; @Brundage2013] and it’s ability to properly reproduce the materials Hugoniot curve, thus accurately modelling shocks, is excellent [@Brundage2013].
\[tab:materials\]
For *condensed states* ($\rho \geq \rho_0$) in region I the pressure is given by $$P_{I,II} = \left( a + \frac{b}{\frac{u}{u_0 \eta^2} + 1} \right) u \rho + A \mu + B \mu^2
\label{condensedandcoldstates}$$ where $\eta = \rho / \rho_0$ is the compression and $\mu = \eta - 1$ is the strain. For very large compression (partial) ionization of the material is described by the Thomas-Fermi model[@Brundage2013].
The expanded states, where the density is smaller than the reference density $\rho_0$, are again divided into three regions. In region II (*expanded cold states*) the material density is low but its internal energy is smaller than the energy at incipient vaporization ($u < u_s$) so it is still a liquid or a solid and the pressure is the same as for the condensed states. For expanded but very cold material the pressure can become negative which does not happen in region I. This corresponds to a tension in the sold material that prevents it from expanding like a gas as low densities. A fluid on the other hand will fragment into small droplets at these low densities and can not be described as a continuum anymore [@Melosh1989]. For this reason we set the pressure to zero if it would become negative and let the material evolve due to gravity only (similar to @Hosono2016). This also prevents the sound speed in the material from becoming unphysical (imaginary, see Appendix \[appendix:soundspeed\] for the calculation of the sound speed) and avoids a numerical instability in SPH that causes unphysical particle clumping and occurs for negative pressures [@Dehnen2012]. It also ensures that the sound speed (used to model artificial viscosity in shocks and for setting the time step) stays non-negative in all regions of the EOS. In the zero pressure region, where the fluid description breaks down, we artificially set the sound speed to a small minimal value, in which case GASOLINE uses a fixed maximum time step (the so called [*base time step*]{}) to evolve the given particle.
If the internal energy is larger than the energy needed for complete vaporization ($u > u_s'$) the material is in region IV (*expanded hot state*) where the pressure is given by $$\begin{gathered}
P_{IV} = a u \rho + \left(\frac{b u \rho}{\frac{u}{u_0 \eta^2} + 1} + A \mu \exp{\left[-\alpha \left( \eta^{-1} - 1 \right)\right]} \right) \\
\exp{\left[-\beta \left( \eta^{-1} - 1 \right)^2\right]}\end{gathered}$$ and the substance is completely vaporized. For very small densities the second term cancels and we asymptotically approximate an ideal gas[^5] with $\gamma = a + 1$. It is important to note that in order to vaporize, the material does not only have to be very hot but also have a low density, so the greatest amount of vapour is not generated during the impact but right after when the material at the impact site expands as mentioned in @Benz1986.
In between those two regions ($u_s < u < u_s'$) are the intermediate states (region III), where the pressure is a linear interpolation $$P_{III} = \frac{P_e \left( u - u_s \right) + P_c \left( u_s' - u \right)}{u_s' - u_s}
\label{eqn:intermediate}$$ between a low density solid/liquid and a vapour phase. This simple mixing rule prevents pressure discontinuities in the intermediate region but of course does not model mixed phases (e.g. liquid/gas) or phase changes which is a key weakness of the Tillotson EOS [@Benz1989; @Canup2004; @Brundage2013]. Especially for Moon-forming impacts, predicting the right amount of vapour generated by the impact is crucial since pressure gradients might play an important role in placing material into orbit [@Canup2004Review]. Another short coming in the context of the present work is that the Tillotson EOS is not thermodynamically complete, meaning that a second equation, e.g., relating internal energy to temperature is missing. For this reason one can not define an entropy function similar to the case on an ideal gas thus being restricted to the evolution of the internal energy in a simulation. As we have seen this does not limit the use of an entropy conserving scheme within SPH.
Calculating the sound speed for the Tillotson equation of state {#appendix:soundspeed}
===============================================================
For numerical hydrodynamics the sound speed is very important as it determines the largest possible time step for the simulation. For SPH the sound speed also enters in the AV (Equation \[eqn:duavdt\]) that determines how much kinetic energy is converted to thermal energy at a shock front (e.g. @Wadsley2004). In order to derive the sound speed for an arbitrary EOS one has to either linearise Euler’s equations and solve the corresponding eigenvalue problem or use the general expression for the sound speed in an ideal fluid $$c^2 = \left. \frac{\partial P}{\partial \rho} \right|_{s}
\label{eqn:soundspeed}$$ as we did for the Tillotson EOS. In regions I and II this gives $$\begin{aligned}
c_{I,II}^2 & = & \left( \Gamma_{I,II} + 1 \right) \frac{P_{I,II}}{\rho} + \frac{A + B \left( \eta^2-1 \right)}{\rho}\nonumber\\
& & + \frac{b}{\omega_0^2} \left(\omega_0 - 1 \right) \left( 2u - \frac{P_{I,II}}{\rho} \right)\\
\Gamma_c & = & A + \frac{b}{\omega_0}\\
\omega_0 & = & \frac{u}{u_0 \eta^2} + 1\end{aligned}$$ while on region IV we obtained $$\begin{aligned}
c_{IV}^2 & = & \left( \Gamma_{IV} + 1 \right) \frac{P_{IV}}{\rho} + \frac{A}{\rho_0} e^{-\left( \alpha z + \beta z^2 \right)}\nonumber\\
& & \left( 1+\frac{\mu}{\eta^2} \left(\alpha + 2 \beta z - \eta \right)\right)+\frac{b \rho u }{\omega_0^2 \eta^2}\nonumber\\
& & e^{- \beta z^2} \left( \frac{2 \beta z}{\rho_0} \omega_0 + \frac{1}{u_0 \rho}\left( \frac{P_{IV}}{\rho}-2u \right) \right)\\
\Gamma_{IV} & = & a + \frac{b}{\omega_0} e^{- \beta z^2}\\
z & = & \frac{1}{\eta} - 1\end{aligned}$$ from (\[eqn:soundspeed\]). In the intermediate states (region III) the sound speed is $$c_{III}^2 = \frac{c_{I,II}^2 \left( u - u_s \right) + c_{IV}^2 \left( u_s' - u \right)}{u_s' - u_s}$$ because the pressure there is a linear interpolation between the cold and hot expanded states (Equation \[eqn:intermediate\]). Unlike in the case of an ideal gas, this expression can be negative in the cold expanded states region of the Tillotson EOS which results in an imaginary sound speed. Since this only occurs when the pressure becomes negative, it does not affect our simulations since we enforce non–negative pressures as discussed in Appendix \[appendix:eos\]. In order to have a well defined time step we simply impose a minimal value for $c_s^2$ in this case.
[^1]: christian.reinhardt@ics.uzh.ch
[^2]: The TillotsonEOS library is freely available under <https://github.com/chreinhardt/tillotson.git>. We also provide our code to create low noise particle representations of planets, `ballic`, at <https://github.com/chreinhardt/ballic.git>.
[^3]: It should be noted that unlike in the case of an ideal gas, isothermal does not mean that the internal energy is constant as there is an additional contribution from the cold curve to the total energy in case of a condensed material EOS (see Appendix \[appendix:eos\] for details). Issues with particles falling below the cold curve due to an unphysical, uniform internal energy profile would also readily provide an explanation for the problems with ANEOS reported in [@Canup2004]
[^4]: Here the Kick-Drift-Kick (KDK) version of the algorithm is described. This means that both the positions and forces are evaluated at the beginning and the end while the velocities are updated in the middle of the time step. This is convenient as the forces are evaluated at positions that are second order accurate.
[^5]: For an ideal gas the pressure is $P = \left( \gamma -1 \right) u \rho$ where $\gamma = C_p/C_v$ is the adiabatic index and $C_p$ and $C_v$ are the heat capacity at constant pressure or volume. For granite, the material used for most simulations in this paper, $a=0.5$ and we approximate an ideal gas with $\gamma=1.5$.
|
---
abstract: 'The cross section of the $\ee$ process is measured with unprecedented precision using data collected with the BESIII detector at $\sqs=\ENERGYAT$, $\ENERGYBT$, $\ENERGYCT$ and $\ENERGYDT$ $\mev$. The non-zero cross section near the $\LLB$ production threshold is cleared. At center-of-mass energies $\sqs=\ENERGYAT$ and $\ENERGYDT$ $\mev$, the higher statistics data enable us to measure the $\Lam$ polar angle distributions. From these, the $\Lam$ electric over magnetic form factor ratios ($\ratios$) are measured for the first time. They are found to be $\RatioA$ and $\RatioD$ respectively, where the first uncertainties are statistical and the second are systematic.'
title: 'Precision measurement of the $\ee$ cross section near threshold'
---
The electromagnetic structure of hadrons, parameterized in terms of electromagnetic form factors (EMFFs), provides a key to understand quantum chromodynamics effects in bound states. The nucleon has been studied rigorously for more than 60 years, but new techniques and the availability of data with larger statistics from modern facilities have given rise to a renewed interest in the field, *e.g.* the proton radius puzzle [@pradius]. Recently, the access to strange and charm hyperon structure by time-like EMFFs provides an additional dimension. Assuming that one-photon exchange dominates the production of spin-1/2 baryons $B$, the cross section of the process $\EE$ $\ra$ $B\bar{B}$ can be parameterized in terms of EMFFs, *i.e.* $G_{E}$ and $G_{M}$, in the following way [@OPEXModel]: $$\sigma_{B\bar{B}}(s)=\frac{4\pi\alpha^{2} C \beta}{3s}|G_{M}(s)|^{2}\big[1+\frac{2m_{B}^{2}c^{4}}{s}\big|\frac{G_{E}(s)}{G_{M}(s)}\big|^{2}\big].
\label{OnePhotonPre}$$ Here, $\alpha$ is the fine-structure constant, $\beta$=$\sqrt{1-4m_{B}^{2}c^{4}/s}$ the velocity of the baryon, $s$ the square of the center-of-mass (CM) energy, and $m_{B}$ is the mass of the baryon. The Coulomb factor $C$ parameterizes the electromagnetic interaction between the outgoing baryon and antibaryon. For neutral baryons, the Coulomb factor is unity, while for point-like charged fermions it reads $C=\varepsilon R$ [@C1; @C2], where $\varepsilon = {\pi\alpha}/{\beta}$ is an enhancement factor resulting in a nonzero cross section at threshold and $R={\sqrt{1-\beta^{2}}}/(1-e^{-\pi\alpha{\sqrt{1-\beta^{2}}}/\beta})$ is the Sommerfeld resummation factor [@C1]. The ratio of EMFFs associated with the polar angle distribution of the baryon can also parameterize the differential production cross section of the corresponding baryon [@OPEXModel].
In the $\EE\ra p\bar{p}$ process, the BaBar collaboration observed a rapid rise of the cross section near threshold, followed by a plateau around 200 MeV above threshold [@babar2]. The BESIII collaboration also observed the cross section enhancement [@ppbarbes]. The non-vanishing cross section near threshold as well as the wide-range plateau have led to various theoretical interpretations, including i) final-state interactions [@TheoFSInter], ii) bound states or meson-like resonances [@TheoRes] and iii) an attractive Coulomb interaction [@Resumfct]. Recently, the BESIII collaboration has observed the non-zero cross section near threshold in the process $\EE\ra\Lambda\bar{\Lambda}$ [@BAM0139]. Naturally, it is also interesting to explore the production behavior of $\lam$, the lightest baryon containing the charm quark. Previously, the Belle collaboration measured the cross section of $\ee$ using the initial-state radiation (ISR) technique [@Belle], but the results suffer from significant uncertainties in CM energy and cross section. Therefore, near $\lam\lambar$ threshold, precise measurements of the production cross section and EMFF ratios are highly desirable.
In this work, the cross section of the reaction $\ee$ is measured at four CM energies: $\sqs=\ENERGYAT$, $\ENERGYBT$, $\ENERGYCT$, and $\ENERGYDT$ $\mev$. At each CM energy, ten Cabibbo-favored hadronic decay modes, $\lam\ra\modeI$, $\modeII$, $\modeIII$, $\modeIV$, $\modeV$, $\modeVI$, $\modeVII$, $\modeVIII$, $\modeIX$, and $\modeX$, as well as the ten corresponding charge-conjugate modes are independently used to reconstruct $\lam$ or $\lambar$. Each mode will produce one measurement of the cross section and the total cross section is obtained from a weighted average over the 20 individual measurements. In addition, the higher statistics data samples at $\sqs=\ENERGYAT$ and $\ENERGYDT$ $\mev$ enable the study of the polar angle distribution of $\Lam$ in the CM system. From these distributions, the ratios between the electric and the magnetic form factors, *i.e.* $\ratios$, are extracted for the first time.
The data samples are collected with the BESIII detector [@BESIII] at BEPCII. The detector has a geometrical acceptance of 93% of the 4$\pi$ solid angle. It contains a small-celled, helium-based main drift chamber (MDC), a time-of-flight system (TOF) based on plastic scintillators, an electromagnetic calorimeter (EMC) made of CsI(Tl) crystals, a muon system (MUC) made of Resistive Plate Chambers, and a superconducting solenoid magnet.
Monte Carlo (MC) simulations based on [geant4]{} [@geant4] are performed to determine detection efficiencies, optimize selection criteria, extract signal shapes and study backgrounds. The $\EE$ collisions are simulated by the [kkmc]{} generator [@KKMC], which takes the beam energy spread and the ISR correction into account. The distribution of the $\Lam$ polar angle is considered in the generator by parameterizing it with the function $f(\theta)~\propto~1~+~\alpha_{\Lam}~\cos^2\theta$. After an iterative procedure, the values of $\alpha_{\Lam}$ at $\sqs$ = $\ENERGYAT$ and $\ENERGYDT$ $\mev$ are obtained from real data (see Table \[AngDisParas\]) and at the remaining CM energies by a linear interpolation.
Using the branching fractions (BR) measured in Ref. [@BAM0162], all tagged $\Lam$ decays are simulated by weighting phase-space events according to the decay behavior observed in real data. The subsequent decays listed by the particle data group (PDG) [@PDG] are modeled with [evtgen]{} [@EVTGEN]. The inclusive MC samples include $\lam\lambar$ pair production, $\ell^{+}\ell^{-}$ ($\ell=\textit{e,~$\mu$,~$\tau$})$ events, open charm processes [@DDsMC], ISR-produced low-mass $\psi$ states and the continuum process $\EE \ra q\bar{q}~(q=\textit{u,~d,~s})$.
Charged tracks as well as the intermediate states $\pi^{0}$, $\kst$, $\Lambda$, $\Sigma^{0}$ and $\Sigma^{+}$ are selected and reconstructed with the same method described in Ref. [@BAM0162].
In the final states of decay modes $\modeV$ and $\modeVII$, potential background from $\Lambda \rightarrow p\pi^{-}$ is eliminated by rejecting events with $M_{p\pi^{-}}$ lying in the mass window (1100, 1125) $\mevcc$, where $M_{p\pi^{-}}$ is the invariant mass of $p\pi^{-}$ combinations in the final state. For the decay mode $\modeX$, the corresponding exclusion window is (1110, 1120) $\mevcc$ due to the smaller observed width of the $M_{p\pi^{-}}$ peak in data. Similarly, background from the intermediate state $\Sigma^{+}$ is removed from the $\modeV$ sample by rejecting events with $M_{p\pi^{0}}$ in the mass window (1170, 1200) $\mevcc$. In modes $\modeVIII$ and $\modeX$, events with $M_{\pi^{+}\pi^{-}}$ within (490, 510) $\mevcc$ are rejected to suppress $\kst$ background.
[fit\_mbc\_at\_4575\_mode1.eps]{}
[fit\_mbc\_at\_4580\_mode1.eps]{}
[fit\_mbc\_at\_4590\_mode1.eps]{}
[fit\_mbc\_at\_4600\_mode1.eps]{}
According to energy and momentum conservation, two discriminating variables, the *energy difference* $\dele$ and *beam-constrained mass* $\mbc$, are utilized to identify the $\Lam$ signals. The energy difference is defined as $\dele~\equiv~E~-~\ebm$, where $E$ is the energy of the $\Lam$ candidate and $\ebm$ the mean energy of the two colliding beams. In each tagged mode, the $\Lam$ candidates are formed by all possible combinations of the final state particles, and only the one with minimum $|\dele|$ is stored. In the following analysis, events are rejected if they fail the $\dele$ requirements specified in Ref. [@BAM0162]. The beam-constrained mass is defined as $\mbc c^{2}\equiv\sqrt{\ebm^{2}-p^{2}c^{2}}$, where $p$ is the momentum of the $\Lam$ candidate. Both $\dele$ and $\mbc$ are calculated in the initial $e^{+}e^{-}$ CM system. In Fig. \[4PointsFig\], the $\mbc$ distributions for $\lamdecay$ at the four CM energies are shown. Clear peaks at the nominal $\lam$ mass are observed. Studies of the inclusive MC samples show that the cross feeds among the ten tagged modes are less than 1.5% and the background shape can be described by the ARGUS function [@Argusf].
Performing an unbinned maximum likelihood fit to each $\mbc$ distribution gives the corresponding event yields, as partly illustrated in Fig. \[4PointsFig\]. The signal shape of the fit is obtained from convolving the $\mbc$ shape of MC simulations with a Gaussian function to compensate a possible resolution difference between data and MC simulations. The background is described by an ARGUS function with the high-end truncation fixed. At $\sqs=\ENERGYDT~\mev$, the parameters of the ARGUS and the Gaussian functions used in the convolution are obtained from the fit. At the remaining CM energies, all parameters obtained at the highest energy, except for the mean of the Gaussian, are used to fix parameters in the new fits. Yields are extracted from the signal region 2276 $\mev$ $<\mbc c^{2}$ $<$ $\ebm$ in each fit. The detection efficiency of each decay mode is evaluated by MC simulations of the $\ee$ process. Figure \[4PointsFig\] gives the efficiencies of mode $\modeI$ at the four CM energies.
The cross section of the $i$-th mode is determined using $$\sigma_{i} = \frac{N_{i}}{\varepsilon_{i}\cdot \mathcal{L}_{\textrm{int}}\cdot f_{\textrm{VP}}\cdot BR_{i}\cdot f_{\textrm{ISR}}},
\label{csformula}$$ where $N_{i}$ and $\varepsilon_{i}$ represent the yield and corresponding detection efficiency. The integrated luminosity $\mathcal{L}_{\textrm{int}}$ is taken from Ref. [@LumXYZ; @LumRscan]. The vacuum polarization (VP) correction factor $f_{\textrm{VP}}$ is calculated to be 1.055 at all four CM energies [@VPfactor]. The $BR_{i}$ represents the product of branching fractions of the $i$-th $\Lam$ decay mode and its subsequent decay(s). The $f_{\textrm{ISR}}$ is the ISR correction factor derived in Ref. [@ISRfactor] and implemented in [kkmc]{}. Since the calculation of $f_{\textrm{ISR}}$ requires the cross section line-shape as input, an iterative procedure has been performed.
------------- ----- ----- ----- ----- ----- ----- ----- ----- -- --
$\modeI$ 3.2 4.6 – – – 0.2 – 6.0
$\modeII$ 1.3 0.5 1.2 – – 0.6 0.2 5.6
$\modeIII$ 1.0 1.0 – 2.5 – 0.8 0.5 6.2
$\modeIV$ 3.0 7.6 – – 1.0 0.6 2.0 8.3
$\modeV$ 1.0 1.8 1.2 – 1.0 1.1 1.0 7.5
$\modeVI$ 1.0 1.0 – 2.5 1.0 0.6 0.6 6.0
$\modeVII$ 2.8 5.3 1.2 – – 1.0 0.5 9.3
$\modeVIII$ 3.0 3.0 – 2.5 – 0.9 0.8 7.9
$\modeIX$ 1.0 1.0 – 2.5 – 1.1 1.7 6.7
$\modeX$ 3.0 4.0 – – 1.0 0.8 0.8 7.4
------------- ----- ----- ----- ----- ----- ----- ----- ----- -- --
: Summary of the reconstruction related, mode-specific, relative systematic uncertainties of the cross section at $\sqs$ = $\ENERGYDT~\mev$, quoted in %.[]{data-label="SysErrs"}
The systematic uncertainties of the cross section can be classified into *reconstruction related* and *general* contributions. The reconstruction related contributions are mode-specific and mainly originate from tracking, PID, reconstruction of intermediate states and total BRs. The uncertainties of $\dele$ and $\mbc$ requirements are negligible after correcting for the difference in resolution between simulated and real data samples. The uncertainties from tracking and PID of charged particles are investigated using control samples from $e^{+}e^{-}~\rightarrow~\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, $K^{+}K^{-}\pi^{+}\pi^{-}$ and $p\bar{p}\pi^{+}\pi^{-}$ collected at $\sqs > 4.0~\gev$ [@BAM0216]. The uncertainties are obtained after weighting according to the momenta of the corresponding final states. Reconstruction uncertainties of $\kst$, $\Lambda$ and $\pi^{0}$ have been found to be 1.2%, 2.5% and 1.0% [@BAM0162]. Statistical uncertainties of detection efficiencies are considered as systematic uncertainties. The dependence of the reconstruction efficiency on the MC model for the ten decay modes also gives a small contribution to the systematic uncertainty [@BAM0162]. Uncertainties originating from the total BRs of the tagged modes are quoted from Refs. [@BAM0162; @PDG]. A summary of the reconstruction related systematic uncertainties are given in Table \[SysErrs\]. The total uncertainty at each energy has been calculated assuming that the values given at $\sqs$ = $\ENERGYDT$ $\mev$ are valid at all CM energies.
\[SysErrsII\]
-- -- ------- -------- -------- ------- ------- -----
shape energy Spread Total
1.2 18.0 3.0 18.6 0.5 1.0
0.6 – 0.2 0.9 0.5 0.7
1.7 – – 1.7 0.5 0.7
2.6 – – 2.6 0.5 1.0
-- -- ------- -------- -------- ------- ------- -----
: Summary of the general relative systematic uncertainties of the cross section originating from the factors $f_{\textrm{ISR}}$, $f_{\textrm{VP}}$ and $\mathcal{L}_{\textrm{int}}$, quoted in %.
The general contributions to the systematic uncertainty originate from uncertainties in $f_{\textrm{ISR}}$, $f_{\textrm{VP}}$ and $\mathcal{L}_{\textrm{int}}$ in Eq. (\[csformula\]) and are the same for all decay modes. The $f_{\textrm{ISR}}$ is obtained using the [kkmc]{} generator which requires a cross section line-shape as input. The line-shape is in turn obtained by an iterative fitting procedure of the cross section data using Eq. . In the fit, the $\ratios$ value at an arbitrary CM energy is assigned by linear interpolation between the two known values listed in Table \[AngDisParas\]. For simplicity, $|G_{M}|$ is assumed to be independent of the CM energy. To precisely describe the data, the $\alpha$ in the Sommerfeld resummation factor is replaced by $\alpha_{s}(=0.25)$. In the line-shape, the cross section at the CM energy region $(2m_{\Lam}c^{2},~\ENERGYAT)$ $\mev$ is obtained from extrapolating the fit; below threshold it vanishes, as shown by the blue solid curve in Fig. \[FitLinShp\]. Four sources of systematic uncertainty from the $f_{\textrm{ISR}}$ are considered: First, the uncertainty of the *calculation model* is studied using a different algorithm mentioned in Ref. [@ConExc]. Second, the uncertainty associated with the input *line-shape* is estimated using different fit functions. Third, the $f_{\textrm{ISR}}$ depends on the CM energy of the $\ee$ process. The uncertainty of the *CM energy* therefore contributes near the threshold. At the lowest energy point, the CM energy is measured to be $\sqs=4574.50\pm0.72$ $\mev$ [@XYZECMS]. Finally, the beam *energy spread*, which has been estimated as $1.55\pm0.18~\mev$, is important near threshold and contributes to the $f_{\textrm{ISR}}$ uncertainty. For the other, higher, energies, the effects from the CM energy uncertainty and the beam energy spread are less than 0.1% and can be neglected due to the flat line-shape of the cross section. The uncertainty of $f_{\textrm{VP}}$ is calculated to be 0.5% at all four CM energies [@VPfactor]. The uncertainty from the integrated luminosity has been found to be 0.7% at $\sqs$ = $\ENERGYBT$ and $\ENERGYCT$ $\mev$ and 1.0% at $\sqs$ = $\ENERGYAT$ and $\ENERGYDT~\mev$ [@LumXYZ; @LumRscan]. A summary of the general contributions to the systematic uncertainties is given in Table \[SysErrsII\].
The cross sections obtained in different decay modes are combined using the method mentioned in Ref. [@AverageData], in which the cross section is given by: $$\sigma =\sum_{i}w_{i}\sigma_{i}~~~\textmd{with}~~~w_{i}=(1/\Delta\sigma_{i}^{2})\bigg/\bigg(\sum_{i}1/\Delta\sigma_{i}^{2}\bigg).
\label{averagex}$$ Here, $w_{i}$ and $\Delta\sigma_{i}$ denote the weight and the total uncertainty, respectively, of the measured cross section $\sigma_{i}$ of mode $i$. The sum is performed over all 20 decay modes of $\lam$ and $\lambar$ [@IndividualCSs]. The combined uncertainty is calculated by $$\Delta\sigma^{2} = \sum_{i,j}w_{i}(\mathbf{M}_{\sigma})_{ij}w_{j},
\label{variance}$$ where $\mathbf{M}_{\sigma}$ represents the covariance matrix of these cross section measurements, in which the correlations between any two measurements $\sigma_{i}$ and $\sigma_{j}$ are considered. The resulting cross sections at the four CM energies are listed in Table \[CroSecSum\] and shown in Fig. \[FitLinShp\] together with the Belle data [@Belle] for comparison.
------------- -------- ------ ------------
$\ENERGYAT$ 47.67 0.45 $\BornCSA$
$\ENERGYBT$ 8.54 0.66 $\BornCSB$
$\ENERGYCT$ 8.16 0.71 $\BornCSC$
$\ENERGYDT$ 566.93 0.74 $\BornCSD$
------------- -------- ------ ------------
: The average cross section of $\ee$ measured at each CM energy, where the uncertainties are statistical and systematic, respectively. The observed cross section can be obtained by multiplying the $f_{\textrm{ISR}}$ and the $\sigma$.[]{data-label="CroSecSum"}
![ Cross section of $\ee$ obtained by BESIII (this work) and Belle. The blue solid curve represents the input line-shape for [kkmc]{} when determining the $f_{\textrm{ISR}}$. The dash-dot cyan curve denotes the prediction of the phase space (PHSP) model, which is parameterized by Eq. \[OnePhotonPre\] but with $C=1$ and flat $|G_{M}|$ with respect to $\sqs$.[]{data-label="FitLinShp"}](extra_line_shape.eps){width="3.33in" height="2.22in"}
[angular\_dis\_at\_4575\_draft.eps]{}
[angular\_dis\_at\_4600\_draft.eps]{}
The data sets collected at $\sqs$ = $\ENERGYAT$ and $\ENERGYDT$ $\mev$ are large enough to perform a detailed study in the CM frame of the $\Lam$ polar angle $\theta_{\Lam}$, which is defined as the angle between the $\Lam$ momentum and the beam direction. The data fulfilling all selection criteria are divided into ten bins in $\cos\theta_{\lam}$. In each $\cos\theta_{\lam}$ bin, the total yield is obtained by summing the yields of all the ten tagged modes. The one-dimensional bin-by-bin efficiency corrections are applied on these total yields. The same procedure is performed by tagging $\lambar$ decay channels. The total yields of $\lam$ and $\lambar$ are combined bin-by-bin, and the shape function $f(\theta)~\propto~(1+\alpha_{\Lam} \cos^{2}\theta)$ is fitted to the combined data, as shown in Fig. \[AngDisFit\]. Table \[AngDisParas\] lists the resulting $\alpha_{\Lam}$ parameters obtained from the fits, as well as the $\ratios$ ratios extracted using the equation $$\label{ratios}
\ratios^{2}(1-\beta^{2})=(1-\alpha_{\Lam})/(1+\alpha_{\Lam}).$$
$\sqs$ ($\mev$) $\alpha_{\Lam}$ $\ratios$
-------------------- --------------------------------------- -------------------------------
$\alpHaA$ $\RatioA$
$\alpHaD$ $\RatioD$
: Shape parameters of the angular distribution and $\ratios$ ratios at $\sqs$ = $\ENERGYAT$ and $\ENERGYDT$ $\mev$. The uncertainties are statistical and systematic, respectively. []{data-label="AngDisParas"}
The systematic uncertainties of the $\alpha_{\Lam}$ considered here are the contributions from the fit range and the bin size. A change of the fit range in $\cos\theta$ from ($-1.0,~1.0$) to ($-0.8,~0.8$) and in the number of bins from 10 to 20 are performed, and the differences in the obtained $\alpha_{\Lam}$ are regarded as the systematic uncertainty. Systematics originating from the model dependencies in the efficiency correction are found to be negligible compared to the statistical uncertainties.
In summary, using data collected at $\sqs=\ENERGYAT$, $\ENERGYBT$, $\ENERGYCT$, and $\ENERGYDT~\mev$ with the BESIII detector, the cross sections of $\ee$ have been measured with high precision, by reconstructing $\lam$ and $\lambar$ independently with ten Cabibbo-favored hadronic decay channels. The most precise cross section measurement is achieved so far at $\sqs=\ENERGYAT~\mev$, which is only $1.6~\mev$ above the threshold. The measured value is ($\BornCSA$) pb, which highlights the enhanced cross section near threshold and indicates the complexity of production behavior of the $\Lam$. At $\sqs=\ENERGYAT$ and $\ENERGYDT~\mev$, the data samples are large enough to study polar angle distributions of $\Lam$ and measure the $\Lam$ form factor ratio $\ratios$ for the first time. These results provide important insights into the production mechanism and structure of the $\Lam$ baryons.
The BESIII collaboration thanks the staff of BEPCII, the IHEP computing center and the supercomputing center of USTC for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11235011, 11335008, 11375205, 11425524, 11625523, 11635010; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1332201, U1532257, U1532258; CAS under Contracts Nos. KJCX2-YW-N29, KJCX2-YW-N45, QYZDJ-SSW-SLH003; 100 Talents Program of CAS; National 1000 Talents Program of China; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11505034, 11575077; National Science and Technology fund; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No. R32-2008-000-10155-0
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---
abstract: 'We investigate the electronic structure of a complex conventional superconductor, ZrB$_{12}$ employing high resolution photoemission spectroscopy and [*ab initio*]{} band structure calculations. The experimental valence band spectra could be described reasonably well within the local density approximation. Energy bands close to the Fermi level possess $t_{2g}$ symmetry and the Fermi level is found to be in the proximity of quantum fluctuation regime. The spectral lineshape in the high resolution spectra is complex exhibiting signature of a deviation from Fermi liquid behavior. A dip at the Fermi level emerges above the superconducting transition temperature that gradually grows with the decrease in temperature. The spectral simulation of the dip and spectral lineshape based on a phenomenological self energy suggests finite electron pair lifetime and a pseudogap above the superconducting transition temperature.'
author:
- Sangeeta Thakur$^1$
- Deepnarayan Biswas$^1$
- Nishaina Sahadev$^1$
- 'P. K. Biswas$^2$'
- 'G. Balakrishnan$^2$'
- Kalobaran Maiti$^1$
title: 'Anomaly in the electronic structure of a BCS superconductor, ZrB$_{12}$'
---
The relationship between pseudogap phase and superconductivity in high temperature superconductors is an issue of discussion for many years now [@Shen; @Statt; @Damascelli]. While one school believes that the pseudogap arises due to some hidden order and/or effects not associated to superconductivity, the other school attributes pseudogap to the electron pair formation above the transition temperature, $T_c$ as a precursor to the superconducting gap that leads to superconductivity upon establishment of the coherence among the electron pairs at $T_c$. Independent of these differences, it is realized that electron correlation plays a significant role in the occurrence of pseudogap phase in these unconventional superconductors [@Superconductor; @Superconductor1]. Several questions are being asked on whether the pseudogap phase is specific to unconventional superconductors, electron correlation or low dimensionality is a necessity for such phase *etc*.
Apart from these issues, the quest of new superconducting compounds resulted into the discovery of new materials such as cubic hexane, $M$B$_6$ [@cab6] and dodecaborides, $M$B$_{12}$ ($M$ = Sc. Y, Zr, La, Lu, Th) those attracted much attention due to their interesting electronic properties [@Matthias]. ZrB$_{12}$ is one such compound exhibiting relatively high superconducting transition temperature of 6 K in the MB$_{12}$ family [@Matthias]. It forms in cubic structure as shown in Fig. 1. In Fig. 1(a), the real lattice positions are shown with the correct bond lengths and atom positions. Although, the structure appears to be complex, it is essentially a rock salt structure constituted by two interpenetrated fcc lattices formed by Zr and B$_{12}$ units as demonstrated in Fig. 1(b) by compressing the B$_{12}$ units for clarity. Various transport and magnetic measurements suggest a Bardeen-Cooper-Schriefer (BCS) type superconductivity in this material, which is termed as conventional superconductivity. Since, Zr atoms are located within the huge octahedral void space formed by the B$_{12}$ units, the superconductivity in these materials could conveniently be explained by the electron-phonon coupling mediated phenomena involving primarily low energy phonon modes associated with the vibration of Zr atoms [@Filippov1].
The electronic properties of ZrB$_{12}$ manifest plethora of conflicts involving the mechanism of the transition. For example, there are conflicts on whether it is a type I or type II superconductor [@Filippov3], whether it possesses single gap [@Daghero] or multiple gaps [@Gasparov2], [*etc.*]{} Interestingly, Gasparov [*et al*]{}. [@Gasparov2] found that superfluid density of ZrB$_{12}$ exhibits unconventional temperature dependence with pronounced shoulder at $T/T_c \sim$ 0.65. In addition, they find different superconducting gap and transition temperatures for different energy bands, and the values of the order parameter obtained for $p$ and $d$ bands are 2.81 and 6.44, respectively. Detailed magnetic measurements reveal signature of Meissner, mixed and intermediate states at different temperatures and magnetic fields [@biswas_thesis]. Some of these variances were attributed to surface-bulk differences in the electronic structure [@jap], the superconductivity at the sample surface [@Tsindlekht1; @Filippov2; @Tsindlekht2] *etc*. Evidently, the properties of ZrB$_{12}$ is complex despite exhibiting signatures of BCS type superconductivity. Here, we probed the evolution of the electronic structure of ZrB$_{12}$ employing high resolution photoelectron spectroscopy. The experimental results exhibit spectral evolution anomalous to conventional type superconductor and signature of pseudogap prior to the onset of superconductivity.
RESULTS {#results .unnumbered}
=======
![(a) XP and He [II]{} valence band spectra of ZrB$_{12}$. The lines show the simulated XP spectrum from the band structure results. Calculated partial density of states of (b) Zr 4$d$ $t_{2g}$ (dashed line) & $e_g$ (solid line) symmetries, and (c) B 2$s$ (solid line) and 2$p$ (dashed line) states.](RevFig2.eps)
Valence band spectra obtained using He [II]{} and Al $K\alpha$ excitation energies are shown in Fig. 2(a) exhibiting three distinct features marked by A, B and C beyond 2 eV binding energies. The intensities between 6-10 eV binding energies represented by B are prominent in the He [II]{} spectrum, while the relative intensity between 3-4 eV binding energies represented by A is enhanced in the $x$-ray photoemission (XP) spectrum. Such change in intensity in the angle integrated spectra may be attributed to the matrix elements associated to different constituent states forming the eigenstates of the valence band, which is a sensitive function of the photon energy [@Yeh; @Yeh1; @Yeh2]. Therefore, the photoemission cross section will vary with the photon energy that can be used to identify the orbital character of the energy bands. Considering this feature of the technique, the spectral feature, B can be attributed to dominant B 2$p$ orbital character and the feature A to Zr 4$d$ orbital character.
Energy band structure of ZrB$_{12}$ has been calculated within the local density approximations (LDA). The calculated partial density of states (PDOS) corresponding to Zr 4$d$ & 5$s$, and B 2$s$ & 2$p$ states are shown in Fig. 2(b) and 2(c), respectively. Zr 5$s$ contribution appears in the energy range of 5 - 7.5 eV above the Fermi level with negligible contributions in the energy window studied here as shown by thick solid line in Fig. 2(b). B 2$s$ contributions appear beyond 6 eV binding energies. Dominant contribution from B 2$p$ PDOS appears between 1.5 to 10 eV binding energies. The center of mass of the B 2$p$ PDOS (considering both occupied and unoccupied parts) appears around 1.9 eV binding energy. The center of mass of the entire Zr 4$d$ PDOS appear around 2.8 eV above $\epsilon_F$, which is about 4.7 eV above the center of mass of the B 2$p$ PDOS. Thus, an estimate of the charge transfer energy could be about 4.7 eV in this system. The ground state is metallic with flat density of states across $\epsilon_F$ arising due to highly dispersive energy bands in this energy range. The density of states at $\epsilon_F$ possess close to 2:1 intensity ratio of the B 2$p$ and Zr 4$d$ orbital character.
The electronic states with $t_{2g}$ and $e_g$ symmetries, shown in Fig. 2(b), exhibit almost overlapping center of mass suggesting negligible crystal field splitting (crystal field splitting $\sim$ 100 meV). The energy bands are shown in Fig. 3. $e_g$ bands are relatively narrow and exhibit a gap at $\epsilon_F$. The intensity at $\epsilon_F$ essentially have highly dispersive $t_{2g}$ symmetry.
Using the above band structure results, we calculated the XP spectrum considering the photoemission cross section of the constituent partial density of states as discussed later in the Method section. The calculated spectrum shown by solid line in Fig. 2(a) exhibits an excellent representation of the experimental XP spectrum. It is clear that the feature, C in the energy range 10-14 eV corresponds to photoemission signal from B 2$s$ levels. The features, A and B arise due to hybridized B 2$p$ - Zr 4$d$ states. The relative contribution from Zr 4$d$ states is higher for the feature A.
In order to probe the electron correlation induced effect on the electronic structure, we have carried out electron density of states calculation using finite electron correlation among the Zr 4$d$ electrons, $U_{dd}$ as well as that among B 2$p$ electrons, $U_{pp}$. In Figs. 4(a) and 4(b), we show the calculated results for Zr 4$d$ and B 2$p$ PDOS with $U_{dd}$ = 0 (solid line) and 6 eV (dashed line) for the Zr 4$d$ electrons. Even the large value of $U_{dd}$ of 6 eV does not have significant influence on the occupied part of the electronic structure, while an energy shift is observed in the unoccupied part. Such scenario is not unusual considering the large radial extension of 4$d$ orbitals and poor occupancy [@ruth]. The inclusion of an electron correlation, $U_{pp}$ of upto 4 eV for B 2$p$ states exhibits insignificant change in the results. Considering Zr being a heavy element, we carried out similar calculations including spin-orbit interactions and did not find significant modification in the results - we have not shown these later results to maintain clarity in the figure. All these theoretical results suggest that consideration of electron correlation within the LSDA+$U$ method has insignificant influence in the occupied part of the electronic structure of this system.
![High resolution spectra near the Fermi level using (a) He [II]{} and (b) He [I]{} photon energies. The symmetrized spectral density of states (SDOS) for (c) He [II]{} and (d) He [I]{} spectra, and SDOS obtained by division of resolution broadened Fermi function for (e) He [II]{} and (f) for He [I]{} spectra.](RevFig5_HR.eps)
We now investigate the spectral changes near $\epsilon_F$ with high energy resolution in Fig. 5. The spectral density of states (SDOS) are calculated by symmetrizing the experimental spectra; $SDOS =
I(\epsilon - \epsilon_F)+I(\epsilon_F - \epsilon)$ and shown in Figs. 5(c) and 5(d). Such an estimation of SDOS is sensitive to the definition of the Fermi level, which is carefully derived at each temperature by the Fermi cutoff in the valence band spectra for silver mounted on the sample holder in electrical connection with the sample and measured using identical experimental conditions. The other effect is the asymmetry across the Fermi level. This can be addressed by obtaining the SDOS by the division of the resolution broadened Fermi-Dirac distribution function. The results are shown in Figs. 5(e) and 5(f) for He [II]{} and He [I]{} spectra, respectively. Clearly, the SDOS obtained following the later procedure exhibits symmetric SDOS across $\epsilon_F$. The 30 K and 10 K data exhibit large noise above $\epsilon_F$ due to the division of negligibly small intensities by small numbers arising from Fermi-Dirac function at these temperatures. These results demonstrate the reliability of SDOS extraction procedures and also indicate signature of particle-hole symmetry in this system.
![High resolution spectral density of states is shown as a function of (a) $|\epsilon - \epsilon_F|^2$, (b) $|\epsilon -
\epsilon_F|^{1/2}$, and (c) $|\epsilon - \epsilon_F|^{0.9}$. Spectral density of states from (d) He [I]{} and (e) He [II]{} spectra. The lines represent the spectral functions simulated to fit the experimental spectra.](Fig6_SDOS.eps)
The spectral lineshape near the Fermi level often provides important information about the thermodynamic properties of the systems. For example, a lineshape dependence of $|\epsilon - \epsilon_F|^2$ corresponds to Fermi liquid behavior [@Shklovskii; @Shklovskii1] - absence of linearity of SDOS with $|\epsilon - \epsilon_F|^2$ shown in Fig. 6(a) indicates a deviation from such a behavior in the present case. The decrease in intensity at $\epsilon_F$ with the decrease in temperature may arise due to the disorder induced localization of the electronic states. Altshuler & Aronov showed that the charge disorder in a correlated system leads to $|\epsilon
- \epsilon_F|^{0.5}$ dependence of the density of states near Fermi level [@Altshuler], which has been proved employing photoemission spectroscopy [@Sarma; @Fujimori]. In the present case, a plot of SDOS with $|\epsilon-\epsilon_{F}|^{0.5}$ shown in Fig. 6(b) exhibits deviation from linearity.
Interestingly, the curvatures at the exponents of 0.5 and 2 are opposite indicating an intermediate exponent for the present system. The simulation of SDOS as a function of $|(\epsilon-\epsilon_{f})|^{\alpha}$ with $\alpha \approx 0.9$ exhibits linear dependence with energy for all the temperatures studied (see Fig. 6(c)). Such a behavior in a conventional superconductor is curious. Most interestingly, in Fig. 6(c), all the data below 150 K superimpose exactly and correspond to the same energy scale, while the data at 300 K exhibit similar energy dependence with a different slope. These results suggest signature of an incipient phase transition at an intermediate temperature.
The SDOS at $\epsilon_F$ is intense and flat at 300 K with large dispersion of the bands and correspond to a good metallic phase consistent with the band structure results. The decrease in temperature introduces a dip at $\epsilon_F$. The dip gradually becomes more prominent with the decrease in temperature. The same scenario is observed in both, He [I]{} and He [II]{} spectra indicating this behavior to be independent of the photon energy used. This dip may be attributed to a pseudogap appearing at low temperatures [@lab6]. Although the probed lowest temperature is slightly above $T_c$, the dip grows monotonically with the decrease in temperature - a large suppression of the spectral weight at the Fermi level is observed at 10 K indicating relation of this dip to the superconducting gap. In order to estimate the energy gap in these spectral functions, we used a phenomenological self-energy, $\Sigma_k(\epsilon)$ following the literature [@Randeria]. $$\Sigma_k(\epsilon) = -i\Gamma_1 + \Delta^2/[(\epsilon + i\Gamma_0)
+\epsilon_k]$$ where $\Delta$ represents the gap size. The first term is the energy independent single particle scattering rate and the second term is the BCS self energy. $\Gamma_0$ represents the inverse electron pair lifetime. The spectral functions, the imaginary part of the Green’s functions, were calculated as follows. $A(\epsilon)=(1/\pi) Im \sum_k G_k(\epsilon) = (1/\pi)\sum_k
\Sigma_k^{\prime\prime}(\epsilon) / [(\epsilon-\epsilon_k-
\Sigma_k^\prime(\epsilon))^2 +
(\Sigma_k^{\prime\prime}(\epsilon))^2]$, where $\Sigma_k^\prime(\epsilon)$ and $\Sigma_k^{\prime\prime}(\epsilon)$ are the real and imaginary part of the self energy. We find that a finite value of $\Gamma_0$ is necessary to achieve reasonable representation of the experimental data. The simulated and experimental spectral density of states are shown by superimposing them in Figs. 6(d) and 6(e) for the He [I]{} and He [II]{} spectra, respectively. Evidently, the simulated spectra provide a remarkable representation of the experimental spectra.
Discussion {#discussion .unnumbered}
==========
The experimental and calculated band structure results suggest significant hybridization between Zr 4$d$ and B 2$p$ states leading to similar energy distribution of the PDOS. Here, the eigenstates are primarily constituted by the linear combination of the Zr 4$d$ and B 2$p$ states. The antibonding bands possess large Zr 4$d$ orbital character and appear in the unoccupied part of the electronic structure [@bndstr1; @bndstr2]. The energy bands below $\epsilon_F$ forming the valence band are the bonding eigenstates consisting of dominant 2$p$ orbital character. The covalency between these states could be attributed to the large radial extension of the 4$d$ orbitals overlapping strongly with the neighboring states as also observed in other 4$d$ systems [@covalency; @covalency1; @covalency2]. The calculated results within the local density approximation are consistent with the experimental spectra - subtle differences between the experiment and theory could not be captured via inclusion of electron correlation and/or consideration of spin-orbit coupling in these LDA calculations.
The band structure results exhibit two important features - (i) the energy bands with $t_{2g}$ symmetry are highly dispersive compared to $e_g$ bands and (ii) the crystal field splitting is negligible. As shown in Fig. 1, the B$_{12}$ dodecahedrons form an octahedra around Zr sites with the center of mass of the B$_{12}$ units at the edge centers of the unit cell. The size of the dodecahedrons are quite large with no boron situating on the unit cell edge. Thus, the overlap of B 2$p$ orbitals with the $t_{2g}$ orbitals is significantly enhanced due to the proximity of borons along the face diagonals, while that with the $e_g$ orbitals is reduced leading to a comparable crystal field potential on both the orbitals. The larger hybridization with the $t_{2g}$ orbitals are also manifested by the large dispersion of the $t_{2g}$ bands spanning across the Fermi level while $e_g$ bands are relatively more localized. From these calculations, the effective valency of Zr was found to vary between (+2.4) to (+2.5) depending on the electron interaction parameters such as correlation strength, spin orbit coupling *etc*.
The Fermi level is located almost at the inflection points of each of the energy band crossing $\epsilon_F$ as shown in Fig. 7. This is verified by investigating the first derivative of the energy bands shown by the symbols in the figure. The arrows in the figure indicate Fermi level crossing and corresponding point in the first derivative curves. Evidently, the band crossing correspond to an extremum in the derivative plot. The inflection point is the point, where the curve changes its curvature that corresponds to a change in the character of the conduction electrons (the Fermi surface) from hole-like to electron-like behavior. This behavior is termed as Lifschitz transition [@lifschitz]. Thus, this material appear to be lying in the proximity of the Lifschitz transition as also observed in various exotic unconventional superconductors such as Fe-pnictides [@liu-nphys]. Due to the lack of cleavage plane and hardness of the sample, the Fermi surface mapping of this sample could not be carried out to verify this theoretical prediction experimentally. It is well known that the band structure calculations captures the features in the electronic structure well in the weakly correlated systems as also found in the present case suggesting possibility of such interesting phenomena in this system. We hope, future studies would help to enlighten this issue further.
![The gap, $\Delta$, inverse pair lifetime, $\Gamma_0$ and single particle scattering rate, $\Gamma_1$ as a function of temperature. The open and close symbols correspond to the fitting of He [I]{} and He [II]{} data, respectively.](RevFig8_par.eps)
The high resolution spectra close to the Fermi level indeed exhibit deviation from Fermi liquid behavior - a suggestive of quantum instability in this system as expected from the above results. R. Lortz [*et al.*]{} [@Filippov1] showed that the resistivity of ZrB$_{12}$ exhibits linear temperature dependence in a large temperature range of 50K - 300K. The resistivity below 50K of the normal phase is complex and does not show $T^2$-dependence. Both, the resistivity and specific heat data exhibit signature of anharmonic mode of Zr-vibrations. All these observations indicate complexity of the electronic properties, different from a typical Fermi-liquid system consistent with the photoemission results.
A dip appears at the Fermi level at temperatures much higher than the superconducting transition temperature reminiscent of a pseudogap feature. In order to investigate this further, we study the fitting parameters simulating the spectral functions in Fig. 8. The energy gap, $\Delta$ is found to be about 7.3 meV and remains almost unchanged in the whole temperature range studied. This gap is significantly larger than the value predicted (1-2 meV) considering BCS behavior. The magnitude of $\Gamma_{1}$, which is a measure of single particle scattering rate, is quite large and found to decrease with the decrease in temperature. On the other hand, $\Gamma_{0}$, the inverse pair lifetime is found to be significant and smaller than 2$\Delta$ suggesting proximity to a BCS limit for this system. $\Gamma_0$ exhibits temperature dependence quite similar to that of $\Gamma_1$. The decrease in $\Gamma_0$ manifests gradual enhancement of the electron pair lifetime with the decrease in temperature that eventually leads to the superconducting phase upon attaining coherence among the pairs. Thus, the temperature in the vicinity of 70 K exhibiting the onset of pair formation could be the critical temperature, $T^*$ in this system. The observation of the signature of an energy gap and finite electron pair lifetime above $T_c$ is quite similar to that observed for under-doped cuprates[@Randeria] and is an indication of precursor effect in this system. Clearly, the electronic structure of this system is complex, appears to be at the crossover of BCS and unconventional behaviors and further studies are required to understand the origin of pseudogap & unusual spectral lineshape.
In summary, we studied the electronic structure of ZrB$_{12}$, predicted to be a complex conventional superconductor employing high resolution photoemission spectroscopy and *ab initio* band structure calculations. The valence band spectra exhibit multiple features with dominant B 2$p$ orbital character close to the Fermi level. The experimental results could be captured reasonably well within the local density approximations. The electronic states close to Fermi level have $t_{2g}$ symmetry and the filling of the bands is in the proximity of Lifschitz transition. The spectral lineshape near the Fermi level in the high resolution spectra exhibit deviation from typical Fermi liquid behavior. A dip at the Fermi level emerges above the superconducting transition temperature and gradually becomes prominent at lower temperatures. The analysis of the spectral functions within the phenomenological descriptions suggests finite electron pair lifetime above $T_c$ as a signature of a precursor effect to the superconducting transition in this system.
Method {#method .unnumbered}
======
### Sample preparation and characterization
Single crystals of ZrB$_{12}$ were grown by floating zone technique in a Crystal Systems Incorporated (CSI) four-mirror infrared image furnace in flowing high purity Ar gas at a pressure of 2 bar [@Geetha]. The quality of the crystal was confirmed and orientation was determined from $x$-ray Laue - diffraction images. The magnetization measurements using a Qunatum Design MPMS magnetometer show a sharp superconducting transition temperature, $T_{c}$ of 6.1 K [@biswas_thesis].
### Photoemission measurements
The photoemission measurements were performed using a Gammadata Scienta R4000 WAL electron analyzer and monochromatic laboratory photon sources. $X$-ray photoemission (XP) and ultraviolet photoemission (UP) spectroscopic measurements were carried out using Al $K\alpha$ (1486.6 eV), He [II]{} (40.8 eV) and He [I]{} (21.2 eV) photon energies with the energy resolution set to 400 meV, 4 meV and 2 meV, respectively. The melt grown sample was very hard and possesses no cleavage plane. Therefore, the sample surface was cleaned using two independent methods; top-post fracturing and scraping by a small grain diamond file at a vacuum better than 3$\times$10$^{-11}$ torr. Both the procedures results to similar spectra with no trace of impurity related signal in the photoemission measurements. The measurements were carried out in analyzer transmission mode at an acceptance angle of 30$^o$ and reproducibility of all the spectra was ensured after each surface cleaning cycle. The temperature variation down to 10 K was achieved by an open cycle He cryostat from Advanced Research systems, USA.
### Band structure calculations
The energy band structure of ZrB$_{12}$ was calculated using full potential linearized augmented plane wave method within the local density approximation (LDA) using Wien 2k software [@wien2k]. The energy convergence was achieved using 512 $k$-points within the first Brillouin zone. The lattice constant of 7.4075 Å was used considering the unit cell shown in Fig. 1 [@Kiev]. In order to introduce electron-electron Coulomb repulsion into the calculation, we have employed LDA+$U$ method with an effective electron interaction strength, $U_{eff}$ ($= U - J$; $J$ = Hund’s exchange integral) setting $J$ = 0 following Anisimov et al.[@anisimov]. Consideration of finite $J$ did not have significant influence on the results. We have carried out the calculations for various values of electron correlation upto 6 eV for Zr 4$d$ electrons and 4 eV for B 2$p$ electrons. Spin-orbit interactions among the Zr 4$d$ electrons are also considered for the calculations.
To calculate the $x$-ray photoemission spectrum, we have multiplied the partial density of states obtained by the band structure calculations by the corresponding photoemission cross sections for Al $K\alpha$ energy. These cross-section weighted PDOS is convoluted by two Lorentzians with the energy dependent full width at half maxima (FWHM) for the hole & electron lifetime broadenings and a Gaussian with FWHM representing the resolution broadening. The sum of these spectral functions provides the representation of the experimental valence band spectrum.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
The authors, K. M. and N. S. acknowledge financial support from the Department of Science and Technology under the Swarnajayanti Fellowship Programme. One of the author, G. B. wishes to acknowledge financial support from EPSRC, UK (EP/I007210/1).
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|
---
author:
- 'Wayne M. Schlingman'
- 'Yancy L. Shirley[^1]'
- 'David E. Bolin'
- Erik Rosolowsky
- John Bally
- Cara Battersby
- 'Miranda K. Dunham'
- 'Timothy P. Ellsworth-Bowers'
- 'Neal J. Evans II'
- Adam Ginsburg
- Guy Stringfellow
title: 'The Bolocam Galactic Plane Survey V: HCO$^+$ and N$_2$H$^+$ Spectroscopy of 1.1 mm Dust Continuum Sources'
---
Introduction
============
####
Stars form out of clouds of dense molecular gas and dust. From detailed studies of nearby molecular clouds, we have developed a picture of how stars with masses typically close to our Sun’s form and evolve onto the main sequence (e.g., Shu, Adams, & Lizano 1987). A corresponding picture does not exist for the highest mass end of the stellar-mass spectrum. This is in part due to high-mass star-forming regions in our Galaxy being at greater distances, and thus being observed at lower spatial resolution, than low-mass regions. High-mass stars also form in highly clustered environments, whereas the well-studied nearby low-mass stars are typically more isolated (e.g. Taurus). Not even the basic formation mechanism of massive-star formation (scaled-up version of monolithic core collapse vs. competitive accretion formation; Shu et al. 1987; Bonnell & Bate 2006; McKee & Ostriker 2007) is yet well agreed upon, especially for the formation of the highest mass stars. It is possible that both processes are important in different regimes of the stellar mass spectrum. For low-mass stars, there are observational indicators of the evolutionary state of the protostar (e.g. $T_{bol}$ – Temperature of a blackbody with a peak at the flux weighted mean frequency of the spectral energy distribution and $\alpha_{IR}$ – IR spectral index, defining the Class 0, I, II, III system; Lada 1987, Evans et al. 2009); a universal evolutionary sequence for high-mass stars is still being developed and that exact ordering of the possible observational indicators (e.g., the presence of a H$_2$O maser or a CH$_3$OH Class I or Class II maser, e.g. Plume et al. 1997; Shirley et al. 2003; De Buizer et al. 2005; Minier et al. 2005; Ellingsen et al. 2007; Longmore et al. 2007; Purcell et al. 2009; Breen et al. 2010) is still debated.
####
One observational aspect that has limited our complete understanding of star formation is that we lack a complete census of the star-forming regions in our own Galaxy and, therefore, a census of their basic properties (size, mass, luminosity). Previous surveys of star-forming regions have been heavily biased. For instance, the earliest studies of dense molecular gas focused on known (optical or radio) H [II]{} regions where an O or B spectral type star had already formed. The discovery and cataloguing of UCH [II]{} regions (e.g. Wood & Churchwell 1989) extended studies to an earlier embedded phase, but still required the presence of a forming high-mass star. Infrared Dark Clouds (IRDCs), clouds of dust and gas that are opaque at mid-infrared wavelengths (i.e. 8 $\mu$m), permitted less-biased studies of star forming regions through the earliest (prestellar) phases and across the stellar mass spectrum (Carey et al. 2009; Peretto & Fuller 2009); however, they were limited to clouds at near kinematic distances and typically observable only in the inner Galaxy ($-60 < \ell < 60$ degrees). Dust continuum observations at far-infrared through millimeter wavelengths provide the least-biased means to survey star forming regions at all embedded evolutionary phases and a wide range of the stellar mass spectrum across the Milky Way Galaxy since the emission is optically thin, always present, and can trace small amounts of mass.
####
In the past decade, new surveys of the Milky Way Galaxy have been made from mid-infrared through millimeter wavelengths. Several Galactic plane surveys are published or currently being observed, including the Bolocam Galactic Plane Survey (BGPS: Aguirre et al. 2011), the APEX Telescope Large Area Survey of the Galaxy (ATLASGAL: Schuller et al. 2009), and the Herschel infrared Galactic Plane Survey (Hi-GAL: Molinari et al. 2010). The goals of these surveys are to look for the precursors to massive star formation in the Galaxy as a whole, without targeting individual regions known to contain forming stars. They are an integral part of completing a full census of star-forming cores and clumps in the Milky Way as they are sensitive to star formation at all stages.
####
The Bolocam Galactic Plane Survey is a 1.1 mm continuum survey of the Galactic plane (Aguirre et al. 2011). Covering 220 square degrees at 33$^{\prime\prime}$ resolution, the BGPS is one of the first large-area, systematic continuum surveys of the Galactic plane in the millimeter regime. The BGPS spans the entire first quadrant of the Galaxy with a latitude range of $|b|< 0.5$ degrees from the Galactic plane and portions of the second quadrant (Aguirre et al. 2011). The survey has detected and catalogued approximately 8400 clumps of dusty interstellar material (Rosolowsky et al. 2010). The BGPS is beam matched to the spectroscopic data we are taking in this paper. This allows us to easily compare the gas and dust in the same phase of star formation. The BGPS data are available in full from the IPAC website[^2].
####
The vast majority of sources detected in the BGPS represents a new population of dense, potentially star-forming clumps in the Milky Way. The basic properties of these objects such as size, mass, and luminosity depend on the distance to the objects. However, since the BGPS observations are continuum observations, they contain no kinematic information. In this paper, we use the line-of-sight velocity (v$_{LSR}$) from a molecular line detection and a model of the Galaxy to determine a kinematic distance. Not only is the v$_{LSR}$ useful, but the line properties themselves can elucidate a number of properties of the dense gas in the clumps (e.g., virial mass, infalling gas, outflows, etc.).
####
Most kinematic surveys of the Milky Way have been performed using low gas density tracers (e.g. H [I]{}: Giovanelli et al. 2005; $^{12}$CO: Dame et al. 2001, GRS($^{13}$CO): Jackson et al. 2006). With these low density tracers, most lines-of-sight in the Galaxy have multiple velocity components. To mitigate this, we choose dense gas tracers that will be excited exclusively in the BGPS clumps. Surveying dense gas has been done before using CS $J=2-1$ toward IRDCs (see Jackson et al. 2008). In this survey, we simultaneously observe two dense gas tracers HCO$^+$ $J=3-2$ and N$_2$H$^+$ $J=3-2$ using the 1 mm ALMA prototype receiver on the Heinrich Hertz Submillimeter Telescope (HHT). The HHT resolution of $\sim 30^{\prime\prime}$ at 1.1 mm is nearly beam-matched to the original BGPS survey, allowing a one-to-one comparison between these dense gas tracers and peak 1.1 mm continuum emission positions. These two molecular tracers have very similar effective excitation densities, n$_{eff}\sim 10^4$ cm$^{-3}$, that are well-matched to the average density derived from the continuum-emitting dust (see Dunham et al. 2010). The effective excitation density for a molecular tracer is defined in Evans (1999) as the density at a given kinetic temperature required to excite a 1 K line for a column density of log$_{10}$ N / $\Delta v = 13.5$. To determine the effective density, we use RADEX, which is a Monte Carlo radiative transfer code (Van der Tak et al. 2007), assuming log$_{10}$ N = 13.5 cm$^{-2}$ and $\Delta v~=~1$ km s$^{-1}$.
####
The chemistry of HCO$^+$ and N$_2$H$^+$ is useful, as these two molecules have opposite chemistries with respect to the CO molecule (Jørgensen et al. 2004). HCO$^+$ is created by CO and N$_2$H$^+$ is destroyed by CO. The formation routes of HCO$^+$ and N$_2$H$^+$ are dominated by the following reactions: $$H_3^+ + CO \rightarrow HCO^+ +H_2$$ $$H_3^+ + N_2 \rightarrow N_2H^+ + H_2$$ $$N_2H^+ + CO \rightarrow N_2 + HCO^+$$ For N$_2$H$^+$ to exist in large quantities, the gas must be cold where CO has frozen out onto dust grains. The ratio of N$_2$H$^+$/HCO$^+$ emission can be a chemical indicator of the amount of dense, cold gas in BGPS clumps. Even with the modest upper energy levels, E$_{u}$/k (HCO$^+$ $J=3-2$) = 25.67 K and E$_{u}$/k (N$_2$H$^+$ $J=3-2$) = 26.81 K, these transitions can still be excited at low temperatures if the density of the gas is high enough. This brings up an interesting conflict for N$_2$H$^+$: chemically, it favors a low kinetic temperature where CO is depleted but higher T$_{kin}$ or higher gas density leads to a stronger excitation of the $J=3-2$ line.
####
In this paper, we present the results for spectroscopic observations of 1882 BGPS clumps. In § 2 we discuss source selection and observing, calibration, and reduction procedures. In § 3 detection statistics, line intensities, velocities and linewidths are analyzed. In § 4 we calculate the kinematic distance to each detected source and determine our size-linewidth relations, clump mass spectra, and present a face-on view of the Milky Way Galaxy based on kinematic distances determined from our sample.
Observations, Calibrations, and Reduction
=========================================
Facility and Setup
------------------
####
Observations were conducted with the Heinrich Hertz Submillimeter Telescope on Mount Graham, Arizona. The data were taken over the course of 44 nights beginning in February 2009 and ending in June 2009. We utilized the ALMA Band-6 dual-polarization sideband-separating prototype receiver in a 4-IF setup (Lauria et al. 2006, ALMA memo \#553). With this setup we simultaneously and separately observe both the upper and lower sidebands (USB and LSB, respectively) in horizontal polarization (H$_{pol}$) and vertical polarization (V$_{pol}$) using two different linearly polarized feeds on the receiver. The receiver was tuned to place the HCO$^+$ $J=3-2$ (267.5576259 GHz) line in the center of the LSB. The IF was set to 6 GHz, which offsets the N$_2$H$^+$ $J=3-2$ line (279.5118379 GHz) in the USB by $+47.47$ km s$^{-1}$. The signals were recorded by the 1 GHz Filterbanks (1 MHz per channel, 512 MHz bandwidth in 4-IF mode; LSB velocity resolution $\Delta v_{ch} = 1.12$ km s$^{-1}$ and USB velocity resolution $\Delta v_{ch} = 1.07$ km s$^{-1}$) in each polarization and sideband pair (V$_{pol}$ LSB, V$_{pol}$ USB, H$_{pol}$ LSB, H$_{pol}$ USB).
\[calibration\]Calibration and Sideband Rejections
--------------------------------------------------
####
Every observing session utilized 3 types of observations to calibrate the velocity offset, temperature scale, and rejection of the sideband separating receiver. The antenna temperature scale T$^*_{A}$ is used at the HHT and is set by the chopper wheel calibration method (Penzias & Burrus 1973). This temperature scale was then converted to T$_{mb} = $T$^*_A / \eta_{mb}$ by observing Jupiter and calculating the main beam efficiency ($\eta_{mb}$; Mangum 1993) $$\eta_{mb} = \frac{ f_{rej} \cdot T^*_{A} (Jupiter)}{J(\nu_s,T_{Jupiter}) - J(\nu_s,T_{CMB})} \cdot \left[ 1-\exp \left(-\;ln\: 2\cdot \frac{\theta_{eq}\theta_{pol}}{\theta^2_{mb}} \right) \right]^{-1} \;\; ,$$ where $J(\nu,T_{B}) = \frac{h \nu/k}{\exp(h \nu/k T_{B} ) - 1}$ is Planck function in temperature units evaluated at the observed frequency $\nu$ and brightness temperature T$_{B}$, T$^*_{A}$ is the average observed temperature of Jupiter in the bandpass, T$_{Jupiter} = 170$ K $\pm 5$ K, T$_{CMB} = 2.73$ K, $\theta_{eq}$ & $\theta_{pol}$ are the daily equatorial and poloidal angular diameters of Jupiter, $\theta_{mb}$ is the HHT FWHM (equal to 28.2$^{\prime\prime}$ LSB and 26.9$^{\prime\prime}$ USB). The sideband rejection correction factor is given by, $$f_{rej} = \left(1+\frac{I(T^*_{A} USB)}{I(T^*_{A} LSB)}\right)^{-1}.$$ We calculate $f_{rej}$, by measuring the integrated intensity of the flux that bleeds over from the LSB into the USB by observing S140, a source with very strong HCO$^+$ $J=3-2$ emission ($T_{mb} = 18$ K). The average rejection was $-13.8$ dB in V$_{pol}$ and $-15.2$ dB in H$_{pol}$. We ignore the difference in atmospheric opacities between 267 GHz and 279 GHz since it is small.
####
We report the computed $\eta_{mb}$ for each observing session in Table \[etamb\] (see Figure \[calibplot\]). Each data point for $\eta_{mb}$ in Figure \[calibplot\] consists of the average of five or more observations of Jupiter each night. There are no apparent trends in $\eta_{mb}$ versus time except on MJD 918 – 920, when $\eta_{mb}$ for V$_{pol}$ is significantly lower than for the rest of the observing sessions. For these dates we choose to use the average $\eta_{mb}$ and treat them independently from the rest of the calibration. A drop in the integrated intensity, I(T$_A^*$), is also seen in the data taken of the two spectral line calibration sources, S140 and W75(OH) (Figure \[calibplot\]), supporting the decision to treat these days separately.
####
We also compared $\eta_{mb}$ for the two polarizations and sidebands against each other (Figure \[polcompjup\]). The two upper panels compare the USB and LSB of each polarizations against each other. These are highly correlated with Spearman’s rank coefficients of $\rho \sim 1$, which is expected, as each linear polarization has its own feed. The Spearman’s rank coefficient is a measure of the monotonic dependence between two variables. The lower panels compare the two polarization feeds of the LSB and two polarization feeds of the USB against each other. These are less well-correlated with Spearman’s rank coefficients of $\rho \sim 0.5$ and show that the variation we see in $\eta_{mb}$ does not come from a systematic effect that affects both polarizations at the same time.
####
We observed each source in the catalog described in § \[catalog\] for 2 minutes total integration time. We position switched between a common OFF position for each 0.5 degrees in Galactic longitude ($\ell$). Each of these OFF positions was observed for 6 minutes to check if the OFF position had any detectable line emission. For a subset of sources near the end of our observations, starting on MJD $-$ 245400 = 986, the integration time was increased to 4 or 10 minutes to compensate for deteriorating weather. Our goal was to keep the baseline rms less than 100 mK ($\Delta v_{ch} \sim 1$ km s$^{-1}$) for as many sources as possible. S140 was also used to calculate the Allan Variance of the ALMA prototype receiver (Schieder & Kramer 2001). Accounting for the Allan Variance, we determined that $\sim 20$ seconds was the optimal switching time between ON and OFF positions for position switching with the 1 mm ALMA prototype receiver.
\[catalog\] Source Selection
----------------------------
####
We selected sources out of a preliminary version of the BGPS source catalog (BOLOCAT, Rosolowsky et al. 2010). We used the BOLOCAT version 0.7 to divide sources into logarithmic flux bins with equal numbers using the 40$^{\prime\prime}$ aperture flux. We point at peak $1.1$ mm continuum positions as listed in the BOLOCAT and we restricted the range of Galactic longitudes to fall between $10^\circ \le \ell \le 100^\circ$. The entire BOLOCAT in this longitude range was divided into logarithmically-spaced flux bins from S$_{1.1mm} = 0.1$ Jy to S$_{1.1mm} \sim 0.4$ Jy in intervals of log$_{10}($S$_{1.1mm}) = 0.1$. All sources greater than S$_{1.1mm} \sim 0.4$ Jy were included for observation ($N=689$). Below this flux, 100 sources per bin were selected at random to comprise a flux-selected set of $N=1,289$ sources from the BOLOCAT v0.7. In addition, we observed all BOLOCAT v0.7 sources in $\ell$ ranges of 10$^{\circ}$ – 11.5$^{\circ}$, 15$^{\circ}$ – 21$^{\circ}$, and 80$^{\circ}$ – 85.5$^{\circ}$. These ranges were chosen due to a combination of overlap with other surveys and observing availability. Sources with $\ell>100^{\circ}$ were taken from the BOLOCAT v1.0 and were only restricted by observability.
####
Near the end of our spectroscopic survey, the BGPS version 1.0 maps and the BOLOCAT v1.0 were released. We recomputed the photometry of our sources at the observed v0.7 positions on the v1.0 maps using the HHT beamsize, this is the data presented in § 3. A correction factor of 1.5 was multiplied to all of the 1.1 mm fluxes (see Aguirre et al. 2011). This calibration factor was determined by the BGPS team to account, in part, for the spatial filtering present in the BGPS v1.0 maps and possible calibration differences between the BGPS and other surveys. This factor brings the BGPS fluxes inline with those of other surveys (e.g. Mőtte et al. 2007).
####
Between versions of the catalogs the algorithms for processing the images were improved and thus source peak continuum positions may have moved or sources may even be removed from the catalog. In § 4 we compare physical properties of the sources and need to know the overall source properties not just the photometry of the locations we observed. We take the nearest source from the v1.0 catalog to the position we observed. The median offset between observed v0.7 positions and v1.0 positions is $6.4^{\prime\prime}$. The vast majority ($83$% ) of v0.7 observed positions lie within 15$^{\prime\prime}$ (1/2 the beam width) of the nearest v1.0 position. Since the median angular size of our observed sources is $\sim 60^{\prime\prime}$, the small positional differences between v0.7 and v1.0 do not significantly affect the physical properties derived from the 1.1 mm emission (e.g., size, mass). No source for which we have resolved the distance ambiguity and derived physical properties for in §4 (Known Distance Sample) has an offset between the catalogs greater than 30$^{\prime\prime}$ (one beam width) when determining physical properties.
####
In the following analysis, we refer to the “Full Sample” of 1882 observed sources and the “Deep Sample” of $707$ sources where the entire BOLOCAT was observed within the longitude ranges described above. Figure \[lvb\] shows the location of the sources we observed in the Full Sample and the sources in the Deep Sample.
\[reduction\] Data Reduction
----------------------------
####
The spectra were reduced using scripts we developed for the CLASS software package[^3]. In 4-IF mode, there are four filterbank spectra for each source. The HCO$^+$ $J=3-2$ spectrum was used to determine the baseline window, typically $\pm$50 – 75 km s$^{-1}$ from the line center, and to determine the line window, typically $\pm$ 10 – 15 km s$^{-1}$ from line center. The two polarizations of HCO$^+$ $J=3-2$ data were then baselined, and averaged together. This averaged spectrum is used to determine the line flag of the observed source and the line flag is determined from the approximate line shape (see Table \[lineflags\]). A flag of *0* means there is no apparent line in the spectrum at any velocity. A flag of *1* indicates a single line in the spectrum and the line exhibits no apparent structure from a line wing or self-absorbed profile. A flag of *2* means there was confusion along the line of sight and multiple velocity components are observed. In this case there is no way of determining which source is associated with the 1.1 mm map without mapping the molecular emission. A flag of *4* means there was a possible line wing. A Gaussian fit is plotted over the data to emphasize any deviations from a Gaussian shape; this helped to accentuate any sources with line profiles with a red or blue line wing, or possibly both. A flag of *5* means the line profile was possibly self-absorbed. Examples of spectra for the various flags are shown in Figure \[sampplots\].
####
The N$_2$H$^+$ $J=3-2$ line is offset $+47.47$ km s$^{-1}$ from the center of the USB; therefore, each of the HCO$^+$ baseline windows is shifted by that offset in order to baseline the N$_2$H$^+$ data. The N$_2$H$^+$ spectra are baselined and averaged in the same manner as above and the line is flagged for its quality and structure.
####
After each spectrum is flagged, it is converted to the T$_{mb}$ scale and the two polarizations are averaged together. Each spectrum is corrected with the corresponding $\eta_{mb}$ given by the date it was observed and its polarization, as explained in §\[calibration\]. The spectra are weighted by their baseline rms values, averaged, and baselined. The resulting combined spectra are used in our analysis.
Analysis Pipeline
-----------------
####
Once the spectrum for each source has been calibrated and averaged, the next step is to measure the line properties. The analysis of all the spectra is performed in IDL using custom and ASTROLIB routines, and all CLASS spectra are exported to an [ASCII]{} file containing the final spectrum. The peak temperature is given by the maximum temperature within the defined line window and the error is the rms of the data outside of that window. The integrated intensity, central velocity, and line width are computed using both an analysis of the 0$^{th}$, 1$^{st}$, and 2$^{nd}$ moments and by fitting a Gaussian model to the spectral line.
### Moment Analysis
####
The moments of a spectral line are calculated using $$M_n = \sum_{i=v_l}^{v_u} T_i v_i^n \Delta v_{ch},
\label{momenteqn}$$ where $n$ is the moment and $i$ represents each channel between the $v_l$ and $v_u$, defining the line window. These moments are then used to compute the integrated intensity, central velocity, and FWHM using $$I(T_{mb}) = M_0$$ $$v_{cen} = \frac{M_1}{M_0}$$ $$v_{FWHM} = \sqrt{ 8 \; \ln{2}} \cdot \left(\frac{M_2}{M_0} - v_{cen}^2 \right)^{1/2}.$$
Moment calculations are sensitive to the rms of the baseline and our generously large line windows. For lower signal-to-noise lines, a small noise feature in the spectrum can drastically change the first moment when using all of the data points in the line window. To compensate for low signal-to-noise we estimate the line center using only data three times the baseline rms in the first moment calculation. This new method returns velocities and widths that more closely match those that are determined by eye than the method using all of the data within the line window. This only has significant effects for low signal-to-noise spectra.
### \[gaussian\]Gaussian Fitting
####
Another method of determining the central velocity and FWHM for a spectral line is to fit a Gaussian model to the line profile. This method has its drawbacks as well, but the main drawback is that it struggles with lines that deviate strongly from Gaussian shaped line profiles. Examples of such are lines with self-absorbed profiles or lines with very prominent line wings, (Figure \[sampplots\] (c) and (d)). The Gaussian fits are computed with the MPFITPEAK function (Markwardt 2009) and return a reduced $\chi^2$. The boundary conditions chosen are the following: $1)$ the baseline is defined to be 0, $2)$ the peak line temperature of the Gaussian is defined to be positive, $3)$ the central velocity of the line must be within the line window, and $4)$ the FWHM must be smaller than the line window. For the starting parameters of the fit, we use the results from the moment analysis.
####
At this point we refine our method of computing the desired quantities for the line shape. After the initial Gaussian fit is completed, the next step is to modify the line window and recompute both the moment analysis and the Gaussian fitting. To modify the line window we used the parameters of the line as determined by the Gaussian fit to center the line window on the line center, v$_{Gauss}$, and extend it by the measured linewidth, $\pm 2 \cdot \sigma_{Gauss}$. If either bound of this new line window extends outside of the original, the original bounding velocity is used for that term. The fit and moments are recomputed but do not yield significant changes for most sources. At this time $\sigma_I$ (the error on the integrated intensity) is calculated using the final “fit” line window. $\sigma_I=\sigma_T \cdot \sqrt{\delta v_{ch}\cdot (v_u-v_l)}$ where $v_{ch}$ is the channel width ($\delta v_{ch} = 1.1$ km/s), $v_u$ and $v_l$ are the upper and lower bounds of the line window, and $\sigma_T$ is the baseline rms. If the line was undetected, flag of *0*, the line window used to estimate the error is $\sim 6$ km s$^{-1}$, within which more than 95% of all measured FWHMs lie (Figure \[linewidthhist\]). The Gaussian fits are used to determine the HCO$^+$ and N$_2$H$^+$ central velocities and the HCO$^+$ linewidths while the zero$^{th}$ moment is used to calculate the integrated intensity. For N$_2$H$^+$ linewidths, we use an IDL script that deals with the hyperfine lines (assuming a Gaussian shape for each hyperfine line) and uses MPFIT to determine the best-fit line profile. For the rest of the paper, linewidths refer to only the Gaussian-fit, observed HCO$^+$ linewidth.
\[analysis\]Detection Statistics and Analysis
=============================================
####
In this section, we discuss the statistics of the detected sources in HCO$^+$ and N$_2$H$^+$ and correlations between integrated intensity for HCO$^+$ and N$_2$H$^+$ with respect to the two sample groups. We also discuss the determined v$_{LSR}$ and the analysis of the line centroids and linewidths.
\[detectionstats\]Detection Statistics
--------------------------------------
####
Each source has two flags, one for HCO$^+$ and one for N$_2$H$^+$; multiple flags are not set for any of the sources/tracer pairs; it is either a detection (1,2,4,5) or a non detection (0). A breakdown of the number of sources with each flag is shown in Figure \[stats\]. Out of a total of 1882 sources observed we detect 1444 (76.7%) in HCO$^+$ and 952 (50.5%) in N$_2$H$^+$ at a $3\sigma$ or greater level. Out of 1444 HCO$^+$ detections, 1119 (77.49%) are single-velocity component detections, 39 (2.70%) are multiple-velocity component detections, 67 (4.64%) have possible line wings, and 219 (15.17%) have a possible self-absorption profile. For the 952 N$_2$H$^+$ detections, 919 (96.53%) are single-velocity component detections, 14 (1.47%) are multiple-velocity component detections, 6 (0.63%) have possible line wings, and 13 (1.37%) have a possible self-absorption profile. Breaking the sources down into the “Deep Sample” where we observed every source in the BOLOCAT v0.7 in certain $\ell$ ranges, we find slightly lower detection rates: 72.6% and 41.2% of the $N=707$ sources are detected in HCO$^+$ and N$_2$H$^+$ respectively (See Figure \[stats\] for the breakdown of flagging statistics for the “Deep Sample”). Detection statistics versus 1.1 mm dust emission are presented in Figure \[percentile\]. For sources in the lowest flux percentile, we detect barely 40% in HCO$^+$. Sources in the highest flux percentile have a 99% detection rate in HCO$^+$ and 88% in N$_2$H$^+$.
####
We find that nearly all (99.6%) of N$_2$H$^+$ detections are associated with an HCO$^+$ detection at the $\ge3\sigma$ level. Only 4 sources that are detected in N$_2$H$^+$ do not have a $3\sigma$ detection in HCO$^+$. These sources are approximately $2\sigma$ detections and show a small amount of HCO$^+$ emission at the correct velocity to be associated with the N$_2$H$^+$ emission. The HCO$^+$ line flag statistics change in a fairly interesting way for the sources with detected N$_2$H$^+$. The percentage of sources showing a possible self-absorbed profile increases from 15.17% to 21.10%. The percentage of single line detections drops by a similar amount. About 1% of sources show possible self-absorption in both HCO$^+$ and in N$_2$H$^+$. For sources that display self absorption in N$_2$H$^+$, 11 of 13 also showed self absorption in HCO$^+$.
####
A recent mapping survey of these two molecular transitions toward a sample of IRDCs has shown that HCO$^+$ and N$_2$H$^+$ $J=3-2$ emission has a very similar extent and morphology to the 1.1 mm emission (Battersby et al. 2010). To completely understand the physical properties of the gas that is excited in these transitions, we require a detailed source model and radiative transfer modeling of multiple transitions in each species. However, from our astrochemical knowledge of these two species, we can make some general statements about the regions where they are excited. HCO$^+$ probes clumps with a wide range of properties. It can exist in warm regions where CO is abundant (e.g., Reiter et al. 2011) and cold regions where CO has frozen out onto dust grains (e.g., Gregersen and Evans 2000). It is possible that HCO$^+$ is depleted by freeze-out in some of these clumps with cold ($T < 20$ K) dense ($n > 10^4$ cm$^{-3}$) gas within the HHT beam (see Tafalla et al. (2006) for examples observed toward low mass cores), although our observations indicate that this mechanism is unlikely to be dominant in BGPS clumps. HCO$^+$ $J=3-2$ emission likely originates in the dense, warm inner regions of these clumps. In contrast, N$_2$H$^+$ is destroyed by CO in the gas phase, and thus N$_2$H$^+$ is most abundant in cold, dense gas where the CO abundance is depleted (Jørgensen et al. 2004). Thus, in star-forming clumps with a strong temperature increase toward the center, N$_2$H$^+$ may only be tracing the outer parts of the clumps where the gas is still relatively dense and cold. This chemical differentiation of N$_2$H$^+$ has been mapped in a few high-mass star-forming regions (Pirogov et al. 2007; Reiter et al. 2011; Busquet et al. 2010), although the differentiation mostly occurs on size scales that are unresolved within our $30^{\prime\prime}$ beam.
####
In nearly 12% of sources, the HCO$^+$ line profiles display apparent self-absorption. For an optically thick line profile, a blue asymmetry (redshifted self-absorption) may be an indication of infalling gas (see Myers et al. 2000); however, the blue asymmetric profile can also be created by rotating and outflowing gas (Redman et al. 2004). For a large sample of sources, it is possible to statistically identify infall in the population by searching for an excess of blue asymmetric profiles. Infall does not create a red asymmetric profile in centrally heated, optically thick gas while rotation and outflow can equally produce both blue and red asymmetric profiles. Surveys of high-mass star-forming regions in HCN $J=3-2$ have shown statistical excesses in blue asymmetric profiles (Wu & Evans 2004). In order to calculate the line asymmetry of our subset of sources with self-absorbed profiles, we must obtain observations of an optically thin isotopologue (H$^{13}$CO$^+$) to discriminate between self-absorption and a cloud with two closely-spaced velocity components along the line of sight. We shall observe this subset of sources in H$^{13}$CO$^+$ $J=3-2$ in a future study.
Integrated Intensity and Peak Line Temperature Analysis
-------------------------------------------------------
### Comparison of Molecular Emission
####
Figures \[ithist\] (a) and (b) show the difference in the distributions of line temperature and integrated intensity for HCO$^+$ and N$_2$H$^+$ $J=3-2$. The HCO$^+$ emission extends to far greater intensities than N$_2$H$^+$, whose distribution seems to be truncated at high intensities. We find, on average, in our Full Sample, HCO$^+$ $J=3-2$ to be 2.18 times as bright as N$_2$H$^+$ $J=3-2$ in integrated intensity I(T$_{mb}$) (Figure \[HCOPvN2HP\] (a)). There are a small number of sources (11.8%, $N=114$) detected in both HCO$^+$ and N$_2$H$^+$ that show stronger N$_2$H$^+$ $J=3-2$ emission than HCO$^+$ $J=3-2$ emission. For these sources 1/3 are self absorbed in HCO$^+$. The integrated intensity of each species is highly correlated with a Spearman’s rank correlation coefficient of $\rho=0.82$. The slope of a linear regression (MPFIT) is $m=0.82$, taking into account errors in x and y directions. The highest intensity points appear to form a tail turning upward on the plot of I(T$_{mb}$ HCO$^+$) versus I(T$_{mb}$ N$_2$H$^+$). This would be expected for the warmest clumps since the N$_2$H$^+$ abundance is expected to decrease in warmer regions, which should be more prevalent toward brighter 1.1 mm sources (§3.2.2).
####
The peak line temperature tells a similar story to the integrated intensities. The upward curving tail of points at the brightest end of the Full Sample is less noticeable for peak line temperature (Figure \[HCOPvN2HP\] (b)). The average ratio of peak line temperatures is T$_{mb}($HCO$^+$ $J=3-2)/$T$_{mb}($N$_2$H$^+$ $J=3-2) = 1.94$ and the correlation coefficient is $\rho = 0.79$. The best fit line has a slope of $m=0.83$.
####
It is interesting that HCO$^+$ and N$_2$H$^+$ $J=3-2$, with their similar effective densities but different chemistries, are so highly correlated. Their similar effective densities should result in their emission being co-spatial; however, their chemical differences should result in differentiation (e.g., Jørgensen et al. 2004; Pirogov et al. 2007). It is likely that any differentiation is unresolved within our beam, which averages over the densities, temperatures, and abundance structure on size scales of a few tenths of a parsec (e.g., Battersby et al. 2010; Reiter et al. 2011).
### Comparison with Millimeter Continuum Emission
####
In Figures \[ITvs1mm\] (a) and (b), we show the integrated intensity of HCO$^+$ $J=3-2$ and N$_2$H$^+$ $J=3-2$ versus the 1.1 mm flux per beam obtained from the BOLOCAT v0.7 positions on the v1.0 BGPS maps (Rosolowsky et al. 2010). The Spearman’s rank coefficient for the two species are $\rho_{HCO^+} = 0.80$ and $\rho_{N_2H^+} = 0.73$. The slopes are $m_{HCO^+}=1.15$ and $m_{N_2H^+} = 1.28$. For HCO$^+$, the high 1.1 mm flux points have a tail that continues curving up toward higher HCO$^+$ emission with increasing 1.1 mm flux. In contrast, the N$_2$H$^+$ $J=3-2$ emission for high 1.1 mm fluxes shows a flattening which is again consistent with N$_2$H$^+$ being less abundant in warm sources. The median ratio of integrated intensity to 1.1 mm emission is 6.32 K km s$^{-1}$ per Jy/beam and 3.27 K km s$^{-1}$ per Jy/beam for HCO$^+$ $J=3-2$ and N$_2$H$^+$ $J=3-2$, respectively.
####
Comparing the peak line temperatures of the molecular emission versus the 1.1 mm dust flux (Figures \[ITvs1mm\] (c) and (d)) leads to similar results as the integrated intensities. The correlations are still significant: $\rho_{HCO^+} = 0.75$ and $\rho_{N_2H^+} = 0.73$. The slopes are lower than for integrated intensity: $m_{HCO^+}=0.88$ and $m_{N_2H^+} = 0.98$. The median ratio of peak line temperature to 1.1 mm emission is 1.76 K per $Jy/beam$ and 1.06 K per $Jy/beam$ for HCO$^+$ $J=3-2$ and N$_2$H$^+$ $J= 3-2$, respectively.
####
We also compare the ratios of the integrated intensities and the peak line temperatures of HCO$^+$ and N$_2$H$^+$ with 1.1 mm flux in Figure \[ratiov1mm\]. Both ratios are uncorrelated with the 1.1 mm dust emission. Surprisingly, there is a wide range in the observed intensity and peak temperature ratios, even for bright sources with fluxes above $1$ Jy. Even in these brightest 1.1 mm sources, the N$_2$H$^+$ $J=3-2$ emission can be strong, indicating significant amounts of unresolved dense, cold ($T < 20$ K) gas within the beam.
Velocities and Linewidths
-------------------------
####
We use the Gaussian fit velocity centers of the HCO$^+$ and N$_2$H$^+$ lines as described in §\[gaussian\] to determine v$_{LSR}$. We plot v$_{LSR}$ versus Galactic longitude in Figure \[lvlsr\] $(a)$. We find the distribution of v$_{LSR}$ in the dense gas tracers is comparable to that of CO 1-0 (Dame et al. 2001). The spread in dense gas velocities is very similar to the spread in CO emission at each $\ell$ when our data is overplotted on the Dame et al. (2001) $v-\ell$ map. The v$_{LSR}$ determined for sources detected in both HCO$^+$ and N$_2$H$^+$ agree very well, see Figure \[lvlsr\] $(b)$. We use only the Gaussian fit HCO$^+$ velocities in §4 to calculate the kinematic distances of sources.
####
Figure \[lvlsr\] $(c)$ shows the FWHM of our detected HCO$^+$ lines versus Galactic longitude. There is no trend with $\ell$ apparent in the sources we have observed. The few bins where the linewidth seems to vary by an appreciable amount have small numbers of sources in them. There is a moderate relationship in the plot of $\Delta v[$HCO$^+]$ versus $I(T_{MB})[$ HCO$^+]$, see Figure \[deltavIhcop\] $(a)$. This is expected, as the integrated area of a line is directly related to the peak temperature multiplied by the FWHM. Given the relationship between $\Delta v[$ HCO$^+]$ and $I(T_{MB})[$ HCO$^+]$ and $S_{1.1mm}$ and $I(T_{MB})[$ HCO$^+]$, it is logical to expect a trend of $\Delta v$ with $S_{1.1mm}$; Figure \[deltavIhcop\] (b) shows this trend. A moderate correlation also exists between the linewidth and the 1.1 mm dust emission per beam at the BOLOCAT v0.7 positions; however, there is large amount of scatter around this trend.
Discussion
==========
Kinematic Distances
-------------------
####
We use the kinematic model of the Galaxy as defined by the parameters determined by Reid et al. (2009) to calculate the near and far distances to BGPS clumps. One thing to note is that the distance determination for Reid et al. (2009) assumes all motions are in the azimuthal direction only and does not account for any radial streaming which is known to exist near $\ell \sim 0$. This model sets the distance from the Galactic center to the Sun to be $R_0 = 8.4\pm0.6$ kpc and the circular rotation speed $\Theta_0=254\pm 16$ km s$^{-1}$ from VLBI parallax measurements. We then use these parameters and the kinematic definition of v$_{LSR}$ to compute the distances to all of our sources with single HCO$^+$ velocity components.
####
The distances for all detected sources are plotted versus Galactic longitude in Figure \[lvdist\]. In the first quadrant ($0^\circ \le \ell < 90^\circ$), a velocity will give two distances that are degenerate. Without further information, we cannot tell if a source is on the near side or the far side of the galaxy. For sources that are known to be within a given region, and thus approximately the same distance, we can quantify the velocity spread of the individual sources. For instance, the spread in v$_{LSR}$ for sources in $109^\circ < \ell < 112^\circ$ is 4.9 km s$^{-1}$. This is one measure of the systematic “random” errors in our v$_{LSR}$ due to intrinsic motion that limits the accuracy of the corresponding distances. Some sources have much larger peculiar motions determined from VBLI parallax, as great as 40 km s$^{-1}$ (Nagayama et al. 2011), but it is not likely the majority of sources will be severely discrepant. Some distribution of distances is expected and the spread in velocities for sources nearby makes an accurate kinematic distance determination difficult. In our distance determination, a cloud with a velocity at or greater than the tangent velocity will be placed at the tangent distance. In § 4.2, we resolve the distance ambiguity for a subsample of our sources.
### Galactocentric Distance
####
The Galactocentric distance is the distance of the source from the Galactic Center and is only dependent on velocity and Galactic longitude of a source and therefore does not have a distance ambiguity. We plot a variety of source properties versus their Galactocentric distance in Figure \[GCdist\]. The distribution of sources clearly traces four major spiral arm structures in the Galaxy. The large peak at $4.5$ kpc corresponds to the molecular ring and, for sources near $\ell = 30^{\circ}$ , the edge of the central bar. The largest concentration of sources is within these two structures. The other structures in order of galactocentric distance are the Sagittarius arm, the local arm, and the Perseus arm.
####
We also plot the observed quantities (linewidth, integrated intensity, and 1.1 mm flux) versus Galactocentric distance. There is a large amount of scatter in each 1.5 kpc bin, and the median values of each quantity are nearly constant except for the bin beyond 10 kpc. The median linewidth, integrated intensity (both HCO$^+$ and N$_2$H$^+$), and 1.1 mm flux are systematically lower for sources beyond 10 kpc compared to smaller Galactocentric distances. This could be due to a bias in the original BGPS observing strategy. Sources at Galactocentric distances greater than 10 kpc are predominately in the second quadrant of the Galaxy. Only a few selected regions (e.g., Gem OB1, G111/NGC7538, IC 1396) were observed by the BGPS in this quadrant. Unlike BGPS observations toward the first quadrant, these second-quadrant heterodyne follow-up observations are not a complete census of sources in the second quadrant and biased toward known star forming regions. It is possible that the observed regions are not entirely representative of the properties of Galactic sources at this distance biasing our results to sources with stronger HCO$^+$ and N$_2$H$^+$ lines. This would make the “true” Galactocentric gradient larger than what we see in Figure \[GCdist\]. Another possibility is that nitrogen and/or carbon metallicity gradients in the Galaxy are becoming important, and that is why HCO$^+$ and N$_2$H$^+$ are becoming weaker, on average, beyond 10 kpc. There is a decreasing trend for both N and C in OB stars when going from the inner galaxy to the outer galaxy (Daflon & Cunha 2004). This same effect may manifest itself in the dense gas as well; however, more complete sampling is needed to understand these effects.
Resolving the Distance Ambiguity
--------------------------------
####
We must determine whether or not a source lies on the near or far side of the tangent distance in order to resolve the distance ambiguity (see Figure \[lvdist\]). We use three conservative methods to break the degeneracy: coincidence with sources with observed maser parallax measurements, coincidence with Infrared Dark Clouds (IRDCs),and correspondence with known kinematic structures in the Galaxy. We then use this subsample of sources where the distance ambiguity has been resolved, the “Known Distance Sample,” to study the properties of BGPS objects. A detailed study of the probability that BGPS sources in the first quadrant lie at the near or far distance is currently being made by Ellsworth-Bowers et al. (in preparation).
####
The most accurate distance determination technique is direct parallax measurements of sources by very long baseline interferometry (e.g., Reid et al. 2009a, 2009b). For a source to be considered associated with a VLBA-determined parallax, we allow for a source position to be different from the VLBA position by up to 30$^{\prime\prime}$ (one beam size). We have 4 sources that are coincident on the sky with a VLBA source but only 3 with a single HCO$^+$ detection. The distances used to these objects are their parallax distances.
####
The next selection criterion we used to determine distance is coincidence with an IRDC. IRDCs are clouds of dust that appear dark against the background of mid-infrared Galactic radiation (for example at 8 $\mu$m). Because these objects appear in front of the majority of Galactic emission, they are assumed to be on the near side of the galaxy. Placing an IRDC is at the near kinematic distance is a good assumption but disregards the fact that a small number of IRDCs could be at the far distance. IRDCs have been catalogued in the Galactic plane from *MSX* mid-infrared observations (Egan et al. 1999; Simon et al. 2006) and most recently from *Spitzer Space Telescope* observations (Peretto & Fuller 2009). Peretto & Fuller (2009) developed a catalog of IRDCs and describe each cloud with an ellipse. We first select sources that lie within the ellipse itself. In reality, IRDCs have a wide range of projected geometries and a simple ellipse is not always the best choice to describe more complicated filamentary shapes. Therefore, we also did a by-eye comparison of Spitzer GLIMPSE (Benjamin et al. 2003) images and BGPS images and made a list of BGPS sources that appeared to be coincident with an IRDC. Both the ellipse-coincidence method and the by-eye method suffer from different biases (e.g., the ellipse shape is too simple and the by-eye method is subjective and depends on image display parameters). We conservatively choose sources that are coincident with an IRDC from both methods to be assumed to be at the near distance ($N=192$).
####
We have also included sources in the Known Distance Sample that do not have distance ambiguities, including sources in the Outer Galaxy ($\ell > 90^\circ$, $N=89$) and sources in the first quadrant that lie at the tangent distance. A caveat is that the method used to compute the distances forces a source to be at the tangent distance if its velocity is larger than allowed in the circular rotation model of the Galaxy. We also include sources that are coincident with the sources in Shirley et al. (2003) and the H [II]{} regions in Kolpak et al. (2003). In addition to all of this we use kinematic information from Dame et al. (2001) to include sources near $\ell \sim 80^\circ$ that have the corresponding velocities of the Cygnus-X region. The other sources in this $\ell$ range have negative velocities, indicating they lie in the outer arms. We do the same analysis and add sources near $20^\circ < \ell< 55^\circ$ that have velocities corresponding to the Outer Arm. We also add a few sources with $\ell \sim 10$ that correspond to the 3 kpc Arm. All of the 529 sources for which the distance has been determined comprise our Known Distance Sample. For the remainder of the paper, we will only use the Known Distance Sample for our analysis of source properties unless otherwise specified. Resolution of the distance ambiguity for all observed sources is beyond the scope of this paper. In a future paper, we will build probability density functions for the distances to BGPS sources by combining dense gas tracers such as HCO$^+$ and N$_2$H$^+$ with extant data sets such as the Galactic Ring Survey ($^{13}$CO ($J=1-0$) for diffuse-gas velocity comparison, H [I]{} Galactic Plane Surveys (VGPS, CGPS, SGPS) for H [I]{} Self-Absoption, models of Galactic molecular gas distribution, and a more refined analysis with IRDCs (Ellsworth-Bowers et al., in preparation).
####
Figure \[disthist\] is a histogram of the heliocentric distances for the Known Distance Sample. These sources include outer Galaxy sources, so the peaks of the distribution do not always correspond to a specific spiral arm. The observed peak of sources at 5 kpc from us does correspond with the Near 3 kpc arm and the edge of the Galactic bar. The median distance of the Known Distance Sample is 2.65 kpc. The large number of sources at a distance of 1-2 kpc comes from sources that are in the range of $\ell = 80^\circ-85^\circ$ and those in the outer galaxy (W3/4/5, NGC 7538). In this area, the intrinsic dispersion of velocities is much larger than the allowable velocities for the kinematic model of the galaxy so we use known average distances to the regions in this $\ell$ range.
####
Compiling this distance information together gives us a look at Galactic structure. Figure \[galpolar\] shows the face-on view of the Milky Way given the kinematic distances we have measured. The sources that lie in the Near Sample are separated from those where we have not resolved the distance ambiguity. The unresolved sources are plotted twice for their near and far distances. Even with the distance ambiguity affecting the majority of our sources one can begin to see strong evidence for Galactic structure including the Near 3 kpc arm, the end of the Galactic bar, as well as the Sagittarius and Perseus Arms. The “molecular ring” is also visible between 3 and 5 kpc from the center of the Galaxy.
Size-Linewidth Relation
-----------------------
####
Several physical properties of the clumps may be derived once the distance to the source is determined. The size of BGPS sources were determined from analysis of the flux distribution in 1.1 mm continuum images (Rosolowsky et al. 2010). The physical size of the object is simply $r = \theta \cdot Distance$, where $\theta$ is the angular radius of the sources. Figure \[raddist\] (a) shows histograms of the deconvolved angular source size as measured in the BOLOCAT v1.0. The median source size for the Known Distance Sample is $\sim 60^{\prime\prime}$, or 0.752 pc. This is a factor of 2.35 larger than the CS $J=5-4$ source sizes from Shirley et al. 2003 (0.32 pc median source size) but inline with Wu et al. 2010 who find median sizes of 0.71 pc in HCN $J=1-0$ 0.77 pc in CS $J=2-1$. This indicates that the typical BGPS clump is of lower density and more extended than the sample of water maser selected clumps traced in CS $J=5-4$ in the Shirley et al. survey. CS $J=5-4$ also has an effective density an order of magnitude higher than HCO$^+$ $J=3-2$ and is tracing the denser gas toward the center (Reiter et al. 2011). The Shirley et al. (2003) CS $J=5-4$ survey was a very heterogeneous sample of sources that spanned a wide range of distances. In comparison to a more homogenous sample at a common distance the continuum survey of the nearby GMC Cygnus X, by Mőtte et al. (2007), find an average source size of 1.2 mm continuum clumps of 0.1 pc, with a HPBW of 11$^{\prime\prime}$; this is a factor of seven times smaller than the median BGPS clumps size.
####
We plot the clump size against the distance to the source (Figure \[raddist\] (b)). There is a very strong relationship directly due to the fact that the physical size grows directly proportional to distance for a given angular size. The line indicates the physical size that is equivalent to our beamsize at each distance. This shows that 89% of sources are spatially resolved by the BGPS beam. We caution that we are not resolving everything within the source as there is clear evidence of unresolved cores within the larger clump structures. Rather, the large size indicates that BGPS objects are, on average, larger and more diffuse structures than have been systematically studied before in high-mass star-formation surveys.
####
The integrated kinematic motions along the line-of-sight determine the observed FWHM linewidth of the clumps. Linewidths may be broadened due to thermal motions, unresolved bulk flows of gas across a cloud, small scale (unresolved) turbulence, and optical depth effects. Thermal broadening for our sources is minimal since $\Delta v_{therm}(FWHM) = 0.612$ km s$^{-1}$ at $T=20$ K. The observed median linewidth of HCO$^+$ for the Known Distance Sample is 2.98 km s$^{-1}$, which indicates supersonic motions within the $30^{\prime \prime}$ beam. The typical linewidth is comparable with the observed linewidths toward high-mass star forming clumps (e.g., Shirley et al. 2003) and much larger than the typical linewidths observed toward nearby low-mass star forming cores (e.g., Rosolowsky et al. 2008). Comparing the distribution of linewidths versus distance shows no trend at all. As a source gets farther away, our beam is averaging over a larger source size; yet, we do not see any systematic increase in linewidth with distance.
####
Optical depth effects may broaden the observed linewidth. This effect may be especially acute for the HCO$^+$ $J=3-2$ line (e.g., $> 11$% of sources have evidence of possible self-absorption). Equation \[widthvtau\] shows that as the optical depth increases, the linewidth increases: $$\frac{\Delta v}{\Delta v_o} = \frac{1}{\sqrt{ln 2}} \; \sqrt{ln(\frac{\tau }{ln(\frac{2}{1 + e^{-\tau}})})}
\label{widthvtau}$$ (Phillips et al. 1979). For $\tau = 10$, the optically thick linewidth to the optically thin linewidth ratio is larger by a factor of 2. For our observed linewidths, which are 10 times the thermal linewidth, even accounting for modest optical depth effects, it is unlikely that optical depth effects can account for the large observed median linewidth. Therefore, we conclude that the dense molecular gas in typical BGPS sources is characterized primarily by supersonic turbulence.
####
The size and linewidth of the Known Distance Sample clumps are directly compared in Figure \[sizelinewidth\] (a). In contrast to the Larson relationship for molecular clouds (Larson 1981) we do not observe a strong size-linewidth relationship in the dense molecular gas for BGPS clumps: $\rho_{Spearman}=0.40$. While there is a very weak trend that generally agrees with Larson’s relationship, the scatter in the data erases our ability to say much about it. Traditionally, studies of the size-linewidth relationship in cores find two different slopes depending on the mass of the objects (Caselli & Myers 1995). For instance, the study of Caselli & Myers (1995) find a very shallow slope of R$^{0.21}$ for high-mass cores and a much steeper slope of R$^{0.53}$ for low-mass cores. Combining these two distributions may partially erase the size-linewidth relationship (Shirley et al. 2003); however, the derived power laws from Caselli & Myers (1995) predict significantly smaller linewidths than the sources in the Known Distance Sample. The lack of a correlation between size and linewidth argues against a universal scaling relationship between the amount of supersonic turbulence in dense clumps and their size.
Mass Calculations
-----------------
####
The clump mass may be calculated from the total 1.1 mm continuum flux for each source in the Known Distance Sample by $$M_{H_{2}} = \frac{S_{1.1}\cdot D^2}{B_\nu(T_{dust}) \cdot \kappa_{dust,1.1}\cdot \frac{1}{100}}
\label{Masseqn}$$ (Hildebrand 1983) where $S_{1.1}$ is the total dust emission from a source, D is the distance to the source, $\kappa_{dust}(1.1mm)=1.14$ cm$^2$ g$^{-1}$ is the dust opacity at 1.1 mm (Ossenkopf & Henning 1994), $B_{1.1mm}$ is the blackbody intensity at 1.1 mm where we initially assume a temperature of $T_{dust} = 20$ K. We also assume a gas-to-dust mass ratio of 100:1 (Hildebrand 1983). Millimeter dust continuum observations are incredibly sensitive to small amounts of mass, as we can detect a 20 K source of 1.1 M$_{\odot}$ at distances of 1 kpc given an average $3\sigma$ flux threshold of 90 mJy at 1.1 mm. Below (§ 4.4.1), we will systematically explore the results of changing the dust temperature distribution of BGPS sources using a Monte Carlo simulation.
####
Figure \[masshistdist\] (a) plots a histogram of the masses of the Known Distance Sample assuming $T_{dust} = 20$ K. The median mass of the sample is 320 M$_\odot$ and the mean is 1648 M$_\odot$. The BOLOCAT is complete to 98% for sources $> 0.4$ Jy; this gives us a completeness limit for masses of 580 M$_\odot$ at a distance of 8.5 kpc (most of our sources are at the near distance). The mass versus distance plot, Figure \[masshistdist\] (b) shows the expected trend of more massive sources at the farthest distances due to the $D^2$ term in the mass calculation. The masses we observed range from 10 $M_{\odot}$ to 10$^5$ $M_{\odot}$. Compared to other samples of high-mass clumps, we probe similar mass ranges as the H$_2$O maser sample of Shirley et al. (2003); although, our observed median mass is smaller. The mass distribution is similar to that observed for the BGPS Galactic Center sample for sources assumed to be at the distance of the Galactic center, 8.5 kpc (Bally et al. 2010). In comparison with clumps in targeted regions in Orion (Johnstone & Bally 2006) and Cygnus-X (Mőtte et al. 2007) probe typical masses of 10s to 100s of $M_{\odot}$, more analogous to core masses. In comparison to IRDCs, (Rathborne et al. 2005, Peretto & Fuller 2010), we appear to probe the same physical properties on the same scales. IRDC fragments tend to agree with masses of clumps in targeted regions, while the overall IRDC properties agree with those found in this study. For example, in Peretto & Fuller (2010) they find mass ranges of a few tenths of solar masses to nearly 10$^{5}$ $M_{\odot}$ and are complete to $\sim 800$ $M_{\odot}$. We are sensitive throughout the entire mass range of these other samples, leading us to believe we are observing objects that span from dense cores to clouds (see Dunham et al. 2010).
####
Using the dust-determined mass and the source size, we compute the volume-averaged number density given a mean free particle weight, $\mu=2.37$ (Figure \[Avgnumberden\] (a)). Assuming a spherical volume for each source. The median value is $n=2.5\times10^3$ cm$^{-3}$, which is within a factor of $3$ of the masses determined in the NH$_3$ survey of Gem OB1, where they find a median volume-averaged density of $n=6.2\times10^3$ cm$^{-3}$ (Dunham et al. 2010a). Compared to the IRDC sample of Peretto & Fuller (2010), we probe the same range of volume-averaged number densities from 100 to 10$^{4}$ $cm^{-3}$. This low volume-averaged number density is a result of the large observed source sizes and, therefore, large volume observed toward BGPS clumps. The volume-averaged density is lower than the typical effective excitation density for HCO$^+$ and N$_2$H$^+$ $J=3-2$ emission. Steep density gradients are known to exist in both low-mass (Shirley et al. 2002) and high-mass (Mueller et al. 2002) star-forming clumps. If we were resolving individual dense knots, we would expect to see a trend of linewidth with volume-averaged density as well (Figure \[Avgnumberden\] (c)). Therefore, BGPS sources tend to be fairly low-density sources with possible compact high-density regions that may be probed with higher angular resolution observations. Some of these clumps of low density gas contain high density regions up to $n=10^5$ cm$^{-3}$, several orders of magnitude higher than the average source properties (Dunham et al. 2011).
####
Comparing the time scales of these clumps shows that the free-fall and crossing time scales are similar at a few times 10$^5$ years. The median free-fall time is $8\times 10^5$ yr; thus, if these clouds were bound, it would take a few hundred thousand years to collapse and begin forming stars. With the masses of these clumps we also compute the virial linewidth as $$\Delta v^{2}_{virial} = \frac{8 \ln{2}\; a\; G\; M_{virial}}{5\; R} \approx \frac{a \cdot\frac{M_{obs}}{209\;M_{\odot}}}{(R\;/1\;pc) (\Delta v_{obs}/1\;km\;s^{-1})^{2}} \; km \; s^{-1}
\label{viriallinewidth}$$ $$a = \frac{1-p/3}{1-2p/5}; p=1.5
\label{virialavalue}$$ This definition includes the correction factor for a power-law density distribution of p=1.5 (Bertoldi & McKee 1992, Shirley et al. 2003). The virial linewidth represents the velocity dispersion from internal motions due to self-gravity. Figure \[sizelinewidth\] (b) shows the virial linewidths are smaller on average than the observed linewidths, about 2–3 km s$^{-1}$ indicating that many of the BGPS clumps are not entirely gravitationally bound. It is likely that there are smaller denser regions within these clumps that are gravitationally bound.
We can also look at the virial parameter in terms of the surface density (see §4.4.1) of the clumps given by $$\alpha_{virial} = \frac{5 \Delta v^2_{obs}} {8\ln{2}\pi G \Sigma R}.
\label{virialparam}$$ This is shown in Figure \[sizelinewidth\] (c). The virial parameter versus mass is also shown in Figure \[sizelinewidth\] (d). This is another way of showing that few of our clouds indicate virialized motion ($\alpha < \sim 1$) and most lie more than a factor of a few above this line. The caveat is we assume the BGPS traces ALL the mass which may not be entirely true. The BGPS does resolve out diffuse emission and the dense gas tracers are missing the low-density gas traced by CO or H [I]{}. It appears that most of the clumps we see are likely not gravitationally bound but do contain dense substructures that likely are.
### \[DiffMass\]Differential Mass Histogram
####
Figure \[masshistdist\] (c) shows the differential mass histogram for the number of sources in bins of log(M). In this parameterization, the mass function takes on the form of $$\frac{dN}{dlog(M)} \sim M^{-(\alpha -1)},
\label{massfunc}$$ where the power-law index is the slope of a line through the histogram. For the observed masses, calculated assuming a dust temperature of $20$ K, we find a slope of $\alpha - 1 = 0.91$. This slope is shallower than the slope of the Salpeter stellar IMF $\alpha - 1 = 1.35$ for dN/dlog(M). Our observed slope is steeper than has been found for the mass distribution of CO clumps $\alpha - 1 = 0.6 - 0.7$ (e.g., Scoville & Sanders 1987). Observations of dense-gas tracers tend to increase the differential mass histogram slope. For instance, Shirley et al. (2003) find a slope of $\alpha -1 = 0.9$ for their cumulative mass function for cores probed by CS $J=5-4$. In comparison to IRDCs (Peretto & Fuller 2010), they find a slope of $\alpha - 1 = .85$ for the total IRDC. This intermediate slope also suggests that we really are looking at the intermediate case of star forming “clumps” rather than clouds or cores.
####
There are several sources of uncertainty in determining the slope of the differential mass histogram which we must characterize. The effect of binning has been shown to cause problems in studies of the stellar mass function and thus will also be problematic with the clump mass spectrum (Ma[í]{}z Apell[á]{}niz & [Ú]{}beda 2005). For instance, the choice of binsize may have a substantial effect on the computed slope of the differential mass histogram. Bins with small numbers of sources dominate the fit if the binwidth is chosen too small, and the number of bins used in a linear regression decreases rapidly if the binwidth is too large. Our data do not span many decades in mass above the the estimated completeness limit in the Known Distance Sample, and thus choosing an appropriate binwidth is difficult. We can circumvent this binning problem by using the maximum-likelihood estimator (MLE) to compute the power-law slope $\alpha$ (Clauset et al. 2007, Swift & Beaumont 2010). The MLE maximizes the likelihood that the data was drawn from a given model. If the data are drawn from the distribution given by equation \[MLEDIST\], then the maximum likelihood estimate of the power-law index is $\hat{\alpha}$ given by equation \[MLE\], where $M_{min}$ is the lower bound of the power law behavior and $n$ is the number of sources with mass greater than $M_{min}$ (Clauset et al. 2007), $$p(x) = \frac{\alpha-1}{M_{min}}\left(\frac{M}{M_{min}}\right)^{-\alpha}
\label{MLEDIST}$$
$$\hat{\alpha} = 1+n\left[\sum_{i=1}^{n} \ln \frac{M_i}{M_{min}} \right]^{-1}.
\label{MLE}$$
####
Another important source of uncertainty is the assumed dust temperature of BGPS sources. We initially assume $T_{dust} = 20$ K for every source in the Known Distance Sample to compute our masses (Merello et al., *in prep*). In reality, BGPS sources have a range of dust temperatures of an unknown distribution. We perform a Monte Carlo simulation of the differential mass distribution to constrain the range $\alpha$ by assuming that the source temperature distribution is approximated by a Normal distribution, characterized by mean $T_{mean}$ and $\sigma(T)$. We then compute the mass of each source with a random temperature drawn from the temperature distribution and the source flux that has an error term drawn randomly from a Normal distribution added in. For each $T_{mean}$ and $\sigma(T)$, the mass histogram is computed $10^4$ times, and we calculate the median value of $\hat{\alpha}$.
####
The variation of $\alpha$ with the temperature distribution is shown in Figure \[montecarlo\] (a). The effect of temperature distribution is significant but not strong. As part of the MLE procedure we estimate the best value of $M_{min}$, the minimum mass used in the power-law fit for each mass distribution. This is determined by minimizing the KS statistic between the best-fit model and $M_{min}$ as a function of $M_{min}$ (Clausset et al. 2007; Swift & Beaumont 2010). The dependence of $M_{min}$ on temperature means that colder temperatures lead to higher masses for a given observed flux. This dependence is also seen in the median of the mass distribution, Figure \[montecarlo\] (b), and the median volume-averaged number density, Figure \[montecarlo\] (c).
####
There are several important caveats that must be considered in this analysis. Foremost, the BGPS pipeline reductions will leave artifacts in the data. The process by which the sky variation is removed from the data will also remove some of the diffuse emission, resulting in spatial filtering of the 1.1 mm brightness distributions (Aguirre et al. 2011). Furthermore, the cataloguing algorithm is also more sensitive to peaked emission than low levels of diffuse emission (Rosolowsky et al. 2010). This means we are not seeing diffuse sources unless they are very bright. We are missing flux from extended emission surrounding the clumps we do see, thus leaving us with a lower limit on the total mass, and it also affects the derived sizes. This survey is flux-limited, which means we also suffer from Malmquist bias. The BGPS is sensitive to different types of sources (i.e. cores and clumps vs. clouds) as the distance increases (see Dunham et al. 2010). The last caveat is that the brightest BGPS clumps (most massive) are associated with H [II]{} regions which heat the dust and may contain a significant amount of free-free emission which would most likely lead to a steepening of the mass spectrum. However, we present one of the first differential mass distribution with statistically significant numbers showing that massive clumps appear to have a shallower slope than nearby low-mass cores.
### Mass Surface Density
####
While the derived quantities discussed above depend explicitly on distance, one quantity that is distance independent is mass surface density. We compute the mass surface density while assuming $T_{dust} = 20K$ from $$\Sigma_{H_{2}} = \frac{M_{H_{2}}}{\pi R^2} = \frac{37.2\cdot S_{Jy}}{\theta^2_{arcsec}} \;\;\; \rm{g} \;\; \rm{cm}^{-2},
\label{surfeq}$$ where $\Sigma$ only depends on the the observed flux, observed source size, and the assumed dust temperature.
####
Figure \[surfacedensity\] (a) shows the mass surface density for the all of the BGPS sources we observed with determined fluxes and sizes from the v1.0 catalog ($N=1684$). Since dust temperature variations affect the derived mass (§\[DiffMass\]), we also perform a Monte Carlo simulation to calculate the median mass surface density for distributions with different $T_{mean}$ and $T_{\sigma}$. The Monte Carlo simulation includes variations of the fluxes by the Normal distribution corresponding to the flux and error on the flux. Figure \[surfacedensity\] (b) shows how the median of the surface density distribution changes with temperature. The median of the $T_{dust} = 20$ K distribution is 0.027 g cm$^{-2}$ for the Full Sample and 0.033 g cm$^{-2}$ for the Known Distance Sample, indicating that the properties derived from the Known Distance Sample are likely representative of the total sample. Both values are significantly smaller than the results from Shirley et al. (2003), where they find a median surface density of 0.605 g cm$^{-2}$. Our median surface density is also much smaller than the 1.0 g cm$^{-2}$ threshold that Krumholz and McKee (2008) require for massive star formation (see Fall et al. 2010).
####
The dependence of the median of surface density distribution on temperature is shown in Figure \[montecarlo\] (d). The median mass surface density only increases by up to a factor of $2.5$ for distributions with colder dust temperatures but they are still significantly below that of Shirley et al. (2003). The smaller mass surface density is likely a result of the larger clump sizes measured for BGPS clumps compared to previous surveys. However, our result does not indicate that these BGPS clumps are not capable of forming massive stars. It is likely that observations with higher spatial resolution will reveal the observed mass surface density approaching 1.0 g cm$^{-2}$ for cores within BGPS clumps. Indeed, the infrared populations of BGPS clumps have recently been characterized by Dunham et al. (2011) who found that many BGPS clumps are in fact forming high-mass stars. They find that 49% of sources that are in the regions where the BGPS overlaps with other mid-IR Galactic plane surveys contain at least one mid-IR source.
Conclusions
===========
####
We used the 10m Heinrich Hertz Telescope to perform spectroscopic follow-up observations of 1882 sources in the Bolocam Galactic Plane Survey. We simultaneously observed HCO$^+$ $J=3-2$ and N$_2$H$^+$ $J=3-2$ emission using the dual-polarization, sideband-separating ALMA prototype receiver. Out of the 1882 observed sources, we detect $\sim 77\%$ of the sources in HCO$^+$ and over 50% in N$_2$H$^+$. Multiple velocity components along the line-of-sight to BGPS clumps in these dense molecular gas tracers are rare. Our detection statistics are somewhat biased toward more detections because this sample includes all of the intrinsically brightest sources.
####
We find a strong correlation between peak temperature and integrated intensity of each dense gas tracer with each other and with the 1.1 mm dust flux. The median ratio of HCO$^+$ integrated intensity to 1.1 mm flux is 5.42 K km s$^{-1}$ per Jy/beam . We find that HCO$^+$ is brighter than N$_2$H$^+$ for the vast majority of sources, with a subset of only 117 sources (12.6% of sources singly detected in N$_2$H$^+$ ) where N$_2$H$^+$ is the brighter of the two dense gas tracers. The ratio of the peak line temperature and integrated intensity of the two molecules does not correlate well with 1.1 mm dust emission.
####
The observed v$_{LSR}$ appear to follow the same distribution with Galactic longitude as $^{13}$CO $J=1-0$ emission observed by Dame et al. (2001). We determine the linewidths from a best-fit Gaussian model and find little change, on average, with Galactic longitude. The median linewidth is $2.98$ km s$^{-1}$, indicating that BGPS clumps are dominated by supersonic turbulence. Linewidths only modestly correlate with 1.1 mm flux.
####
We compute kinematic distances for all detected sources and are able to break the distance ambiguity for 529 of our detections in the first and second quadrants using IRDC coincidence, VLBA-determined parallax source coincidence, or proximity to the tangent velocity or known kinematic structures. Using the set of sources of known distance, we compute the radius, mass, and average density of the sources. We find the median source size to be 0.752 pc at a median distance of 2.65 kpc, typically larger than source sizes observed in published surveys of high-mass star-forming clumps and cores. Comparing linewidth to the physical size of the source, we do not find any compelling evidence for a size-linewidth relation in our data. We find our sources lie above the relationships found by Caselli and Myers (1995) and have too small of a correlation to say much about Larson’s relationship (Larson 1981).
####
For an assumed dust temperature of 20K we find a median mass of $\sim 300$ M$_\odot$, a low median volume-averaged number density of $2.4\times 10^3$ cm$^{-3}$, and a median mass surface density of $0.03$ g cm$^{-2}$. The similarity of the median mass surface density between the full sample and the Known Distance Sample indicates that the sources in the Known Distance Sample are characteristic of the Full Sample. Compared to published surveys of high mass star formation, BGPS clumps tend to be larger and less dense on average. We also analyzed the variation in median mass and volume density using a Monte Carlo simulation of cores with various dust temperature distributions. From the differential clump mass histogram, we find a power-law slope (dN/dlogM) that is intermediate between the slope derived for diffuse CO clumps and the stellar IMF. Finally, a comparison of the virial linewidth to the observed linewidth indicates that many of the BGPS clumps in this survey have motions consistent with not being gravitationally bound.
####
In the future, we plan to complete observations of the BGPS catalog for all sources with $\ell \ge 10^\circ$ in the dense molecular gas tracers HCO$^+$ and N$_2$H$^+$ with the HHT. This will be the largest systematic survey of dense molecular gas in the Milky Way. A series of follow-up observations are currently underway to characterize the physical properties (density, temperature) and evolutionary state of BGPS clumps.
We would like to thank the operators (John Downey, Patrick Fimbres, Sean Keel, and Bob Moulton) and the staff of the Heinrich Hertz Telescope for their excellent assistance through numerous observing sessions. Yancy L. Shirley is funded by NSF Grant AST-1008577.
[*Facilities:*]{}
Appendix A
==========
In this appendix, we derive the relationship between $\sigma_T$ and $\sigma_I$ for a Gaussian line profile.
Assume that the spectral line is well described by a Gaussian function: $$T(v) = T_{pk} \cdot e^{\left(-4\;ln(2) \cdot v^2/\Delta v_{FWHM}^2 \right)}$$ Where T$_{pk}$ is the peak line temperature and $\Delta v_{FWHM}$ is the FWHM linewidth. The integrated intensity of this line is thus defined by the integral of temperature with velocity over the line window. $$\int T(v)\; dv = T_{pk} \cdot \Delta v_{FWHM} \cdot \sqrt{\frac{\pi}{4\;ln(2)}}$$ We may then ask the question, if we have a non-detection in integrated intensity, then how many $\sigma_T$ (baseline rms) correspond to $3\sigma_I$? Lets rewrite T$_{pk}$ for the undetected Gaussian line as: $$T_{pk} = N \cdot \sigma_{T}$$ The standard formula for the uncertainty of the integrated intensity as given by $$\sigma_{I} = \sigma_{T} \cdot \sqrt{\Delta v_{ch} \cdot \Delta v_{int}}$$ where $\Delta v_{ch}$ is the velocity resolution of each channel in the spectrum, and $\Delta v_{int}$ is the velocity interval integrated over which encompasses the entire line, the line window. We choose a line window that integrates over 99% of the area in the line which is 2 times the 3$\sigma$ velocity width of the Gaussian line. The $\frac{1}{e}$ velocity width is related to the FWHM by: $$v(\frac{1}{e}) = \frac{\Delta v_{FWHM}}{\sqrt{4\;ln(2)}}$$ Thus $2\cdot 3\sigma_v$ (the full width of the line, in both directions) is: $$\Delta v_{int} = 6 \cdot \Delta v_{FWHM} \cdot \sqrt{4\;ln(2)}$$ Therefore, the ratio of I to $\sigma_I$ is $$\frac{I}{\sigma_{I}} = \left(\frac{N \cdot \sigma_{T} \cdot \Delta v_{FWHM} \sqrt{\frac{\pi}{4\;ln(2)}}}{\sigma_{T} \cdot \sqrt{\Delta v_{ch}} \cdot \sqrt{\frac{6 \cdot \Delta v_{FWHM}}{\sqrt{4\;ln(2)}}}} \right)$$ Setting the ratio $\frac{I}{\sigma_{I}}$ equal to 3 and solving for N gives $$N = 6 \cdot \sqrt{\frac{3 \cdot \sqrt{ln(2)}}{ \pi}} \cdot \sqrt{\frac{\Delta v_{ch}}{\Delta v_{FWHM}}}$$ which is approximately $$N \sim 4.88 \cdot \sqrt{\frac{\Delta v_{ch} }{\Delta v_{FWHM}}}$$ For example, if we have a barely resolved line with $\Delta v_{ch} = 1$ and $\Delta v_{FWHM}=1$, a 4.88$\sigma$ detection in temperature of a Gaussian line would be a 3$\sigma$ detection in integrated intensity.
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![The two upper panels, (a) and (b), plot $\eta_{mb}$ determined from the USB and LSB for each polarization. The two lower panels, (c) and (d), compare the two polarization feeds against each sideband. The Spearman’s rank coefficients are printed in the upper left corner of each plot.[]{data-label="polcompjup"}](f2.pdf){width=".8\textwidth"}
![40$^{\prime\prime}$ aperture fluxes for all of the BOLOCAT V1.0, solid blue line, and 40$^{\prime\prime}$ aperture fluxes for the BOLOCAT v0.7 positions that were observed with the HHT as determined from the v1.0 maps, dotted red line.[]{data-label="bolocatvobs"}](f3.pdf){width=".6\textwidth"}
![Sample HCO$^+$ spectra of the 4 main flag types. (a) flag = 1 – single detection. (b) flag = 2 – multiple detections. (c) flag = 4 – Line wing(s). (d) flag = 5 – Self-absorbed spectrum.[]{data-label="sampplots"}](f5a.pdf "fig:"){width=".49\textwidth"} ![Sample HCO$^+$ spectra of the 4 main flag types. (a) flag = 1 – single detection. (b) flag = 2 – multiple detections. (c) flag = 4 – Line wing(s). (d) flag = 5 – Self-absorbed spectrum.[]{data-label="sampplots"}](f5b.pdf "fig:"){width=".49\textwidth"}\
![Sample HCO$^+$ spectra of the 4 main flag types. (a) flag = 1 – single detection. (b) flag = 2 – multiple detections. (c) flag = 4 – Line wing(s). (d) flag = 5 – Self-absorbed spectrum.[]{data-label="sampplots"}](f5c.pdf "fig:"){width=".49\textwidth"} ![Sample HCO$^+$ spectra of the 4 main flag types. (a) flag = 1 – single detection. (b) flag = 2 – multiple detections. (c) flag = 4 – Line wing(s). (d) flag = 5 – Self-absorbed spectrum.[]{data-label="sampplots"}](f5d.pdf "fig:"){width=".49\textwidth"}
![A histogram of HCO$^+$ FWHM determined from the Gaussian fit to the spectrum. The Gaussian fits were computed using the MPFIT package also described in §\[gaussian\]. The median of the distribution of FWHM is 2.9 km s$^{-1}$. As a comparision, the median of the FWHM from the second moment calculation is 2.3 km s$^{-1}$. The Gaussian linewidths for N$_2$H$^+$ are broadened due to hyperfine splitting and are not shown here. Fitting the hyperfine structure of N$_2$H$^+$ leads to a median FWHM of 3.5 km s$^{-1}$ and are shown as the red curve.[]{data-label="linewidthhist"}](f7.pdf){width=".6\textwidth"}
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![To see how detection statistics change with 1.1 mm flux we plot the percentage of detected sources versus the percentile of its flux (e.g. the 25% of sources, by number, with the lowest flux, etc.). The highest percentile has a 99% detection rate in HCO$^+$ and a 88% detection rate of N$_2$H$^+$.[]{data-label="percentile"}](f9.pdf){width=".49\textwidth"}
![(a) A histogram of integrated intensity of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. These histograms are not scaled with respect to each other. They are logarithmically binned. Note the lack of sources with bright N$_2$H$^+$ $J=3-2$ emission and that HCO$^+$ sources are on average brighter than the N$_2$H$^+$. The median $3\sigma_{I(HCO^+)} = 0.37$ K km s$^{-1}$ and median $3\sigma_{I(N_2H^+)} = 0.41$ K km s$^{-1}$. (b)A histogram of peak line temperatures of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. These histograms are not scaled with respect to each other. They are logarithmically binned. While it is apparent from the T versus T plots that HCO$^+$ has a typically higher line temperature it is clear from these distributions there are two different cutoffs for the peak line temperature of HCO$^+$ and N$_2$H$^+$. The average $3\sigma_{T_{pk}(HCO^+)} = 0.16$ K and the average $3\sigma_{T_{pk}(N_2H^+)} = 0.17$ K.[]{data-label="ithist"}](f10a.pdf "fig:"){width=".49\textwidth"} ![(a) A histogram of integrated intensity of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. These histograms are not scaled with respect to each other. They are logarithmically binned. Note the lack of sources with bright N$_2$H$^+$ $J=3-2$ emission and that HCO$^+$ sources are on average brighter than the N$_2$H$^+$. The median $3\sigma_{I(HCO^+)} = 0.37$ K km s$^{-1}$ and median $3\sigma_{I(N_2H^+)} = 0.41$ K km s$^{-1}$. (b)A histogram of peak line temperatures of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. These histograms are not scaled with respect to each other. They are logarithmically binned. While it is apparent from the T versus T plots that HCO$^+$ has a typically higher line temperature it is clear from these distributions there are two different cutoffs for the peak line temperature of HCO$^+$ and N$_2$H$^+$. The average $3\sigma_{T_{pk}(HCO^+)} = 0.16$ K and the average $3\sigma_{T_{pk}(N_2H^+)} = 0.17$ K.[]{data-label="ithist"}](f10b.pdf "fig:"){width=".49\textwidth"}
![A comparison of the integrated intensities and line temperatures of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. The light blue circles represent the 952 sources that have both an HCO$^+$ and N$_2$H$^+$ detection and the blue contours overlaid represent the density of points on the plot. The light red circles are HCO$^+$ detections and 3$\sigma$ N$_2$H$^+$ upper limits. The red contours represent these points. The green circles are detections in N$_2$H$^+$ that do not have a corresponding 3$\sigma$ HCO$^+$ detection and are upper limits for HCO$^+$. The solid line plotted is the 1-1 line and the dotted line represents the best fit to the data. (a) A comparison of the integrated intensities (K km s$^{-1}$) of HCO$^+$ and N$_2$H$^+$. The slope of the best fit line is $m=0.82$. (b) A comparison of the Peak line temperatures in Kelvin of HCO$^+$ and N$_2$H$^+$. The average source has a higher line temperature (T$_{mb}$) in HCO$^+$ than N$_2$H$^+$ by a factor of $\sim 2$. The slope of the bestfit line is $m=0.83$.[]{data-label="HCOPvN2HP"}](f11a.pdf "fig:"){width=".49\textwidth"} ![A comparison of the integrated intensities and line temperatures of HCO$^+$ and N$_2$H$^+$ for the “Full Sample” of sources. The light blue circles represent the 952 sources that have both an HCO$^+$ and N$_2$H$^+$ detection and the blue contours overlaid represent the density of points on the plot. The light red circles are HCO$^+$ detections and 3$\sigma$ N$_2$H$^+$ upper limits. The red contours represent these points. The green circles are detections in N$_2$H$^+$ that do not have a corresponding 3$\sigma$ HCO$^+$ detection and are upper limits for HCO$^+$. The solid line plotted is the 1-1 line and the dotted line represents the best fit to the data. (a) A comparison of the integrated intensities (K km s$^{-1}$) of HCO$^+$ and N$_2$H$^+$. The slope of the best fit line is $m=0.82$. (b) A comparison of the Peak line temperatures in Kelvin of HCO$^+$ and N$_2$H$^+$. The average source has a higher line temperature (T$_{mb}$) in HCO$^+$ than N$_2$H$^+$ by a factor of $\sim 2$. The slope of the bestfit line is $m=0.83$.[]{data-label="HCOPvN2HP"}](f11b.pdf "fig:"){width=".49\textwidth"}
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![$(a)$ The HCO$^+$ v$_{LSR}$ determined from the Gaussian fit for each detected source versus Galactic longitude. The envelope formed by the largest velocity at each $\ell$ represents the tangent velocity. $(b)$ The comparison of v$_{LSR}$ of HCO$^+$ and N$_2$H$^+$. The scatter in the data is 0.71 km s$^{-1}$ which is smaller than the width of a channel. $(c)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the Galactic longitude of each source. We also plot the median FWHM within bins in $\ell$ to emphasize any overall trends. There does not appear to be any strong relationship with $\ell$.[]{data-label="lvlsr"}](f14a.pdf "fig:"){width=".98\textwidth"}\
![$(a)$ The HCO$^+$ v$_{LSR}$ determined from the Gaussian fit for each detected source versus Galactic longitude. The envelope formed by the largest velocity at each $\ell$ represents the tangent velocity. $(b)$ The comparison of v$_{LSR}$ of HCO$^+$ and N$_2$H$^+$. The scatter in the data is 0.71 km s$^{-1}$ which is smaller than the width of a channel. $(c)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the Galactic longitude of each source. We also plot the median FWHM within bins in $\ell$ to emphasize any overall trends. There does not appear to be any strong relationship with $\ell$.[]{data-label="lvlsr"}](f14b.pdf "fig:"){width=".49\textwidth"} ![$(a)$ The HCO$^+$ v$_{LSR}$ determined from the Gaussian fit for each detected source versus Galactic longitude. The envelope formed by the largest velocity at each $\ell$ represents the tangent velocity. $(b)$ The comparison of v$_{LSR}$ of HCO$^+$ and N$_2$H$^+$. The scatter in the data is 0.71 km s$^{-1}$ which is smaller than the width of a channel. $(c)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the Galactic longitude of each source. We also plot the median FWHM within bins in $\ell$ to emphasize any overall trends. There does not appear to be any strong relationship with $\ell$.[]{data-label="lvlsr"}](f14c.pdf "fig:"){width=".49\textwidth"}
![ $(a)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the integrated intensity of the HCO$^+$ line. There is a moderate correlation, with a Spearman’s Rank Coefficient of $\rho = 0.60$. $(b)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the 1.1 mm flux per beam from the BOLOCAT v0.7 positions, light blue circles. There is a lot of scatter but there is a moderate correlation between linewidth and 1.1 mm dust emission, $\rho=0.54$. []{data-label="deltavIhcop"}](f15a.pdf "fig:"){width=".49\textwidth"} ![ $(a)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the integrated intensity of the HCO$^+$ line. There is a moderate correlation, with a Spearman’s Rank Coefficient of $\rho = 0.60$. $(b)$ The HCO$^+$ linewidth determined from the Gaussian fit versus the 1.1 mm flux per beam from the BOLOCAT v0.7 positions, light blue circles. There is a lot of scatter but there is a moderate correlation between linewidth and 1.1 mm dust emission, $\rho=0.54$. []{data-label="deltavIhcop"}](f15b.pdf "fig:"){width=".49\textwidth"}
![Kinematic distance versus Galactic longitude for all sources detected and do not have a flag of *2*. The near and far distances are shown in blue circles and red boxes respectively. This version of the plot accentuates the number of sources that lie near the tangent points at each $\ell$.[]{data-label="lvdist"}](f16.pdf){width=".6\textwidth"}
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![A histogram of Heliocentric Distance for the Known Distance Sample of sources with the distance ambiguity resolved. The median of the distribution is 2.65 kpc. The spike of sources that lies around 1 kpc are sources that are in the range of $\ell = 80-85^\circ$ and sources in the outer Galaxy. We use other distance measurements for sources in these regions.[]{data-label="disthist"}](f18.pdf){width=".6\textwidth"}
![Face on view of Galactic structure. Using the kinematic distances determined along with the Galactic longitude we make a polar plot presenting the face on view of the Milky Way. Sources in the Known Distance Sample are plotted as green triangles. In the first quadrant of the galaxy, sources with an unresolved distance ambiguity are plotted in blue circles and red boxes for the near and far distance respectively. These points are plotted twice to represent the distance ambiguity for those not associated with an object that breaks the degeneracy. The large orange circle represents the Galactic Center and the Sun is located at the origin.[]{data-label="galpolar"}](f19.pdf){width=".6\textwidth"}
![$(a)$ The histogram of the source sizes, in pc, as determined from the distance and the angular size in BOLOCAT v1.0 for the sources in the Known Distance Sample. The median source size is 0.752 pc and is shown by the dotted line. $(b)$ shows the radius of each source versus the distance from us. This trend is primarily from the fact the radius is a function of distance for a given source size. The line plotted is the source size that corresponds to our beamsize at each distance. []{data-label="raddist"}](f20a.pdf "fig:"){width=".49\textwidth"} ![$(a)$ The histogram of the source sizes, in pc, as determined from the distance and the angular size in BOLOCAT v1.0 for the sources in the Known Distance Sample. The median source size is 0.752 pc and is shown by the dotted line. $(b)$ shows the radius of each source versus the distance from us. This trend is primarily from the fact the radius is a function of distance for a given source size. The line plotted is the source size that corresponds to our beamsize at each distance. []{data-label="raddist"}](f20b.pdf "fig:"){width=".49\textwidth"}
![(a) Size linewidth relationship of the 529 sources that have determined distances. Overplotted is the Larson relationship from Larson (1981), black line. While it runs through the middle of our data, there is not enough of a trend to determine anything about it. Also plotted are the two powerlaws from Caselli and Myers (1995), $R^{.21}$ (light red line) for high mass and $R^{.53}$ (light blue line) for low mass. (b) Virial linewidth versus observed linewidth shows that our clumps are not dominated by motions due to self gravity with the majority of sources with observed linewidths larger than the virial linewidth. (c & d) Observed linewidth ($1/e$-width) vs surface density / mass compared with the virial parameter $\alpha$. The colored lines represent different values of the virial parameter. This plot also shows that few of the clumps have a virial parameter less than 1 and are fully gravitationally bound. []{data-label="sizelinewidth"}](f21a.pdf "fig:"){width=".49\textwidth"} ![(a) Size linewidth relationship of the 529 sources that have determined distances. Overplotted is the Larson relationship from Larson (1981), black line. While it runs through the middle of our data, there is not enough of a trend to determine anything about it. Also plotted are the two powerlaws from Caselli and Myers (1995), $R^{.21}$ (light red line) for high mass and $R^{.53}$ (light blue line) for low mass. (b) Virial linewidth versus observed linewidth shows that our clumps are not dominated by motions due to self gravity with the majority of sources with observed linewidths larger than the virial linewidth. (c & d) Observed linewidth ($1/e$-width) vs surface density / mass compared with the virial parameter $\alpha$. The colored lines represent different values of the virial parameter. This plot also shows that few of the clumps have a virial parameter less than 1 and are fully gravitationally bound. []{data-label="sizelinewidth"}](f21b.pdf "fig:"){width=".49\textwidth"}\
![(a) Size linewidth relationship of the 529 sources that have determined distances. Overplotted is the Larson relationship from Larson (1981), black line. While it runs through the middle of our data, there is not enough of a trend to determine anything about it. Also plotted are the two powerlaws from Caselli and Myers (1995), $R^{.21}$ (light red line) for high mass and $R^{.53}$ (light blue line) for low mass. (b) Virial linewidth versus observed linewidth shows that our clumps are not dominated by motions due to self gravity with the majority of sources with observed linewidths larger than the virial linewidth. (c & d) Observed linewidth ($1/e$-width) vs surface density / mass compared with the virial parameter $\alpha$. The colored lines represent different values of the virial parameter. This plot also shows that few of the clumps have a virial parameter less than 1 and are fully gravitationally bound. []{data-label="sizelinewidth"}](f21c.pdf "fig:"){width=".49\textwidth"} ![(a) Size linewidth relationship of the 529 sources that have determined distances. Overplotted is the Larson relationship from Larson (1981), black line. While it runs through the middle of our data, there is not enough of a trend to determine anything about it. Also plotted are the two powerlaws from Caselli and Myers (1995), $R^{.21}$ (light red line) for high mass and $R^{.53}$ (light blue line) for low mass. (b) Virial linewidth versus observed linewidth shows that our clumps are not dominated by motions due to self gravity with the majority of sources with observed linewidths larger than the virial linewidth. (c & d) Observed linewidth ($1/e$-width) vs surface density / mass compared with the virial parameter $\alpha$. The colored lines represent different values of the virial parameter. This plot also shows that few of the clumps have a virial parameter less than 1 and are fully gravitationally bound. []{data-label="sizelinewidth"}](f21d.pdf "fig:"){width=".49\textwidth"}
![$(a)$ Histogram of observed masses as determined from the 1.1 mm dust emission for the Known Distance Sample. The median of this distribution is 320 $M_{\odot}$ and the mean is 1272 $M_{\odot}$. Our completeness limit is $\sim 140$ M$_{\odot}$ shown by the black dotted line based on a source of 1 Jy (above which we observe every object in the BOLOCAT) at our median distance. $(b)$ The distribution of masses versus the distance from us. The main trend in this plot is due to distance dependence of the conversion from flux to mass, $M \propto D^2$. The black dotted line represents the mass corresponding to our minimum flux versus distance. The red dotted line represents a 1 Jy source versus distance. (c) Differential mass histogram with dlogM = 0.3 (binsize) assuming $T_d = 20K$. The black dotted line is our completness limit, see (a). The slope of the line is fit from the largest mass peak of the distribution to the highest masses. The slope determined for this “binned” differential mass histogram is $m=-0.70$ which is not the slope we determine from the MLE analysis of $m=-0.91$. This difference is directly caused by the fact we have binned the data and will change based on the way one choses the binsize.[]{data-label="masshistdist"}](f22a.pdf "fig:"){width=".49\textwidth"} ![$(a)$ Histogram of observed masses as determined from the 1.1 mm dust emission for the Known Distance Sample. The median of this distribution is 320 $M_{\odot}$ and the mean is 1272 $M_{\odot}$. Our completeness limit is $\sim 140$ M$_{\odot}$ shown by the black dotted line based on a source of 1 Jy (above which we observe every object in the BOLOCAT) at our median distance. $(b)$ The distribution of masses versus the distance from us. The main trend in this plot is due to distance dependence of the conversion from flux to mass, $M \propto D^2$. The black dotted line represents the mass corresponding to our minimum flux versus distance. The red dotted line represents a 1 Jy source versus distance. (c) Differential mass histogram with dlogM = 0.3 (binsize) assuming $T_d = 20K$. The black dotted line is our completness limit, see (a). The slope of the line is fit from the largest mass peak of the distribution to the highest masses. The slope determined for this “binned” differential mass histogram is $m=-0.70$ which is not the slope we determine from the MLE analysis of $m=-0.91$. This difference is directly caused by the fact we have binned the data and will change based on the way one choses the binsize.[]{data-label="masshistdist"}](f22b.pdf "fig:"){width=".49\textwidth"}\
![$(a)$ Histogram of observed masses as determined from the 1.1 mm dust emission for the Known Distance Sample. The median of this distribution is 320 $M_{\odot}$ and the mean is 1272 $M_{\odot}$. Our completeness limit is $\sim 140$ M$_{\odot}$ shown by the black dotted line based on a source of 1 Jy (above which we observe every object in the BOLOCAT) at our median distance. $(b)$ The distribution of masses versus the distance from us. The main trend in this plot is due to distance dependence of the conversion from flux to mass, $M \propto D^2$. The black dotted line represents the mass corresponding to our minimum flux versus distance. The red dotted line represents a 1 Jy source versus distance. (c) Differential mass histogram with dlogM = 0.3 (binsize) assuming $T_d = 20K$. The black dotted line is our completness limit, see (a). The slope of the line is fit from the largest mass peak of the distribution to the highest masses. The slope determined for this “binned” differential mass histogram is $m=-0.70$ which is not the slope we determine from the MLE analysis of $m=-0.91$. This difference is directly caused by the fact we have binned the data and will change based on the way one choses the binsize.[]{data-label="masshistdist"}](f22c.pdf "fig:"){width=".49\textwidth"}
![ (a) The volume-averaged number density for the “Known Distance Sample”. The median value is $2.4\times10^3$ cm$^{-3}$ (red dotted line). We are complete to $\sim 1300$ $cm^{-3}$ (black dashed line). (b) $\Delta v$ versus the average density. (c) The average volume density of a source versus distance. We are able to measure higher densities for closer sources because source size increases with distance given our finite beamsize. Only nearby sources can be fully resolved. The red line indicates our volume-averaged number density completness vs distance for a $\sim 140$ M$_{\odot}$ object, Mass completeness limit, with size equal to the beamsize, minimum resolved source size. (d) The clump free-fall timescale and the clump crossing time scales. The median free-fall time is $\sim 8\times 10^5$ years. The median crossing time is $\sim 5 \times 10^5$ years.[]{data-label="Avgnumberden"}](f23a.pdf "fig:"){width=".49\textwidth"} ![ (a) The volume-averaged number density for the “Known Distance Sample”. The median value is $2.4\times10^3$ cm$^{-3}$ (red dotted line). We are complete to $\sim 1300$ $cm^{-3}$ (black dashed line). (b) $\Delta v$ versus the average density. (c) The average volume density of a source versus distance. We are able to measure higher densities for closer sources because source size increases with distance given our finite beamsize. Only nearby sources can be fully resolved. The red line indicates our volume-averaged number density completness vs distance for a $\sim 140$ M$_{\odot}$ object, Mass completeness limit, with size equal to the beamsize, minimum resolved source size. (d) The clump free-fall timescale and the clump crossing time scales. The median free-fall time is $\sim 8\times 10^5$ years. The median crossing time is $\sim 5 \times 10^5$ years.[]{data-label="Avgnumberden"}](f23b.pdf "fig:"){width=".49\textwidth"}\
![ (a) The volume-averaged number density for the “Known Distance Sample”. The median value is $2.4\times10^3$ cm$^{-3}$ (red dotted line). We are complete to $\sim 1300$ $cm^{-3}$ (black dashed line). (b) $\Delta v$ versus the average density. (c) The average volume density of a source versus distance. We are able to measure higher densities for closer sources because source size increases with distance given our finite beamsize. Only nearby sources can be fully resolved. The red line indicates our volume-averaged number density completness vs distance for a $\sim 140$ M$_{\odot}$ object, Mass completeness limit, with size equal to the beamsize, minimum resolved source size. (d) The clump free-fall timescale and the clump crossing time scales. The median free-fall time is $\sim 8\times 10^5$ years. The median crossing time is $\sim 5 \times 10^5$ years.[]{data-label="Avgnumberden"}](f23c.pdf "fig:"){width=".49\textwidth"} ![ (a) The volume-averaged number density for the “Known Distance Sample”. The median value is $2.4\times10^3$ cm$^{-3}$ (red dotted line). We are complete to $\sim 1300$ $cm^{-3}$ (black dashed line). (b) $\Delta v$ versus the average density. (c) The average volume density of a source versus distance. We are able to measure higher densities for closer sources because source size increases with distance given our finite beamsize. Only nearby sources can be fully resolved. The red line indicates our volume-averaged number density completness vs distance for a $\sim 140$ M$_{\odot}$ object, Mass completeness limit, with size equal to the beamsize, minimum resolved source size. (d) The clump free-fall timescale and the clump crossing time scales. The median free-fall time is $\sim 8\times 10^5$ years. The median crossing time is $\sim 5 \times 10^5$ years.[]{data-label="Avgnumberden"}](f23d.pdf "fig:"){width=".49\textwidth"}
![(a) The variation of the power law index with choice of temperature distribution. (b) A Monte Carlo simulation of the median mass with variations in temperature. The units are in M$_{\odot}$. (c) A Monte Carlo simulation of the median volume-averaged number density with variations in temperature. The units are in cm$^{-3}$. (d) A Monte Carlo simulation of the median surface density for variations in $T_{dust}$. The units are g cm$^{-2}$.[]{data-label="montecarlo"}](f24a.pdf "fig:"){width=".49\textwidth"} ![(a) The variation of the power law index with choice of temperature distribution. (b) A Monte Carlo simulation of the median mass with variations in temperature. The units are in M$_{\odot}$. (c) A Monte Carlo simulation of the median volume-averaged number density with variations in temperature. The units are in cm$^{-3}$. (d) A Monte Carlo simulation of the median surface density for variations in $T_{dust}$. The units are g cm$^{-2}$.[]{data-label="montecarlo"}](f24b.pdf "fig:"){width=".49\textwidth"}\
![(a) The variation of the power law index with choice of temperature distribution. (b) A Monte Carlo simulation of the median mass with variations in temperature. The units are in M$_{\odot}$. (c) A Monte Carlo simulation of the median volume-averaged number density with variations in temperature. The units are in cm$^{-3}$. (d) A Monte Carlo simulation of the median surface density for variations in $T_{dust}$. The units are g cm$^{-2}$.[]{data-label="montecarlo"}](f24c.pdf "fig:"){width=".49\textwidth"} ![(a) The variation of the power law index with choice of temperature distribution. (b) A Monte Carlo simulation of the median mass with variations in temperature. The units are in M$_{\odot}$. (c) A Monte Carlo simulation of the median volume-averaged number density with variations in temperature. The units are in cm$^{-3}$. (d) A Monte Carlo simulation of the median surface density for variations in $T_{dust}$. The units are g cm$^{-2}$.[]{data-label="montecarlo"}](f24d.pdf "fig:"){width=".49\textwidth"}
![(a) The surface density for the entire sample assuming $T_{dust} = 20$ K. The median value is 0.027 g cm$^{-2}$. The completeness limit for the entire sample is 0.051 g cm$^{-2}$ which is based on a source of 1 Jy coupled with the beamsize (black dashed line). (b) The surface density histogram for the Known Distance Sample. The median value is 0.033 g cm$^{-2}$. The completeness limit for the Known Distance Sample is 0.016 g cm$^{-2}$ which is based on mass completeness and the median size of a clump (black dashed line).[]{data-label="surfacedensity"}](f25a.pdf "fig:"){width=".49\textwidth"} ![(a) The surface density for the entire sample assuming $T_{dust} = 20$ K. The median value is 0.027 g cm$^{-2}$. The completeness limit for the entire sample is 0.051 g cm$^{-2}$ which is based on a source of 1 Jy coupled with the beamsize (black dashed line). (b) The surface density histogram for the Known Distance Sample. The median value is 0.033 g cm$^{-2}$. The completeness limit for the Known Distance Sample is 0.016 g cm$^{-2}$ which is based on mass completeness and the median size of a clump (black dashed line).[]{data-label="surfacedensity"}](f25b.pdf "fig:"){width=".49\textwidth"}
[lcccc]{} $\eta_{mb}$ – Average & 0.81(0.04) & 0.81(0.03) & 0.70(0.03) & 0.70(0.04)\
$\eta_{mb}$ – MJD 918 – 920 & 0.64(0.01) & 0.64(0.02) & 0.64(0.01) & 0.64(0.02)\
\[etamb\]
[cl]{} 0 & No Detection: There is no visible line\
1 & Single Detection: One line visible with no visible structure\
2 & Multiple Detections: More than one line is visible in the spectrum\
4 & Obvious Line Wings: One line that shows a possible line wing(s)\
5 & Self-absorbed Profile: One line that shows a possible self-absorbed profile\
9 & Unusable Data: Both polarizations have defects that make the spectrum unusable\
\[lineflags\]
[lcccccccccccccccccccc]{} 010.000+008 & 18:07:28.5 & -20:14:14.2 & 10.014 & +0.084 & 5.2 & 0.246(0.061) & 0 & (0.069) & (0.324) & & & 0 & (0.083) & (0.211) & & & & & 0\
010.016-039 & 18:09:16.9 & -20:28:04.3 & 10.017 & -0.398 & 137.1 & 0.102(0.067) & 1 & 0.498(0.077) & 1.955(0.447) & 3.045 & 11.405 & 0 & (0.098) & (0.248) & & & 1.9191 & 14.6252 & 0\
010.029-035 & 18:09:08.5 & -20:26:24.8 & 10.026 & -0.356 & 7.8 & 0.375(0.050) & 0 & (0.088) & (0.228) & & & 0 & (0.103) & (0.260) & & & & & 0\
010.045-042 & 18:09:26.0 & -20:27:29.2 & 10.043 & -0.424 & 105.6 & 0.171(0.051) & 1 & 0.488(0.033) & 1.521(0.071) & 3.573 & 10.542 & 0 & (0.040) & (0.103) & & & 1.8064 & 14.7366 & 0\
010.056-021 & 18:08:39.0 & -20:20:49.9 & 10.051 & -0.210 & 7.7 & 0.227(0.049) & 0 & (0.055) & (0.142) & & & 0 & (0.046) & (0.117) & & & & & 0\
010.072-041 & 18:09:25.3 & -20:25:42.6 & 10.068 & -0.408 & 3 & 0.441(0.058) & 4 & 1.080(0.070) & 4.906(0.349) & 3.678 & 11.442 & 1 & 0.441(0.088) & 1.362(0.182) & 3.943 & 60.028 & 1.9171 & 14.6246 & 8\
010.067-018 & 18:08:38.7 & -20:19:00.7 & 10.077 & -0.194 & 2.4 & 0.439(0.055) & 0 & (0.091) & (0.480) & & & 1 & 0.281(0.092) & 0.730(0.191) & 2.355 & 77.192 & & & 0\
010.080-043 & 18:09:34.0 & -20:25:44.6 & 10.084 & -0.438 & 4.2 & 0.268(0.058) & 0 & (0.069) & (0.399) & & & 0 & (0.090) & (0.227) & & & & & 1\
010.071+004 & 18:07:48.9 & -20:11:50.5 & 10.087 & +0.034 & 13.1 & 0.145(0.051) & 0 & (0.076) & (0.196) & & & 0 & (0.086) & (0.217) & & & & & 0\
010.103-000 & 18:08:01.2 & -20:12:19.2 & 10.104 & -0.012 & 2 & 0.315(0.048) & 1 & 0.840(0.084) & 3.896(0.442) & 4.651 & 41.581 & 1 & 0.719(0.071) & 3.595(0.146) & 4.306 & 90.107 & 4.3500 & 12.1894 & 0\
010.107-042 & 18:09:32.3 & -20:24:03.5 & 10.105 & -0.418 & 4.6 & 0.454(0.073) & 1 & 1.699(0.064) & 7.047(0.340) & 3.516 & 11.327 & 1 & 1.154(0.091) & 4.465(0.189) & 3.567 & 60.271 & 1.8981 & 14.6418 & 0\
010.114-007 & 18:08:16.7 & -20:13:41.2 & 10.113 & -0.076 & 7.2 & 0.216(0.066) & 0 & (0.065) & (0.167) & & & 0 & (0.086) & (0.217) & & & & & 0\
010.134-037 & 18:09:25.7 & -20:20:54.9 & 10.139 & -0.370 & 27.7 & 0.720(0.085) & 1 & 1.954(0.052) & 7.521(0.136) & 4.646 & 13.833 & 1 & 0.472(0.055) & 1.526(0.115) & 3.586 & 60.732 & 2.1921 & 14.3459 & 0\
010.144-040 & 18:09:35.5 & -20:21:25.5 & 10.149 & -0.408 & 2.2 & 0.662(0.071) & 2 & (0.043) & (0.144) & & & 1 & 0.844(0.044) & 3.943(0.111) & 5.655 & 59.950 & & & 8\
010.168-034 & 18:09:26.6 & -20:19:18.4 & 10.164 & -0.360 & 8 & 3.717(0.091) & 4 & 4.431(0.050) & 20.571(0.131) & 6.859 & 14.019 & 1 & 1.295(0.049) & 6.322(0.125) & 5.177 & 61.020 & 2.2097 & 14.3270 & 0\
010.191-038 & 18:09:36.5 & -20:18:43.8 & 10.191 & -0.390 & 4.9 & 0.598(0.087) & 1 & 1.628(0.044) & 6.012(0.093) & 4.211 & 10.894 & 1 & 1.774(0.041) & 8.501(0.103) & 4.870 & 58.357 & 1.8331 & 14.7022 & 0\
010.195-023 & 18:09:05.5 & -20:13:50.1 & 10.204 & -0.244 & 10.4 & 0.207(0.059) & 1 & 0.858(0.072) & 2.930(0.418) & 2.760 & 13.161 & 0 & (0.094) & (0.238) & & & 2.1055 & 14.4289 & 0\
010.208+004 & 18:08:01.0 & -20:05:02.3 & 10.209 & +0.048 & 185.9 & 0.156(0.058) & 0 & (0.067) & (0.175) & & & 0 & (0.073) & (0.184) & & & & & 0\
010.207-031 & 18:09:24.6 & -20:15:39.2 & 10.213 & -0.324 & 2.9 & 2.331(0.095) & 5 & 0.983(0.030) & 5.151(0.090) & 9.071 & 9.434 & 5 & 1.813(0.054) & 10.322(0.157) & 7.326 & 59.428 & 1.6415 & 14.8926 & 8\
010.225-020 & 18:09:00.5 & -20:11:40.1 & 10.226 & -0.209 & 3.8 & 1.422(0.056) & 5 & 1.293(0.047) & 6.012(0.122) & 5.269 & 12.295 & 1 & 1.616(0.050) & 7.548(0.127) & 4.441 & 59.642 & 2.0004 & 14.5328 & 8\
\[datatable\]
[llccc]{} I(N$_2$H$^+$) K km s$^{-1}$ & I(HCO$^+$) K km s$^{-1}$ & 0.33 & 0.823 & 0.82\
T$_{mb}$(N$_2$H$^+$) K & T$_{mb}$(HCO$^+$) K & 0.21 & 0.83 & 0.79\
I(HCO$^+$) K km s$^{-1}$& S$_\nu$(1.1mm) Jy/Beam & .78 & 1.15 & 0.80\
I(N$_2$H$^+$) K km s$^{-1}$ & S$_\nu$(1.1mm) Jy/Beam & 0.55 & 1.28 & 0.73\
T$_{mb}$(HCO$^+$) K & S$_\nu$(1.1mm) Jy/Beam & 0.21 & 0.89 & 0.75\
T$_{mb}$(N$_2$H$^+$) K & S$_\nu$(1.1mm) Jy/Beam & -0.01 & 0.99 & 0.73\
\[fitsum\]
[lcccccccccccc]{} 010.000+008 & 10.014 & 0.084 & 6.5 & 12 & 790 & 2.683 & 3.436 & 752.6 & 568.2 & 0.016 & 16.78 & 13.066\
010.072-041 & 10.068 & -0.408 & 1.917 & 2 & 249 & 3.678 & 10.511 & 4560.1 & 1326.8 & 0.02 & 10.981 & 4.889\
010.207-031 & 10.213 & -0.324 & 1.641 & 2 & 492 & 9.071 & 23.717 & 28188 & 6662.8 & 0.072 & 4.9 & 1.452\
010.225-020 & 10.226 & -0.209 & 2 & 2 & 695 & 5.269 & 8.055 & 14106.7 & 3270.1 & 0.05 & 6.995 & 3.557\
010.399-020 & 10.403 & -0.202 & 1.844 & 2 & 107 & 4.975 & 21.303 & 4570.7 & 5260 & 0.037 & 5.515 & 1.725\
010.622-043 & 10.621 & -0.441 & 6.5 & 12 & 917 & 3.345 & 1.34 & 1708.2 & 26650.2 & 0.224 & 2.45 & 3.055\
010.624-033 & 10.625 & -0.338 & 6.5 & 12 & 5063 & 4.725 & 2.527 & 5085.7 & 1037.4 & 0.045 & 12.419 & 11.276\
010.631-050 & 10.631 & -0.51 & 6.5 & 12 & 1648 & 2.975 & 2.061 & 1695.1 & 1123.8 & 0.033 & 11.932 & 11.997\
010.653-012 & 10.651 & -0.126 & 3.468 & 2 & 134 & 6.264 & 44.083 & 1112.8 & 1505.3 & 0.017 & 10.31 & 2.241\
010.670-016 & 10.665 & -0.161 & 3.506 & 2 & 132 & 5.326 & 26.62 & 1099 & 2681.3 & 0.025 & 7.725 & 2.161\
010.672-022 & 10.671 & -0.22 & 3.503 & 2 & 552 & 3.621 & 5.289 & 3482.5 & 1926.3 & 0.033 & 9.114 & 5.721\
010.686-030 & 10.687 & -0.308 & 6.5 & 2 & 1709 & 2.318 & 1.459 & 1371.6 & 658.9 & 0.023 & 15.583 & 18.621\
010.722-033 & 10.726 & -0.332 & 6.5 & 12 & 2119 & 3.176 & 2.102 & 2495.4 & 948.3 & 0.032 & 12.989 & 12.932\
010.739-012 & 10.743 & -0.126 & 3.433 & 2 & 571 & 5.564 & 13.11 & 3964.7 & 1558.3 & 0.029 & 10.133 & 4.04\
010.742-029 & 10.745 & -0.295 & 6.5 & 12 & 467 & 1.176 & 0.855 & 497.6 & 749.7 & 0.017 & 14.609 & 22.81\
010.761-019 & 10.752 & -0.2 & 3.671 & 2 & 554 & 3.082 & 4.709 & 2430.5 & 1033.7 & 0.022 & 12.441 & 8.276\
010.802-038 & 10.799 & -0.38 & 6.5 & 12 & 434 & 2.918 & 5.589 & 501.1 & 723.8 & 0.016 & 14.868 & 9.078\
010.983-036 & 10.985 & -0.37 & 6.5 & 12 & 1478 & 2.945 & 2.902 & 1148.6 & 471.6 & 0.018 & 18.42 & 15.608\
010.997-007 & 10.989 & -0.084 & 3.456 & 2 & 992 & 2.6 & 1.944 & 4472.3 & 1649.3 & 0.036 & 9.85 & 10.197\
011.050-004 & 11.053 & -0.04 & 3.506 & 2 & 91 & 1.733 & 4.189 & 903.2 & 1703 & 0.017 & 9.693 & 6.836\
011.061-009 & 11.063 & -0.096 & 3.427 & 2 & 680 & 3.005 & 3.621 & 2925 & 1293.8 & 0.027 & 11.121 & 8.435\
\[datatable2\]
[^1]: Adjunct Astronomer at the National Ratio Astronomy Observatory
[^2]: http://irsa.ipac.caltech.edu/data/BOLOCAM\_GPS/
[^3]: http://www.iram.fr/IRAMFR/GILDAS/
|
---
abstract: 'Determining the mechanism behind the current cosmic acceleration constitutes a major question nowadays in theoretical physics. If the dark energy route is taken, this problem may potentially bring to light new insights not only in Cosmology but also in high energy physics theories. Following this approach, we explore in this paper some cosmological consequences of a new time-dependent parameterization for the dark energy equation of state (EoS), which is a well behaved function of the redshift $z$ over the entire cosmological evolution, i.e., $z\in [-1,\infty)$. This parameterization allows us to divide the parametric plane $(w_0,w_1)$ in defined regions associated to distinct classes of dark energy models that can be confirmed or excluded from a confrontation with current observational data. A statistical analysis involving the most recent observations from type Ia supernovae, baryon acoustic oscillation peak, Cosmic Microwave Background shift parameter and Hubble evolution $H(z)$ is performed to check the observational viability of the EoS parameterization here proposed.'
address: '$^{1}$Observatório Nacional, 20921-400, Rio de Janeiro - RJ, Brasil'
author:
- 'E. M. Barboza Jr.$^{1}$[^1]'
- 'J. S. Alcaniz$^{1,2}$[^2]'
title: A parametric model for dark energy
---
Introduction
============
Over the last decade, a considerable number of high quality observational data have transformed radically the field of cosmology. Results from distance measurements of type Ia supernovae (SNe Ia) [@Riess; @Astier] combined with Cosmic Microwave Background (CMB) observations [@wmap] and the Large-Scale Structure (LSS) data [@lss; @Eisenstein] seem to indicate that the simple picture provided by the standard cold dark matter scenario is not enough. These observations are usually explained by introducing a new hypothetical energy component with negative pressure, the so-called dark energy or *quintessence*, usually characterized by the equation of state (EoS) parameter $w \equiv p/\rho$, i.e., the ratio between the dark energy pressure to its energy density (see, e.g., [@rev] for some recent reviews).
Among the many candidates for dark energy, the energy density associated with the quantum vacuum or the cosmological constant ($\Lambda$) emerges as the simplest and the most natural possibility. However, this interpretation of the cosmological term brings to light an unsettled situation in the Particle Physics/Cosmology interface, in which the cosmological upper bound ($\rho_{\Lambda} \lesssim 10^{-47}$ GeV$^4$) differs from theoretical expectations ($\rho_{\Lambda} \lesssim 10^{71}$ GeV$^4$) by more than 100 orders of magnitude [@Weinberg]. Thus, although $\Lambda$ may be able to explain the majority of observations available so far, if dark energy is in fact associated with the vacuum energy density, we should look for an explanation for this enormous discrepancy between theory and observation.
In this regard, many proposals have appeared in the literature trying to solve this problem, but so far no reasonable explanation, if there is one, was obtained. Thus, despite the beauty and simplicity of the cosmological term, other proposals, even if not so attractive, should be explored. In the context of the General Theory of Relativity, for instance, models with a time-varying cosmological term [@atcdm], irreversible processes (e.g., cosmological matter creation [@mc]), barotropic fluids (e.g., Chapligyn Gas [@Chapligyn]) and, dynamical scalar fields $\Phi$ (e.g., quintessence [@Quintessence], phantom fields [@Phantom] and quintom [@Quintom]), are some of those alternatives[^3].
Phenomenologically, it is usual to explore some possible time-dependent parameterizations to describe the dark energy EoS. In this concern, Taylor series-like parameterizations $$\label{TaylorExpantion}
w(z)=\sum_{n=0}w_n\,x_n(z),
$$ where $w_n$’s are constants to be fixed by observations and $x_n(z)$’s are functions of redshift, are among the most commonly discussed and, depending on the allowed values for $w_n$’s, they may have quintessence ($-1\leq w(z)\leq1$) and phantom fields ($w(z)<-1$) as special cases. Among the parameterizations based on series expansion we can quote the following first order expansions: $$\begin{aligned}
\label{OtherParameterizations}
w(z) = \; \left\{
\begin{tabular}{l}
$w_0 + w_1 z$ \, \quad \quad \quad \quad \mbox{(redshift)} \quad\, \hspace{0.2cm}\cite{Huterer}\\
\\
$w_0+w_1z/(1+z)$ \quad \mbox{(scale factor)} \,
\cite{Chevallier} \\
\\
$w_0+w_1\ln(1+z)$ \quad \mbox{(logarithmic)} \quad \cite{Efstathiou}
\end{tabular}
\right.
\nonumber\end{aligned}$$
The first parameterization represents a good fit for low redshifts, but presents a problematic behavior for high redshifts. For example, it fails to explain the estimated ages of high-$z$ objects [@Friaca]. The second one solves this problem, since $w (z)$ is a well behaved function for $z\gg1$ and recovers the linear behavior in $z$ at low redshifts. The latter was built empirically to adjust some quintessence models at $z\lesssim4$. It is worth mentioning that it is difficult to obtain the above parameterizations from scalar field dynamics since they are not limited functions, i.e., the EoS parameter does not lie in the interval defined by $w = \frac{{\dot{\Phi}^2/2} - V(\Phi)}{{\dot{\Phi}^2}/{2} + V(\Phi)}$, where $V(\Phi)$ is the field potential. This amounts to saying that when extended to the entire history of the universe, $z\in [-1,\infty)$, the three parameterizations above are divergent functions of the redshift[^4]. However, since the dark energy dominance is a very recent phenomena, this particular aspect is not usually taken as important because it is always possible to obtain a quintessence-like behavior as a particular approximation when $z$ is not too large. Even so, one can suspect that the information that can be obtained about dark energy from these parameterizations may be compromised. In fact, to avoid the ambiguities and uncertainty that can be contained in these parameterizations, it is desirable a parameterization that can be extended to entire expansion history of the Universe, so that the constraints obtained from the scalar field behavior can also be applied.
In this paper, in order extend the range of applicability of the dark energy EoS, we study some cosmological consequences of a new phenomenological parameterization for this quantity. We discuss the classification of this expression in the parametric space $w_0 - w_1$, and explore its main observational features. We test the viability of this new dark energy scenario from the most recent distance measurements from type Ia supernovae (SNe Ia), measurements of the baryonic acoustic oscillations (BAO), the shift parameter of the cosmic microwave background and measurements of the rate of the cosmic expansion $H(z)$.
Parameterization
================
In this paper, we will consider the following parameterization for the dark energy EoS: $$\label{MyParameterization}
w(z)=w_0+w_1\frac{z(1+z)}{1+z^2},$$ where $w_0$ is the EoS value at present time (the subscript and superscript zero denotes the present value of a quantity) and $w_1=dw/dz\vert_{z=0}$ gives a measure of how time-dependent is the dark energy EoS. This parameterization has the same linear behavior in $z$ at low redshifts presented by the parameterizations discussed above and has the advantage of being a limited function of $z$ throughout the entire history of the Universe.
For a flat Friedmann-Robertson-Walker universe, it is straightforward to show from the continuity equation for each component, $\dot{\rho}_i=3\,\rho_i\,\dot{z}[1+w_i(z)]/(1+z)$, that the dark energy density $\rho_{X}$ for parameterization (\[MyParameterization\]) evolves as $$\label{DE-Density}
f(z)\equiv\frac{\rho_{X}}{\rho_{X}^{\,\,0}}=(1+z)^{3(1+w_0)}(1+z^2)^{3w_1/2}\;.$$ Note that, if Eq. (\[MyParameterization\]) should be always valid, the parameters $w_0$ and $w_1$ should be constrained by $$\label{EarlyTimeConstraint}
w_0+w_1<0\;,$$ so that the dark energy is always subdominant at $z\gg1$.
In this background, the Friedman equation for a dark matter/dark energy dominated universe reads $$\label{FriedmannEquation}
H^2\equiv\Big(\frac{\dot{a}}{a}\Big)^2=H_0^2[\Omega_m^0(1+z)^3+(1-\Omega_m^0)\,f(z)]\;,$$ where $a$ is the cosmological scale factor, $H$ is the Hubble parameter and $\Omega_i^0=\rho_i^0/\rho_c^0$ ($\rho_c^0=3\,H_0^2/8\,\pi\,G$) is density parameter of the $i$th-component ($m \equiv$ baryonic + dark matter).
The deceleration parameter is given by $$\label{DeccelerationParameter}
q(z)=-\frac{\ddot{a}}{a\,H^2}=\frac{1}{2}[1+3\,w(z)\,\Omega_X]\;.$$ By solving the above equation for $w(z)$ at $z=0$ we obtain $$\label{GeneralConstraint}
w_0=\frac{2\,q_0-1}{3(1-\Omega_m^{\,0})} \quad \mbox{or} \quad w_0 <-\frac{1}{3(1-\Omega_m^{\,0})}\;,$$ since $q_0<0$, as indicated by current observations [@Riess]. Note that, for $\Omega_m^{\,0}=0.27\pm 0.04$ [@wmap], it is possible obtain a model-independent bound on the current value of the dark energy EoS, i.e., $w_0<-0.43$.
The $w_0 - w_1$ plane
---------------------
By differentiating Eq. (\[MyParameterization\]) with respect to $z$, we find that $w(z)$ has absolute extremes at $z_{\pm}=1\pm\sqrt{2}$ corresponding, respectively, to $w_-=w(z_-)=w_0-0.21w_1$ and $w_+=w(z_+)=w_0+1.21w_1$. For $w_1>0$ $(<0)$, $w_-$ is a minimum (maximum) and $w_+$ is a maximum (minimum). Since quintessence and phantom scalar fields EoS are limited by $-1\leq w(z)\leq1$ and $w(z)<-1$, respectively, the region occupied in the $(w_0,w_1)$ plane by these fields can be easily determined by imposing that the maximum and minimum of $w(z)$ satisfy these limits. Thus, for quintessence fields we find the following bounds $$-1\leq w_0-0.21w_1\quad {\mbox{and}} \quad w_0+1.21w_1\leq1 \quad ({\mbox{if}} \quad w_1>0) \;,$$ and $$-1\leq w_0+1.21w_1\quad {\mbox{and}} \quad w_0-0.21w_1\leq1 \quad ({\mbox{if}} \quad w_1<0)\;.$$ whereas for phantom fields we obtain $$w_1<-(1+w_0)/1.21 \quad ({\mbox{if}} \quad w_1>0)\;,$$ and $$w_1>(1+w_0)/0.21 \quad ({\mbox{if}} \quad w_1<0)\;.$$
Figure 1 shows different classes of dark energy models in the $(w_0,w_1)$ plane that arises from parameterization (\[MyParameterization\]). The forbidden region represents the constraint (\[EarlyTimeConstraint\]) while the decelerating region is limited by the upper bound (\[GeneralConstraint\]) with $\Omega_m^{\,0}=0.27$, i.e., $w_0<-0.43$. The blank regions indicate models that at some point of the cosmic evolution, $z\in [-1,\infty)$, have switched or will switch from quintessence to phantom behaviors or vice-versa. Clearly, Parameterization (\[MyParameterization\]) provides a simple way to classify distinct DE models in the $(w_0,w_1)$ plane. This is a direct consequence from the fact that (\[MyParameterization\]) is a well behaved bounded function along the entire history of the Universe. The standard $\Lambda$CDM scenario corresponds to the intersection point between quintessence ($w > -1$) and phantom models ($w < -1$).
0.1in
Observational constraints
=========================
In the previous Section, we have defined the regions occupied by different classes of dark energy models derived from parameterization (\[MyParameterization\]) in the plane $(w_0,w_1)$. Now, we will test the viability of these scenarios by using the most recent cosmological data, namely, $115$ distance measurements of SNe Ia from the Supernova Legacy Survey (SNLS) [@Astier], measurements of the baryonic acoustic oscillations (BAO) from SDSS [@Eisenstein], the CMB shift parameter as given by the WMAP team [@wmap] and estimates of the Hubble parameter $H(z)$ obtained from ages of high-$z$ galaxies [@Simon] (for more details on the statistical analysis discussed below we refer the reader to Ref. [@refs]).
SNe Ia observations
-------------------
The predicted distance modulus for a supernova at redshift $z$, given a set of parameters $\mathbf{P}$, is $$\label{dm}
\mu_p(z|\mathbf{P}) = m - M = 5\,\mbox{log} d_L + 25,$$ where $m$ and $M$ are, respectively, the apparent and absolute magnitudes, and $d_L$ stands for the luminosity distance (in units of megaparsecs), $$\label{LuminosityDistance}
d_L(z;{\bf P})=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime};{\bf P})}\;,$$ where $H(z; {\bf P})$ is given by Eq. (\[FriedmannEquation\]).
We estimated the best fit to the set of parameters $\mathbf{s}$ by using a $\chi^{2}$ statistics, with $$\chi^{2}_{SNe} = \sum_{i=1}^{N}{\frac{\left[\mu_p^{i}(z|\mathbf{P}) -
\mu_o^{i}(z)\right]^{2}}{\sigma_i^{2}}},$$ where $\mu_p^{i}(z|\mathbf{P})$ is given by Eq. (\[dm\]), $\mu_o^{i}(z)$ is the extinction corrected distance modulus for a given SNe Ia at $z_i$, and $\sigma_i$ is the uncertainty in the individual distance moduli. Since we use in our analyses the SNLS collaboration sample (see [@Astier] for details), $N = 115$.
0.2in 0.1in
Baryonic acoustic oscillations
------------------------------
As well known, the acoustic peaks in the cosmic microwave background (CMB) anisotropy power spectrum is an efficient way for determining cosmological parameters (e.g., [@wmap]). Because the acoustic oscillations in the relativistic plasma of the early universe will also be imprinted on to the late-time power spectrum of the non-relativistic matter [@peeblesyu], the acoustic signatures in the large-scale clustering of galaxies yield additional tests for cosmology.
In particular, the characteristic and reasonably sharp length scale measured at a wide range of redshifts provides an estimate of the distance-redshift relation, which is a geometric complement to the usual luminosity-distance from SNe Ia. Using a large spectroscopic sample of 46,748 luminous, red galaxies covering 3816 square degrees out to a redshift of $z=0.47$ from the Sloan Digital Sky Survey, Eisenstein et al. [@Eisenstein] have successfully found the peaks, described by the ${\cal{A}}$-parameter, i.e., $$\label{BAO}
\mathcal{A}\equiv\sqrt{\Omega_m^0}\Big[\frac{H_0^2}{z_{\ast}\,H^{2}(z_{\ast};{\bf P})}\int_0^{z_{\ast}}\frac{H_0\,dz}{H(z;{\bf P})}\Big]^{2/3},$$ where $z_{\ast}=0.35$ is the redshift at which the acoustic scale has been measured.
CMB shift parameter
-------------------
The shift parameter $R$ which determines the whole shift of the CMB angular power spectrum is given by [@b] $$\label{ShiftParameter}
\mathcal{R}\equiv\sqrt{\Omega_m^0}\int_0^{z_{\mathrm ls}}\frac{H_0\,dz}{H(z;{\bf P})},$$ where the $z_{\mathrm ls} = 1089$ is the redshift of the last scattering surface, and the current estimated value for this quantity is ${\mathcal R}_{\mathrm{obs}} = 1.70\pm 0.03 $ [@wang].
Hubble Expansion
----------------
In our joint analysis we also use 9 determinations of the Hubble parameter as a function of redshift, as given in Ref. [@Simon]. These determinations, based on differential age method, relates the Hubble parameter $H(z)$ directly to measurable quantity $dt/dz$ and can be achieved from the recently released sample of old passive galaxies from Gemini Deep Deep Survey (GDDS) [@Abraham] and archival data [@Dunlop]. The use of these data to constrain cosmological models is interesting because, differently of luminosity distance measures, the Hubble parameter is not integrated over (see [@Simon] for more details). To perform this test we minimize the quantity $$\chi^{2}_{H} = \sum_{i=1}^{9}{\frac{\left[H^{i}(z|\mathbf{P}) - H_{obs}^{i}(z)\right]^{2}}{\sigma_i^{2}}},$$ where the predicted Hubble evolution for parameterization (\[MyParameterization\]) is given by Eq. (\[FriedmannEquation\]).
Analyses and Discussions
------------------------
### Parameterization (1)
Figure (2) shows the main results of our analyses. In order to compare the theoretical frame with the observational constraints discussed above the three dimensional parameter space $(\Omega_m^0,w_0,w_1)$ has been projected into the plane $(w_0,w_1)$. In Fig. (2a) we show confidence intervals (68.3%, 95.4% and 99.7%) in this parametric space $(w_0,w_1)$ for parameterization (\[MyParameterization\]). The best-fit values for these parameters are $w_0=-1.11$ and $w_1=0.43$ whereas at 68.3% (c.l.) they lie, respectively, in the intervals $-1.35 \leq w_0 \leq -0.86$ and $-0.33 \leq w_1 \leq 0.91$. Note that no DE model is preferred or ruled out by observations, although the largest portion of the confidence contours lies into the blanc region (indicating models that have switched or eventually will switch from quintessence to phantom behaviors or vice-versa). At $99\%$ C.L. we also have $0.21\leq\Omega_m^0\leq0.33$ so that the decelerated region is limited, in accordance with (\[GeneralConstraint\]), by the constraint $w_0<-0.42$ and the possibility of a decelerated universe today is almost completely excluded. For the best-fit values discussed above, the transition redshift $z_t$, at which the Universe switches from deceleration to acceleration, occurs at $z_t \simeq 0.58$.
### Scale factor Parameterization
For the sake of comparison, we employ the same analysis to scale factor parameterization $w_0+w_1z/(1+z)$ [@Chevallier]. Note that this parameterization has an absolute extreme in $w_{\infty}=w(z=\infty)=w_0+w_1$. For $w_1>0$, $w_{\infty}$ is a maximum whereas for $w_1<0$ it is a minimum. Thus, the region occupied by phantom fields is determined by the constraints $w_{\infty}<-1$ and $w_1>0$, whereas similar constraints cannot be obtained for the quintessence case. The region occupied by phantom fields in the context of this parameterization and the confidence intervals for the statistical analysis discussed above are displayed in the Fig. 3. At 68.3% c.l., we found $w_0=-1.14^{+0.31}_{-0.24}$, $w_1=0.84^{+0.65}_{-1.59}$ and $\Omega_m^0=0.27\pm0.03$. As can be seen, at this confidence level, the possibility of an early dark energy dominance is not completely excluded.
Finally, we also note that for this parameterization, the continuity equation solves as $$\label{ScaleFactorDE}
f_{SF}(z)=(1+z)^{3(1+w_0+w_1)}e^{-3w_1\,z/(1+z)}$$ so that it must also be submitted to the constraint (\[EarlyTimeConstraint\]). Therefore, when $z\to-1$ ($a\to\infty$), $f_{SF}(z)$ blows up if $w_1>0$ while $f(z)$, given by Eq. (1), blows up if $w_0<-1$. Thus, the roles of the parameters $w_0$ and $w_1$ are inverted in these scenarios, in the sense that while for Parameterization (1) the fate of the Universe is dictated by the equilibrium part ($w_0$), for the scale factor parameterization the future of the Universe is driven by the time-dependent term $w_1$.
Final Remarks
=============
The recent cosmic expansion history has the potential to greatly extend our physical understanding of the Universe. This in turn is closed related to the origin and nature of the mechanism behind the current cosmic acceleration, for instance, if it is associated with a new component of energy or large-scale modifications of gravity.
In this paper, by assuming a hypothetical component of dark energy as the fuel that drives the acceleration of the Universe, we have proposed and studied some theoretical and observational aspects of a new parameterization for this dark energy EoS, as given by Eq. (\[MyParameterization\]). This parameterization is a well-behaved, bounded function of the redshift throughout the entire cosmic evolution, which allows us to study the effects of a time varying EoS component to the distant future of the Universe at $z = -1$ as well as back to the last scattering surface of the CMB. We have classified different classes of dark energy models in the parametric space ($w_0 - w_1$) and studied their theoretical and observational consequences. In order to check the observational viability of the phenomenological scenario proposed here, we also have performed a joint statistical analysis involving some of the most recent cosmological measurements of SNe Ia, BAO peak, CMB shift parameter and the Hubble expansion $H(z)$. From a pure observational perspective, we have shown that both quintessence and phantom behaviours are fully acceptable regimes, although the largest portion of the confidence contours arising from these observations lies in the region of models that have crossed or will eventually cross these regimes at some point of the cosmic evolution.
The authors are grateful to R. Silva for many useful suggestions and a critical reading of the manuscript. E.M.B. Jr. is partially supported by CNPq. JSA is supported by CNPq under Grants 304569/2007-0 and 485662/2006-0.
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[^1]: E-mail: edesio@on.br
[^2]: E-mail: alcaniz@on.br
[^3]: Out of the context of the General Relativity, some other attractive approaches to the dark energy problem, such as brane-world models [@brane] and f(R) derived cosmologies [@fr] have also been recently explored.
[^4]: As an example, the scale factor parameterization above blows up exponentially in the future as $z \rightarrow -1$ for $w_1 > 0$ — see, e.g., [@teg] for a discussion.
|
---
author:
- 'A. Deltuva'
- 'A. C. Fonseca'
- 'R. Lazauskas'
title: 'Faddeev equation approach for three-cluster nuclear reactions'
---
Introduction
=============
Nuclear collision experiments, performed at ion accelerators, are a very powerful tool to study nuclear properties at low and intermediate energies. In order to interpret accumulated experimental data appropriate theoretical methods are necessary enabling the simultaneous description of the available elastic, rearrangement and breakup reactions.
Regardless of its importance, the theoretical description of quantum-mechanical collisions turns out to be one of the most complex and slowly advancing problems in theoretical physics. If during the last decade accurate solutions for the nuclear bound state problem became available, full solution of the scattering problem (containing elastic, rearrangement and breakup channels) remains limited to the three-body case.
The main difficulty is related to the fact that, unlike the bound state wave functions, scattering wave functions are not localized. In configuration space one is obliged to solve multidimensional differential equations with extremely complex boundary conditions; by formulating the quantum-mechanical scattering problem in momentum space one has to deal with non-trivial singularities in the kernel of multivariable integral equations.
A rigorous mathematical formulation of the quantum mechanical three-body problem in the framework of non relativistic dynamics has been introduced by Faddeev in the early sixties [@Fad_60], in the context of the three-nucleon system with short range interactions. In momentum space these equations might be slightly modified by formulating them in terms of three-particle transition operators that are smoother functions compared to the system wave functions. Such a modification was proposed by Alt, Grassberger, and Sandhas [@alt:67a] (AGS).
Solutions of the AGS equations with short range interactions were readily obtained in the early seventies. As large computers became available progress followed leading, by the end eighties, to fully converged solutions of these equations for neutron-deuteron ($n$-$d$) elastic scattering and breakup using realistic short range nucleon-nucleon ($N$-$N$) interactions. Nevertheless the inclusion of the long range Coulomb force in momentum space calculations of proton-deuteron ($p$-$d$) elastic scattering and breakup with the same numerical reliability as calculations with short range interactions alone, only become possible in the last decade.
Significant progress has been achieved [@deltuva:05a; @deltuva:05d] by developing the screening and renormalization procedure for the Coulomb interaction in momentum space using a smooth but at the same time sufficiently rapid screening. This technique permitted to extend the calculations to the systems of three-particles with arbitrary masses above the breakup threshold [@deltuva:06b; @deltuva:07d].
However it has taken some time to formulate the appropriate boundary conditions in configuration space for the three-body problem [@Merkuriev_71; @Merkuriev_74; @MGL_76] and even longer to reformulate the original Faddeev equations to allow the incorporation of long-range Coulomb like interactions [@Merkuriev_80; @Merkuriev_81]. Rigorous solution of the three-body problem with short range interactions has been achieved just after these theoretical developments, both below and above breakup threshold. On the other hand the numerical solution for the three-body problem including charged particles above the three-particle breakup threshold has been achieved only recently. First it has been done by using approximate Merkuriev boundary conditions in configuration space [@kievsky:97]. Nevertheless this approach proved to be a rather complex task numerically, remaining unexplored beyond the $p$-$d$ scattering case, but not yet for the $p$-$d$ breakup.
Finally, very recently configuration space method based on complex scaling have been developed and applied for $p$-$d$ scattering [@lazauskas:11a]. This method allows to treat the scattering problem using very simple boundary conditions, equivalent to the ones employed to solve the bound-state problem.
The aim of this lecture is to present these two recently developed techniques, namely the momentum-space method based on screening and renormalization as well as the configuration-space complex scaling method. This lecture is structured as follows: the first part serves to introduce theoretical formalisms for momentum space and configuration space calculations; in the second part we present some selected calculations with an aim to test the performance and validity of the two presented methods.
Momentum-space description of three-particle scattering {#sec:p}
=======================================================
We describe the scattering process in a system of three-particles interacting via pairwise short-range potentials $v_\alpha$, $\alpha=1,2,3$; we use the odd-man-out notation, that is, $v_1$ is the potential between particles 2 and 3. In the framework of nonrelativistic quantum mechanics the center-of-mass (c.m.) and the internal motion can be separated by introducing Jacobi momenta $$\begin{aligned}
\label{eq:Jacobi}
\vec{p}_\alpha & = &\frac{m_{\gamma} \vec{k}_\beta - m_{\beta} \vec{k}_\gamma }
{m_{\beta} + m_{\gamma} }, \\
\vec{q}_\alpha & = &
\frac{m_{\alpha} (\vec{k}_\beta + \vec{k}_\gamma) -
(m_{\beta} + m_{\gamma}) \vec{k}_\alpha }
{m_{\alpha} + m_{\beta} + m_{\gamma} }, $$ with ($\alpha \beta \gamma $) being cyclic permutations of (123); $\vec{k}_\alpha$ and $m_{\alpha}$ are the individual particle momenta and masses, respectively. The c.m. motion is free and in the following we consider only the internal motion; the corresponding kinetic energy operator is $H_0$ while the full Hamiltonian is $$\label{eq:H}
H = H_0 + \sum_{\alpha=1}^3 v_\alpha .$$
Alt, Grassberger, and Sandhas equations
---------------------------------------
We consider the particle $\alpha$ scattering from the pair $\alpha$ that is bound with energy $ \epsilon_\alpha$. The initial channel state $|b_{\alpha}\vec{q}_\alpha\rangle$ is the product of the bound state wave function $|b_\alpha \rangle$ for the pair $\alpha$ and a plane wave with the relative particle-pair $\alpha$ momentum $\mathbf{q}_\alpha$; the dependence on the discrete quantum numbers is suppressed in our notation. $|b_{\alpha}\vec{q}_\alpha\rangle$ is the eigenstate of the corresponding channel Hamiltonian $H_\alpha = H_0 + v_\alpha$ with the energy eigenvalue $E= \epsilon_\alpha + q^2_\alpha/2M_\alpha$ where $M_\alpha$ is the particle-pair $\alpha$ reduced mass. The final channel state is the particle-pair state in the same or different configuration $|b_{\beta}\vec{q}_\beta\rangle$ in the case of elastic and rearrangement scattering or, in the case of breakup, it is the state of three free particles $|\vec{p}_{\gamma}\vec{q}_\gamma\rangle$ with the same energy $E= p_\gamma^2/2\mu_\gamma + q_\gamma^2/2M_\gamma $ and pair $\gamma$ reduced mass $\mu_\gamma$; any set of Jacobi momenta can be used equally well for the breakup state.
The stationary scattering states [@schmid:74a; @gloeckle:83a] corresponding to the above channel states are eigenstates of the full Hamiltonian; they are obtained from the channel states using the full resolvent $G = (E+i0-H)^{-1}$, i.e., $$\begin{aligned}
\label{eq:psi_a}
|b_\alpha \vec{q}_\alpha \rangle^{(+)} & = &
i0 G |b_\alpha \vec{q}_\alpha \rangle, \\
\label{eq:psi_0}
|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} & = &
i0 G |\vec{p}_\alpha\vec{q}_\alpha \rangle.\end{aligned}$$ The full resolvent $G$ may be decomposed into the channel resolvents $G_\beta = (E+i0-H_\beta)^{-1}$ and/or free resolvent $G_0 = (E+i0-H_0)^{-1}$ as $$G = G_\beta + G_\beta \bar{v}_\beta G ,$$ with $\beta=0,1,2,3$ and $ \bar{v}_\beta = \sum_{\gamma=1}^3 \bar{\delta}_{\beta \gamma} v_\gamma$ where $\bar{\delta}_{\beta \gamma} = 1-{\delta}_{\beta \gamma}$. Furthermore, the channel resolvents $$G_\beta = G_0 + G_0 T_\beta G_0 ,$$ can be related to the corresponding two-particle transition operators $$T_\beta = v_\beta + v_\beta G_0 T_\beta ,$$ embedded into three-particle Hilbert space. Using these definitions Eqs. (\[eq:psi\_a\]) and (\[eq:psi\_0\]) can be written as triads of Lippmann-Schwinger equations $$\begin{aligned}
\label{eq:psi_LS}
|b_\alpha \vec{q}_\alpha \rangle^{(+)} & = {} &
\delta_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle
+ G_\beta \bar{v}_\beta |b_\alpha \vec{q}_\alpha \rangle^{(+)} , \\
|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} & = {} &
(1+ G_0 T_\beta ) |\vec{p}_\alpha\vec{q}_\alpha \rangle
+ G_\beta \bar{v}_\beta |\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} ,\end{aligned}$$ with $\alpha$ being fixed and $\beta =1,2,3$; they are necessary and sufficient to define the states $|b_\alpha \vec{q}_\alpha \rangle^{(+)}$ and $|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)}$ uniquely. However, in scattering problems it may be more convenient to work with the multichannel transition operators $U_{\beta \alpha}$ defined such that their on-shell elements yield scattering amplitudes, i.e., $$\label{eq:U-V}
U_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle =
\bar{v}_\beta |b_\alpha \vec{q}_\alpha \rangle^{(+)}.$$ Our calculations are based on the AGS version [@alt:67a] of three-particle scattering theory. In accordance with Eq. (\[eq:U-V\]) it defines the multichannel transition operators $U_{\beta \alpha}$ by the decomposition of the full resolvent $ G$ into channel and/or free resolvents as $$\label{eq:G-U}
G = \delta_{\beta \alpha} G_\alpha + G_\beta U_{\beta \alpha} G_\alpha .$$ The multichannel transition operators $U_{\beta \alpha}$ with fixed $\alpha$ and $\beta = 1,2,3$ are solutions of three coupled integral equations $$\label{eq:AGSnsym_a}
U_{\beta \alpha} = \bar{\delta}_{\beta \alpha} G_0^{-1} +
\sum_{\gamma=1}^3 \bar{\delta}_{\beta \gamma} T_{\gamma} G_0 U_{\gamma \alpha}.$$ The transition matrix $U_{0 \alpha} $ to final states with three free particles can be obtained from the solutions of Eq. (\[eq:AGSnsym\_a\]) by quadrature, i.e., $$\label{eq:AGSnsym_b}
U_{0 \alpha} = G_0^{-1} + \sum_{\gamma=1}^3 T_{\gamma} G_0 U_{\gamma \alpha}.$$
The on-shell matrix elements $\langle b_{\beta} \vec{q}'_\beta |U_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle$ are amplitudes (up to a factor) for elastic ($\beta = \alpha$) and rearrangement ($\beta \neq \alpha$) scattering. For example, the differential cross section for the $\alpha + (\beta\gamma) \to \beta + (\gamma\alpha)$ reaction in the c.m. system is given by $$\label{eq:dcsab}
\frac{d \sigma_{\alpha \to \beta}}{d \Omega_\beta} =
(2\pi)^4 M_\alpha M_\beta \frac{q'_\beta}{q_\alpha}
| \langle b_{\beta} \vec{q}'_\beta |U_{\beta \alpha}
|b_\alpha \vec{q}_\alpha \rangle|^2.$$ The cross section for the breakup is determined by the on-shell matrix elements $\langle \vec{p}'_{\gamma} \vec{q}'_\gamma |U_{0 \alpha}
|b_\alpha \vec{q}_\alpha \rangle$. Thus, in the AGS framework all elastic, rearrangement, and breakup reactions are calculated on the same footing.
Finally we note that the AGS equations can be extended to include also the three-body forces as done in Ref. [@deltuva:09e].
Inclusion of the Coulomb interaction
------------------------------------
The Coulomb potential $w_C$, due to its long range, does not satisfy the mathematical properties required for the formulation of standard scattering theory as given in the previous subsection for short-range interactions $v_\alpha$. However, in nature the Coulomb potential is always screened at large distances. The comparison of the data from typical nuclear physics experiments and theoretical predictions with full Coulomb is meaningful only if the full and screened Coulomb become physically indistinguishable. This was proved in Refs. [@taylor:74a; @semon:75a] where the screening and renormalization method for the scattering of two charged particles was proposed. We base our treatment of the Coulomb interaction on that idea.
Although we use momentum-space framework, we first choose the screened Coulomb potential in configuration-space representation as $$\label{eq:wr}
w_R(r) = w_C(r)\; e^{-(r/R)^n} ,$$ and then transform it to momentum-space. Here $R$ is the screening radius and $n$ controls the smoothness of the screening. The standard scattering theory is formally applicable to the screened Coulomb potential $w_R$, i.e., the Lippmann-Schwinger equation yields the two-particle transition matrix $$\label{eq:tr}
t_R = w_R + w_R g_0 t_R ,$$ where $g_0$ is the two-particle free resolvent. It was proven in Ref. [@taylor:74a] that in the limit of infinite screening radius $R$ the on-shell screened Coulomb transition matrix (screened Coulomb scattering amplitude) $\langle \mathbf{p}'| t_R | \mathbf{p} \rangle$ with $p'=p$, renormalized by an infinitely oscillating phase factor $z_R^{-1}(p) = e^{2i\phi_R(p)}$, approaches the full Coulomb amplitude $\langle \mathbf{p}'| t_C | \mathbf{p} \rangle$ in general as a distribution. The convergence in the sense of distributions is sufficient for the description of physical observables in a real experiment where the incoming beam is not a plane wave but wave packet and therefore the cross section is determined not directly by the scattering amplitude but by the outgoing wave packet, i.e., by the scattering amplitude averaged over the initial state physical wave packet. In practical calculations [@alt:02a; @deltuva:05a] this averaging is carried out implicitly, replacing the renormalized screened Coulomb amplitude in the $R \to \infty$ limit by the full one, i.e., $$\label{eq:taylor2}
\lim_{R \to \infty} z_R^{-1}(p)
\langle \mathbf{p}'| t_R | \mathbf{p} \rangle \to
\langle \mathbf{p}'| t_C | \mathbf{p} \rangle.$$ Since $z_R^{-1}(p)$ is only a phase factor, the above relations indeed demonstrate that the physical observables become insensitive to screening provided it takes place at sufficiently large distances $R$ and, in the $R \to \infty$ limit, coincide with the corresponding quantities referring to the full Coulomb. Furthermore, renormalization by $ z_{R}^{-\frac12}(p_i)$ in the $R \to \infty$ limit relates also the screened and full Coulomb wave functions [@gorshkov:61], i.e., $$\label{eq:gorshkov}
\lim_{R \to \infty} (1 + g_0 t_R) |\mathbf{p} \rangle z_R^{-\frac12}(p)
= |\psi_C^{(+)}(\mathbf{p}) \rangle.$$
The screening and renormalization method based on the above relations can be extended to more complicated systems, albeit with some limitations. We consider the system of three-particles with charges $z_\alpha$ of equal sign interacting via pairwise strong short-range and screened Coulomb potentials $v_\alpha + w_{\alpha R}$ with $\alpha$ being 1, 2, or 3. The corresponding two-particle transition matrices are calculated with the full channel interaction $$\label{eq:TR}
T^{(R)}_\alpha = (v_\alpha + w_{\alpha R}) +
(v_\alpha + w_{\alpha R}) G_0 T^{(R)}_\alpha,$$ and the multichannel transition operators $U^{(R)}_{\beta \alpha}$ for elastic and rearrangement scattering are solutions of the AGS equation $$U^{(R)}_{\beta \alpha} = \bar{\delta}_{\beta \alpha} G_0^{-1}
+ \sum_{\gamma=1}^3 \bar{\delta}_{\beta \gamma}
T^{(R)}_\gamma G_0 U^{(R)}_{\gamma \alpha} ;
\label{eq:Uba}$$ all operators depend parametrically on the Coulomb screening radius $R$.
In order to isolate the screened Coulomb contributions to the transition amplitude that diverge in the infinite $R$ limit we introduce an auxiliary screened Coulomb potential $W^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}$ between the particle $\alpha$ and the center of mass (c.m.) of the remaining pair. The same screening function has to be used for both Coulomb potentials $w_{\alpha R}$ and $W^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}$. The corresponding transition matrix $$\label{eq:Tcm}
T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R} = W^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R} +
W^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R} G^{(R)}_{\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R} ,$$ with $ G^{(R)}_{\alpha} = (E+i0-H_0-v_\alpha - w_{\alpha R})^{-1}$ is a two-body-like operator and therefore its on-shell and half-shell behavior in the limit $R \to \infty$ is given by Eqs. (\[eq:taylor2\]) and (\[eq:gorshkov\]). As derived in Ref. [@deltuva:05a], the three-particle transition operators may be decomposed as $$\begin{aligned}
U^{(R)}_{\beta \alpha} &=& \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}
+ [1 + T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\beta R} G^{(R)}_{\beta}] \tilde{U}^{(R)}_{\beta\alpha} [1 + G^{(R)}_{\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}] \quad
\label{eq:U-T} \\
&=& \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}
+ (U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}).
\label{eq:U-T2}\end{aligned}$$ where the auxiliary operator $\tilde{U}^{(R)}_{\beta\alpha}$ is of short range when calculated between on-shell screened Coulomb states. Thus, the three-particle transition operator $U^{(R)}_{\beta \alpha}$ has a long-range part $\delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}$ whereas the remainder $U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}$ is a short-range operator that is externally distorted due to the screened Coulomb waves generated by $[1 + G^{(R)}_{\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}]$. On-shell, both parts do not have a proper limit as $R \to \infty$ but the limit exists after renormalization by an appropriate phase factor, yielding the transition amplitude for full Coulomb $$\begin{aligned}
\nonumber
&& \langle b_\beta \mathbf{q}'_\beta | U^{(C)}_{\beta \alpha}
|b_\alpha \mathbf{q}_\alpha\rangle =
\delta_{\beta \alpha}
\langle b_\alpha \mathbf{q}'_\beta |T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha C}
|b_\alpha \mathbf{q}_\alpha \rangle \\ & &
+ \lim_{R \to \infty} [ Z^{-\frac{1}{2}}_{\beta R}(q'_\beta)
\langle b_\beta \mathbf{q}'_\beta |
( U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R})
|b_\alpha \mathbf{q}_\alpha \rangle
Z^{-\frac{1}{2}}_{\alpha R}(q_\alpha) ]. \quad
\label{eq:UC2}\end{aligned}$$ The first term on the right-hand side of Eq. (\[eq:UC2\]) is known analytically [@taylor:74a]; it corresponds to the particle-pair $\alpha$ full Coulomb transition amplitude that results from the implicit renormalization of $T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}$ according to Eq. (\[eq:taylor2\]). The $R \to \infty$ limit for the remaining part $( U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R})$ of the multichannel transition matrix is performed numerically; due to the short-range nature of this term the convergence with the increasing screening radius $R$ is fast and the limit is reached with sufficient accuracy at finite $R$; furthermore, it can be calculated using the partial-wave expansion. We emphasize that Eq. (\[eq:UC2\]) is by no means an approximation since it is based on the obviously exact identity (\[eq:U-T2\]) where the $R \to \infty$ limit for each term exists and is calculated separately.
The renormalization factor for $R \to \infty $ is a diverging phase factor $$Z_{\alpha R}(q_\alpha) = e^{-2i \Phi_{\alpha R}(q_\alpha)},$$ where $\Phi_{\alpha R}(q_\alpha)$, though independent of the particle-pair relative angular momentum $l_\alpha$ in the infinite $R$ limit, may be realized by $$\label{eq:phiRl}
\Phi_{\alpha R}(q_\alpha) = \sigma_{l_\alpha}^{\alpha}(q_\alpha) -
\eta_{l_\alpha R}^{\alpha}(q_\alpha),$$ with the diverging screened Coulomb phase shift $\eta_{l_\alpha R}^{\alpha}(q_\alpha)$ corresponding to standard boundary conditions and the proper Coulomb one $\sigma_{l_\alpha}^{\alpha}(q_\alpha)$ referring to the logarithmically distorted proper Coulomb boundary conditions. For the screened Coulomb potential of Eq. (\[eq:wr\]) the infinite $R$ limit of $\Phi_{\alpha R}(q_\alpha)$ is known analytically, $$\label{eq:phiRlln}
\Phi_{\alpha R}(q_\alpha)=\mathcal{K}_{\alpha}(q_\alpha)[\ln{(2q_\alpha R)} - C/n],$$ where $C \approx 0.5772156649$ is the Euler number and $\mathcal{K}_{\alpha}(q_\alpha) = \alpha_{e.m.}z_\alpha \sum_\gamma
\bar{\delta}_{\gamma\alpha} z_\gamma M_\alpha/q_\alpha$ is the Coulomb parameter with $\alpha_{e.m.} \approx 1/137$. The form of the renormalization phase $\Phi_{\alpha R}(q_\alpha)$ to be used in the actual calculations with finite screening radii $R$ is not unique, but the converged results show independence of the chosen form of $\Phi_{\alpha R}(q_\alpha)$.
For breakup reactions we follow a similar strategy. However, the proper three-body Coulomb wave function and its relation to the three-body screened Coulomb wave function is, in general, unknown. This prevents the application of the screening and renormalization method to the reactions involving three free charged particles (nucleons or nuclei) in the final state. However, in the system of two charged particles and a neutral one with $z_\rho = 0$, the final-state Coulomb distortion becomes again a two-body problem with the screened Coulomb transition matrix $$T_{\rho R} = w_{\rho R} + w_{\rho R} G_0 T_{\rho R}.$$ This makes the channel $\rho$, corresponding to the correlated pair of charged particles, the most convenient choice for the description of the final breakup state. As shown in Ref. [@deltuva:05d], the AGS breakup operator $$\label{eq:U0a}
U^{(R)}_{0\alpha} = {}
G_0^{-1} + \sum_{\gamma=1}^3 T^{(R)}_{\gamma} G_0 U^{(R)}_{\gamma \alpha} ,$$ can be decomposed as $$\label{eq:U0t}
U^{(R)}_{0\alpha} = {} (1 + T_{\rho R} G_{0})
\tilde{U}^{(R)}_{0\alpha} (1 + G^{(R)}_{\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R}),$$ where the reduced operator $\tilde{U}^{(R)}_{0\alpha}(Z)$ calculated between screened Coulomb distorted initial and final states is of finite range. In the full breakup operator $U^{(R)}_{0 \alpha}(Z)$ the external distortions show up in screened Coulomb waves generated by $(1 + G^{(R)}_{\alpha} T^{{\mathrm{c\!\:\!.m\!\:\!.}}}_{\alpha R})$ in the initial state and by $(1 + T_{\rho R} G_{0})$ in the final state; both wave functions do not have proper limits as $R \to \infty$. Therefore the full breakup transition amplitude in the case of the unscreened Coulomb potential is obtained via the renormalization of the on-shell breakup transition matrix $ U^{(R)}_{0 \alpha}$ in the infinite $R$ limit $$\langle \mathbf{p}'_\rho \mathbf{q}'_\rho | U^{(C)}_{0 \alpha}
|b_\alpha \mathbf{q}_\alpha \rangle =
\lim_{R \to \infty} [ z^{-\frac{1}{2}}_{\rho R}(p'_\rho)
\langle \mathbf{p}'_\rho \mathbf{q}'_\rho | U^{(R)}_{0 \alpha}
|b_\alpha \mathbf{q}_\alpha \rangle Z_{\alpha R}^{-\frac{1}{2}}(q_\alpha )],
\label{eq:UC1a}$$ where $\mathbf{p}'_\rho$ is the relative momentum between the charged particles in the final state, $\mathbf{q}'_\rho$ the corresponding particle-pair relative momentum, and $$\label{eq:phiRp}
z_{\rho R}(p'_\rho) = e^{-2i\kappa_\rho(p'_\rho)[\ln{(2p'_\rho R)} - C/n]} ,$$ the final-state renormalization factor with the Coulomb parameter $\kappa_\rho(p'_\rho)$ for the pair $\rho$. The limit in Eq. (\[eq:UC1a\]) has to be performed numerically, but, due to the short-range nature of the breakup operator, the convergence with increasing screening radius $R$ is fast and the limit is reached with sufficient accuracy at finite $R$. Thus, to include the Coulomb interaction via the screening and renormalization method one only needs to solve standard scattering theory equations.
Practical realization
---------------------
We calculate the short-range part of the elastic, rearrangement, and breakup scattering amplitudes (\[eq:UC2\]) and (\[eq:UC1a\]) by solving standard scattering equations (\[eq:Uba\]), (\[eq:Tcm\]), and (\[eq:U0a\]) with a finite Coulomb screening radius $R$. We work in the momentum-space partial-wave basis [@deltuva:phd], i.e., we use three sets\
$|p_\alpha q_\alpha \nu_\alpha \rangle \equiv
|p_\alpha q_\alpha (l_\alpha \{ [L_\alpha(s_\beta s_\gamma)S_\alpha]
I_\alpha s_\alpha \} K_\alpha) { J M} \rangle$ with $(\alpha,\beta,\gamma)$ being cyclic permutations of (1,2,3). Here $s_\alpha$ is the spin of particle $\alpha$, $L_\alpha$ and $l_\alpha$ are the orbital angular momenta associated with $p_\alpha$ and $q_\alpha$ respectively, whereas $S_\alpha$, $I_\alpha$, and $K_\alpha$ are intermediate angular momenta that are coupled to a total angular momentum $J$ with projection $M$. All discrete quantum numbers are abbreviated by $\nu_\alpha$. The integration over the momentum variables is discretized using Gaussian quadrature rules thereby converting a system of integral equations for each $J$ and parity $\Pi = (-)^{L_\alpha +l_\alpha}$ into a very large system of linear algebraic equations. Due to the huge dimension those linear systems cannot be solved directly. Instead we expand the AGS transition operators (\[eq:Uba\]) into the corresponding Neumann series $$\label{eq:neumann}
U^{(R)}_{\beta \alpha} = \bar{\delta}_{ \beta \alpha } G_0^{-1} +
\sum_{\gamma=1}^3 \bar{\delta}_{ \beta \gamma } T^{(R)}_\gamma
\bar{\delta}_{ \gamma \alpha }
+ \sum_{\gamma=1}^3 \bar{\delta}_{ \beta \gamma } T^{(R)}_\gamma G_0
\sum_{\sigma=1}^3 \bar{\delta}_{\gamma \sigma } T^{(R)}_\sigma
\bar{\delta}_{\sigma \alpha}
+ \cdots ,$$ that are summed up by the iterative Pade method [@chmielewski:03a]; it yields an accurate solution of Eq. (\[eq:Uba\]) even when the Neumann series (\[eq:neumann\]) diverges. Each two-particle transition operator $ T^{(R)}_{\gamma}$ is evaluated in its proper basis $|p_\gamma q_\gamma \nu_\gamma \rangle$, thus, transformations between all three bases are needed. The calculation of the involved overlap functions $ \langle p_\beta q_\beta \nu_\beta |p_\alpha q_\alpha \nu_\alpha \rangle$ follows closely the calculation of three-nucleon permutation operators discussed in Refs. [@deltuva:phd; @gloeckle:83a]. A special treatment [@chmielewski:03a; @deltuva:phd] is needed for the integrable singularities arising from the pair bound state poles in $ T^{(R)}_{\gamma}$ and from $G_0$. Furthermore, we have to make sure that $R$ is large enough to achieve (after renormalization) the $R$-independence of the results up to a desired accuracy. However, those $R$ values are larger than the range of the nuclear interaction resulting in a slower convergence of the partial-wave expansion. As we found in Ref. [@deltuva:05a], the practical success of the screening and renormalization method depends very much on the choice of the screening function, in our case on the power $n$ in Eq. (\[eq:wr\]). We want to ensure that the screened Coulomb potential $w_R$ approximates well the true Coulomb one $w_C$ for distances $r<R$ and simultaneously vanishes rapidly for $r>R$, providing a comparatively fast convergence of the partial-wave expansion. As shown in Ref. [@deltuva:05a], this is not the case for simple exponential screening $(n =1)$ whereas the sharp cutoff $(n \to \infty)$ yields slow oscillating convergence with the screening radius $R$. However, we found that values of $3 \le n \le 8$ provide a sufficiently smooth and rapid screening around $r=R$. The screening functions for different $n$ values are compared in Ref. [@deltuva:05a] together with the results demonstrating the superiority of our optimal choice: using $3 \le n \le 8$ the convergence with the screening radius $R$, at which the short range part of the amplitudes was calculated, is fast enough such that the convergence of the partial-wave expansion, though being slower than for the nuclear interaction alone, can be achieved and there is no need to work in a plane-wave basis. Here we use $n=4$ and show in Figs. \[fig:Rad\]and \[fig:Radb\] few examples for the $R$-convergence of the $\alpha$-deuteron scattering observables calculated in a three-body model $(\alpha,p,n)$; the nuclear interaction is taken from Ref. [@deltuva:06b]. The convergence with $R$ is impressively fast for both $\alpha$-deuteron elastic scattering and breakup. In addition we note that the Coulomb effect is very large and clearly improves the description of the experimental data, especially for the differential cross section in $\alpha$-deuteron breakup reaction. This is due to the shift of the $\alpha p$ $P$-wave resonance position when the $\alpha p$ Coulomb repulsion is included that leads to the corresponding changes in the structure of the observables.
![ Differential cross section and deuteron vector analyzing power $iT_{11}$ of the $\alpha d$ elastic scattering at 4.81 MeV deuteron lab energy as functions of the c.m. scattering angle. Convergence with the screening radius $R$ used to calculate the short-range part of the amplitudes is studied: $R= 5$ fm (dotted curves), $R= 10$ fm (dash-dotted curves), and $R= 15$ fm (solid curves). Results without Coulomb are given by dashed curves. The experimental data are from Refs. [@bruno:80; @gruebler:70a]. []{data-label="fig:Rad"}](Rd48c.eps)
![ Fivefold differential cross section of the $\alpha d$ breakup reaction at 15 MeV $\alpha$ lab energy for several combinations of $\alpha$ and proton scattering angles as function of the final-state energy variable $S$ with $dS = (dE_\alpha^2 + dE_p^2)^{1/2}$. Convergence with the screening radius $R$ is studied: $R= 10$ fm (dotted curves), $R= 15$ fm (dash-dotted curves), and $R= 20$ fm (solid curves). Results without Coulomb are given by dashed curves. The experimental data are from Ref. [@koersner:77]. []{data-label="fig:Radb"}](Ra15.eps)
In addition to the internal reliability criterion of the screening and renormalization method — the convergence with $R$ — we note that our results for proton-deuteron elastic scattering [@deltuva:05b] agree well over a broad energy range with those of Ref. [@kievsky:01a] obtained from the variational configuration-space solution of the three-nucleon Schrödinger equation with unscreened Coulomb potential and imposing the proper Coulomb boundary conditions explicitly.
Configuration space {#sec:r}
===================
In contrast to the momentum-space representation, the Coulomb interaction has a trivial expression in configuration space and thus may seem to be easier to handle. However the major obstacle for configuration-space treatment of the scattering problem is related with the complexity of the wave function asymptotic structure, which strongly complicates once three-particle breakup is available. Although for short range interactions the analytical behavior of the breakup asymptote of the configuration space wave function is well established, this is not a case once long range interactions (like Coulomb) are present. Therefore a method which enables the scattering problem to be solved without explicit use of the wave function asymptotic form is of great importance. The complex scaling method has been proposed [@Nuttal_csm; @CSM_71] and successfully applied to calculate the resonance positions [@Moiseyev] by using bound state boundary conditions. As has been demonstrated recently this method can be extended also for the scattering problem [@CSM_Curdy_04; @Elander_CSM]. We demonstrate here that this method may be also successfully applied to solve three-particle scattering problems which include the long-range Coulomb interaction together with short range optical potentials.
Faddeev-Merkuriev equations
---------------------------
Like in the momentum space formalism described above Jacobi coordinates are also used in configuration space to separate the center of mass of the three-particle system. One has three equivalent sets of three-particle Jacobi coordinates $$\begin{aligned}
\mathbf{x}_{\alpha } &=&\sqrt{\frac{2m_{\beta }m_{\gamma }}{(m_{\beta
}+m_{\gamma })m}}(\mathbf{r}_{\gamma }-\mathbf{r}_{\beta }) , \\
\mathbf{y}_{\alpha } &=&\sqrt{\frac{2m_{\beta }(m_{\beta }+m_{\gamma })}{(m_{\alpha }+m_{\beta }+m_{\gamma })m}}(\mathbf{r}_{\alpha }-\frac{m_{\beta }\mathbf{r}_{\beta }+m_{\gamma }\mathbf{r}_{\gamma }}{m_{\beta }+m_{\gamma }}) ,
\nonumber\end{aligned}$$here $r_{\alpha }$ and $m_{\alpha }$ are individual particle position vectors and masses, respectively. The choice of a mass scale $m$ is arbitrary. The three-particle problem is formulated here using Faddeev-Merkuriev (FM) equations [@Merkuriev_80]: $$\begin{aligned}
(E-H_{0}-\sum_{\kappa =1}^{3}w_{i}^{l})\psi _{\alpha }=(v_{\alpha
}+w_{\alpha }^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma })\nonumber , \\
(E-H_{0}-\sum_{\kappa =1}^{3}w_{i}^{l})\psi _{\beta }=(v_{\beta }+w_{\beta
}^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma }) , \\
(E-H_{0}-\sum_{\kappa =1}^{3}w_{i}^{l})\psi _{\gamma }=(v_{\gamma
}+w_{\gamma }^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma }) , \nonumber$$where the Coulomb interaction is split in two parts (short and long range), $w_{\alpha }=w_{\alpha }^{s}+w_{\alpha }^{l}$, by means of some arbitrary cut-off function $\chi _{\alpha }(x_{\alpha },y_{\alpha })$: $$w_{\alpha }^{s}(x_{\alpha },y_{\alpha })=w_{\alpha }(x_{\alpha })\chi
_{\alpha }(x_{\alpha },y_{\alpha })\qquad w_{\alpha }^{l}(x_{\alpha
},y_{\alpha })=w_{\alpha }(x_{\alpha })[1-\chi _{\alpha }(x_{\alpha
},y_{\alpha })]$$This cut-off function intends to shift the full Coulomb interaction in the $w_{\alpha }^{s}$ term if $x_{\alpha }$ is small, whereas the $w_{\alpha }^{l}$ term acquires the full Coulomb interaction if $x_{\alpha }$ becomes large and $y_{\alpha }<x_{\alpha }$. The practical choice of function $\chi _{\alpha }(x_{\alpha },y_{\alpha })$ has been proposed in [@Merkuriev_80]: $$\chi _{\alpha }(x_{\alpha },y_{\alpha })=\frac{2}{[1+exp{(\frac{[x_{\alpha }/x_0]^\mu}{1+y_{\alpha }/y_0})}]} ,$$with free parameters $x_{0},y_{0}$ having size comparable with the charge radii of the respective binary systems; the value of parameter $\mu$ must be larger than 1 and is usually set $\mu\approx2$. In such a way the so-called Faddeev amplitude $\psi _{\alpha }$ intends to acquire full asymptotic behavior of the binary $\alpha -(\beta \gamma )$ channels, i.e: $$\begin{aligned}
\psi _{\alpha }(\mathbf{x}_{\alpha },\mathbf{y}_{\alpha }\rightarrow \infty
)=\delta _{\kappa ,\alpha }\psi _{\alpha }^{i_{\kappa }}(\mathbf{x}_{\alpha
})\phi _{\alpha }^{i_{\kappa },in}(\mathbf{y}_{\alpha })&+&\sum_{j_{\alpha
}}f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }.\mathbf{y}_{\alpha })\psi
_{\alpha }^{j_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{j_{\alpha
},out}(\mathbf{y}_{\alpha })\nonumber \\
&+&A_{i_{\kappa }}(\mathbf{x}_{\alpha },\mathbf{y}_{\alpha })\Phi _{i_{\kappa }}^{out}(\mathbf{\rho }) ,\end{aligned}$$ where the hyperradius is $\rho =\sqrt{x_{\alpha }^{2}+y_{\alpha }^{2}}$. An expression $\varphi _{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi
_{\alpha }^{i_{\kappa },in}(\mathbf{y}_{\alpha })$ represents the incoming wave for particle $\alpha $ on pair $(\beta \gamma )$ in the bound state $i_{\alpha }$, with $\varphi _{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })$ representing the normalized wave function of bound state $i_{\alpha }$. This wave function is a solution of the $(E-H_{0}-w_{\alpha }-v_{\alpha
}-W_{\alpha }^{c.m.})$ two-body Hamiltonian. The $\phi _{\alpha }^{j_{\alpha },out}(\mathbf{y}_{\alpha })$ and $\Phi _{i_{\kappa }}^{out}(\mathbf{\rho }_{\alpha
})$ represent outgoing waves for binary and three-particle breakup channels respectively. In the asymptote, one has the following behavior: $$\begin{aligned}
\varphi _{\alpha }^{i_{\alpha }}(x_{\alpha } &\rightarrow &\infty )\propto
\exp (-k_{i_{\alpha }}x_{\alpha }) , \nonumber \\
\phi _{\alpha }^{i_{\alpha },out}(y_{\alpha } &\rightarrow &\infty )\propto
\exp (iq_{i_{\alpha }}y_{\alpha }) , \\
\Phi _{i_{\alpha }}^{out}(\rho &\rightarrow &\infty )\propto \exp (iK\rho ) ,
\label{eq:assf}\end{aligned}$$ with $k_{i_{\alpha }}=\sqrt{-\varepsilon _{_{i_{\alpha }}}m}$ representing momentum of 2-body bound state $i_{\alpha }$ with a negative binding energy $\varepsilon _{_{i_{\alpha }}}$; $q_{i_{\alpha }}=\sqrt{(E-\varepsilon
_{_{i_{\alpha }}})m}$ is relative scattering momentum for the $\alpha -(\beta
\gamma )$ binary channel, whereas $K=\sqrt{mE}$ is a three-particle breakup momentum (three-particle breakup is possible only if energy value $E$ is positive).
When considering particle’s $\alpha $ scattering on the bound state $i_{\alpha
}$ of the pair $(\beta \gamma )$, it is convenient to separate readily incoming wave $\psi _{\alpha }^{i_{\alpha },in}=\psi _{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{i_{\alpha },in}(\mathbf{y}_{\alpha
})$, by introducing: $$\begin{aligned}
\psi _{\alpha }^{i_{\alpha },out} &=&\psi _{\alpha }^{i_{\alpha }}-\psi
_{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{i_{\alpha
},in}(\mathbf{y}_{\alpha }) , \\
\psi _{\beta }^{i_{\alpha },out} &=&\psi _{\beta }^{i_{\alpha }}\qquad \beta
\neq \alpha , \nonumber\end{aligned}$$ Then Faddeev-Merkuriev equations might be rewritten in a so-called driven form: $$\begin{aligned}
(E-H_{0}-\sum_{\kappa =1}^{3}w_{\kappa }^{l})\psi _{\alpha
}^{out}&=&(v_{\alpha }+w_{\alpha }^{s})(\psi _{\alpha }^{out}+\psi _{\beta
}^{out}+\psi _{\gamma }^{out})+\left[ \sum_{\kappa =1}^{3}w_{\kappa
}^{l}-w_{\alpha }-W_{\alpha }^{c.m.}\right] \psi _{\alpha }^{in} ,
\nonumber \\
(E-H_{0}-\sum_{\kappa =1}^{3}w_{\kappa }^{l})\psi _{\beta }^{out}&=&(v_{\beta
}+w_{\beta }^{s})(\psi _{\alpha }^{out}+\psi _{\beta }^{out}+\psi _{\gamma
}^{out}+\psi _{\alpha }^{in}) , \\
(E-H_{0}-\sum_{\kappa =1}^{3}w_{\kappa }^{l})\psi _{\gamma
}^{out}&=&(v_{\gamma }+w_{\gamma }^{s})(\psi _{\alpha }^{out}+\psi _{\beta
}^{out}+\psi _{\gamma }^{out}+\psi _{\alpha }^{in}) . \nonumber
\label{eq:drive_FM}\end{aligned}$$ In this expression index of the incoming state $i_{\alpha }$ has been omitted in all Faddeev component expressions $\psi _{\alpha }^{in}$ and $\psi _{\alpha }^{out}$.
Complex scaling
---------------
Next step is to perform the complex scaling operations i.e. scale all the distances $x $ and $y$ by a constant complex factor $e^{i\theta },$ so that both $Re(e^{i\theta })$ and $Im(e^{i\theta })$ are positive (angle $\theta $ must be chosen in the first quartet in order to satisfy this condition). The complex scaling operation, in particular, implies that the analytical continuation of the interaction potentials is performed: $v_{\alpha }(x_{\alpha }e^{i\theta })$ and $w_{\alpha }(x_{\alpha }e^{i\theta })$. Therefore the complex scaling method may be used only if these potentials are analytic. It is easy to see that the solutions of the complex scaled equations coincide with the ones obtained without complex scaling but to which the complex scaling operation is applied: $\left[ \psi (x_{\alpha },y_{\alpha })\right] ^{CS}=\psi (x_{\alpha
}e^{i\theta },y_{\alpha }e^{i\theta })$.
Namely, it is easy to demonstrate that all the outgoing wave functions of eq.(\[eq:assf\]) becomes exponentially bound after the complex scaling operation: $$\begin{aligned}
\left[ \varphi _{\alpha }^{i_{\alpha }}(x_{\alpha }\rightarrow \infty )\right] ^{CS} &\propto &\exp (-k_{i_{\alpha }}x_{\alpha }\cos \theta ) , \nonumber\\
\left[ \phi _{\alpha }^{i_{\alpha },out}(y_{\alpha }\rightarrow \infty )\right] ^{CS} &\propto &\exp (-q_{i_{\alpha }}y_{\alpha }\sin \theta ) , \\
\left[ \Phi _{i_{\alpha }}^{out}(\rho \rightarrow \infty )\right] ^{CS}
&\propto &\exp (-K\rho \sin \theta ) . \nonumber\end{aligned}$$
Nevertheless an incoming wave diverges in $y_{\alpha }$ after the complex scaling: $$\left[ \phi _{\alpha }^{i_{\alpha },out}(y_{\alpha }\rightarrow \infty )\right] ^{CS}\propto \exp (+q_{i_{\alpha }}y_{\alpha }\sin \theta ) .$$ However these terms appear only on the right hand sides of the driven Faddeev-Merkuriev equation (\[eq:drive\_FM\]) being pre-multiplied with the potential terms and under certain conditions they may vanish outside of some finite (resolution) domain $x_{\alpha }\in \lbrack 0,x^{\max
}]$ and $y_{\alpha }\in \lbrack 0,y^{\max }]$. Let us consider the long range behavior of the term $\left[ (v_{\beta }+w_{\beta }^{s})\psi _{\alpha }^{in}\right] ^{CS}$. Since the interaction terms $v_{\beta }$ and $w_{\beta }^{s}$ are of short range, the only region the former term might not converge is along $y_{\beta }$ axis in $(x_{\beta },y_{\beta })$ plane, i.e. for $x_{\beta }\ll y_{\beta }$. On the other hand $x_{\alpha }(\mathbf{x}_{\beta }\mathbf{,y}_{\beta })\approx \sqrt{m_{\gamma }/(m_{\gamma
}+m_{\beta })}\sqrt{M/(m_{\gamma }+m_{\alpha })}y_{\beta }$ and $y_{\alpha }(\mathbf{x}_{\beta }\mathbf{,y}_{\beta })\approx \sqrt{m_{\beta }/(m_{\gamma
}+m_{\beta })}\sqrt{m_{\alpha }/(m_{\gamma }+m_{\alpha })}y_{\beta }$ under condition $x_{\beta }\ll y_{\beta }$. Then one has: $$\small{
\left[ (v_{\beta }+w_{\beta }^{s})\psi _{\alpha }^{i_{\alpha },in}\right]
^{CS}_{x_{\beta }\ll y_{\beta }}\propto \exp\left(-k_{i_{\alpha }}\sqrt{\frac{m_{\gamma }M}{(m_{\gamma }+m_{\beta })(m_{\gamma }+m_{\alpha })}}y_{\beta
}\cos \theta +q_{i_{\alpha }}\sqrt{\frac{m_{\alpha }m_{\beta }}{(m_{\gamma }+m_{\beta })(m_{\gamma }+m_{\alpha })}}y_{\beta }\sin \theta \right)} .$$ This term becomes bound to finite domain in $(x_{\beta },y_{\beta }) $ plane, if condition: $$\tan \theta <\sqrt{\frac{m_{\gamma }M}{m_{\alpha }m_{\beta }}}\frac{k_{i_{\alpha }}}{q_{i_{\alpha }}}=\sqrt{\frac{m_{\gamma }M}{m_{\alpha
}m_{\beta }}}\sqrt{\frac{\left\vert B_{_{i_{\alpha }}}\right\vert }{E+\left\vert B_{_{i_{\alpha }}}\right\vert }} ,
\label{max_theta}$$ is satisfied. This implies that for rather large scattering energies $E$, above the break-up threshold, one is obliged to use rather small complex scaling parameter $\theta $ values.
The term $\left[\sum_{\kappa =1}^{3}w_{\kappa }^{l}-w_{\alpha }-W_{\alpha
}^{c.m.}\right] \psi _{\alpha }^{i_{\alpha },in}$, in principle, is not exponentially bound after the complex scaling. It represents the higher order corrections to the residual Coulomb interaction between particle $\alpha $ and bound pair $(\beta \gamma )$. These corrections are weak $o(1/y^{2})$ and might be neglected by suppressing this term close to the border of the resolution domain. Alternative possibility might be to use incoming wave functions, which account not only for the bare $\alpha -(\beta
\gamma )$ Coulomb interaction but also takes into account higher order polarization corrections.
Extraction of the scattering observables is realized by employing Greens theorem. One might demonstrate that strong interaction amplitude for $\alpha -(\beta \gamma )$ collision is: $$f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }.\mathbf{y}_{\alpha })=-\frac{m}{q_{j_{\alpha}}}\int
\int \left[ (\psi _{\alpha }^{j_{\alpha },in})^*\right] ^{CS}(\overline{v}_{\alpha }+\overline{w}_{\alpha }-W_{\alpha }^{c.m.})^{CS}\left[ \Psi
_{i_{\kappa }}\right] ^{CS}e^{6i\theta }d^{3}\mathbf{x}_{i}d^{3}\mathbf{y}_{i} \label{3b_amp_nc} ,$$with $\left[ \Psi _{i_{\kappa }}\right] ^{CS}=\left[ \psi _{\alpha
}^{i_{\kappa },out}+\psi _{\beta }^{i_{\kappa },out}+\psi _{\gamma
}^{i_{\kappa },out}+\psi _{\alpha }^{i_{\kappa },in}\right] ^{CS}$ being the total wave function of the three-body system. In the last expression the term containing product of two incoming waves is slowest to converge. Even stronger constraint than eq.(\[max\_theta\]) should be implied on complex scaling angle in order to make this term integrable on the finite domain. Nevertheless this term contains only the product of two-body wave functions and might be evaluated without using complex scaling prior to three-body solution. Then the appropriate form of the integral (\[3b\_amp\_nc\]) to be used becomes: $$\begin{aligned}
f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }.\mathbf{y}_{\alpha })
&=&-\frac{m}{q_{j_{\alpha}}}\int \int \left[ (\psi _{\alpha }^{j_{\alpha },in})^*\right] ^{CS}(\overline{v}_{\alpha }+\overline{w}_{\alpha }-W_{\alpha }^{c.m.})^{CS}\left[
\Psi _{i_{\kappa }}-\psi _{\alpha }^{j_{\alpha },in}\right] ^{CS}e^{6i\theta
}d^{3}\mathbf{x}_{i}d^{3}\mathbf{y}_{i} \nonumber\\
&&-\frac{m}{q_{j_{\alpha}}}\int \int (\psi _{\alpha }^{j_{\alpha },in})^*(\overline{v}_{\alpha }+\overline{w}_{\alpha }-W_{\alpha }^{c.m.})\psi _{\alpha }^{j_{\alpha
},in}d^{3}\mathbf{x}_{i}d^{3}\mathbf{y} .\end{aligned}$$
Application to three-body nuclear reactions
===========================================
The two methods presented in sections \[sec:p\] and \[sec:r\] were first applied to the proton-deuteron elastic scattering and breakup [@deltuva:05a; @deltuva:05d; @deltuva:09e; @lazauskas:11a]. The three-nucleon system is the only nuclear three-particle system that may be considered realistic in the sense that the interactions are given by high precision potentials valid over a broad energy range. Nevertheless, in the same way one considers the nucleon as a single particle by neglecting its inner quark structure, in a further approximation one can consider a cluster of nucleons (composite nucleus) to be a single particle that interacts with other nucleons or nuclei via effective potentials whose parameters are determined from the two-body data. A classical example is the $\alpha$ particle, a tightly bound four-nucleon cluster. As shown in Figs. \[fig:Rad\] and \[fig:Radb\] and in Ref. [@deltuva:06b], the description of the $(\alpha,p,n)$ three-particle system with real potentials is quite successful at low energies but becomes less reliable with increasing energy where the inner structure of the $\alpha$ particle cannot be neglected anymore. At higher energies the nucleon-nucleus or nucleus-nucleus interactions are modeled by optical potentials (OP) that provide quite an accurate description of the considered two-body system in a given narrow energy range; these potentials are complex to account for the inelastic excitations not explicitly included in the model space. The methods based on Faddeev/AGS equations can be applied also in this case, however, the potentials within the pairs that are bound in the initial or final channel must remain real. The comparison of the two methods based on the AGS and FM equations will be performed in section \[sec:compare\] for such an interaction model with OP.
In the past the description of three-body-like nuclear reactions involved a number of approximate methods that have been developed. Well-known examples are the distorted-wave Born approximation (DWBA), various adiabatic approaches [@johnson:70a], and continuum-discretized coupled-channels (CDCC) method [@austern:87]. Compared to them the present methods based on exact Faddeev or AGS equations, being more technically and numerically involved, have some disadvantages. Namely, their application in the present technical realization is so far limited to a system made of two nucleons and one heavier cluster. The reason is that the interaction between two heavier cluster involves very many angular momentum states and the partial-wave convergence cannot be achieved. The comparison between traditional nuclear reaction approaches and momentum-space Faddeev/AGS methods for various neutron + proton + nucleus systems are summarized in section \[sec:cdcc\].
On the other hand, the Faddeev and AGS methods may be more flexible with respect to dynamic input and thereby allows to test novel aspects of the nuclear interaction not accessible with the traditional approaches. Few examples will be presented in section \[sec:nonloc\].
Numerical comparison of AGS and FM methods {#sec:compare}
------------------------------------------
As an example we consider the $n+p+^{12}C$ system. For the $n$-$p$ interaction we use a realistic AV18 model [@wiringa:95a] that accurately reproduces the available two-nucleon scattering data and deuteron binding energy. To study not only the $d+{}^{12}$C but also $p+{}^{13}$C scattering and transfer reactions we use a $n$-$^{12}$C potential that is real in the $^2P_\frac{1}{2}$ partial wave and supports the ground state of $^{13}C$ with 4.946 MeV binding energy; the parameters are taken from Ref. [@nunes:11b]. In all other partial waves we use the $n$-$^{12}$C optical potential from Ref. [@CH89] taken at half the deuteron energy in the $d+{}^{12}$C channel. The $p$-$^{12}$C optical potential is also taken from Ref. [@CH89], however, at the proton energy in the $p+{}^{13}$C channel. We admit that, depending on the reaction of interest, other choices of energies for OP may be more appropriate, however, the aim of the present study is comparison of the methods and not the description of the experimental data although the latter are also included in the plots.
We consider $d+{}^{12}$C scattering at 30 MeV deuteron lab energy and $p+{}^{13}$C scattering at 30.6 MeV proton lab energy; they correspond to the same energy in c.m. system. First we perform calculations by neglecting the $p$-$^{12}$C Coulomb repulsion. One observes a perfect agreement between the AGS and FM methods. Indeed, the calculated S-matrix elements in each three-particle channel considered (calculations have been performed for total three-particle angular momentum states up to $J=13$) agree within three digits. Scattering observables converge quite slowly with $J$ as different angular momentum state contributions cancel each other at large angles. Nevertheless, the results of the two methods are practically indistinguishable as demonstrated in Fig. \[fig:dC-noC\] for $d+{}^{12}$C elastic scattering and transfer to $p+{}^{13}$C.
Next we perform the full calculation including the $p$-$^{12}$C Coulomb repulsion; we note that inside the nucleus the Coulomb potential is taken as the one of a uniformly charged sphere [@deltuva:06b]. Once again we obtain good agreement between the AGS and FM methods. However, this time small variations up to the order of 1% are observed when analyzing separate $S$-matrix elements, mostly in high angular momentum states. This leads to small differences in some scattering observables, e.g., differential cross sections for $d+{}^{12}$C elastic scattering (at large angles where the differential cross section is very small) and for the deuteron stripping reaction $d+{}^{12}$C $ \to p+{}^{13}$C shown in Fig. \[fig:dC\]. The $p+{}^{13}$C elastic scattering observables presented in Fig. \[fig:pC\] converge faster with $J$. As a consequence, the results of the two calculations are indistinguishable for the $p+{}^{13}$C elastic cross section and only tiny differences can be seen for the proton analyzing power at large angles. In any case, the agreement between the AGS and FM methods exceeds both the accuracy of the data and the existing discrepancies between theoretical predictions and experimental data.
![ Comparison of momentum- (solid curves) and configuration-space (dashed-dotted curves) results for the deuteron-${}^{12}$C scattering at 30 MeV deuteron lab energy. Differential cross sections for elastic scattering and stripping are shown neglecting the Coulomb interaction.[]{data-label="fig:dC-noC"}](dC30noC.eps)
![ Comparison of momentum- (solid curves) and configuration-space (dashed-dotted curves) results for the deuteron-${}^{12}$C scattering at 30 MeV deuteron lab energy. Differential cross sections for elastic scattering and stripping are shown, the former in ratio to the Rutherford cross section $d\sigma_R/d\Omega$. The experimental data are from Refs. [@perrin:77; @dC30p].[]{data-label="fig:dC"}](dC30.eps)
![ Comparison of momentum- (solid curves) and configuration-space (dashed-dotted curves) results for the proton-${}^{13}$C elastic scattering at 30.6 MeV proton lab energy. Differential cross section divided by the Rutherford cross section and proton analyzing power are shown. The experimental data are from Ref. [@pC30].[]{data-label="fig:pC"}](pC30.eps)
Comparison with traditional nuclear reaction approaches {#sec:cdcc}
-------------------------------------------------------
The method based on the momentum-space AGS equations has already been used to test the accuracy of the traditional nuclear reaction approaches; limitations of their validity in energy and kinematic range have been estalished. The distorted-wave impulse approximation for breakup of a one-neutron halo nucleus ${}^{11}$Be on a proton target has been tested in Ref. [@crespo:08a] while the adiabatic-wave approximation for the deuteron stripping and pickup reactions ${}^{11}$Be$(p,d){}^{10}$Be, ${}^{12}$C$(d,p){}^{13}$C, and ${}^{48}$Ca$(d,p){}^{49}$Ca in Ref. [@nunes:11b]. However, one of the most sophisticated traditional approaches is the CDCC method [@austern:87]. A detailed comparison between CDCC and AGS results is performed in Ref. [@deltuva:07d]. The agreement is good for deuteron-${}^{12}$C and deuteron-${}^{58}$Ni elastic scattering and breakup. In these cases nucleon-nucleus interactions were given by optical potentials; thus, there was no transfer reaction. A different situation takes place in proton-${}^{11}$Be scattering where ${}^{11}$Be nucleus is assumed to be the bound state of a ${}^{10}$Be core plus a neutron. In this case, where the transfer channel $d + {}^{10}$Be is open, the CDCC approach lacks accuracy as shown in Ref. [@deltuva:07d]. The semi-inclusive differential cross section for the breakup reaction $p + {}^{11}$Be $\to p + n + {}^{10}$Be was calculated also using two CDCC versions where the full scattering wave function was expanded into the eigenstates of either the $n + {}^{10}$Be (CDCC-BU) or the $p+n$ (CDCC-TR) pair. Neither of them agrees well with AGS over the whole angular regime as shown in Fig. \[fig:cdcc\]. It turns out that, depending on the ${}^{10}$Be scattering angle, the semi-inclusive breakup cross section is dominated by different mechanisms: at small angles it is the proton-neutron quasifree scattering whereas at intermediate and large angles it is the neutron-${}^{10}$Be $D$-wave resonance. However, a proper treatment of proton-neutron interaction in CDCC-BU and of neutron-${}^{10}$Be interaction in CDCC-TR is very hard to achieve since the wave function expansion uses eigenstates of a different pair. No such problem exists in the AGS method that uses simultaneously three sets of basis states and each pair is treated in its proper basis.
![Semi-inclusive differential cross section for the breakup reaction $p + {}^{11}$Be $\to p + n + {}^{10}$Be at lab energy of 38.4 MeV/nucleon. Results obtained with AGS and CDCC methods are compared.[]{data-label="fig:cdcc"}](ags-cdcc.eps)
Beyond standard dynamic models {#sec:nonloc}
------------------------------
The standard nucleon-nucleus optical potentials employed in three-body calculations have central and, eventually, spin-orbit parts that are local. This local approximation yields a tremendous simplification in the practical realization of DWBA, CDCC and other traditional approaches that are based on configuration-space representations where the use of nonlocal optical potentials was never attempted. However, nonlocal optical potentials do not yield any serious technical difficulties in the momentum-space representation. Thus, they can be included quite easily in the AGS framework employed by us.
There are very few nonlocal parametrizations of the optical potentials available. We take the one from Refs. [@giannini; @giannini2] defined in the configuration space as $$\label{eq:vnl}
v_{\gamma}(\vec{r}',\vec{r}) = H_c(x)[V_c(y) + iW_c(y)] +
2\vec{S_\gamma}\cdot \vec{L_\gamma} H_s(x) V_s(y) ,$$ with $x = |\vec{r}'-\vec{r}|$ and $y=|\vec{r}'+\vec{r}|/2$. The central part has real volume and imaginary surface parts, whereas the spin-orbit part is real; all of them are expressed in the standard way by Woods-Saxon functions. Some of their strength parameters were readjusted in Ref. [@deltuva:09b] to improve the description of the experimental nucleon-nucleus scattering data. The range of the nonlocality is determined by the functions $H_i(x) = (\pi \beta_i^2)^{-3/2} \exp{(-x^2/\beta_i^2)}$ with the parameters $\beta_i$ being of the order of 1 fm.
A detailed study of nonlocal optical potentials in three-body reactions involving stable as well as weakly bound nuclei, ranging from ${}^{10}$Be to ${}^{40}$Ca, is carried out in Ref. [@deltuva:09b]. In order to isolate the nonlocality effect we also performed calculations with a local optical potential that provides approximately equivalent description of the nucleon-nucleus scattering at the considered energy. The nonlocality effect turns out to be very small in the elastic proton scattering from the bound neutron-nucleus system and of moderate size in the deuteron-nucleus scattering. However, the effect of nonlocal proton-nucleus optical potential becomes significant in deuteron stripping and pickup reactions $(d,p)$ and $(p,d)$; in most cases it considerably improves agreement with the experimental data. Examples for $(d,p)$ reactions leading to ground and excited states of the stable nucleus ${}^{17}$O and one-neutron halo nucleus ${}^{15}$C are presented in Figs. \[fig:Odp\] and \[fig:Cdp\]. We note that in these transfer reactions the proton-nucleus potential is taken at proton lab energy in the proton channel while the neutron-nucleus potential has to be real in order to support the respective bound states.
![ Differential cross section for $(d,p)$ reaction on ${}^{16}$O at 36 MeV deuteron lab energy leading to ${}^{17}$O nucleus in the ground state $5/2^+$ (top) and first excited state $1/2^+$ (bottom). Predictions of nonlocal (solid curve) and local (dashed curve) optical potentials (OP) are compared with the experimental data from Ref. [@dO25-63].[]{data-label="fig:Odp"}](dOp36nl.eps)
![ Differential cross section for $(d,p)$ reaction on ${}^{14}$C at 14 MeV deuteron lab energy leading to one-neutron halo nucleus ${}^{15}$C in the ground state $1/2^+$ (top) and first excited state $5/2^+$ (bottom). Curves as in Fig. \[fig:Odp\] and the experimental data are from Ref. [@d14C14p].[]{data-label="fig:Cdp"}](dCp14.eps)
Another extension beyond the standard dynamic models includes the AGS method using energy-dependent optical potentials Although such calculations don’t correspond to a rigorous Hamiltonian theory, they may shed some light on the shortcomings of the traditional nuclear interaction models. A detailed discussion of the calculations with energy-dependent optical potentials is given in Ref. [@deltuva:09a].
Summary
=======
We have presented the results of three-body Faddeev-type calculations for systems of three particles, two of which are charged, interacting through short-range nuclear plus the long-range Coulomb potentials. Realistic applications of three-body theory to three-cluster nuclear reactions — such as scattering of deuterons on a nuclear target or one-neutron halo nucleus impinging on a proton target — only became possible to address in recent years when a reliable and practical momentum-space treatment of the Coulomb interaction has been developed. After the extensive and very complete study of $p$-$d$ elastic scattering and breakup, the natural extension of these calculations was the application to complex reactions such as $d$-${}^{4}$He, $p$-${}^{17}$O, ${}^{11}$Be-$p$, $d$-${}^{58}$Ni and many others using a realistic interaction such as AV18 between nucleons, and optical potentials chosen at the appropriate energy for the nucleon-nucleus interactions. The advantage of three-body calculations vis-à-vis traditional approximate reaction methods is that elastic, transfer, and breakup channels are treated on the same footing once the interaction Hamiltonian has been chosen. Another advantage of the three-body Faddeev-AGS approach is the possibility to include nonlocal optical potentials instead of local ones as commonly used in the standard nuclear reaction methods; as demonstrated, this leads to an improvement in the description of transfer reactions in a very consistent way across different energies and mass numbers for the core nucleus.
Although most three-body calculations have been performed in momentum space over a broad range of nuclei from ${}^{4}$He to ${}^{58}$Ni and have encompassed studies of cross sections and polarizations for elastic, transfer, charge exchange, and breakup reactions, coordinate space calculations above breakup threshold are coming to age using the complex scaling method. We have demonstrated here that both calculations agree to within a few percent for all the reactions we have calculated. This is a very promising development that may bring new light to the study of nuclear reactions given that the reduction of the many-body problem to an effective three-body one may be better implemented and understood by the community in coordinate space rather than in momentum space. On the other hand, compared to DWBA, adiabatic approaches, or CDCC, the Faddeev-type three-body methods are computationally more demanding and require greater technical expertise rendering them less attractive to analyze the data. Nevertheless, when benchmark calculations have been performed comparing the Faddeev-AGS results with those obtained using CDCC or adiabatic approaches, some discrepancies were found in transfer and breakup cross sections depending on the specific kinematic conditions. Therefore the Faddeev-AGS approach is imminent in order to calibrate and validate approximate nuclear reaction methods wherever a comparison is possible.
The work of A.D. and A.C.F. was partially supported by the FCT grant PTDC/FIS/65736/2006. The work of R.L. was granted access to the HPC resources of IDRIS under the allocation 2009-i2009056006 made by GENCI (Grand Equipement National de Calcul Intensif). We thank the staff members of the IDRIS for their constant help.
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---
abstract: 'A complete study for the production of neutral ($h^0,H^0, A^0 (=\phi^0_i)$) and charged Higgs ($H^\pm$) bosons at electron-photon colliders is presented in the context of the minimal supersymmetric standard model. A particular choice of the non-linear $R_{\xi}$-gauge is used to evaluate the amplitudes of the reaction $e\gamma \to e \phi^0_i$. The resulting cross section indicates that it will be possible to detect a signal from the neutral Higgs bosons for most regions of parameter space at the future linear colliders with $\sqrt{s}=500$ GeV through the reaction $e\gamma \to e \phi^0_i$. This reaction also offers the interesting possibility to measure the Higgs mass through the detection of the outgoing electron. The production of the charged Higgs boson ($H^+$) through the reaction $e\gamma \to \nu_e H^\pm$ has in general smaller values for the cross section, which seems more difficult to observe.'
address: |
$^1$SISSA-ISAS, via Beirut 2-4 I-34013, Trieste, Italy\
$^2$Instituto de Física, BUAP, A.P. J-48, 72570 Puebla, Pue. México\
$^3$Dpto. de Física, CINVESTAV-IPN, México D.F.\
$^4$ Facultad de Ciencias Físico Matem' aticas, BUAP, A. P. 1152,Puebla, Pue., M' exico
author:
- 'U. Cotti$^{1,3}$, J.L. Diaz-Cruz$^2$[^1] and J.J. Toscano$^4$'
title: 'Production of neutral and charged Higgs bosons of the MSSM at the future $\bf e\gamma$ colliders '
---
Introduction
============
The search for Higgs bosons at future colliders has become the focus of extensive studies, because of its importance as a test of the mechanism of electroweak symmetry breaking [@kane:90]. The detection of the full spectrum of scalars seems necessary in order to determine the nature of the physics that lies beyond the [*standard model*]{} (SM). Among the extensions of the SM, supersymmetry has received increasing attention, not only because of its aesthetical properties as a field theory, but also because the naturalness problem of the SM can be alleviated within the so-called [*minimal supersymmetric standard model*]{} (MSSM) [@Martin:1997ns].
The MSSM contains two Higgs doublets, whose physical spectrum includes a charged pair $(H^\pm)$, two neutral CP-even scalars ($h^0$, $H^0$), and one pseudoscalar ($A^0$). The MSSM Higgs sector is determined at tree-level by two parameters, which are nowadays chosen as the $A^0$ mass and $\tan\beta$ (the ratio of the vevs. of the two Higgs doublets). This in turn fixes the values of the neutral Higgs mixing angle $\alpha$ and the remaining Higgs masses, which obey the tree-level relations, $m_{h^0} \leq m_Z \leq m_{H^0}$, $m_{h^0}\leq m_{A^0} \leq m_{H^0}$, $m_W \leq m_{H^\pm}$. However, these relations are substantially modified when the effect of radiative corrections is included [@Okada:1991vk; @Haber:1991aw; @Ellis:1991nz]. In particular, it makes possible that $m_Z\leq m_{h^0}$, $m_{A^0} \leq m_{h^0}$, and for some regions of parameters $m_{h^0}$ could even reach a value of about 130 GeV [@Kunszt:1992qe; @Zhang:1998bm; @Heinemeyer:1998jw].
Current LEP2 limits on the Higgs masses are of about 90 GeV for the light scalar $(h^0)$, and, depending on the value of $\tan\beta$, up to about 85 GeV for $A^0$ [@PepeAltarelli:1999ii]. However LEP2 will be able to cover the region up to $m_{h^0} \lesssim 110$ GeV [@Ellis:1999bk]. Recently, it was found that Tevatron can be used to test a significant portion of parameter space through the reaction $p \bar{p}\to Wh^0 + X$ [@Carena:1998gk], moreover the possibility to perform b-tagging with a high efficiency has opened the window to detect the mode $p{\bar p} \to b{\bar b} h^0 + X$ at LHC [@Dai:1996rn] and it will allow to test the large $\tan\beta$ region of parameters at Tevatron [@Balazs:1999nt]. On the other hand, it has been shown [@Spira:1997dg] that LEP2 plus LHC results will be able to cover almost all the MSSM Higgs sector parameter space, and through a combination of the reactions $ p \bar{p} \to t \bar{t} +h^0 (\to \gamma \gamma) + X $, $ p \bar{p} \to h^0 (\to ZZ^*) + X $ [@Gunion:1992er; @Diaz-Cruz:1993hc; @Dittmar:1997ss] and weak boson fusion [@Plehn:1999nw], the full region will be covered. At future linear colliders like NLC [@Kuhlman:1996rc], TESLA [@Accomando:1997wt] or JLC [@Tauchi:1998ty], it will be possible to search for neutral and charged Higgs bosons, through the production reactions $e^+e^- \rightarrow h^0 Z, h^0 \gamma$ [@Haber:1996qb; @Barroso:1986et] and $e^+e^- \rightarrow H^+ H^-$ [@Komamiya:1988rs].
The future linear colliders can also operate in the $e\gamma$ mode [@Giacomelli:1993ip]; in this case the production of the SM Higgs boson has been studied through the $\gamma\gamma \rightarrow \phi^0$ mechanism [@Eboli:1993jt; @Hernandez:1996br], and also with the full 1-loop two-body reaction $e \gamma \rightarrow \phi^0_i e$ . The production of the pseudoscalar Higgs boson $A^0$ through the reaction $e \gamma \rightarrow A^0 e$ has been studied too [@Dicus:1996in], using the photon pole approximation. Even though, these reactions can occur at tree-level, they receive the main contributions at one-loop level where the heavy particles of the model play an important role. These processes could be used to measure the couplings: $\phi^0\gamma^*\gamma$, $\phi^0Z^*\gamma$ [@Gabrielli:1997ix; @Banin:1999ap; @Gabrielli:1999qi], which constitute important one-loop predictions of the theory, and are also sensitive to the effects of new physics. Moreover, because of the possibility to measure the electron momenta, this reaction offers the possibility to determine the Higgs mass with high precision.
In this paper we present a detailed analysis of the production of the neutral and charged Higgs of the MSSM at $\gamma e^-$ colliders through the reactions $e\gamma \to e \phi^0_i$ ($\phi^0_i=h^0,H^0,A^0$) and $e\gamma \to \nu H^\pm$, at the one-loop level. Our goal is to determine the regions of parameter space where a signal is detectable, and also to find out where it will be possible to distinguish between the MSSM and SM Higgs signals. We shall assume that the super-partners are heavy, and thus decouple from the amplitudes [@Dobado:1998tr], however the effect of squarks will be included in the Higgs effective potential using the approximations presented in Ref. [@Barbieri:1991tk].
The organization of the paper is as follows: Section 2 discusses the production of neutral Higgs bosons. It includes a discussion of the region of parameter space that can be excluded, and also on the determination of the Higgs mass. Section 3 is devoted to the production of the charged Higgs, whereas the conclusions of our work are presented in Section 4. Details about the non-linear gauge used in the evaluation of the 1-loop amplitudes as well as the explicit formulae for the various amplitudes are presented in the Appendices.
The neutral Higgses
===================
We now proceed to present the results of our calculation of the amplitudes of the reactions: $$\begin{aligned}
\gamma
(k_1)+e^-(p_1) & \rightarrow & \phi^0(k_2)+e^-(p_2),\end{aligned}$$ where $\phi^0$ denotes any of $h^0,H^0,A^0$, and we have also displayed the notation for incoming and outgoing momenta. We have organized the calculation according to the $\rm U_{em}(1)$ gauge invariance, thus the diagrams are grouped as follows:
1. three-point diagrams characterized by the $\phi^0\gamma^*\gamma$ and by the $\phi^0Z^*\gamma$ coupling (Fig. 1-a) ;
2. Z and W-mediate box diagram and its related triangle graphs (Fig. 1-b,c,d),
The triangle graphs related to the $Z$- and $W$-mediated box diagrams are the one-loop $\phi^0 e^*e$ 3-point functions. These groups of diagrams are finite and gauge invariant by themselves. The group (1) receives contributions from loops of charged fermions, $W$ gauge boson, and charged Higgs boson $H^{\pm}$. On the other hand, the group (2) is sensitive to the $\phi^0
W^+W^-$ and $\phi^0 ZZ$ vertices, respectively. For $A^0$ there are contributions from fermionic triangles only, thus only group (1) appear.
In the linear gauge, the reaction $e\gamma \to e \phi^0_i$ receives also contributions coming from the reducible diagrams with the $Z^*\gamma$ self-energy, which is an extra complication because one needs to perform a renormalization of this term. However, in the nonlinear gauge this term is absent as consequence of the explicit $\rm U_{em}(1)$ gauge symmetry in the W, goldstone and ghost sectors. Notice that there are no contributions coming from the charged Higgs boson to the group (2). This happens because we use the approximation $m_e=0$ and also because there is no $\gamma W^{\pm}H^{\pm}$ vertex at tree-level.[^2]
Production of $h^0,H^0$
-----------------------
The Mandelstam variables used in this calculation are defined by $s=(k_1+p_1)^2$, $t=(k_1-k_2)^2$, and $u=(k_1-p_2)^2$. In addition, $\epsilon^{\mu}(k_1,\lambda_1)$ will denote the $\gamma$ polarization vector. We have evaluated the amplitudes using dimensional regularization, with the help of the program FeynCalc [@Mertig:1990an] and the numerical libraries FF [@vanOldenborgh:1990yc; @vanOldenborgh:1990wn].
The result for the total amplitude of the reaction $e \gamma \to e h^0 (H^0)$ can be written as: $${\cal M}={\cal M}_{\gamma}+{\cal M}_Z+{\cal M}^{\rm box}_Z
+{\cal M}^{\rm box}_W,$$ where ${\cal M}_\gamma$, ${\cal M}_Z$, ${\cal M}^{\rm box}_Z$, and ${\cal M}^{\rm box}_W$ correspond to the sets of diagrams (1), (2), (3), and (4), respectively. The amplitudes coming from the $\phi^0 \gamma^* \gamma$ and $\phi^0Z^* \gamma$ couplings can be written as follows: $${\cal M}_{\gamma,Z}
=
\frac{i\alpha^2m_W}{4s^3_wc^4_w}
\bar{u}(p_2)\gamma^\nu(a_{\gamma,Z}-b_{\gamma,Z}\gamma_5)
u(p_1)\epsilon^{\mu}(k_1,\lambda_1)F_{\gamma,Z}
(k_1\cdot k_2g_{\mu\nu}-k_{2\mu}k_{1\nu}),$$ where $$F_{\gamma}=\frac{4s^2_wc^4_w}{m^2_Wt}
\left(
\sum_f 2f_{\phi}N_c Q^2_f F^{1/2}_f + v_{\phi}F^1_{\gamma}+s_{\phi}F^0
\right),$$ $$F_Z=\frac{c^4_w}{m^2_W(m^2_Z-t)}
\left(
-\sum_f\frac{f_{\phi}C^f_VN_cQ_f}{c^2_w}F^{1/2}_f
+v_{\phi}F^1_Z-\frac{s_{\phi}c_{2w}}{2c^2_w}F^0
\right),$$ with $a_{\gamma}=1$, $b_{\gamma}=0$, $a_Z=1-4s^2_w$, $b_Z=1$, $t_w = \frac{s_w}{c_w}$, and $c_{2w}=c^2_w-s^2_w$. The coefficients $f_{\phi}$, $v_{\phi}$, and $s_{\phi}$ characterize the $\phi^0 \bar{f}f$, $\phi^0 ZZ$, and $\phi^0 H^{\pm}H^{\mp}$ couplings in the MSSM, and are given by: $$f_\phi=
\left\{
\begin{array}{ll}
\frac{\sin \alpha}{\sin \beta}, & \textrm{$H^0 \bar{u}u$}\\
\frac{\cos \alpha}{\cos \beta}, & \textrm{$H^0 \bar{d}d$}\\
\frac{\cos \alpha}{\sin \beta}, & \textrm{$h^0 \bar{u}u$}\\
-\frac{\sin \alpha}{\cos \beta}, & \textrm{$h^0 \bar{d}d$},
\end{array}
\right.$$ $$v_\phi=
\left\{
\begin{array}{ll}
\cos(\beta-\alpha), & \textrm{$\phi^0=H^0$}\\
\sin(\beta-\alpha), & \textrm{$\phi^0=h^0$},
\end{array}
\right.$$ $$s_\phi=
\left\{
\begin{array}{ll}
\cos(\beta-\alpha)-\frac{1}{2c^2_w}\cos2\beta \cos(\beta+\alpha), &
\textrm{$\phi^0=H^0$}\\
\sin(\beta-\alpha)+\frac{1}{2c^2_w}\cos2\beta \sin(\beta+\alpha), &
\textrm{$\phi^0=h^0$}.
\end{array}
\right.$$ The functions $F^{1/2}_f$, $F^0$, $F^1_{\gamma}$ and $F^1_Z$ arising from fermion, scalar and gauge boson loops are given in the Appendix B.
The amplitude for the contributions of the $Z$-mediated box diagram, including the related $\phi^0e^{*-}e^-$ triangle graphs, is the following: $$\begin{aligned}
{\cal M}^{\rm box}_Z
&=&
\frac{i\alpha^2m_W}{4s^3_wc^4_w}\bar{u}(p_2)\gamma^{\nu}
\left(a_Z-\gamma_5\right)^2u(p_1)\epsilon^\mu(k_1,\lambda_1) \nonumber
\\
&
\times
&
\left[-A(t,s,u)\left(k_1\cdot p_1g_{\mu\nu}-p_{1\mu}k_{1\nu}\right)
+A(t,u,s)(k_1\cdot p_2g_{\mu\nu}-p_{2\mu}k_{1\nu})\right],\end{aligned}$$ where the functions $A(t,s,u)$ and $A(t,u,s)$ are presented in the Appendix B.
The amplitude for the $W$-mediated box diagrams and its related triangle graphs is given by: $$\begin{aligned}
{\cal M}^{\rm box}_W
&=&
\frac{i\alpha^2m_W}{4s^3_wc^4_w}
\bar{u}(p_2)\gamma^{\nu}(1-\gamma_5)^2u(p_1)\epsilon^{\mu}(k_1,\lambda_1)
\left\{
A_{12}(t,s,u)
\right.
\nonumber \\
&
\times
&
\left.
(k_1\cdot p_1g_{\mu\nu}-p_{1\mu}k_{1\nu})
-A_{21}(t,s,u)(k_1\cdot p_2 g_{\mu\nu}-p_{2\mu}k_{1\nu})
\right\},\end{aligned}$$ where the functions $A_i$ and $A_{ij}$ are given in the Appendix B.
After squaring the total amplitude, the corresponding cross-section for this process is given by: $$\hat{\sigma}
=
\frac{1}{16\pi s^2}\int_{m^2_{\phi}-s}^{0}dt
\arrowvert \overline{\cal M} \arrowvert^2,$$ where $$\arrowvert \overline{\cal M}\arrowvert^2
=
\frac{\alpha^4m^2_W}{64s^6_wc^8_w}(-t)
\left[
\left(s^2+u^2\right)F_{\gamma Z}+s^2F_s+u^2F_u
\right],$$ with $$F_{\gamma Z}
=
\arrowvert F_\gamma \arrowvert^2 + a\arrowvert
F_Z \arrowvert^2+2a_ZRe\left(F_ZF^*_{\gamma}\right),$$ $$\begin{aligned}
F_s
&=&
\arrowvert A_{12}(t,s,u)\arrowvert^2
+c\arrowvert
A(t,s,u)\arrowvert^2 +2Re
\left\{
[A_{12}(t,s,u)-2aA(t,s,u)]F^*_{\gamma}
\right.
\nonumber \\
&+&
\left.
\left[
4bA_{12}(t,s,u)-2dA\left(t,s,u\right)
\right]F^*_Z-2A_{12}\left(t,s,u\right)A^*\left(t,s,u\right)
\right\},\end{aligned}$$ $$F_u=F_s(s \leftrightarrow u),$$ where we have defined $a=1+a^2_z$, $b=1+a_z$, $c=1+6a^2_z+a^4_z$, and $d=a_z(a^2_z+3)$.
Finally, in order to obtain the total cross-section $(\sigma_T)$, one needs to convolute $\hat{\sigma}$ with the photon distribution, namely: $$\sigma_T
=
\frac{1}{S}\int_{m^2_{\phi}}^{0.83S}f_{\gamma}(\frac{s}{S})\hat{\sigma}(s)ds,$$ where $S$ denotes the squared c.m. energy of the $e^+e^-$-system, and the photon distribution is given by: $$f_{\gamma}
=
\frac{1}{D(\xi)}
\left[
1-x + \frac{1}{1-x} - \frac{4x}{\xi(1-x)}
+ \frac{4x^2}{\xi^2(1-x)^2}
\right],$$ where $$D(\xi)
=
\left(
1-\frac{4}{\xi}-\frac{8}{\xi^2}
\right)
\log(1+\xi)
+\frac{1}{2}+\frac{8}{\xi}-\frac{1}{2(1+\xi)^2}.$$ Notice that, as in Ref. [@Gabrielli:1997ix], one should use, instead of the photon distribution, an exactly monochromatic initial photon beam which can help in distinguish the phisical effects related to the particular collision process from details depending on the final realization of the laser beam.
To evaluate the cross-section for $h^0$ and $H^0$, we have taken $m_t= 175\;$ GeV, and the values for the electroweak parameters given in the table of particle properties [@Caso:1998tx]. In fact, as it was discussed in Ref. [@Gabrielli:1997ix], the contributions of the boxes can be neglected. The cross section for $h^0$ is about 4.2 fb for $m_{A^0}>200$ GeV and $E_{c.m.}=500$ GeV, as can be seen from Fig. 2. With the expected luminosity at the future linear colliders [@Kuhlman:1996rc; @Accomando:1997wt; @Tauchi:1998ty] of about 50 fb$^{-1}$/yr, it will be possible to observe up to about 210 $h^0 + e$ events, which should allow to study the properties of the Higgs boson. Among them, it will be interesting to test its spin and CP-even nature by studying the angular distributions, however this aspect is beyond the goal of the present work, where we are mainly interested in determining if the MSSM Higgs sector can be tested at electron-photon colliders. Total cross-sections for the production of the heavy Higgs boson $H^0$ as a function of the pseudoscalar mass $m_{A^0}$ and for different values of $\tan\beta$ are shown in Fig. 3. These graphs present the total cross-section as a function of the pseudoscalar mass ($m_{A^0}$) for several values of $\tan\beta$ for c.m. energy of 500 GeV.
Production of $A^0$
-------------------
On the other hand the amplitude for the production of the pseudoscalar contains only contributions from the fermionic triangles. Thus, the amplitude takes the form: $${\cal M} = {\cal M}_\gamma +{\cal M}_Z,$$ where $${\cal M}_\gamma
=
\frac{- i \alpha^2 Q_f^2 N_C f_\beta m_W}{s_w}
\bar{u}(p_2)\gamma^\nu u(p_1) \epsilon^\mu (k_1,\lambda_1)
\frac{4 m_f^2}{m_W^2} C_0 \left(t, m_A^2, m_f^2 \right)
\left(
\frac{\epsilon_{\mu\nu\alpha\beta}k_1^\alpha k_2^\beta}{t}
\right),$$ and $${\cal M}_Z
=
\frac{i \alpha^2 Q_f C_V^f N_C f_\beta m_W}{4 s^3_w c^2_w}
\bar{u}(p_2) \gamma^\nu
\left(
C_V^e - C_A^e \gamma_5
\right)
u(p_1)\epsilon^\mu(k_1,\lambda_1)
\frac{4 m_f^2}{m_W^2} C_0 \left(t, m_A^2, m_f^2 \right)
\left(
\frac{\epsilon_{\mu\nu\alpha\beta}k_1^\alpha k_2^\beta}{t - m_Z^2}
\right),$$ with $C_0 \left(t, m_A^2, m_f^2 \right)=
C_0 \left(t, m_A^2, 0, m_f^2, m_f^2, m_f^2 \right)$ the Passarino-Veltman three-point scalar function written in the notation of the FeynCalc program and $$f_\beta=\left\{
\begin{array}{ll}
\cot\beta,&\textrm{$f=u$}\\
\tan\beta,&\textrm{$f=d$},
\end{array}
\right.$$ The total squared amplitude is $$\begin{aligned}
\arrowvert \overline{\cal M}\arrowvert^2
&
=
&
\frac{\alpha^4 Q_f^4 N_C^2 f_\beta^2}{s^2_w}
\frac{-t}{m_A} \frac{s^2 + u^2}{t^2}
\left[
1-
\frac{C_V^e C_V^f}{2 s_w^2 c_w^2 Q_f}
\frac{t}{t - m_Z^2}
\right.
\\
&
+
&
\left.
{C_V^f}^2
\frac{{C_V^e}^2 + {C_V^f}^2}{16 s_w^4 c_w^4 Q_f^2}
\left(\frac{t}{t - m_Z^2}\right)^2
\right]
\arrowvert
2 m_f^2 C_0\left(t, m_A^2, m_f^2 \right)
|^2.\end{aligned}$$ To obtain the total cross-section we use the expressions written previously for the CP-even Higgs bosons (Eqs. 13-15). Results are shown in Fig. 4, where we plot again the total cross-section as a function of the pseudoscalar mass ($m_{A^0}$), for several values of $\tan\beta$ and for c.m. energy of 500 GeV. It can be noted that the cross-section for $A^0$ is smaller than the one resulting for $h^0/H^0$. Thus, in this case it will be more difficult to detect the signal.
Backgrounds and exclusion contours
----------------------------------
The final signature for the reaction $e\gamma \to e \phi^0_i$ depends on the decay of the Higgs boson. For $h^0$, the dominant mode is into $b\bar b$, whereas for $H^0$ and $A^0$ this decay can also be relevant for some regions of parameter space. To evaluate the B.R. of the Higgs bosons we shall assume that the decays into SUSY modes (i.e. charginos, neutralinos, sfermions) are not allowed and take the relevant equation for the decay widths from [@kane:90]. Our results are in agreement with the ones obtained in the literature, for instance in ref. [@Spira:1995rr]. Thus, we shall concentrate on the signature coming from the decay $\phi^0_i \to b\bar b$. In this case the main background comes from $e\gamma \to e b\bar{b}$, which receives contribution from 8 graphs at tree level; we have evaluated numerically this processes using CompHep [@Boos:1994xb; @Baikov:1997zr]. Following Dicus et al. [@Dicus:1996in], we have also imposed a cut on the angular distribution of the outgoing electron ($|\cos\theta| < 0.98$, relative to the incident photon) that reduces significantly the background, while retaining most of the signal rate. To determine the region of parameter space, which is taken as the plane $\tan\beta$-$m_{A^0}$, where the Higgs signal is detectable, we have proceeded as follows: for each value of $m_{A^0}$ we evaluate the Higgs masses and the cross-sections,then we find the value for $\tan\beta$ where the cross-section of the signal is above the background at the 3 and $5\sigma$ level.
In Fig. 5, we show the regions in the plane $\tan\beta$-$m_{A^0}$ where the cross-section coming from $h^0$ has detectable values. The region to the right from the heavy line is where the signal is detectable, above backgrounds at the 3$\sigma$ level. The dashed line denotes the contour at the 5$\sigma$ level. It can be seen that the signal is above the backgrounds for a significant region of parameter space.
On the other hand, Fig. 6 shows the region where the cross-section from $H^0$ reaches detectable values. Again, The region to the left from the heavy (dashed) line is where the signal is detectable at the 3 (5) $\sigma$ level. It can be noticed that this region covers the sector of parameter space where it is more difficult to detect the light $h^0$. Thus, the cross-sections for $h^0/A^0$ and $H^0/A^0$ play a complementary role in providing a detectable signal for the full parameter space. Moreover, we also notice from the superposition of Figs. 5 and 6, that there is a small region where the two Higgs boson signals can be detected, which will allow to distinguish clearly between the MSSM and the SM Higgs sectors.
Finally, we also want to stress the fact that this reaction offers a unique opportunity to obtain a clean measurement of the Higgs mass, thanks to the possibility to measure the outgoing electron momentum. The mass of the Higgs boson is related to the maximal energy of the electron as: $$m_{h^0} = \sqrt{s} - E_{max}.$$ Thus, the precision attained for the electron energy will translate into a good determination of the Higgs mass.
On the other hand, the largest values of the cross-section for $A^0$ are obtained for large values of $\tan\beta$ (which is usually assumed to be at most of order 50); however, we find that even in this case the resulting cross-section does not seem to give a detectable signal. Moreover, for large values of $\tan\beta$ there appears a mass-degeneracy between $A^0$ and $h^0$ or $H^0$, which will make difficult to distinguish the individual signals. In this case, it will be neccesary to optimize the cuts to be able to detect the pseudoscalar $A^0$, and to separate it from the largest signals coming from $h^0$ and $H^0$. Otherwise, one would have to add the respective signals; for instance, one could imposed the criteria that whenever the difference between the Higgs masses is less than 5 GeV (a conservative criteria given the possibility to obtain a precise measurement of the Higgs mass), the individual contributions to the signal must be combined. We find that this only helps to enlarge the exclusion contours for $\tan\beta$ above 30.
Another option to detect $A^0$ (and a heavy $H^0$ too) is to use the planned second and third stages of the future e-gamma colliders [@Kuhlman:1996rc; @Accomando:1997wt; @Tauchi:1998ty], which could reach an energy of 1 TeV and 2 TeV, with integrated luminosities that could reach 125 and 500 $fb^{-1}$, respectively.
The charged Higgs
=================
Now, turning to the production of the charged Higgs, we observe that it can also proceed through the 1-loop reaction: $$\begin{aligned}
\gamma(k_1)+e^-(p_1) &\to& H^-(k_2)+\nu_e(p_2).\end{aligned}$$
The diagrams encountered in the calculation of the $e^- \gamma \rightarrow \nu H^- $ are shown in Fig. \[diagrch\]. They include: triangle graphs with bosons and fermions in the loop (Fig. \[diagrch\]a), seagull-type graphs with bosonic contributions (Fig. \[diagrch\]b-c), and finally those with self-energy insertions (Fig. \[diagrch\]d, \[diagrch\]e, \[diagrch\]f). Fig. \[diagrch\]f gives a vanishing contribution for massless external fermions.
Production of $H^\pm$
---------------------
Our result for the total amplitude is written as:
$$\begin{aligned}
{\cal M}
&
=
&
\frac{i \alpha^2}{2 \sqrt{2} {\rm s}^3_w m_W (t - m^2_W)}
\bar{u}(p_2)
\gamma^\nu
(1-\gamma_5)
u(p_1) \epsilon^\mu (k_1,\lambda_1)
\nonumber
\\
&
\times
&
\left[
\left(
V_f + V_{H^+ \phi^0} + V_{W \phi^0}
\right)
\left(
k_{2\mu} k_{1\nu} -g_{\mu \nu}k_1 \cdot k_2
\right)
+ i A_f \epsilon_{\mu \nu \alpha \beta} k^\alpha_1 k^\beta_2
\right]\end{aligned}$$
where $V_f$, $V_{H^+ \phi^0}$, $V_{W \phi^0}$ and $A_f$ denote the contribution from the different sets of graphs shown in Fig. \[diagrch\] and are given in the Appendix C.
Fig. \[plotchsig\] shows the cross-section for our process for several values of $\tan\beta$ (= 1.5, 5, 10). The cross section is small, about 0.5 fb for $m_{H^\pm} = 200$ GeV and $\rm E_{c.m.}=500$ GeV, which can give 25 events with an expected future linear colliders luminosities of 50 fb$^{-1}$/yr. In Fig. \[plotchall\] we compare the cross section from $e \gamma \rightarrow H^- \nu$, with the pair production $e^+ e^- \rightarrow H^+ H^-$, $e \gamma \rightarrow H^+ H^- e$ and $\gamma \gamma \rightarrow H^+ H^-$. The production mechanisms $e^+ e^- \rightarrow H^+ H^-$, $\gamma \gamma \rightarrow H^+ H^-$, have been discussed in the literature[@Bowser-Chao:1993ji]. On the other hand, the reaction $e \gamma \rightarrow H^+ H^- e$ is evaluated using the Williams-Weizsacker approximation (for the second photon) and we use the sub-reaction $\gamma \gamma \rightarrow H^+ H^-$. It can be seen that the single production dominates only for large Higgs masses i.e. for values that lay beyond the threshold for pair production.
Backgrounds
-----------
In this paper we only show (Fig. \[plotchsig\]) the results for fixed values of $\tan\beta= 1.5,5,10$, and as can be appreciated it turns out that the resulting cross-section has smaller results, which hardly seem detectable. In order to determine if the signal is detectable, one needs to consider the potential backgrounds, which depend on the Higgs mass, since this determines the decay signatures. For instance, if $m_{H^\pm} < m_t + m_b$, the dominant decay mode is $H^+ \rightarrow \tau \nu$ which reaches a branching ratio of order 1 for most values of $\tan \beta$. In this case the background will come from the production $\gamma e \rightarrow W^* \nu$, which for masses $m_{H^\pm} \approx m_W$ will be much larger than the signal (it reaches $\sigma \simeq 4 \rm pb$) and probably will not allow detection. For masses somehow larger, both the signal and the background will be more suppressed and the question of detectability will depend on the experimental ability to identify the decay mode and to reconstruct the charged Higgs mass.
For heavier Higgs masses, $M_{H^\pm} > m_t + m_b$, the dominant Higgs decay is into $t+\bar{b}$, and in this case one needs to compare the signal with the background arising from single top production, $e \gamma \rightarrow \nu \bar{t} b$ for $M_{H^\pm} > m_t + m_b$, which according to the results of Boos et al. [@Boos:1997rd], has $\sigma \simeq 15 \rm fb$ for $\sqrt{s} = 500$ GeV and $m_t = 175$GeV. Thus we can see that at the level of total cross-section the background is again larger than the signal. However if one makes a cut in the invariant mass of the $t-b$ system, then it will be possible to reduce the background.
Conclusions
===========
We have studied the production of the neutral CP-even and charged Higgs boson of the MSSM ($h^0,H^0, H^\pm$) at future $e \gamma$ colliders, through the reactions $e\gamma \rightarrow e h^0, eH^0, e A^0, \nu H^-$. The amplitudes are evaluated using a non-linear $R_{\xi}$-gauge, which greatly simplifies the calculation. The resulting cross section indicates that it is possible to detect the light neutral Higgs boson ($h^0$) for most values of parameters. On the other hand, detection of the heavy neutral Higgs bosons $H^0$ seems possible only for light values of $m_{A^0}$. We have determined the regions in the plane $\tan\beta$-$m_{A^0}$ where the signal is above the backgrounds. The cross-sections for $h^0$ and $H^0$ play a complementary role, since the region where $H^0$ reaches detectable values occurs precisely in the region where it is more difficult to detect the light $h^0$. Thus, both reactions allow to cover the full plane $\tan\beta-m_{A^0}$ with at least one detectable signal. However, it is found that the possibility to distinguish the MSSM from the SM case, through detection of both $h^0$ and $H^0$ signals, occurs only for a limited region of parameter space.
On the other hand, the results for the pseudoscalar and the charged Higgs boson have smaller values of the cross-section, which seems difficult to detect.
We acknowledge financial support from CONACYT and SNI (México). U.C. whish to acknowledge useful discussions with Borut Bajc and Goran Senjanović.
The non-linear gauge
====================
In the evaluation of the above processes, we have used a Feynman-t’Hooft version of the nonlinear gauge that greatly simplify the calculation. In order to appreciate the advantages that offer a nonlinear gauge, we find convenient to compare it with the conventional linear gauge for the SM. It is well known that in a linear gauge only the charged fermion sector presents explicit electromagnetic gauge symmetry ($\rm U_{em}(1)$), since the gauge fixing procedure used for the SU(2) sector destroys manifest $\rm U_{em}(1)$ symmetry in the charged gauge boson ($W^\pm$) and ghosts sectors ($c,\bar{c}$), when they are considered separately, because the gauge functional used to define the $W$-propagator does not transform covariantly under the $\rm U_{em}(1)$ symmetry. It follows that instead of obeying naive Ward identities, these sectors are related through Ward-Slavnov identities. In order to obtain a finite and $\rm U_{em}(1)$-invariant result for higher-order (loop) calculations, one must sum over the contributions arising from the $W$ gauge boson, the charged ghosts, and the $W^\pm G^\mp$ combined effects, $G^\pm$ denotes the would-be Goldstone boson. On the other hand, the functional used to define the $W$-propagator in the non-linear gauge contains the electromagnetic covariant derivative. Thus, the $\rm U_{em}(1)$ symmetry is respected by each charged sector of the SM. It follows that a finite and $\rm U_{em}(1)$ gauge invariant result is obtained for each type of diagram containing a given kind of charged particles. Moreover, the number of diagrams involved is considerably reduced because there are no $W^\pm G^\mp$ combined effects. In the SM, the functionals that define the $W$, $Z$, and $\gamma$ propagators are given by: $$\begin{aligned}
f^+ & = & \bar{D}_\mu W^{+\mu}-i\xi m_WG^+, \nonumber \\
f^Z & = & \partial_\mu Z^\mu - \xi m_Z G^0, \\
f^A & = & \partial_\mu A^\mu, \nonumber \end{aligned}$$ where $D_\mu = \partial_\mu - ig'B_\mu$, with $B_\mu = -s_w Z_\mu + c_w A_\mu $, $G^0$ is the neutral would-be Goldstone. $\xi$ is the gauge parameter, which in general is different for each of the three gauge bosons. However, in the non-linear Feynman-t’Hooft version of this gauge, one takes $\xi=1$ for all sectors. The gauge fixing Lagrangian is given by: $${\cal L}_{GF}
=
-\frac{1}{\xi}f^+f^--\frac{1}{2\xi}(f^Z)^2-\frac{1}{2\xi}(f^A)^2,$$ which in turn removes the $W^\pm G^\mp \gamma$ and $W^\pm G^\mp Z$ vertices from the Higgs kinetic energy term. Notice that the $f^+f^-$ term is $\rm U_{em}(1)$ gauge invariant, which implies that the Fadeev-Popov ${\cal L}_{FP}$ Lagrangian is also $\rm U_{em}(1)$-invariant. It is also possible to proceed further with this scheme and remove more un-physical vertices, such as $\phi^0 W^\pm G^\mp \gamma(Z)$ and $G^0 W^\pm G^\mp \gamma(Z)$. This procedure requires to define the functionals $f^\pm$ nonlinearly both in the vector and scalar parts. However, for the present purposes it is sufficient to use the scheme presented above.
In this case only the couplings in the gauge and ghost sectors are modified. We shall give the lagrangian for these sectors in a form that clarifies the role of the electromagnetic gauge invariance. For this, we define the derivative: $\hat{D}_\mu= \partial_\mu -i g W^3_\mu$, with $W^3_\mu= c_w Z_\mu + s_w A_\mu$. This operator shares with the derivative $D$ defined above the property of containing the e.m. covariant derivative. After adding the gauge-fixing lagrangian, one obtains the couplings for the gauge bosons ($W,Z,A$), which are contained in:
$$\begin{aligned}
{\cal L}&=&-\frac{1}{2}( \hat{D}_\mu W^+_\nu-\hat{D}_\nu W^+_\mu)^\dagger
( \hat{D}^\mu W^{+\nu}-\hat{D}^\nu W^{+\mu})
- ig (s_w F_{\mu\nu}+c_w Z_{\mu\nu}) W^{-\mu}W^{+\nu} \nonumber \\
& & - ig
\left[
s_w F_{\mu\nu}+c_w Z_{\mu\nu} + i\frac{g}{2}
\left(
W^-_\mu W^+_\nu - W^-_\nu W^+_\mu
\right)
\right]
W^{-\mu}W^{+\nu} \nonumber \\
& &
-\frac{1}{\xi}\left(
\bar{D}_\mu W^{+\mu}
\right)
\left(
\bar{D}_\nu W^{+\nu}
\right)^\dagger
+ m^2_W W^-_\mu W^{+\mu} \nonumber \\
& &
- \frac{1}{4} Z_{\mu\nu} Z^{\mu\nu}
+ \frac{1}{2} m^2_Z Z_\mu Z^\mu
- \frac{1}{2\xi}
\left(
\partial_\mu Z^\mu
\right)^2
- \frac{1}{4} F_{\mu\nu} F_{\mu\nu}
- \frac{1}{2\xi}
\left(\partial_\mu A^\mu\right)^2\end{aligned}$$
where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, $Z_{\mu\nu}=\partial_\mu Z_\nu-\partial_\nu Z_\mu$. Notice that, apart from the gauge fixing term $\partial_\mu A^\mu$, this lagrangian is $\rm U_{em}(1)$ invariant.
On the other hand, the ghost sector is substantially modified in the non-linear gauge; however, since the gauge-fixing functionals are U(1) invariant, the charged part of this sector is also invariant. We present now the corresponding Fadeev-Popov lagrangian, written in such a way that $\rm U_{em}(1)$ invariance is explicit, namely: $$\begin{aligned}
{\cal L}_{FPG}
&=&
-\bar{c}^-
\left[
\bar{D}_\mu \hat{D}^\mu + \xi m_W
\left(
m_W + \phi^0 + iG^0
\right)
\right]
c^+ \nonumber \\
& &
-igc_W \bar{c}^-\bar{D}_\mu
\left(
c_Z W^{+\mu}
\right)
-ie \bar{c}^-\bar{D}_\mu(c_{\gamma} W^{+\mu}) \nonumber\\
& &
-i\frac{gs^2_W}{c_W} W^{+\mu}\bar{c}^-
\left(
\partial_\mu c_Z
\right)
+ie W^{+\mu}\bar{c}^-
\left(
\partial_\mu c_\gamma
\right) \nonumber \\
& &
-igc_W W^{+\mu}
\left(
\partial_\mu \bar{c}_Z
\right)
c^- -ie W^{+\mu}
\left(
\partial_\mu \bar{c}_\gamma
\right) c^- \nonumber \\
& &
-g c_{2w} \xi m_Z G^+
\left(
\bar{c}^- c_Z +\bar{c}_Z c^-
\right)
-e \xi m_W G^+
\left(
\bar{c}^- c_\gamma +\bar{c}_\gamma c^-
\right) + h.c. \nonumber \\
& &
-\bar{c}_Z
\left[ \Box + \xi m_Z
\left(
m_Z + \phi^0
\right)
\right]
c_Z - \bar{c}_\gamma \Box c_\gamma \end{aligned}$$ where $c^\pm (\bar{c}^\pm), c_Z (\bar{c}_Z), c_\gamma (\bar{c}_\gamma)$ denote the pairs of ghosts associated with the $W,Z,A$ gauge bosons respectively. The phase-convention for the charged ghost is $(c^+)^\dagger=c^-$ and $(\bar{c})^\dagger=\bar{c}^-$. In addition, one has: $\phi^0=\cos(\alpha-\beta) H^0+\sin(\alpha-\beta) h^0$, where $h^0,H^0$ denote the light and heavy neutral Higgs bosons of the MSSM. Notice that neither the charged Higgs ($H^\pm$) nor the CP-odd Higgs bosons ($A^0$) appear in this lagrangian, which can be understood if one reminds that these fields do not appear in the definition of the gauge fixing terms, and also because they do not receive a v.e.v..
Finally, we would like to mention the properties of the couplings of gauge bosons with the higgs and goldstone bosons. First, in this gauge the couplings $W^+ G^- \gamma$ and $W^+ G^-Z$ are absent by construction, whereas the couplings $G^+ G^- Z (\gamma), H^+H^- Z (\gamma), \phi^0 W^+W^-$ and $W^+H^- \phi^0$ do not depend on the gauge-fixing procedure. However, the couplings $\phi^0 G^+ G^-$ or $\phi^0 c^+c^-$ can depend on the choice of the gauge-fixing terms, when both the vector and scalars sectors are chosen non-linearly.
In conclusion, the Feynman rules needed for our calculation, which are depending of the gauge fixing procedure, arise from the lagrangians of Eqs. (A3-A4) written above.
Loop functions associated to neutral scalars {#a2}
============================================
In this appendix we present the various loop functions coming from the $e^-\gamma \to e^-\phi^0$ processes. The functions arising from fermion, scalar and gauge boson loops, which characterize the $\phi^0 \gamma^* \gamma$ and $\phi^0 Z^*\gamma$ couplings, are given by:
$$\begin{aligned}
F^{1/2}_f
&
=
&
\frac{4m^2_f}{m^2_{\phi}-t}
\left\{
1-\frac{1}{2}
\left(
m^2_{\phi}-t-4m^2_f
\right)C_0\left(t,m^2_{\phi},m^2_f\right)
\right.
\nonumber
\\
&
+
&
\left.
\frac{t}{m^2_{\phi}-t}
\left[
B_0\left(m^2_\phi,m^2_f\right)-B_0\left(t,m^2_f\right)
\right]
\right\},\end{aligned}$$
$$F^0=-\frac{4m^2_W}{m^2_{\phi}-t}
\left\{
1+2m^2_{H^{\pm}}C_0\left(t,m^2_{\phi},m^2_{H^\pm}\right)
+\frac{t}{m^2_{\phi}-t}\left[B_0\left(m^2_{\phi},m^2_{H^\pm}\right)
-B_0\left(t,m^2_{H^\pm}\right)\right]
\right\},$$
$$\begin{aligned}
F^1_{\gamma}
&=&
-\frac{4m^2_W}{m^2_{\phi}-t}
\left\{
3+\frac{m^2_{\phi}}{2m^2_W}+
\left[
3\left(1-\frac{m^2_{\phi}}{2m^2_W}\right)+\frac{2t}{m^2_W}
\right]
2m^2_WC_0\left(t,m^2_{\phi},m^2_W\right)
\right.
\nonumber \\
&+&
\left.
\left(3+\frac{m^2_\phi}{2m^2_W}\right)\left(\frac{t}{m^2_\phi-t}\right)
\left[
B_0
\left(
m^2_{\phi},m^2_W
\right)-B_0
\left(t,m^2_W\right)
\right]
\right\},\end{aligned}$$
$$\begin{aligned}
F^1_Z
&=&
-4\left(3-t^2_w\right)m^2_WC_0\left(t,m^2_{\phi},m^2_W\right)
+\frac{2m^2_W}{m^2_\phi-t}
\left[
5+\frac{m^2_\phi}{2m^2_W}
-\left(1+\frac{m^2_{\phi}}{2m^2_W}\right)t^2_w
\right]
\nonumber \\
&
\times
&
\left\{
1 +
2m^2_WC_0(t,m^2_{\phi},m^2_W)
+\frac{t}{m^2_\phi-t}
\left[
B_0\left(m^2_{\phi},m^2_W\right)-B_0\left(t,m^2_W\right)
\right]
\right\},\end{aligned}$$
where $C_0(t,m^2_{\phi},m^2_i)=C_0(0,t,m^2_{\phi},m^2_i,m^2_i,m^2_i)$, $B_0(m^2_{\phi},m^2_i)=B_0(m^2_{\phi},m^2_i,m^2_i)$ and $B_0(t,m^2_i)=B_0(t,m^2_i,m^2_i)$ are Passarino-Veltman three- and two-point scalar functions written in the notation of the FeynCalc program.
The loop functions coming from $Z$-mediated box diagram, including the related $\phi^0 e^{*-}e^-$ triangle graphs, are given by:
$$\begin{aligned}
A(t,s,u)
&=&
\frac{1}{2st}
\left\{
\frac{s-m^2_Z}{s}\left[m^2_Z(s+u)-su\right]D_0(1,2,3,4)
+(s-m^2_Z)\left[C_0(1,2,4)
\right.
\right.
\nonumber \\
&+&
\left.
\frac{u}{s}C_0(1,2,3)-\frac{t+u}{s}C_0(2,3,4)
+\frac{1}{s}\left(t-s-\frac{2m^2_Z st}{(s+t)(s-m^2_Z)}\right)
C_0(1,3,4)\right]
\nonumber \\
&+&
\left.
\frac{2t}{s+t}
\left[
B_0(3,4)-B_0(1,3)
\right]
\right\}.\end{aligned}$$
$A(t,u,s)$ is obtained from $A(t,s,u)$ by means of the interchange $s \leftrightarrow u$. The arguments of the scalar functions are given by: $$\begin{aligned}
D_0(1,2,3,4)
&=&
D_0\left(0,s,m^2_{\phi},u,0,0,m^2_e,m^2_e,m^2_Z,m^2_Z\right),\nonumber \\
C_0(1,2,4)
&=& C_0\left(0,s,0,m^2_e,m^2_e,m^2_Z\right), \nonumber \\
C_0(1,2,3)
&=& C_0\left(0,0,u,m^2_e,m^2_e,m^2_Z\right), \nonumber \\
C_0(2,3,4)
&=&
C_0\left(s,0,m^2_\phi,m^2_Z,m^2_e,m^2_Z\right), \nonumber \\
C_0(1,3,4)
&=&
C_0\left(0,m^2_\phi,u,m^2_e,m^2_Z,m^2_Z\right), \nonumber \\
B_0(3,4)
&=&
B_0\left(m^2_\phi,m^2_Z,m^2_Z\right), \nonumber \\
B_0(1,3)
&=&
B_0\left(u,m^2_e,m^2_Z\right).\end{aligned}$$
Finally, the loop functions associated to the $W$-mediated box diagrams and its related triangle graphs are the following:
$$\begin{aligned}
A_{12}(t,s,u) &=& 2c^4_w\left[A_1(t,s,u)+A_2(t,u,s)\right], \nonumber \\
A_{21}(t,s,u) &=& 2c^4_w\left[A_2(t,s,u)+A_1(t,u,s)\right],\end{aligned}$$
with $$\begin{aligned}
A_1(t,s,u)
&=&
\frac{1}{2st}
\left\{
\frac{s-m^2_W}{s}\left[m^2_W(s+u)+st\right]D_0(1,2,3,4)
\right. \nonumber \\
&+&
(s-m^2_W)\left[C_0(2,3,4)
-\frac{t}{s}C_0(1,3,4)-\frac{s+u}{s}C_0(1,2,3)
\right.
\nonumber \\
&+&
\left.
\left.
\frac{u+t}{s}C_0(1,2,4)\right]
+\frac{2t}{s+u}\left[B_0(1,2)-B_0(1,3)\right]
\right\},\end{aligned}$$ $$\begin{aligned}
A_2(t,s,u)
&=&
\frac{1}{2tu}
\left\{
\left[
\frac{t+u-m^2_W}{u}\left(m^2_W(s+u)-st\right)-2m^2_Wt
\right]
D_0(1,2,3,4)
\right.
\nonumber \\
&+&
\left(t+u-m^2_W\right)
\left[
\frac{s}{u}C_0(2,3,4)
+\frac{t}{u}C_0(1,3,4)
\right.
\nonumber \\
&-&
\left.
\frac{s+u}{u}C_0(1,2,3)+\frac{u^2-2ut-t^2}{u(t+u)}C_0(1,2,4)
\right]
\nonumber \\
&+&
\left.
\frac{2t}{t+u}\left[B_0(2,4)-B_0(1,2)\right]
+\frac{2t}{s+u}\left[B_0(1,3)-B_0(1,2)\right]
\right\}.\end{aligned}$$ The scalar functions have the following arguments: $$\begin{aligned}
D_0(1,2,3,4)
&=&
D_0\left(0,0,0,m^2_\phi,t,s,m^2_W,0,m^2_W,m^2_W\right),\nonumber \\
C_0(2,3,4)
&=&
C_0\left(0,0,s,0,m^2_W,m^2_W\right), \nonumber \\
C_0(1,3,4)
&=& C_0\left(0,0,t,m^2_W,0,m^2_W\right), \nonumber \\
C_0(1,2,3)
&=&
C_0\left(t,0,m^2_\phi,m^2_W,m^2_W,m^2_W\right), \nonumber \\
C_0(1,2,4)
&=&
C_0\left(0,s,m^2_\phi,m^2_W,0,m^2_W\right), \nonumber \\
B_0(1,2)
&=&
B_0\left(m^2_{\phi},m^2_W,m^2_W\right), \nonumber \\
B_0(1,3)
&=&
B_0\left(t,m^2_W,m^2_W\right), \nonumber \\
B_0(2,4)
&=&
B_0\left(s,0,m^2_W\right).\end{aligned}$$ The expressions for $A_1(t,u,s)$ and $A_2(t,u,s)$ can be obtained from the respective $A_1(t,s,u)$ and $A_2(t,s,u)$ through the interchange $s \leftrightarrow u$.
Loop functions associated to charged scalars
============================================
In this appendix we shall write the explicit formulae for the functions $V_f$, $V_{H^\pm \phi^0}$, $V_{W \phi^0}$ and $A_f$ using a notation that, hopefully, facilitates its use by the interested reader. Our convention for the scalar functions $B_0$, $C_0$ is the same as in the previous Appendix. In this case, the contributions arising from loops with fermions, scalars and gauge bosons, have the following expression:
$$\begin{aligned}
V_f
&
=
&
\sum_f
\frac{N_c}{t - m_{H^\pm}^2}
\left\{
\left(
Q_u + Q_d
\right)
\left(
m_d^2 \tan\beta + m_u^2 \cot\beta
\right)
\right.
\nonumber
\\
&+&
\left.
2Q_d
\left[
m_d^2
\left(
m_d^2 \tan\beta + m_u^2 \cot\beta
\right)
-
\left(
t - m_{H^\pm}^2
\right)
m_d^2 \tan\beta
\right]
C_0
\left(
0, m^2_{H^\pm}, t, m_d^2, m_d^2, m_u^2
\right)
\right.
\nonumber
\\
&+&
\left.
2Q_u
\left[
m_u^2
\left(
m_d^2 \tan\beta + m_u^2 \cot\beta
\right)
-
\left(
t - m_{H^\pm}^2
\right)
m_u^2 \cot\beta
\right]
C_0
\left(
0, m^2_{H^\pm}, t, m_u^2, m_u^2, m_d^2
\right)
\right.
\nonumber
\\
&+&
\left.
\left[
\frac{m_d^2 \tan\beta + m_u^2 \cot\beta}{t - m_{H^\pm}^2}
\left(
m_u^2 - m_d^2
+
\left(
Q_u + Q_d
\right)
m_{H^\pm}^2
\right)
\right.
\right.
\nonumber
\\
&+&
\left.
\left.
2
\left(
Q_d m_d^2 \tan\beta + Q_u m_u^2 \cot\beta
\right)
\right]
\left[
B_0
\left(
t, m_d^2, m_u^2
\right)
-
B_0
\left(
m_{H^\pm}^2, m_d^2, m_u^2
\right)
\right]
\right.
\nonumber
\\
&+&
\left.
m_u m_d
\frac{m_u^2 - m_d^2}{m_{H^\pm}^2}
\left[
B_0
\left(
m_{H^\pm}^2, m_d^2, m_u^2
\right)
-
B_0
\left(
0, m_d^2, m_u^2
\right)
\right]
\right\}, \end{aligned}$$
and $$\begin{aligned}
A_f
&
=
&
\sum_f
N_c
\left\{
Q_d m_d^2 \tan\beta
C_0
\left(
0, m^2_{H^\pm}, t, m_d^2, m_d^2, m_u^2
\right)
-
Q_u m_u^2 \cot\beta
C_0
\left(
0, m^2_{H^\pm}, t, m_u^2, m_u^2, m_d^2
\right)
\right.
\nonumber
\\
&+&
\left.
\frac{m_d^2 \tan\beta + m_u^2 \cot\beta}{t - m_{H^\pm}^2}
\left[
B_0
\left(
t, m_d^2, m_u^2
\right)
-
B_0
\left(
m_{H^\pm}^2, m_d^2, m_u^2
\right)
\right]
\right\},\end{aligned}$$ $$\begin{aligned}
V_{H^+ \phi^0}
&
=
&
\sum_{\phi^0 = h^0, H^0}
\left(
-v_{\phi^0} s_{\phi^0}
\right)
\frac{m_W^2}{t - m^2_{H^\pm}}
\left\{
1
+
2m^2_{H^\pm}C_0
\left(
0, t, m^2_{H^\pm}, m^2_{H^\pm}, m^2_{H^\pm}, m^2_{\phi^0}
\right)
\right. \nonumber
% \\[2ex]
\\
&+&
\left.
\left(
\frac{m^2_{\phi^0} - m^2_{H^\pm} - t}{t - m^2_{H^\pm}}
\right)
\left[
B_0
\left(
t, m^2_{H^\pm}, m^2_{\phi^0}
\right)
-
B_0
\left(
m^2_{H^\pm}, m^2_{H^\pm}, m^2_{\phi^0}
\right)
\right]
\right. \nonumber
% \\[2ex]
\\
&+&
\left.
\left(
\frac{m^2_{\phi^0} - m^2_{H^\pm}}{m^2_{H^\pm}}
\right)
\left[
B_0
\left(
0, m^2_{H^\pm}, m^2_{\phi^0}
\right)
-
B_0
\left(
m^2_{H^\pm}, m^2_{H^\pm}, m^2_{\phi^0}
\right)
\right]
\right\},\end{aligned}$$ $$\begin{aligned}
V_{W \phi^0}
&
=
&
\sum_{\phi^0 = h^0, H^0}
\left(
-\sigma_{\phi_0} v_{\phi_0}
\right)
\frac{m_W^2}{t - m^2_{H^\pm}}
\left\{
\frac{m^2_{H^\pm} - m^2_{\phi^0} + m^2_W}{2 m^2_W}
\right. \nonumber
% \\[2ex]
\\
&+&
\left.
\left(
3 m^2_{H^\pm} - m^2_{\phi^0} + m^2_W - 2t
\right)
C_0
\left(
0, m^2_{H^\pm}, t, m^2_W, m^2_W, m^2_{\phi^0}
\right)
\right. \nonumber
\\
&+&
\left.
\frac{\left(
m^2_{H^\pm} - m^2_{\phi^0} - m^2_W
\right)
\left(
m^2_{\phi^0_i} - m^2_W -t
\right)
-4 m^2_W
\left(
m^2_{H^\pm} -t
\right)}%
{2 m^2_W
\left(
t - m^2_{H^\pm}
\right)}
\right. \nonumber
\\
&\times&
\left.
\left[
B_0
\left(
t, m^2_W, m^2_W
\right)
-
B_0
\left(
m^2_{H^\pm}, m^2_W, m^2_W
\right)
\right]
\right. \nonumber
\\
&+&
\left.
\frac{\left(
m^2_{\phi^0} - m^2_W
\right)
\left(
m^2_{H^\pm} - m^2_{\phi^0} + m^2_W
\right)
}{2 m^2_W m^2_{H^\pm}}
\left[
B_0
\left(
0, m^2_W, m^2_{\phi^0}
\right)
-
B_0
\left(
m^2_{H^\pm}, m^2_W, m^2_{\phi^0}
\right)
\right]
\right\},\end{aligned}$$ where $\sigma_{H^0} = -v_{h^0}$, $\sigma_{h^0} = -v_{H^0}$ and $\phi^0_1 = H^0$, $\phi^0_2 = h^0$.
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[^1]: e-mail: ldiaz@sirio.ifuap.buap.mx
[^2]: We could think on possible contributions arising from reducible diagrams that include the $\gamma^*\phi^0$ and $Z^*\phi^0$ self-energies (with the virtual fields tied to the electronic line). However, it can be shown that these terms always vanish.
|
---
abstract: 'Advances in Web technology enable personalization proxies that assist users in satisfying their complex information monitoring and aggregation needs through the repeated querying of multiple volatile data sources. Such proxies face a scalability challenge when trying to maximize the number of clients served while at the same time fully satisfying clients’ complex user profiles. In this work we use an abstraction of complex execution intervals (CEIs) constructed over simple execution intervals (EIs) represents user profiles and use existing offline approximation as a baseline for maximizing completeness of capturing CEIs. We present three heuristic solutions for the online problem of query scheduling to satisfy complex user profiles. The first only considers properties of individual EIs while the other two exploit properties of all EIs in the CEI. We use an extensive set of experiments on real traces and synthetic data to show that heuristics that exploit knowledge of the CEIs dominate across multiple parameter settings.'
author:
- |
Haggai Roitman[^1]\
IBM Research\
Haifa University Campus, Mount Carmel\
Haifa 31905, Israel\
haggai@il.ibm.com
- |
Avigdor Gal\
Technion - Israel Institute of Technology\
Haifa 32000, Israel\
avigal@ie.technion.ac.il
- |
Louiqa Raschid\
University of Maryland\
College Park, MD 20742\
louiqa@umiacs.umd.edu\
title: Query Scheduling in the Presence of Complex User Profiles
---
Introduction
============
Advances in Web technology now enable the creation of personalization proxies that assist users in satisfying their complex information monitoring and aggregation needs. A proxy actively decides when it needs to query information streams using pull-based technology to satisfy clients’ complex user profiles. Example platforms include personalization portals and news aggregation applications (*e.g.,* MyYahoo![^2], Feedly[^3]), which provide a single point of access, services for continuously refreshing profiles, and tools for integration via a *mashup*[^4] of data extracted from multiple heterogeneous data sources. Proxies are required to query multiple streams of events in a timely manner to satisfy both the characteristics of servers, *e.g.,* intensity of updates, and the complex user profiles of clients. This results in a scalability challenge when the proxy tries to satisfy millions of clients.
Push-based solutions to satisfy complex user needs exist, e.g., WebSocket, WebRTC, HTTP/2 Push, etc. Yet, data collection with push-based solutions requires a high cost [@CherniackBBCCXZ03], especially if such push-based technology is not natural to the Web environment. Pull-based solutions have considered only simple monitoring solutions (*e.g.*, [@PANDEY2004]) that cannot satisfy complex user needs. Lately, such complex requirements have been also introduced in the context of Web and Social streams search, discovery, and analysis (*e.g.*, [@BansalK07; @paul2016social; @Roitman2008MDC; @Sakaki:2010; @LaFleur:2015; @woodall2017]) which require the collection of data from multiple (possibly interrelated) pull-only sources (e.g., RSS feeds).
Following [@ROITMAN2008b] , complex user profiles are represented in this work as a set of *complex execution interval*s (CEIs). A CEI is an extension of a simple execution interval (EI) [@ROITMAN2008] which defines periods of time during which the proxy has to probe the corresponding resources to satisfy the profile. Querying multiple data sources is represented by combining individual EIs to construct CEIs, possibly over a set of resources. Each EI in a CEI should be monitored once for the CEI to be satisfied (or captured"). A proxy schedules queries to corresponding resources in these intervals to satisfy the profile.
![Motivating example: personal business portal.[]{data-label="Fig:ex"}](portal_new.jpg){width="3.35in"}
We illustrate a personalized portal in Figure \[Fig:ex\]. A business analyst identifies data sources that fit her needs, *e.g.*, the *CNN Breaking News*[^5] website, *CNN Money.com*,[^6] and *Mish’s Global Economic Trend Analysis*[^7] blog. The data sources can be specified as URLs of online Web feeds (*e.g.,* RSS, Atom) as shown by the RSS symbols in Figure \[Fig:ex\]. Alternatively, *Web Scraping*[^8] technology allows her to delineate content that is to be extracted. Most Web feeds are available via pull-only access protocols, *e.g.*, via `HTTP GET` requests. However, some feeds may be pushed to the user, with the appropriate registration, using proprietary technology (*e.g.,* Google Alerts service[^9]).
The analyst constructs a wide perspective by integrating data from multiple business and market data sources. She is interested in querying *CNN Breaking News* and *CNN Money.com* when a periodic querying of *Mish’s Global Economic Trend Analysis* blog detects that a new post in the blog contains the word *%oil%*. This will be translated into the three queries in Figure \[Fig:ex\]. The analyst is willing to accept a delay of up to two minutes in probing `MishBlog` and a delay of 10 minutes for the other two feeds.
According to [@HONG2005], 55% of Web feeds are updated hourly. Further, about 80% of the feeds have an average size smaller than 10 KB, suggesting that items are promptly removed from the feeds. These statistics on refresh frequency and volatility illustrate the challenges faced in satisfying millions of complex profiles.
Our research goal is to support a new generation of solutions to address complex user profiles in (primarily) pull-only settings.
Previously, [@ROITMAN2008b; @ROITMAN2009] has introduced the problem of capturing CEIs and showed that an offline solution is of high polynomial complexity. An approximate **offline** solution was presented in [@ROITMAN2009] as a baseline for maximizing completeness. In this paper, we present three **online** heuristics that can scale to capture millions of CEIs. S-EDF is a simple extension of the well-known Early Deadline First policy and only considers individual EIs while making a schedule. MRSF considers the number of EIs in a CEI that have not been scheduled and M-EDF considers the deadlines of all the EIs in the CEI that have not been scheduled; thus, both exploit the properties of CEIs. Using an extensive empirical analysis of real and synthetic data, we demonstrate the dominance of the heuristics that exploit the properties of CEIs.
The paper is organized as follows: In Section \[sec:model and problem definition\] we summarize the model of [@ROITMAN2008a; @ROITMAN2009] and define the problem. Section \[sec:solution\] presents the heuristic solution. We present experiments in Section \[sec:experiments\] and in Section \[sec:related\] we describe related work. Section \[sec:conclusions\] concludes with future work.
Model and Problem Definition {#sec:model and problem definition}
============================
We summarize the model of complex user profiles presented in [@ROITMAN2008b; @ROITMAN2009], for completeness sake. *Clients* query *Servers* through *proxies*. A server manages resources and can be *queried* (pull-based) by the proxy on behalf of its clients. We discuss the three building blocks of our model, namely *client profiles*, *execution intervals*, and *schedules*, and then define the problem.
Profiles and complex execution intervals
----------------------------------------
A client’s complex user profile is translated into a set of resources and *complex execution intervals* (CEI). An execution interval (EI) [@ROITMAN2008] defines periods of time during which the resource must be probed. A profile combines individual EIs, possibly over a set of resources, to construct a CEI. Each EI in a CEI should be monitored once for the CEI to be satisfied (or captured"). Details of a query language to express complex user profiles is presented in [@ROITMAN2008b]. We use pseudo continuous queries in our examples. We expect that a proxy will provide tools, *e.g.*, Web scraping, to provide intuitive interfaces to clients.
Formally, let $\mathcal{R}=\{r_{1},r_{2},...,r_{n}\}$ be a set of $n$ resources and let $\mathcal{T}=\left( T_{1},T_{2},...,T_{K}\right) $ be an epoch with $K$ chronons.[^10] We assume the proxy manages a set of client profiles $\mathcal{P}=\{p_{1},p_{2},\ldots ,p_{m}\}$. A client profile $p=\left\{ \eta |\eta =\langle I_{1},I_{2},\ldots ,I_{t}\rangle
\right\} $ is a collection of CEIs. A CEI $\eta $ contains several EIs, where each EI $I$ is associated with a resource $r\in \mathcal{R}$ and $I$ contains a start and finish chronon $I=[T_{s},T_{f}];T_{s},T_{f}\in \mathcal{T};T_{s}\leq {T_{f}}$. An interesting class of profiles, we denote by $\mathcal{P}^{[1]}$, are profiles for which any EI $I$ of any CEI has a width of *exactly one chronon*. This class serves later in our analysis.
![A hierarchical illustration of the model’s main concepts.[]{data-label="fig:concepts"}](hierarchy.jpg){width="3.0in"}
Profiles, CEIs, and EIs construct a *hierarchy* as seen in Figure [fig:concepts]{}. Two CEIs within the same profile, or two EIs within the same CEI, are *siblings*. To model *profile complexity*, we denote by $rank(p)$ the maximal number of execution intervals in any CEI $\eta \in {p}$ $=\max_{\eta \in {p}}\left\{ \left\vert \eta \right\vert \right\} $, where $\left\vert \eta \right\vert $ is the number of execution intervals in $\eta $. The definition is easily extended to a set of profiles $\mathcal{P}$ as follows: $rank(\mathcal{P})=\max_{p\in \mathcal{P}}\left\{ rank(p)\right\} $.
The beginning of an interval is determined by an update event at a resource or a temporal event (*e.g.*, every three minutes). In the case that the server will *push* the update event, or for a temporal event, the beginning of the interval is deterministic. A proxy may also need to predict an update event using an update model and stochastic modeling [@GAL2000b]. The window (length) of the interval is determined with respect to the stream of update events, (*e.g.*, update = overwrite), or as a temporal event (*e.g.*, within five minutes of the beginning of the interval). For example, a profile for Web scraping over Web feeds requires that published items be collected before the server overwrites them; this is a stochastic event.
![EIs for the profile of the case study example[]{data-label="fig:exp3"}](ex3.jpg){width="3.5in"}
The CEIs of our motivating example (see Figure \[Fig:ex\]) are illustrated in Figure \[fig:exp3\]. Probing the `MishBlog` feed every 3 minutes, with a possible slack of 2 minutes, is represented by the first set of EIs, labeled `T1`. We note that by registering to a Pub/Sub system, the proxy may be informed of updates to Mish’s blog. However, the proxy still has to query to get the updated blog. For posts on Mish’s blog that contain the keyword `oil`, EIs will be scheduled to probe the other two resources, labeled `T2` and `T3`, respectively. In this example, some CEIs will only have a rank of 1 (when updates on Mish’s blog do not include `oil`) while others will have a rank of 3.
EIs of the same or different profiles may overlap in time; two cases of overlap are interesting. When EIs of different resources overlap (*inter-resource overlap*), then they are all candidates for being simultaneously queried by the proxy. This can lead to *congestion* when the available budget is low. When EIs associated with an *identical* resource overlap (*intra-resource overlap*), there is the potential to exploit this overlap to build an efficient query schedule. While delaying a query on a resource could lead to more efficient schedules, a delay may also result in failing to capture an EI, *e.g.*, there is congestion-based competition in a future chronon. The special case of *no intra-resource overlap* is of theoretical interest since it allows us to present some bounds [@ROITMAN2009].
Schedules {#schedules and metrics}
---------
A data delivery schedule $S=\{s_{i,j}\}_{i=1,...,n;j=1,...,K}$ ($n$ resources and $K$ chronons) assigns $s_{i,j}=1 $ if resource $r_{i}\in \mathcal{R}$ should be queried by the proxy at chronon $T_{j}\in \mathcal{T}$, else $s_{i,j}=0$. We denote by $\mathbb{S}$ the set of all possible schedules.
The expression $\mathbb{I}(I,S)$ indicates whether a schedule $S$ successfully captures (*i.e.*, some resource $r_{i}$ is queried during) the EI $I$. Given a profile $p$, a CEI $\eta \in {p}$, and an EI $I\in \eta $ that refers to resource $r_{i}\in \mathcal{R}$, we have the following: $$\mathbb{I}(I,S)=\left\{
\begin{array}{cc}
1, & \exists T_{j}\in {I}:s_{i,j}=1 \\
0, & otherwise\end{array}\right.$$
We extend $\mathbb{I}(I,S)$ to describe capturing a CEI as follows: Given a profile $p$ and a CEI $\eta \in {p}$, we say that $\eta $ is *captured* by schedule $S\in \mathbb{S}$ if $\mathbb{I}(\eta ,S)=\prod_{I\in \eta }\mathbb{I}(I,S)=1$.
Problem statement {#sec:problem}
-----------------
We assume that the proxy has a limited amount of resources that can be consumed for querying. In this paper we consider a constraint similar to the one used in prior works of Web Monitoring [PANDEY2004]{} and Web Crawlers [@WOLF2002], where at each chronon $T_{j}\in \mathcal{T}$ the proxy can query up to $C_{j}$ resources. This constraint is represented by a budget vector $\vec{C}=\left(
C_{1},C_{2},\ldots ,C_{K}\right) $.
Given a set of client profiles $\mathcal{P}=\{p_{1},p_{2},...,p_{m}\}$, the proxy objective is to *maximize gained completeness*, *i.e.*, the number of CEIs from $\mathcal{P}$ that are captured given the budget $\vec{C}
$. A CEI is successfully captured once *all* of its execution intervals are captured. Every CEI $\eta \in {p}$ that is successfully captured by the proxy schedule (indicated by $\mathbb{I}(\eta ,S)=1$) increases the gained completeness.
Given a schedule $S\in \mathbb{S}$, the *gained completeness* (denoted `GC` in short) from monitoring $\mathcal{P}$ during $\mathcal{T}$ according to $S$ is calculated as follows (where $\left\vert p\right\vert $ denotes the number of CEIs in profile $p$): $$\mathtt{GC}(\mathcal{P},\mathcal{T},S)=\frac{\sum_{p\in \mathcal{P}}\sum_{\eta \in {p}}\mathbb{I}(\eta ,S)}{\sum_{p\in {\mathcal{P}}}\left\vert
p\right\vert } \label{eq:gain compliteness}$$
Formally, the problem of query scheduling in the presence of complex user profiles is defined by the following constrained optimization problem.
\[maximization problem\] Given a set of profiles $\mathcal{P}$ and an epoch $\mathcal{T}$: $$\begin{array}{l}
\emph{maximize}\ \texttt{GC}(\mathcal{P},\mathcal{T},S) \\
\emph{s.t.}\ \sum_{i=1}^{n}s_{i,j}\leq {C_{j}},\ \forall {j=1,2,\ldots ,K}\end{array}$$
Previously, [@ROITMAN2009] presented two offline solutions to Problem \[maximization problem\]. In an offline setting, the proxy is provided with **all** CEIs in $\mathcal{P}$ for $K$ chronons **in advance** and has to determine the schedule $S$ of probing resources in $\mathcal{R}$. The offline solutions are of high polynomial complexity [@ROITMAN2009]. Yet, they provide a baseline of optimal performance.
Query Scheduling Policies {#sec:solution}
=========================
We now present three heuristic solutions to the problem of query scheduling in the presence of complex user profiles. CEIs are not known a priori and proxy decision making is done online. In our example, the EIs for querying the `MishBlog` feed can be determined in advance, but query scheduling for the other two streams, (*CNN Breaking News* and *CNN Money.com*) depends on the contents of the first feed. At every chronon $T_{j}$, the proxy may receive a set of new CEIs. The proxy then has to decide which resources in $\mathcal{R}$ to probe, while considering the set of all candidate CEIs, including those submitted prior to $T_{j}$, which have not been completely captured yet, and the new set of CEIs. We denote the set of all candidate CEIs at chronon $T_{j}$ as $\emph{cands}(\eta )$ and the union bag of all their EIs (termed *candidate EIs*) as $\emph{cands}(I)=\biguplus_{\eta _{q}\in \emph{cands}(\eta )}\eta _{q}$. The bag notation ($\biguplus $) is used due to intra-resource overlaps.
Policies {#sec:online policies}
--------
To determine which candidate EIs in $\emph{cands}(I)$ to choose the proxy uses *policies*. At chronon $T_{j}$, a policy $\Phi $ considers $\emph{cands}(I)$ and the budget $C_{j}$, and returns up to $C_{j}$ EIs to probe. Such policies can be efficiently implemented.
Each of the policies we propose can be executed in either a *non-preemptive* or *preemptive* manner. Non-preemptive policies do not allow new candidate CEIs to be scheduled for monitoring at chronon $T_{j}$ if previously probed CEIs need to be probed at $T_{j}$. Therefore, a non-preemptive policy $\Phi$ first selects $I\in \emph{cands}(I)$ that belongs to previously probed CEIs. Then, if there is any budget left, $\Phi$ selects EIs from the newly introduced CEIs. It is worth noting that even the non-preemptive policies do not guarantee a successful capture of a CEI that has been probed at least once. If the number of previously probed CEIs exceeds the bandwidth budget, some of these CEIs may be dropped.
Policies can be classified according to the amount of information about candidate CEIs they use. We propose a three level classification, as follows.
[**Individual EI level**]{}: An individual EI level policy utilizes only the local properties of a *single EI* without considering the parent CEI or sibling EIs.As a representative of this level we suggest the *Single Interval Early Deadline First* (or `S-EDF` in short) policy, $\Phi _{\mathtt{S}\text{\texttt{-}}\mathtt{EDF}}$. This policy is modeled on the well known EDF policy [@Liu73], prefering EIs that have the earliest deadline. Given an execution interval $I$ and a chronon $T$, the deadline is calculated (in terms of number of remaining chronons) as follows: $$\mathtt{S}\text{\texttt{-}}\mathtt{EDF}(I,T)=I.T_{f}-T+1$$Proposition \[prop:SEDFOptimality\] holds for this policy.
\[prop:SEDFOptimality\] Given $\mathcal{P}$ without intra-resource overlap, and $rank(\mathcal{P})=1$, the policy $\Phi_{\mathtt{S}\text{\texttt{-}}\mathtt{EDF}}$ is optimal.
WIC [@PANDEY2004], a well-known monitoring solution for the Web, can also be classified as an individual EI level policy. We note that WIC was designed to address a different optimization goal compared to our Problem \[maximization problem\]). WIC’s solution balances completeness with timeliness, providing a bound of 2-competitiveness for its optimization goal. WIC defines a utility for each EI and picks those EIs with the maximum accumulated utility for probing in each chronon. For our experiments in Section \[sec:experiments\] we implemented WIC (details provided later) and compared the performance of WIC with the online policies from this section.
[**Rank level**]{}: A rank level policy bases its decision on profile complexity by considering the rank of the parent CEI. As a representative of this level we suggest the *Minimal Residual Stub First* (`MRSF`) policy, $\Phi _{\mathtt{MRSF}}$. This policy prefers EIs that belong to parent CEIs with a minimal number of EIs left to be captured. The intuition behind this policy is that a CEI with less EIs remaining to probe has a higher probability of success. Formally, given an EI $I$, $I\in
\eta $ and $\eta \in {p}$, then the `MRSF` value is calculated as follows:$$\mathtt{MRSF}(I)=rank(p)-\sum_{I^{\prime }\in \eta }\mathbb{I}(I^{\prime
},S)$$where $I^{\prime }$ iterates over all EIs in $\eta $. The following proposition provides a bound for the performance of this policy.
Given $\mathcal{P}$ without intra-resource overlap and $rank(\mathcal{P})=k$, the $\Phi_{\texttt{MRSF}}$ policy is $l$-competitive, where: $$l=\max_{\eta\in\mathcal{P}}(\sum_{I\in\eta}\left|I\right|).$$
[**Multi-EI level**]{}: A multi-EI level policy utilizes the properties of all EIs of a parent CEI (including sibling EIs). As a representative, we suggest the *Multi Interval EDF* (`M-EDF`) policy, $\Phi _{\mathtt{M}\text{\texttt{-}}\mathtt{EDF}}$, which prefers execution intervals that have the minimal `M-EDF` value, calculated as follows (for $I\in \eta $): $$\mathtt{M}\text{\texttt{-}}\mathtt{EDF}(I,T)=\sum_{I^{\prime }\in \eta
}\left( \mathtt{S}\text{\texttt{-}}\mathtt{EDF}\left( I^{\prime },T\right)
\cdot \left[ 1-\mathbb{I}\left( I^{\prime },S\right) \right] \right)$$
The `M-EDF` value combines the EDF values of execution interval $I$ and its siblings that were not captured by chronon $T$. For each $I^{\prime }
$ (iterating over all EIs in the CEI) if the EI is not yet active (chronon $T<I^{\prime }.T_{s}$), then the EDF value is calculated with $T=0$, taking into account the full length of the EI. The intuition is to favor CEIs with less total remaining chronons so as to increase their probability of capture, leading to a gain in completeness.
The following proposition discusses the class $\mathcal{P}^{[1]}$ of profiles. Recall that we denote by $\mathcal{P}^{[1]}$ a set of profiles for which any EI $I$ of any CEI has a width of *exactly one chronon*.
\[prop. MRSF equals MS-EDF\] For problem instances with $\mathcal{P}^{[1]}$ profiles the `M-EDF` policy is equivalent to the `MRSF` policy.
![Policies illustration[]{data-label="policies"}](policies_simple.jpg){width="3.35in"}
\[example policies\] Figure \[policies\] illustrates the value each policy assigns to a candidate CEI with four execution intervals. At chronon T, `S-EDF` counts the number of remaining chronons until the end of the execution interval ($5$). `MRSF` counts the number of remaining EIs in the CEI ($4$). Finally, `M-EDF` accumulates the number of chronons of all remaining EIs ($22$). The values each policy assigns are given at the bottom of the figure.
![Competing CEIs illustration[]{data-label="fig:competingPolicies"}](policies2.jpg){width="2.7in"}
\[example policies1\] Figure \[fig:competingPolicies\] illustrates the decision making of each of the policies, when faced with multiple complex execution intervals and a bandwidth constraint. Here, we have two candidates, the first ($CEI_1$) with four required EIs (marked in black) and the second ($CEI_2$) with three EIs (marked in gray). Dashed lines connect a parent complex execution interval to its children. Assume the current time is $T$ and only one EI can be selected ($C_T=1$). At $T$ the first two EIs of $CEI_1$ were successfuly captured (see the $\surd$ signs near the top of the figure) yet we allow preemption. A decision needs to be taken whether to probe $EI_1$ or $EI_2$.
`S-EDF` counts the number of remaining chronons until the end of the current EI, yielding $5$ for $EI_1$ and $6$ for $EI_2$. Therefore, the `S-EDF` policy will stick with $CEI_1$.
`MRSF` counts the number of remaining EIs. With $2$ remaining EIs for $CEI_1$ and $3$ remaining EIs for $CEI_2$, `MRSF` will stick with $CEI_1$, choosing $EI_1$.
Finally, `M-EDF` accumulates the number of chronons in all the remaining EIs. We have $19$ chronons for $CEI_1$ and $16$ for $CEI_2$. Therefore, $CEI_1$ will be preempted, since `M-EDF` favors $EI_2$.
Experiments {#sec:experiments}
===========
Datasets and Experiment Setup {#sec:dataSet}
-----------------------------
### Datasets
We used two data streams of update events in our experiments. The first is a real-world trace of 732 auctions for Intel, IBM, and Dell laptop computers[^11]. To obtain the traces, we used an RSS feed retrieved for a search query on a popular e-commerce website to obtain timestamps of updates and we reconstructed the real stream. Each auction lasted 3 days with a total of 11,150 bids in the datasets. We also used a synthetic data stream that was generated using a Poisson based update model; the parameter $\lambda$ controls the update intensity of each resource.
### Profiles and CEIs {#sec:profile generation}
We used a profile template to specify complex user needs and to generate multiple profile instances. $\mathtt{AuctionWatch(}k\mathtt{)}$" is a sample template that monitors the prices of $k$ auctions and notifies the user after *a new bid is posted in all $k$ auctions*. The start of an EI to monitor an auction will be prediced from an update model constructed from the traces. The length of each EI can be specified as `overwrite` or `window(W)`. The `overwrite` requests *every new bid to be delivered before the next update occurs and overwrites the last published bid*.
The `window(W)` requests * every new bid to be delivered within a window of `W` chronons from the time the bid was posted*.
We generated up to $m$ profile instances from a template using a 2-stage process and 2 Zipf distributions. Recall that $rank(\mathcal{P})=k$ corresponds to $k$ EIs in a CEI. We determine the rank of each profile instance according to a Zipf$(\beta,k)$ distribution, where $k$ is specified in the template. $\beta=0$ implies a random selection or a uniform distribution $U[1,k]$, while a positive $\beta$ value produces more profiles whose CEIs include less ($<$ $k$) EIs. This stage models a variance of the complexity of a profile (termed *intra-user preferences*).
In a second step, given some profile of rank $k$, we use a Zipf$(\alpha,n)$ distribution to select a set of resources. $\alpha=0$ implies a random selection or a uniform distribution $U[1,n]$, while positive $\alpha$ value implies a preference towards popular" resources. This stage models *inter-user preferences* and imitates the way popular resources are chosen by users. For Web Feeds the value of $\alpha$ was estimated to be $1.37$ [@HONG2005].
### Experimental Setup
We implemented a simulation-based environment to test the different solutions. Given a profile template and an update event stream, we generate $m$ profile instances and their CEIs. The proxy receives as input at each chronon the set of CEIs that overlap in that chronon. We repeated each execution $10$ times and recorded the average performances. Table \[controlled parameters\] summarizes the control parameters. We report on two metrics, namely *completeness* (as given in Eq. \[eq:gain compliteness\]) and *runtime costs*, the execution time normalized over the total number of EIs that must be captured.
For comparison we implemented WIC [@PANDEY2004]. We used a straightforward implementation of the algorithm presented in [@PANDEY2004], setting `urgency` to be uniform (that is, for each resource $r_{i}$ and chronon $T$ $\mathtt{urgency}_{i}(T)=1$), `life` to be either `overwrite` or\
`time-window-append(W)` [@PANDEY2004] (which corresponds to our `window(W)`), and $p_{ij}$ set to $1$ if resource $r_{i}$ has an update at chronon $T_{j}$, otherwise we set $p_{ij}=0$. While we note that WIC does not handle complex CEIs, for certain parameter settings, *i.e.*, with $rank(\mathcal{P})=1$, $\mathtt{W}=0$, and with no intra-resource overlap, both WIC and `S-EDF` provide an optimal schedule [@PANDEY2004]. Therefore both `S-EDF` and WIC are used as a baseline of simple EI policies for comparison. In addition, we implemented an offline approximation for the problem, as introduced in [@ROITMAN2009] , which serves as another baseline for evaluating the performance of the proposed policies.
The various policies and the experimental setup were implemented in Java, using JDK version 7. The JVM was initiated with a heap-memory size of $1.00GB$.
Comparative policy performance {#sec:proof of concept}
------------------------------
![Online policies with and without preemption[]{data-label="fig:poc"}](poc.jpg){width="3.35in"}
We have two versions of each of the online policies, namely with and without preemption. To compare them, we used the real-world trace and the profile template $\mathtt{AuctionWatch}$$(3)$ (each profile will monitor up to $3$ auctions). We also used a window of `W`=20, *i.e.*, the new bid must be reported within 20 chronons. Finally, we used a budget of $C=2$ probes per chronon. We compared the gained completeness of each online policy using various parameter settings, with and without preemption. Figure \[fig:poc\] reports on the results with $400$ auction resources, $1590$ CEIs and $3599$ simple EIs.
Typically `MRSF` and `M-EDF` perform better then `S-EDF` over a range of experiment settings. This improved performance can be attributed to the use of more knowledge about the rank of the profile and the number of EIs to be captured. Our complete set of experiments with `S-EDF` indicates that it performs better without preemption for $C=1$, while for $C>1$, the preemptive version is better. The other two policies, `MRSF` and `M-EDF`, perform better with preemption in all but a few cases. These results were consistent for most of the parameter settings that were tested, with a difference of up to $20\%$ in completeness between the preemptive and non-preemptive versions of each online policy. We label the policies `(P)`“ to denote the preemptive version and `(NP)`” for non-preemptive.
Policies vs. the Baseline
-------------------------
We compared the performance of the proposed policies and WIC, with the offline approximation solution, for different parameter settings. We used the real trace and the $\mathtt{AuctionWatch}$$(k)$ profile template with $\mathtt{W}=0$, *i.e.*, the client requires an *immediate* probing of each EI as soon as a bid is posted.
We first examine $\mathcal{P}^{[1]}$ profiles. To generate these profiles and CEIs, we use $C=1$ and avoid intra-resource overlap by ensuring that each EI of a CEI refers to a distinct resource. For this parameter setting, the offline approximation guarantees a $2k$-approximation [@ROITMAN2009] (for $rank(\mathcal{P}^{[1]})=k$). \[prop. MRSF equals MS-EDF\], both `MRSF(P)` and `M-EDF(P)` policies perform the same and we only report on `MRSF(P)`.
![Online policies compared with the baseline[]{data-label="fig:off_vs_on_rank"}](baseline.jpg){width="3.0in"}
Figure \[fig:off\_vs\_on\_rank\] reports on the results, where the $rank(\mathcal{P})$ varies from 1 to 5. For this experiment, if $rank(\mathcal{P})=k$, then all CEIs that were generated for that problem instance have exactly $k$ EIs. Figure \[fig:off\_vs\_on\_rank\] and experiments with other settings show that both `M-EDF(P)` and `MRSF(P)` policies typically dominate the offline approximation, the `S-EDF` policy (preemptive or not), and WIC. The `MRSF(P)` outperforms the offline approximation by up to $10\%$. The `S-EDF` policy does not dominate the offline approximation yet it dominates WIC, as is the offline approximation. We observe that complex profiles with higher rank force the proxy to devote more probes to each CEI and completeness decreases for all policies.
We can compute an upper bound on the optimal completeness for $rank(\mathcal{P})=1$ by using `S-EDF` optimal completeness for the same problem instance. We can also determine a worst case upper bound on the optimal completeness for $rank(\mathcal{P})>1$. To do that, for every $rank(\mathcal{P})$ level, we measure the completeness in terms of single EIs that are captured (*i.e.*, assuming that $rank(\mathcal{P})=1$); This measure provides a worst case upper bound on the optimal performance. This bound is illustrated by the dashed line in Figure \[fig:off\_vs\_on\_rank\]. Therefore, we observe that the completeness achieved by the complex policies is actually very high for every profile complexity level (from optimal completeness for $rank(\mathcal{P})=1$ to at least $77\%$ of the optimal completeness for $rank(\mathcal{P})=5$). To conclude, the complex policies dominate both the offline approximation and the best possible simple policies available in the literature, and provide good performance guarantees.
Runtime scalability {#sec:scalability}
-------------------
\[sec:scale\]
![Scalability analysis[]{data-label="fig:scale uniform utility"}](scalability_new.jpg){width="2.75in"}
We next performed scalability analysis and measured the runtime of the offline approximation and the online policies aggregated over a run with $K=1000$ chronons. The results are given in Figure \[fig:scale uniform utility\]. The runtime is given per a single EI processing unit. We ran the experiments with increasing number of profiles, generating as a result an increasing number of execution intervals. The number of profiles and the update intensity that were used are given on top of the curves, and on the lower right side of the graphs respectively. First we compared the offline approximation and online policies with small workloads ($\lambda=20$ and 100-500 profiles; see Figure \[fig:scale uniform utility\](1)). We can see that the offline approximation performs much worse then the online policies. We further continued to investigate the scalability of the online policies and increased the workload using 2.5 times higher updates intensity and increased the number of profiles up to 2500 (see Figure \[fig:scale uniform utility\](2)). We can see that there is still a linear trend in the policies runtime behavior, suggesting that for real problem instances the online policies are scalable and more robust compared to the offline approximation.
Workload analysis {#sec:on vs. off}
-----------------
![Workload analysis (update intensity)[]{data-label="fig:workload:lambda"}](workload1.jpg){width="2.7in"}
We next studied the effect of different workload settings on the gained completeness of policies. For this purpose we controlled two parameter settings, namely the average updates intensity per resource (given by $\lambda$), and number of profiles ($m$) (where the other parameters are those described in Table \[controlled parameters\]). It is worth noting that for this analysis we used a strict budget allocation of $C=1$. In Section \[sec:budget\] we show how a more lenient budget improves performance.
Figure \[fig:workload:lambda\] contains the results of this analysis for increasing update intensity. In general, we observe that both the `MRSF(P)` and `M-EDF(P)` policies are much better then `S-EDF(NP)` policy for all workload parameter settings that were used. We also observe that the `MRSF(P)` performs slightly better then `M-EDF(P)`. Finally, as the average update intensity increases, there are more updates to each resource, and thus, each profile requires to capture more CEIs, resulting in decreased gained completeness. The effect of number of profiles the proxy has to consider, on the policies performance (not presented here due to space considerations) is similar.
The impact of user preferences {#sec:prefs}
------------------------------
![Impact of user preferences[]{data-label="fig:prefs"}](preferences_new.jpg){width="2.7in"}
We now report on the impact of user preferences on performance. For this analysis, we have used various settings of $\alpha$ (inter-user preferences), and $\beta$ (intra-user preference), where the other parameters were set for the baseline values of Table \[controlled parameters\]. See Section \[sec:profile generation\] for the definition of these two parameters.
Figure \[fig:prefs\](1) shows the impact of inter-user preferences. As $\alpha$ increases, there is less random selection of resources in each profile, with more execution intervals coming from popular resources. The online policies gain more completeness due to more opportunities to capture intra-resource overlapping execution intervals of popular resources. Furthermore, we observe that the `S-EDF(NP)` policy is dominated by the two other policies.
Figure \[fig:prefs\](2) shows the impact of intra-user preferences. As $\beta$ increases, users prefer less complex profiles. The impact, as observed in Figure \[fig:off\_vs\_on\_rank\] as well, is an increase in performance. We further observe that both `MRSF(P)` and `M-EDF(P)` policies still outperform the `S-EDF(NP)` policy, with a slight variation between the first two. This may be attributed to the fact that there are still CEIs that require to probe more then one execution interval, where `S-EDF(NP)` was shown in Figure \[fig:off\_vs\_on\_rank\] to be dominated.
Effect of budgetary limitations {#sec:budget}
-------------------------------
![Budget limitations[]{data-label="fig:budget"}](budget.jpg){width="2.7in"}
We studied the effect of budgetary limitations on the different policies. So far we have used a **strict budgetary allocation** of $C=1$. We now show the impact of additional budget on performance. The results are given in Figure \[fig:budget\]. We observe that as the proxy budget increases, allowing it to probe more resources per chronon, a remarkable increase in performance is achieved. In particular, `MRSF(P)` policy utilizes the budget much better then the `S-EDF(P)` policy. We conclude that the aggregated view of `MRSF(P)` and `M-EDF(P)` policies utilizes the budget better.
Related Work {#sec:related}
============
Many contemporary applications, such as Web crawlers and monitors [@PANDEY2004; @WOLF2002], news feeds aggregators [@AltinelBCKLMMNSS07; @Gruhl06; @LaFleur:2015] and social streams monitors [@agarwal2014time; @Mathioudakis:2010; @osborne2014real; @paul2016social; @Sakaki:2010; @woodall2017], may require nowadays complex access patterns to multiple Web sources. We can classify these applications based on their information gathering strategy.
With push based systems, data is pushed to the system and the research focus is mainly on aspects of efficient data processing, where load shedding techniques [@Chi05] can be applied to determine the portions of the data that must be processed. Examples are publish-subscribe (pub-sub) (*e.g.,* [@DemersGHRW06; @DiaoRF04]), stream processing (*e.g.,* [@ArasuBBDIRW03; @CherniackBBCCXZ03]), and complex event processing (CEP) (*e.g.,* [@WuDR06; @DemersGPRSW07]). Pub-sub systems such as the ONYX system [@DiaoRF04] allow the registration of complex requirements at servers and focus mainly on the trade-off between data processing efficiency and the expressiveness of the queries that can be processed by the system. Stream processing systems are also push-based in nature and focus mainly on smart filtering and load shedding techniques. Complex event processing (CEP) systems (e.g., [@DemersGPRSW07]) assume the pushing of a stream of raw events and focus mainly on efficient complex event identification.
In our work, we assume a pull based solution; there is a need to collect the data, *e.g.,* via periodic queries. In the presence of stream data that is available via pull-only access (*e.g.,* Web feeds), complex queries must cross multiple streams. Example systems include query processing in sensor networks (*e.g.,* [@DeshpandeGMHH04; @Diao07]), continuous queries (CQ) and Web monitoring (*e.g.,* [@LiuPT00WebCQ; @Garg04; @PANDEY2004; @ECKSTEIN2007]), Grid and Web services query processing (*e.g.,* [@ZhangARS05; @Srivastava06]), and mashups of Web sources (*e.g.,* [@AltinelBCKLMMNSS07]). Current pull based monitoring is unable to handle complex data needs over multiple data sources. For example, current works in CQ and Web monitoring such as WIC [@PANDEY2004] handle only simple single resource monitoring tasks that are assumed to be independent of each other.
Conclusions and Future work {#sec:conclusions}
===========================
In this work we presented efficient query scheduling online policies for the satisfaction of complex data needs in pull based environments that involve volatile data. Using intensive experiments we analyzed the performance of these policies under different settings and showed that even under restrictive budget constraints they can perform well. We further showed that utilizing additional profile structures can assist to improve performance significantly.
The** **problem presented in this work can be extended in several interesting directions. First, we do not consider the varying costs of probing resources. These variations may be due to computational costs, *e.g.*, extracting a stock price may be cheaper than searching for a keyword in a blog, the bandwidth needed to download results of varying size, monetary charges at the servers, *etc*. In this case, queries will not consume the same budget. A second extension is to consider more general profile satisfaction constraints given as client profile utilities. Such utilities can help to construct better prioritized policies.
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[^1]: Work was done while the author was still a PhD student in the Technion - Israel Institute of Technology.
[^2]: <https://my.yahoo.com/>
[^3]: <https://feedly.com>
[^4]: See http://www.programmableweb.com/ for mashup examples.
[^5]: <http://edition.cnn.com/>
[^6]: <http://money.cnn.com/>
[^7]: http://globaleconomicanalysis.blogspot.com/
[^8]: <http://en.wikipedia.org/wiki/Web_scraping>.
[^9]: [www.google.com/alerts](www.google.com/alerts)
[^10]: A chronon is an indivisible unit of time.
[^11]: <http://git.annonymous.com>.
|
---
author:
- 'Hui Xu,'
- 'Lei Ming,'
- 'Yeuk-Kwan E. Cheung,'
title: 'Dynamics of scalar fields in an expanding/contracting cosmos at finite temperature'
---
Introduction
============
It is widely believed that our universe starts with a hot big bang, which is considered as the beginning of the radiation dominated era in the cosmic history. Many properties of the cosmos that we observe today can be understood as the results of quantum processes, which would be typically out of equilibrium, in the hot and dense plasma [@Gamow:1946eb; @Kolb:1990vq] that filled the universe after the big bang. Prior to the radiation dominated era, there was a period of accelerated cosmic expansion known as inflation [@Starobinsky:1980te; @Guth:1980zm; @Linde:1981mu]. At the end of inflation the universe is cold and empty; all energy is stored in the zero mode of the inflaton field. One mechanism for setting up the “hot big bang” initial conditions of a radiation dominated universe is “reheating” [@Kofman:1994rk; @Shtanov:1994ce; @Drewes:2013iaa], in which the universe is “reheated" from a complete vacuum by the energy transfer from the inflaton to other degrees of freedom, e.g. dark matter particles and elementary particles that made up the Standard Model of Particle Physics. The study of the quantum dissipative effects in the early universe, therefore, has profound implications on the studies of matter production and thus on the thermal history of our universe.
The thermal history of the early universe is an important theoretical basis to determine the abundance of thermal relics. It thus plays an important role in distinguishing or excluding cosmological models. Studies of the particle dynamics in the early universe uncover crucial details within and beyond the Standard Model. We are interested in the thermal production of particles from plasma [@Berera:2004kc; @Graham:2008vu], dissipation effects of fields in medium [@Lee:1999iv; @deOliveira:1997jt], cosmological freeze-out processes [@Joyce:1996cp; @Joyce:1997fc] and their imprints on early universe physics.
As we have mentioned above, matter production is via the relaxation of inflaton into scalar, gauge and fermionic quantum fields in a large thermal bath [@Berera:2004kc; @Yokoyama:2004pf; @Anisimov:2008dz; @Bartrum:2014fla]. Inflaton, in the standard model of cosmology, is a scalar and responsible for an epoch of exponential expansion to produce a flat, homogeneous and isotropic universe free of topologically stable relics like monopoles and cosmic strings. Therefore, scalar fields, despite being the simplest, play important roles within or beyond the Standard Models of Particle Physics and Modern Cosmology. The existence of scalar fields in the Standard Model of Particle physics has been firmly established by high precision experiments conducted at the Large Hadron Collider in 2012, where the Higgs boson [@Aad:2012tfa; @Chatrchyan:2012xdj] plays a pivotal role of giving mass to elementary particles in the standard model. In addition, scalar fields can be candidates for dark energy [@Wetterich:1987fm; @ArmendarizPicon:2000dh; @Copeland:2006wr] or dark matter [@McDonald:1993ex; @Burgess:2000yq; @Bento:2000ah]. Scalar fields also play important roles in the bounce universe which is an alternative way to address how our current universe comes about. In this model a contraction phase exists prior to the “birth” of our presently observed universe, see [@Brandenberger:2016vhg; @Battefeld:2014uga] for recent reviews. Our study focuses on the scalar field dynamics in hot medium: it thus founds the basis to follow up on the particle productions in these bounce models [@Cai:2011zx; @Lin:2010pf; @Allen:2004vz; @Khoury:2001wf; @Cai:2013kja; @Loewenfeld:2009aw; @Li:2011nj], where the Hubble parameter can be set to positive or negative values, and the bounce process is driven by two or more scalar fields.
This paper builds on an earlier study of the finite-temperature effects in a thermal bath, carried out by Y. K. E. Cheung, M. Drewes, J. U Kang and J. C. Kim [@Cheung:2015iqa], to further establish the rigorous theoretical framework to precisely study the evolution and interactions of elementary fields in the inflationary cosmology background or in a bounce universe. In [@Cheung:2015iqa], the authors have made progress towards a quantitative understanding of the non-equilibrium dynamics of scalar fields in the non-trivial background of the early universe with a high temperature, large energy density and a rapid cosmic expansion. They calculated the effective action and quantum dissipative effects of a cosmological scalar field in this background. The analytic expressions for the effective potential and damping coefficient are presented using a simple scalar model with quartic interaction. In this paper, we extend their efforts on building this theoretical framework by incorporating a non-zero Hubble parameter in the analysis and obtain temperature dependent expression of the damping coefficient to first order in $H$. In this way one can properly address the questions of how the hot primordial plasma may have been created after inflation [@Drewes:2010pf] and whether there are observable features of the reheating process [@Drewes:2015coa; @Drewes:2017fmn].
Our study of the non-equilibrium process in an early universe starts from the action of scalar fields $\phi$ and $ \chi $. The non-equilibrium process is non-Markovian. That is, the evolution of the fields in late times depends on all the previous states, and the non-Markovian effects are contained in a “memory integral” in the Kadanoff-Baym equations. We shall use the closed-time-path (CTP) formalism [@Schwinger:1960qe; @Keldysh:1964ud], the so-called “in-in formalism.” The “in-in formalism" is made to deal with such finite temperature problems in out-of-thermal-equilibrium processes. The non-equilibrium nature renders the usual “in-out formalism” ineffective.
Unlike the usual zero-temperature quantum field theory, this paper considers both the thermal corrections and quantum corrections. To be more specific, we first derive the equation of motion, up to the first order in $ H $, from the effective action of $\phi$. The effective potential and the dissipation coefficient (characterising the dissipation of the energy from $\phi$ to the plasma) rely on the self-energy and the corrections to the quartic coupling constant. The calculations of these quantities using Feynman diagrams make up the major part of work reported in this article. If we set $H$ to 0, our results match up with those obtained, in [@Cheung:2015iqa], using a Minkowski-space propagator in loops. In addition we observe non-trivial features that are not revealed in flat spacetime.
Exposing scalar fields to a high temperature and a rapid cosmic expansion is an important setup for understanding the non-equilibrium dynamics of scalar fields in cosmology. Under the condition of cosmic expansion at high temperature, the matter production process is out of equilibrium. If there is an effective potential that is not steep, the matter fields take a long time to reach equilibrium. Our focus is the back reaction [@Vachaspati:2018llo; @Calzetta:1986ey] on the primary particles from their decay products. Based on the previous work done using Minkowski-space propagators in loop diagrams, we further their studies of scalar field dynamics in the early universe evolution by incorporating the effect of cosmic expansion. There are infinitely many back reactions, and thus it is important to generalise the leading-order re-summation results to higher orders.
Although elementary particles in our universe consist of fermions and gauge bosons, we still expect our toy model with two scalar fields to serve as a good playground for studying the early universe physics. The earliest decay process involves only scalars, because the creation of fermions can be assumed to happen in the subsequent decay chain or inelastic scatterings as the transition to bosonic states is usually Bose-enhanced [@Drewes:2013iaa]. When the background temperature is higher than the oscillation frequency, the dissipation rate arising from the interactions with fermions is suppressed due to Pauli blocking, whereas it is enhanced for interactions with bosons due to the induced effect. In a future work, we will consider the direct coupling of scalars to fermions and gauge bosons.
This paper is organised as follows: In section \[sec:assumptions\], we explain our working assumptions. We sketch the prerequisites for field theory calculations at finite temperature to establish notations, which is followed by the derivation of the EoM in section \[sec:EoM\]. We demonstrate the calculations of self-energy and obtain the correction to the quartic coupling constant in section \[sec:computation\]. Section \[sec:discussion\] briefly summarises the main results obtained in this paper and concludes with a short discussion. The detailed calculations of the Feynman diagrams and the relevant formulae needed in the calculations are contained in the appendices.
Assumptions and Prerequisites {#sec:assumptions}
=============================
We study a scalar field, denoted by $\phi$, in a de Sitter space with the Hubble parameter, $H$, taken to be nonzero. The scalar field, $\phi$, interacts with another scalar field, $ \chi $, which plays the role of the cosmic background in our model. The mass of the thermal bath, $\chi$, is assumed to be less than half the mass of $\phi $. $\phi$ can be decomposed into its thermal average, $ \varphi $, and fluctuations, denoted by $ \eta $: $ \phi=\varphi+\eta $. The fields, $\eta $ and $ \chi $, are assumed to be in thermal equilibrium initially. We assume the reaction between $ \varphi $ and other degrees of freedom is weak enough to allow for the application of perturbation theory and the neglecting of back reaction of $\varphi$ on the $ \eta $ and $ \chi $ fields. In this way $\eta$ and $\chi$ can assume thermal equilibrium in the evolution of $ \varphi $. In order that we can expand the equation of motion to first order in $ H $ and simplify the calculations, we further assume $ H < m_\phi, m_\chi\ll T $($ m_\phi $ and $ m_\chi $ are the masses of the $ \phi $ field and the $ \chi $ field, respectively) and $ T $ is inversely proportional to the scale factor as the universe expands. There exists a prolonged period of time in our universe when these conditions are satisfied.
We use the closed-time-path formalism to perform the calculations. The time ordering and integration path $C$ in the complex time plane starts from $ t_i $ on the real axis and runs to real $t_f$, then back to $ t_i $ and ends at $ t_i-i\beta $, as depicted in Fig. \[fig:path\].
![The path of integration in the complex-time plane[]{data-label="fig:path"}](time_path){width="10cm"}
We take $ t_i \to -\infty$ and $ t_f\to\infty $ and denote the upper section of $ C $ which runs forward in time by $ C_1 $ and the lower section which runs backward in time by $ C_2 $. And a general scalar field $ \xi(x) $ that lies on $ C_1/C_2 $ is labelled as, respectively, $ \xi_1(x) $/$ \xi_2(x) $ .
In order to obtain the dynamical information of such a scalar field $ \xi $, we need to know the two-point correlation functions which are defined as follows, $$\begin{aligned}
\begin{array}{rclrcl} \Delta_{ab}(x,y)&=&\langle T_C\xi_a(x)\xi_b(y) \rangle & \quad (a,b &=&1,2),\\
\Delta^>(x,y) &=& \langle\xi(x)\xi(y)\rangle, &\quad \Delta^<(x,y) &=&\langle\xi(y)\xi(x)\rangle,\\
\Delta^-(x,y) &=& i\left[\Delta^>(x,y)-\Delta^<(x,y)\right],
&\quad \Delta^+(x,y) &=&\frac{1}{2}\left[\Delta^>(x,y)+\Delta^<(x,y)\right],
\end{array}
\end{aligned}$$ where $ T_C $ indicates the time ordering along the path $ C $ in Fig. \[fig:path\]. For a real scalar field $ \xi $, we see that $ \Delta^{>}(x,y)={(\Delta^<(x,y))}^{*} $ and that $$\label{propagator_identity}
\begin{aligned}
&\Delta_{11}(x,y)=\theta(x^0-y^0)\Delta^{>}(x,y)+\theta(y^0-x^0)\Delta^{<}(x,y),\\
&\Delta_{22}(x,y)=\theta(x^0-y^0)\Delta^{<}(x,y)+\theta(y^0-x^0)\Delta^{>}(x,y),\\
&\Delta_{12}(x,y)=\Delta^{<}(x,y),\\
&\Delta_{21}(x,y)=\Delta^{>}(x,y).
\end{aligned}$$
The 1-loop propagators of a scalar field $ \xi $ with effective mass $ M_\xi $ in a de-Sitter space, $$\label{metric}
g_{\mu\nu}=\text{diag}(1,-a(t)^2,-a(t)^2,-a(t)^2),$$ where $a(t)= e^{Ht}$, were obtained by solving the Kadanoff-Baym equations [@Drewes:2012qw]. The result in [@Drewes:2012qw] is expressed in terms of conformal time, with a time dependent mass $ m(t) $. To obtain the corresponding result in terms of the cosmic time, we first carry out the usual replacement: $$t\to -\frac{1}{Ha(t)}~.$$ The next two replacements can be inferred by comparing the free spectral function in de Sitter spacetime (the detailed calculation of which is presented in Appendix \[appendix:free-spectral-function\]) with the corresponding flat-spacetime propagator [@Drewes:2012qw]: $$\xi(x)\to \xi(x)/a(t),\quad m(t)^2\to \left(M_\xi^2-2H^2\right)a(t)^2~.$$
We thus start our calculations with the following propagators in de Sitter spacetime: $$\left\{
\begin{aligned}
&\Delta^{-}(\boldsymbol{p},t_1,t_2)=\dfrac{\displaystyle\sin\left(\int_{t_2}^{t_1}dt'\Omega_\xi(t')\right)\exp\left(-\frac{1}{2}\left|\int_{t_2}^{t_1}dt' \Gamma_\xi(t')\right|\right)}{a(t_1)^{3/2}a(t_2)^{3/2}\sqrt{\Omega_\xi(t_1)\Omega_\xi(t_2)}}\\
&\Delta^{+}(\boldsymbol{p},t_1,t_2)= \left[1+2f(t_B)\right] \cdot \dfrac{\displaystyle\cos\left(\int_{t_2}^{t_1}dt'\Omega_\xi(t')\right)\exp\left(-\frac{1}{2}\left|\int_{t_2}^{t_1}dt' \Gamma_\xi(t')\right|\right)}{2a(t_1)^{3/2}a(t_2)^{3/2}\sqrt{\Omega_\xi(t_1)\Omega_\xi(t_2)}}~,
\end{aligned}
\right.$$ where $ \Gamma_\xi $ is the decay rate of $ \xi $, $\Omega_\xi$ is given by $$\Omega_\xi(t)=\sqrt{\boldsymbol{p}^2/a(t)^2+M_\xi^2-2H^2}~,$$ with $ t_B= \min(t_1,t_2) $, and $ f(t) $ satisfies a Markovian equation [@Drewes:2012qw].
If we assume that a scalar field $ \xi $ is in (approximate) thermal equilibrium as the universe expands, and can be characterized by an effective temperature $ 1/\beta(t) $ which depends on time, then by imposing the KMS relations, the propagators become $$\label{propagator+-}
\left\{
\begin{aligned}
\Delta^{-}(\boldsymbol{p},t_1,t_2)=&\dfrac{\displaystyle\sin\left(\int_{t_2}^{t_1}dt'\Omega_\xi(t')\right)\exp\left(-\frac{1}{2}\left|\int_{t_2}^{t_1}dt' \Gamma_\xi(t')\right|\right)}{a(t_1)^{3/2}a(t_2)^{3/2}\sqrt{\Omega_\xi(t_1)\Omega_\xi(t_2)}}\\
\Delta^{+}(\boldsymbol{p},t_1,t_2)=&\dfrac{1}{2a(t_1)^{3/2}a(t_2)^{3/2}\sqrt{\Omega_\xi(t_1)\Omega_\xi(t_2)}}\exp\left(-\frac{1}{2}\left|\int_{t_2}^{t_1}dt' \Gamma_\xi(t')\right|\right)\\
&\times\left\{\displaystyle\exp\left(-i\int_{t_2}^{t_1}dt'\Omega_\xi(t')\right)\left[\frac{1}{2}+f\left(\Omega_\xi\left(t_B\right)+\frac{i\Gamma_\xi\left(t_B\right)}{2}\right)\right]\right.\\
&\left.+\exp\left(i\int_{t_2}^{t_1}dt'\Omega_\xi(t')\right)\left[\frac{1}{2}+f\left(\Omega_\xi\left(t_B\right)-\frac{i\Gamma_\xi\left(t_B\right)}{2}\right)\right]\right\}~,
\end{aligned}
\right.$$ where $ f $ becomes the Bose distribution function: $$f(x)=\frac{1}{e^{\beta(t)x}-1}~.$$ Here the inverse temperature $ \beta(t) $ is proportional to the scale factor by our assumptions: $ \beta(t)=(\beta_0/a_0)a(t) $ with $ \beta_0 $ being the inverse temperature when the scale factor is $ a_0 $. For simplicity, we will denote $ \beta_0/a_0 $ by $ \gamma $ in the following calculations. For more details about field theory at finite temperature, the readers are referred to [@Bellac:2011kqa; @Drewes:2012qw].
Derivation of the EoM {#sec:EoM}
=====================
As we have mentioned in Introduction, our system consists of a scalar field $ \phi $ and the background plasma collectively denoted by $ \chi $. We use the model to study the early universe dynamics: it gives us clues on how the fields behave in an expanding or contracting universe. In particular we wish to capture how their behaviour differs as one goes beyond using the Minkowski propagator in computing the quantum and thermal corrections. A general renormalizable action for such a system in a de-Sitter space, whose potential energy is bounded from below, is of the form: $$\label{eqn:action}
\begin{aligned}
S=&-\int d^4x\sqrt{-g}\left\{\frac{1}{2}\phi\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)+m_\phi^{2}\right]\phi+\frac{\lambda}{4!}\phi^4\right.\\
&\left.+\frac{1}{2}\chi\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)+m_\chi^{2}\right]\chi+\frac{\lambda'}{4!}\chi^4+\frac{h}{4}\phi^2\chi^2\right\}.
\end{aligned}$$ The action is furthermore invariant under diffeomorphism and $ \phi(x)\to -\phi(x) $.
The effective action for $ \varphi $ (which can be used to obtain the equation of motion of $ \varphi $) has the same linear symmetries as the original action [@Weinberg:1996kr], it can be written in the form, $$\label{effective_action}
\begin{aligned}
\Gamma&=-\frac{1}{2}\int_C d^4x_1\sqrt{-g}\varphi(x_1)\left[\frac{1}{\sqrt{-g}}\partial_{1\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{1\nu}\right)+m_\phi^{2}\right]\varphi(x_1)\\
&\hphantom{=}-\frac{1}{2}\int_C \sqrt{-g(x_1)}\sqrt{-g(x_2)}d^4x_1d^4x_2\Pi(x_1,x_2)\varphi(x_1)\varphi(x_2)\\
&\hphantom{=}-\int_C\sqrt{-g(x_1)}\sqrt{-g(x_2)}d^4x_1d^4x_2\frac{1}{4!}\left[\lambda\frac{\delta^4(x_1-x_2)}{\sqrt{-g(x_1)}}+\widetilde{\Pi}(x_1,x_2)\right]\varphi(x_1)^2\varphi(x_2)^2,
\end{aligned}$$ up to the fourth power of $ \varphi $. In doing so we have only considered terms up to 1-loop in the quartic part, and the time integration is along the path $C$ shown in Fig. \[fig:path\]. $ \Pi $ is the self-energy, and $ \widetilde{\Pi} $ the correction to the quartic coupling constant, the computation of which will be presented in the next section. Similar to the case of propagators, we will denote $ \Pi(x_1,x_2) $ as $ \Pi_{ab}(x_1,x_2) $ when $ x_1^0 $ lies on $C_a$ and $x_2^0$ lies on $ C_b $ ($a,b=1\, \text{or}\, 2$). Different from the situation in flat spacetime, here we do not have time translation symmetry: $ \Pi(x_1,x_2),\widetilde{\Pi}(x_1,x_2) $ depend not only on $ t_1-t_2 $, but on $ t_1+t_2 $ as well.
From (eq. \[effective\_action\]) we obtain the equation of motion for $ \varphi $, $$\begin{gathered}
\begin{aligned}
\frac{\delta\Gamma}{\delta\varphi(x_1)}&=e^{3Ht_1}\left(\partial_{0}^{2}+3H\partial_0-e^{-2Ht_1}\nabla^2+m_\phi^2\right)\varphi(x_1)+\int_C dx_2 e^{3H(t_1+t_2)}\Pi(x_1,x_2)\varphi(x_2)\\
&\hphantom{=}+\frac{\lambda}{3!}\varphi(x_1)^3e^{3Ht_1}+\frac{1}{3!}\varphi(x_1)\int_C dx_2e^{3H(t_1+t_2)}\widetilde{\Pi}(x_1,x_2)\varphi(x_2)^2\,=\, 0~. \end{aligned}\end{gathered}$$ We restrict ourselves to the case in which the only non-vanishing Fourier mode of $ \varphi(x) $ is $ \varphi(\boldsymbol{q}=0) $, then in (spatial) momentum space, the EoM simplifies to $$\begin{gathered}
\label{zeroModeEoM}
\begin{aligned}
\left(\partial_{0}^{2}+3H\partial_0+m_\phi^2\right)\varphi(t_1)+\int_C dt_2 e^{3Ht_2}\Pi(t_1,t_2)\varphi(t_2)&\\
\hphantom{=}+\frac{\lambda}{3!}\varphi(t_1)^3+\frac{1}{3!}\varphi(t_1)\int_C dt_2e^{3Ht_2}\widetilde{\Pi}(t_1,t_2)\varphi(t_2)^2 &=0~.
\end{aligned}\end{gathered}$$ This non-local equation is still hard to solve. To make progress the Hubble parameter $ H $ is assumed to be small and the system is in pseudo-equilibrium during the process. This is a valid hypothesis for the most of the physical applications we have in mind. We can hence simplify the equation by expanding $e^{3Ht_2} \approx1+3Ht_2 $ to first order in $ H$; while the adiabatic assumption is realised as, $ \varphi(t_2)\approx \varphi(t_1)+\dot{\varphi}(t_1)(t_2-t_1)$. Altogether, $$\begin{aligned}
\left(\partial _{0}^{2}+3H\partial _0+m_\phi^2\right)\varphi(t_1)+\int_{-\infty}^{\infty}dt_2(1+3Ht_2)\Pi^R(t_1,t_2)\left[\varphi(t_1)+\dot{\varphi}(t_1)(t_2-t_1)\right]&\\
+\frac{\lambda}{3!}\varphi(t_1)^3 +\frac{1}{3!}\varphi(t_1)\int_{-\infty}^{\infty}dt_2(1+3Ht_2)\widetilde{\Pi}^R(t_1,t_2)\left[\varphi(t_1)^2+2\varphi(t_1)\dot{\varphi}(t_1)(t_2-t_1)\right] &=0 ~.
\end{aligned}$$ Let $\Pi^{R}(t_1,t_2) $ denote the retarded propagator: $$\Pi^{R}(t_1,t_2)=\Pi_{11}(t_1,t_2)-\Pi_{12}(t_1,t_2),$$ and likewise, $$\widetilde{\Pi}^R(t_1,t_2)=\widetilde{\Pi}_{11}(t_1,t_2)-\widetilde{\Pi}_{12}(t_1,t_2);$$ and the Fourier transformation of $\Pi^R(t_1,t_2)$ and $ \widetilde{\Pi}^R(t_1,t_2) $ (with respect to $t=t_1-t_2$) be denoted by $\kappa(t_1,\omega) $ and $\widetilde{\kappa}(t_1,\omega) $: $$\begin{aligned}
&\kappa(t_1,\omega)\equiv \int dt\,\,\Pi^R(t_1,t_1-t)e^{i\omega t},\\
&\widetilde{\kappa}(t_1,\omega)\equiv \int dt\,\,\widetilde{\Pi}^R(t_1,t_1-t)e^{i\omega t}~.
\end{aligned}$$ The equation of motion then becomes, $$\label{eqn: effective_EoM}
\begin{aligned}
&\ddot{\varphi}(t_1)+\Bigg\{\left[m_\phi^2+(1+3Ht_1){\ensuremath{\kappa}}(t_1,\omega=0)-3H\left(-i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\right)\right]\varphi(t_1)\\
&\hphantom{\ddot{\varphi}(t_1)+}
+\frac{1}{6}\left[\lambda+(1+3Ht_1){\ensuremath{\widetilde{\kappa}}}(t_1,\omega=0)-3H\left(-i{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right)\right]\varphi(t_1)^3\Bigg\}\\
&+\Bigg\{3H+(1+3Ht_1)\left(-i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\right)+3H{
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\\
&\hphantom{+}
+\frac{1}{3}\left[(1+3Ht_1)\left(-i{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right)+3H{\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right]\varphi(t_1)^2\Bigg\}\,\dot{\varphi}(t_1) =0~.
\end{aligned}$$ From this equation of motion we see that the “potential” (terms in the first curly bracket) which is time-dependent for $ \varphi $, is determined by $${\ensuremath{\kappa}}(t_1,\omega=0), ~~{\ensuremath{\widetilde{\kappa}}}(t_1,\omega=0), ~~{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}, ~~{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi} ;$$ whereas the dissipation coefficient (terms in the second curly bracket) relies on $${
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}, ~~{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}, ~~{
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}, ~~{\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi} .$$ We will obtain these expressions in the following section.
Computation of the potential and the dissipation coefficient {#sec:computation}
============================================================
In this section we shall calculate the self-energy and the correction to the self-coupling constant of $\phi$, together with the first and second derivatives of them.
Computation of ${\ensuremath{\kappa}}(t_1)$ {#sec:kappa}
-------------------------------------------
The leading order contribution to $ \Pi^R(t_1,t_2) $, and $ \kappa(t_1) \equiv \kappa(t_1,\omega=0)$, is given by the tadpole diagram, shown in Fig. \[fig:tadpole\], with $ \eta $ or $ \chi $ running in the loop.
(-1,0) – (0,0) node \[align=center, below\][$ t_1=t_2 $]{} – (1,0); (0,0.5) circle(0.5);
We observe that $\Pi_{12}=0 $ because $ t_1 $ and $ t_2 $ must be identical. Using (eq. \[propagator\_identity\]) we obtain, $$\label{eqn:tadpole}
\begin{aligned}
\Pi^{R}(t_1,t_2)&=\Pi_{11}(t_1,t_2)\\
&=\frac{1}{2}\delta(t_1-t_2)\sum_{\xi=\eta,\chi}\left[g_{\xi}\int \frac{d^3p}{(2\pi)^3}(\Delta_\xi)_{11}(\boldsymbol{p},t_1,t_2)\right]\\
&=\frac{1}{2}\delta(t_1-t_2)\sum_{\xi=\eta,\chi}\left[g_{\xi}\int \frac{d^3p}{(2\pi)^3}\mathrm{Re}(\Delta_\xi)_>(\boldsymbol{p},t_1,t_2)\right]~,
\end{aligned}$$ with $ g_\eta\equiv\lambda, g_\chi\equiv h $. The integral is computed in Appendix \[appendix: bose distribution integral\], $$\label{eqn:Pit1t2}
\begin{aligned}
\Pi^R(t_1,t_2)&=\delta(t_1-t_2)\left\{\frac{1}{2\pi^2\beta(t_1)^2}\sum_{\xi=\eta,\chi}g_{\xi}h_3 \left[M_\xi\beta(t_1)\right]\right\}~.
\end{aligned}$$ $ M_\eta$ and $M_\chi $ denote, respectively, the effective masses for $\eta$ and $\chi$, and $ h_3 $ is given by, $$\label{eqn: h_3}
h_3(y)
=\frac{\pi^2}{12}-\frac{\pi y}{4}-\left(\frac{\gamma_E}{8}-\frac{1}{16}\right)y^2-\frac{y^2}{8}\log\frac{y}{4\pi}
+\sum_{m=1}^{\infty}\frac{(-1)^{m+1}}{2^{m+3}}\frac{(2m-1)!!}{(m+1)!}\frac{\zeta(2m+1)}{(2\pi)^{2m}}y^{2m+2}~,
$$ with $ \gamma_E $ being the Euler constant. Fourier transforming (eq. \[eqn:Pit1t2\]) yields, $$\label{eqn:kappa}
\begin{aligned}
{\ensuremath{\kappa}}(t_1)&=\frac{1}{2\pi^2\beta(t_1)^2}\sum_{\xi=\eta,\chi}g_{\xi}h_3\left[M_\xi\beta(t_1)\right]\\
&\approx \frac { \lambda + h } { 24 \gamma ^ { 2 } } \left( 1 - 2 H t _ { 1 } \right).
\end{aligned}$$
Computation of ${\ensuremath{\widetilde{\kappa}}}(t_1) $
--------------------------------------------------------
The leading order contribution to $ \widetilde{\Pi}_R(t_1,t_2) $, and $ \widetilde{\kappa} (t_1)\equiv \widetilde{\kappa}(t_1,\omega=0)$, is obtained from the “fish” diagram, shown in Fig. \[fig:fish\],
(v1); (v2); (e1); (e2); (e3); (e4); (e1)–(v1); (v1)–(e2); (v2)–(e3); (v2)–(e4); ; ;
with the internal lines corresponding to two $ \eta $ or two $ \chi $ free propagators: $$\label{eqn:fish}
\begin{aligned}
\widetilde{\Pi}_R(t_1,t_2)
= &\widetilde{\Pi}_{11}(t_1,t_2)-\widetilde{\Pi}_{12}(t_1,t_2)\\
= &\frac{i}{2}\sum_{\xi=\eta,\chi}\int(-ig_\xi)^2\frac{d^3q}{(2\pi)^3}\Big[(\Delta_\xi)_{11}
(\boldsymbol{q},t_1,t_2)(\Delta_\xi)_{11}(-\boldsymbol{q},t_1,t_2)\\
& -(\Delta_\xi)_{12}(\boldsymbol{q},t_1,t_2)(\Delta_\xi)_{12}(-\boldsymbol{q},t_1,t_2)\Big]\\
= &\theta(t_1-t_2)\sum_{\xi=\eta,\chi}g_{\xi}^{2}\int\frac{d^3q}{(2\pi)^3}\mathrm{Im}
\left[(\Delta_\xi)_>(\boldsymbol{q},t_1,t_2)^2\right].
\end{aligned}$$
In order to perform the integral over 3-momentum and carry out Fourier transformation over $ t=t_1-t_2 $, we first expand the integrand in (eq. \[eqn:fish\]) to first order in $ H $, $$\label{propagator squared}
\begin{aligned}
& (\Delta_{\xi})_>(\boldsymbol{q},t_1,t_2)^2\\
\approx & \frac{1}{4\omega_{\xi q}^{2}}\left[(1-2Ht_1\overline{\gamma}_0)+Ht\overline{\gamma}_1+2iHt_1t\overline{\gamma}_2-iHt^2\overline{\gamma}_2\right]\left(1+f\left(\omega_{\xi q}\right)\right)^2e^{-2i\omega_{\xi q} t}\\
&+(\omega_{\xi q}\to -\omega_{\xi q})-\frac{1}{2\omega_{\xi q}^2}\left[(1+Ht_1\gamma'_0)+Ht\gamma'_1\right](1+f(\omega_{\xi q}))f(\omega_{\xi q}),
\end{aligned}$$ where $ \omega_{\xi q}=\sqrt{\boldsymbol{q}^2+M_\xi^2} $ and $$\left\{
\begin{aligned}
& \overline{\gamma}_0(\omega_{\xi q})=3+\frac{\gamma M_\xi^2}{\omega_{\xi q}}f(\omega_{\xi q})-\frac{\omega_{\xi q}^2-M_\xi^2}{\omega_{\xi q}^2}\\
& \overline{\gamma}_1(\omega_{\xi q})=3+2\frac{\gamma M_\xi^2}{\omega_{\xi q}}f(\omega_{\xi q})-\frac{\omega_{\xi q}^2-M_\xi^2}{\omega_{\xi q}^2}\\
& \overline{\gamma}_2\left(\omega_{\xi q}\right)=\frac{\omega_{\xi q}^2-M_\xi^2}{\omega_{\xi q}}\\
&\gamma'_0(\omega_{\xi q})=-6+2\frac{\omega_{\xi q}^2-M_\xi^2}{\omega^2_{\xi q}}-\left(1+2f(\omega_{\xi q})\right)\frac{\gamma M_\xi^2}{\omega_{\xi q}}\\
& \gamma'_1(\omega_{\xi q})=3+\frac{\gamma M_\xi^2}{\omega_{\xi q}}\left(1+2f(\omega_{\xi q})\right)-\frac{\omega^2_{\xi q}-M_\xi^2}{\omega_{\xi q}^2}
\end{aligned}
\right. ~.$$
For a general function $K$ given by $$K= K_0(\omega_{\xi q}) + t K_1(\omega_{\xi q}) +t^{2} K_2(\omega_{\xi q})$$ with $K_0(\omega_{\xi q})$, $K_1(\omega_{\xi q}) $, $K_2(\omega_{\xi q})$ being generic rational functions of $ \omega_{\xi q} $, we obtain an explicit expression of its Fourier transformation in Appendix \[appendix: Fourier of general form\]: $$\label{general_form_real_part}
\begin{aligned}
& \operatorname{Re} F^{\omega}\left[ \theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left[K_{0}\left(\omega_{{\xi q}}\right)+t K_{1}\left(\omega_{{\xi q}}\right)+t^{2} K_{2}\left(\omega_{{\xi q}}\right)\right] e^{-2 i \omega_{{\xi q}} t}\right]\\
=&\left.I_{re}\left[K_{0}\right]\right|_{\alpha_\xi=0}+\frac{\partial}{\partial \omega} \left.I_{im}\left[K_{1}\right]\right|_{\alpha_\xi=0}-\frac{\partial^{2}}{\partial \omega^{2}} \left.I_{re}\left[K_{2}\right]\right|_{\alpha_\xi=0}~,
\end{aligned}$$ where $ \alpha_\xi $, which will be defined and used in the following section, is related to the decay rate of a $\xi$ field. $I_{r e}\left[K\left(\omega_{{\xi q}}\right)\right]$, and $I_{im}\left[K\left(\omega_{{\xi q}}\right)\right]$ are given as follows, $$\label{definition_Ire_Iim}
\left\{
\begin{aligned}
I_{r e}\left[K\left(\omega_{{\xi q}}\right)\right]=&\frac{1}{(2 \pi)^{3}} \int_{M_\xi}^{\infty} d \omega_{{\xi q}} 4 \pi \omega_{{\xi q}} \sqrt{\omega_{{\xi q}}^{2}-M_\xi^2}\Bigg\{\operatorname{Re} K\left(\omega_{{\xi q}}\right) \frac{2 \omega_{{\xi q}}}{\omega^{2}-4 \omega_{{\xi q}}^{2}}\\
&+\operatorname{Im} K\left(\omega_{{\xi q}}\right)\left[\frac{\alpha_\xi}{2\omega_{{\xi q}}\left(\omega+2 \omega_{{\xi q}}\right)^{2}}+\frac{\alpha_\xi}{2\omega_{{\xi q}}\left(\omega-2 \omega_{{\xi q}}\right)^{2}}\right]\Bigg\}~,\\
I_{im}\left[K\left(\omega_{{\xi q}}\right)\right]=&\frac{1}{(2 \pi)^{3}} \int_{M_\xi}^{\infty} d \omega_{{\xi q}} 4 \pi \omega_{{\xi q}} \sqrt{\omega_{{\xi q}}^{2}-M_\xi^2}\Bigg\{\operatorname{Im} K\left(\omega_{{\xi q}}\right) \frac{\omega}{\omega^{2}-4 \omega_{{\xi q}}^{2}}\\
&-\operatorname{Re} K\left(\omega_{{\xi q}}\right)\left[\frac{\alpha_\xi}{2\omega_{{\xi q}}\left(\omega-2 \omega_{{\xi q}}\right)^{2}}-\frac{\alpha_\xi}{2\omega_{{\xi q}}\left(\omega+2 \omega_{{\xi q}}\right)^{2}}\right]\Bigg\}~.
\end{aligned}\right.$$
Combining (eq. \[propagator squared\]) and (eq. \[definition\_Ire\_Iim\]) we obtain, $$\begin{aligned}
&\sum_{\xi}g_\xi^2\operatorname{Re} F^{\omega} \left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}} (\Delta_{\xi})_>(\boldsymbol{q},t_1,t_2)^2\right]\\
=&\frac{1}{2}\sum_{\xi}g_{\xi}^2\left\{I_{re}\left[\frac{1}{2\omega_{\xi q}^{2}}(1-2Ht_1\overline{\gamma}_0(\omega_{\xi q}))\left(1+f(\omega_{\xi q})\right)^2-(\omega_{\xi q}\to -\omega_{\xi q})\right]\right.\\
&+\frac{\partial}{\partial \omega}I_{im}\left[\frac{H}{2\omega_{\xi q}^{2}}(\overline{\gamma}_1(\omega_{\xi q})+2it_1\overline{\gamma}_2(\omega_{\xi q}))\left(1+f(\omega_{\xi q})\right)^2+(\omega_{\xi q}\to -\omega_{\xi q})\right]\\
&\left.+\frac{\partial^2}{\partial \omega^2}I_{re}\left[iH \frac{1}{2\omega_{\xi q}^{2}}\overline{\gamma}_2(\omega_{\xi q})\left(1+f(\omega_{\xi q})\right)^2-(\omega_{\xi q}\to -\omega_{\xi q})\right]\right\}~.
\end{aligned}$$ Setting $ \omega=0 $ and performing the integrals, we arrive at, $$\label{eqn:kappatilde}
\begin{aligned}
\tilde{\kappa}(t_1)&=\left.\sum_{\xi}g_\xi^2\operatorname{Re} F^{\omega} \biggg[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}} (\Delta_{\xi})_>(\boldsymbol{q},t_1,t_2)^2\right|_{\omega=0}\biggg]\\
&=-\sum_{\xi=\eta,\chi}g_{\xi}^{2}(1-4Ht_1)\frac{1}{32\pi M_\xi\gamma}~.
\end{aligned}$$ If we set $ H=0 $, we obtain the familiar results in flat spacetime, $${\ensuremath{\widetilde{\kappa}}}(t_1)\approx -\frac{\lambda^2T}{32\pi M_\eta}-\frac{h^2T}{32\pi M_\chi}~.$$ using Minkowski-space propagators in loop diagrams.
Computation of [11$\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}$$\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}$]{} {#sec:kappa_tilde_derivative}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As mentioned in [@Cheung:2015iqa], the leading contribution to ${\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}$ comes from the fish diagram, Fig. \[fig:fish\], but with the one-loop corrected ($ \eta-$ or $ \chi-$) propagators. Such propagators rely on the decay rates, which has been calculated in [@Cheung:2015iqa] in flat spacetime: $$\label{Gamma_in_CDKK}
\Gamma_\chi=\frac{\lambda'^2+3h^2}{256\pi^3\gamma^2\omega_\chi}\equiv \frac{\alpha_\chi}{\omega_\chi},
\hspace{0.8cm}
\Gamma_\eta=\frac{\lambda^2+3h^2}{256\pi^3\gamma^2\omega_\eta}\equiv \frac{\alpha_\eta}{\omega_\eta}.$$ In the present situation, since we have assumed the system to be in pseudo-equilibrium and we do not consider higher order corrections, we replace (eq. \[Gamma\_in\_CDKK\]) by $$\Gamma_\chi(t')=\frac{\lambda'^2+3h^2}{256\pi^3\beta(t')^2\Omega_\chi(t')}~,
\hspace{0.8cm}
\Gamma_\eta(t')=\frac{\lambda^2+3h^2}{256\pi^3\beta(t')^2\Omega_\eta(t')}~.$$
Similar to (eq. \[general\_form\_real\_part\]), we have $$\label{general_form_cross_part}
\begin{aligned}
&F^{\omega}\left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left(K_{0}+t K_{1}+t^{2} K_{2}\right) e^{- \frac{\alpha_\xi}{\omega_{q}} t}\right]\\
=&\int \frac{d^3q}{(2\pi)^3}\left(\mathrm{Im}K_0\frac{\omega_q \alpha_\xi}{\alpha_\xi^2+\omega^2\omega^2_q}+\mathrm{Im}K_1\frac{\partial}{\partial \omega}\frac{\omega\omega^2_q}{\alpha_\xi^2+\omega^2\omega^2_q}-\mathrm{Im}K_2\frac{\partial^2}{\partial \omega^2}\frac{\omega_q \alpha_\xi}{\alpha_\xi^2+\omega^2\omega^2_q}\right)\\
&+i\int \frac{d^3q}{(2\pi)^3}\left(\mathrm{Im}K_0\frac{\omega\omega^2_q}{\alpha_\xi^2+\omega^2\omega^2_q}-\mathrm{Im}K_1\frac{\partial }{\partial \omega}\frac{\omega_q\alpha_\xi}{\alpha_\xi^2+\omega^2\omega^2_q}-\mathrm{Im}K_2\frac{\partial ^2}{\partial \omega^2}\frac{\omega\omega^2_q}{\alpha_\xi^2+\omega^2\omega^2_q}\right)
\end{aligned}$$ and $$\label{general_form_imaginary_part}
\begin{aligned}
& \operatorname{Im} F^{\omega} \left\{\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left[K_{0}\left(\omega_{{\xi q}}\right)+t K_{1}\left(\omega_{{\xi q}}\right)+t^{2} K_{2}\left(\omega_{{\xi q}}\right)\right] e^{-2 i \omega_{{\xi q}} t}\right\}\\
=&I_{i m}\left[K_{0}\right]-\frac{\partial}{\partial \omega} I_{r e}\left[K_{1}\right]-\frac{\partial^{2}}{\partial \omega^{2}} I_{i m}\left[K_{2}\right],
\end{aligned}$$ where $ I_{re} $ and $ I_{im} $ are defined in (eq. \[definition\_Ire\_Iim\]), and the derivation of which is shown in Appendix \[appendix: Fourier of general form\]. By expanding the propagator to the first order in $ H $ and making the following peak approximation, $$f(\omega_{\xi q})\to \frac{1}{\gamma \omega_{\xi q}}, \int_{m}^{\infty}\to \int_{m}^{1/\gamma},$$ and then with the help of (eq. \[general\_form\_cross\_part\]) and (eq. \[general\_form\_imaginary\_part\]) to perform the Fourier transformations, to the first order in $H$ and to the lowest orders in $\alpha_\chi$ and $\alpha_\eta$ we get, $$\label{kappa_tilde_first_derivative}
\begin{aligned}
&-i {\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\\
=&-h^2\left[-\frac{32\pi\gamma}{\lambda'^2+3h^2}\left(1+\log\frac{M_\chi\gamma}{2}\right)+Ht_1\frac{32\pi\gamma}{3\left(\lambda'^2+3h^2\right)}\left(5+9\log\frac{M_\chi \gamma}{2}\right)-H\frac{2^{13}\pi^4\gamma^2}{\left(\lambda'^2+3h^2\right)^2}\right]\\
&-\lambda^2\left[-\frac{32\pi\gamma}{\lambda^2+3h^2}\left(1+\log\frac{M_\chi\gamma}{2}\right)+Ht_1\frac{32\pi\gamma}{3\left(\lambda^2+3h^2\right)}\left(5+9\log\frac{M_\chi\gamma}{2}\right)-H\frac{2^{13}\pi^4\gamma^2}{\left(\lambda^2+3h^2\right)^2}\right]~.
\end{aligned}$$
Computation of ${
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}$
----------------------------------------------------------------------------------------------------------------
(v1); (v2); (e1); (e2); (e1)–(v1) node \[align=left,below\][$ t_1\quad$]{}; (v2)node\[align=right,below\][$\quad\, t_2 $]{}–(e2); (v1)–(v2); ; ;
$ {
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
} $ is determined by the imaginary part of the self-energy [@Cheung:2015iqa] whose leading order contribution, for soft momenta, comes from the sunset diagram in Fig. \[fig:sunset\]. $$\begin{aligned}
&\Pi^R(t_1,t_2)\\
=&h^2\theta(t_1-t_2)\int \frac{d^3 k}{(2\pi)^3}\frac{d^3l}{(2\pi^3)}\mathrm{Im}\left[(\Delta_\chi)_>(\boldsymbol{k},t_1,t_2)(\Delta_\chi)_>(\boldsymbol{l},t_1,t_2)(\Delta_\eta)_>(\boldsymbol{k}+\boldsymbol{l},t_1,t_2)\right]~.
\end{aligned}$$
In Appendix \[appendix: C3\] we show that we can express the derivative of $ \kappa(-\omega) $ in the following form, where the derivatives of $ I_s, I_c, J_s, J_c $ are given by equations (\[I\_s\]), (\[I\_c\]), (\[J\_s\]), (\[J\_c\]), $$\label{self_imaginary}
\begin{aligned}
& -i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\\
=&-\frac{h^2}{8}\Bigg\{ \left( 1 - 9 H t _ { 1 } \right) \left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 0 } \right] \right| _ { \omega = 0 } + H t _ { 1 }\left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 1 } \right] \right| _ { \omega = 0 } - H t _ { 1 }\left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 2 } \right] \right| _ { \omega = 0 } - 2 H t _ { 1 }\left. \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } } I _ { c } \left[ \gamma _ { 3 } \right] \right|_ { \omega = 0 }\\
& -\frac { 9 } { 2 } H\left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 0 } \right] \right| _ { \omega = 0 } - \frac { 1 } { 2 } H \left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 1 } \right] \right|_ { \omega = 0 } + H \left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 2 } \right] \right| _ { \omega = 0 } - H\left. \frac { \partial } { \partial \omega } J _ { c } \left[ \gamma _ { 3 } \right] \right|_ { \omega = 0 }\Bigg\}~.
\end{aligned}$$ This derivation is not much different from the calculation of the imaginary part of the self-energy performed in [@Cheung:2015iqa]. But the analogous integrals in de Sitter spacetime are more complicated: we do not have delta functions resulted from momentum conservation, which in turn greatly simplify the subsequent calculations. Our strategy is to calculate each term in the above equation, with the assumption that $ M_\chi/M_\eta\ll 1 $, to obtain analytical results of the integrals.
Let us now calculate the first line in the curly bracket in (eq. \[self\_imaginary\]), which is dominated by regions where $ \omega_{\chi k},\omega_{\chi l}\ll 1/\gamma $, since in these regions the Bose distribution function has a peak. We can then make an approximation, $ f(x)\approx 1/(\gamma x)-1/2 \quad (\text{for } x\ll 1/\gamma)$, to simplify the integrals. Since $ \gamma_0,\gamma_1,\gamma_2 $ ( at least to the zeroth order in $ M_\eta\gamma $) do not change when all three arguments change simultaneously, while $ \gamma_3 $ changes sign, the temperature dependent part in $ \partial /\partial \omega I_s[\gamma_{0,1,2}]|_{\omega=0} $ and in $ \partial^2 /\partial^2 \omega I_c[\gamma_3] |_{\omega=0}$ appears in the following forms: $$\begin{aligned}
& \left(1+f(\omega_{\eta q})\right)f(\omega_{\chi k})f(\omega_{\chi l})-f(\omega_{\eta q})\left(1+f(\omega_{\chi k})\right)\left(1+f(\omega_{\chi l})\right)\approx \frac{1}{\gamma^2}\left(\frac{1}{\omega_{\chi k}\omega_{\chi l}}-\frac{1}{\omega_{\eta q}\omega_{\chi k}}-\frac{1}{\omega_{\eta q}\omega_{\chi l}}\right),\\
& \left(1+f(\omega_{\eta q})\right)f(\omega_{\chi k})f(\omega_{\chi l})+f(\omega_{\eta q})\left(1+f(\omega_{\chi k})\right)\left(1+f(\omega_{\chi l})\right)\approx \frac{1}{\gamma^3\omega_{\eta q}\omega_{\chi k}\omega_{\chi l}}~.
\end{aligned}$$ We thus conclude that $ \partial /\partial \omega I_s[\gamma_{0,1,2}]|_{\omega=0} $ is much smaller than $ \partial^2 /\partial^2 \omega I_c[\gamma_3] |_{\omega=0}$ and the former can be safely neglected.
Considering $- \left. \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } } I _ { c } \left[ \gamma _ { i } \right] \right|_ { \omega = 0 }$ it is the sum of the following three integrals (eq. \[I\_c\]), $$\label{ic1}
\begin{aligned}
&- \frac { 1 } { 2 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l}\int _ { M_\chi t _ { 0 m } } ^ { M_\chi t _ { 0 p } } d \omega_{\chi k} \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } }\Big[\gamma _ { i } \left( - \omega + \omega_{\chi k} + \omega_{\chi l}- \omega_{\chi k} , - \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
& - \gamma _ { i } \left( \omega - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\Big]\bigg|_{\omega=0},
\end{aligned}$$ $$\label{ic2}
\begin{aligned}
& - \frac { 1 } { 2 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l}M_\chi^2(t_{1p})^2\frac{\partial}{\partial \omega_{\chi k}}\Big[\gamma_i(\omega_{\chi k}+\omega_{\chi l},-\omega_{\chi k},-\omega_{\chi l})\left(1+f\left(\omega_{\chi k}+\omega_{\chi l}\right)\right)f\left(\omega_{\chi k}\right)f\left(\omega_{\chi l}\right)\\
&-\gamma_i\left(-\omega_{\chi k}-\omega_{\chi l},\omega_{\chi k},\omega_{\chi l}\right)f\left(\omega_{\chi k}+\omega_{\chi l}\right)\left(1+f\left(\omega_{\chi k}\right)\right)\left(1+f\left(\omega_{\chi l}\right)\right)\Big]\bigg|_{\omega_{\chi k}\to M_\chi t_{0p}},
\end{aligned}$$ $$\label{ic3}
\begin{aligned}
&\frac { 1 } { 2 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l}M_\chi^2(t_{1m})^2\frac{\partial}{\partial \omega_{\chi k}}\Big[\gamma_i(\omega_{\chi k}+\omega_{\chi l},-\omega_{\chi k},-\omega_{\chi l})\left(1+f\left(\omega_{\chi k}+\omega_{\chi l}\right)\right)f\left(\omega_{\chi k}\right)f\left(\omega_{\chi l}\right)\\
&-\gamma_i\left(-\omega_{\chi k}-\omega_{\chi l},\omega_{\chi k},\omega_{\chi l}\right)f\left(\omega_{\chi k}+\omega_{\chi l}\right)\left(1+f\left(\omega_{\chi k}\right)\right)\left(1+f\left(\omega_{\chi l}\right)\right)\Big]\bigg|_{\omega_{\chi k}\to M_\chi t_{0m}}~.
\end{aligned}$$ It is easy to check that $ M_\chi t_{0p} $ grows rapidly with $ \omega_{\chi l} $: $ M_\chi t_{0p}\sim\frac{M_\eta^2}{M_\chi^2}\omega_{\chi l} $ while $ M_\chi t_{0m} $ decreases with $ \omega_{\chi l} $. Therefore for the integral (eq. \[ic2\]) the rational-function approximation of the Bose distribution function is inappropriate. The integrand will indeed be negligible due to the large exponential in the denominator. On the other hand, it is viable to use the rational-function approximation in the integral (eq. \[ic3\]), which turns out to be much larger than (eq. \[ic2\]). For the integral (eq. \[ic1\]), we can see that it would not be of much difference in the order of magnitude if we replace the partial derivative $ \partial /\partial \omega $ by $ \partial /\partial \omega_{\chi k} $. After carrying out this replacement, it can be written as $ [(\ref{ic2})/(t_{1p})^2-(\ref{ic3})/(t_{1m})^2]$. The first term can be neglected because (eq. \[ic2\]) is small and $ t_{1p} $ is large. We can also neglect the second term because $ t_{1m} $ is much greater than 1 when $ \omega_{\chi l} $ is small, and we can make a rough estimate of the ratio of the contribution of $ t_{1m} $ to that of $ 1 $, $$\dfrac{\displaystyle\int_{m}^{M}t_{1m}d\omega_{\chi l}}{\displaystyle\int_{M_\chi}^{M_\eta}d\omega_{\chi l}}\approx \frac{M_\eta}{6M_\chi}~.$$ It shows that (eq. \[ic1\]) is indeed much smaller than (eq. \[ic3\]) and therefore we will only keep (eq. \[ic3\]).
With the above approximations performed (eq. \[I\_c\]) can be integrated out: $$\label{I_result}
\begin{aligned}
&- \left. \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } } I _ { c } \left[ \gamma _ { 3} \right] \right|_ { \omega = 0 }\\
\approx&\frac { 1 } { ( 2 \pi ) ^ { 3 } }\int _ { M_\chi } ^ { \infty } d \omega_{\chi l}M_\chi ^ { 2 } \left( t _ { 1 m } \right) ^ { 2 } \frac { \partial } { \partial \omega_{\chi k} } \left[ \gamma _ { 3 } \left( \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right)\frac{1}{\gamma^3\left(\omega_{\chi k}+\omega_{\chi l}\right)\omega_{\chi k}\omega_{\chi l}}\right]\bigg|_{\omega_{\chi k}\to M_\chi t_{0m}}\\
\approx &\frac{T^3}{4\pi^3M_\eta^2}\left[\frac{M_\eta^8}{(M_\eta^4+4M_\chi^2T^2)^2}-\frac{M_\eta^4}{M_\eta^4+4M_\chi^2T^2}-\frac{4M_\chi^2M_\eta^4T^2}{(M_\eta^4+4M_\chi^2T^2)^2}+\frac{8M_\chi^4T^2}{M_\eta^2(M_\eta^4+4M_\chi^2T^2)}\right.\\
&\left.+\log\left(\frac{1}{4}\right)-1+\log\left(\frac{M_\eta^6}{M_\chi^4\sqrt{M_\eta^4+4M_\chi^2T^2}}\right)\right]~.
\end{aligned}$$ The accuracy of this approximation can be seen in some examples in Table \[table:icd\], the error is around 3%.
$ T $ $ M_\eta $ $ M_\chi $ analytic result numerical result error
------------------ ------------ ------------ --------------------- --------------------- ------------
$ 1\times 10^5 $ $ 500 $ $ 1 $ $7.01\times10^{8}$ $ 7.30\times10^8 $ $-4\%$
$ 1\times 10^5 $ $ 1000 $ $ 1 $ $ 2.03\times 10^8 $ $ 1.99\times 10^8 $ 2%
$ 1\times 10^4 $ $ 100 $ $ 1 $ $ 1.20\times 10^7 $ $ 1.19\times 10^7 $ $ 0.9\% $
$ 1\times 10^4 $ $ 100 $ $ 0.5 $ $ 1.45\times 10^7 $ $ 1.51\times 10^7 $ $ -3.8\% $
: Comparison of analytic and numerical results: here we choose some numerical values for the parameters and list the analytic results, which is determined by (eq. \[I\_result\]), and the numerical results, which is numerically computed from the first line in the curly bracket in (eq. \[self\_imaginary\]), corresponding to each of the choices.[]{data-label="table:icd"}
Next we compute $ \partial /\partial \omega J_c[\gamma_3]|_{\omega=0} $ and $ \partial /\partial \omega J_s[\gamma_{0,1,2}]|_{\omega=0} $, for which we assume that: $\lambda, \lambda', h\ll M_\chi \gamma$. As we have mentioned in Section \[sec:assumptions\], in order for the calculations to be carried out perturbatively, $ \lambda,\lambda', h $ should be small enough and, in particular, smaller than any dimensionless quantity that can be constructed using dimensionful dynamical quantities in the model. There are eight terms in $\partial /\partial \omega J_c[\gamma_i]|_{\omega=0}$, with different signs $ \chi_q,\chi_k,\chi_l $ before $ \omega_{\eta q},\omega_{\chi k},\omega_{\chi l} $. We first compute the terms in $\partial /\partial \omega J_c[\gamma_3]|_{\omega=0}$ that correspond to $ \chi_q=1,\chi_k=-1,\chi_l=-1 $ and $ \chi_q=-1,\chi_k=1,\chi_l=1 $: $$\label{principle_gamma3}
\begin{aligned}
& 6 \frac { 1 } { ( 2 \pi ) ^ { 4 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi k} d \omega_{\chi l} \int _ { - 1 } ^ { 1 } d w \frac { \sqrt { \omega_{\chi k} ^ { 2 } - M_\chi ^ { 2 } } \sqrt { \omega_{\chi l} ^ { 2 } - M_\chi ^ { 2 } } \left( \omega _ { \eta q } -\omega_{\chi k} - \omega_{\chi l} \right) ^ 4} { \omega _ { \eta q } \left[\left( \omega _ { \eta q } - \omega_{\chi k} - \omega_{\chi l} \right) ^2+\alpha_\chi^2\left(1/(2\omega_{\chi k})+1/(2\omega_{\chi l})\right)^2\right] ^4} \gamma _ { 3} \left( \omega _ { \eta q } , -\omega_{\chi k} , -\omega_{\chi l} \right)\\
&\times [(1+f(\omega_{\eta q}))(1+f(-\omega_{\chi k}))(1+f(-\omega_{\chi l}))+(1+f(-\omega_{\eta q}))(1+f(\omega_{\chi k}))(1+f(\omega_{\chi l}))]~.
\end{aligned}$$ These are the dominant contribution to the principle part of the sunset diagram. The second line in the above formula has a peak when $\omega_{\chi k},\omega_{\chi l}\ll 1/\gamma$, allowing us to approximate it by, $$\frac{1}{4}+\frac{1}{\gamma^2}\left(\frac{1}{\omega_{\chi k} \omega_{\chi l}}
-\frac{\omega_{\chi k}+\omega_{\chi l}}{\omega_{\chi k} \omega_{\chi l} \omega_{\eta q}}\right)~,$$ and to cut down the upper limit of integration to $ 1/\gamma $. Let us now consider the following term in the integral: $$\label{fraction_part}
G(\omega_{\chi k},\omega_{\chi l},w)=\frac{g(\omega_{\chi k},\omega_{\chi l},w)^4}{\left[g(\omega_{\chi k},\omega_{\chi l},w)^2+\alpha_\chi^2\left(\dfrac{1}{2\omega_{\chi k}}+\dfrac{1}{2\omega_{\chi l}}\right)^2\right]^4},$$ where $$\begin{aligned}
g(\omega_{\chi k},\omega_{\chi l},w)&\equiv \omega_{\eta q}-\omega_{\chi k}-\omega_{\chi l}\\
&=\sqrt{\omega_{\chi k}^2+\omega_{\chi l}^2+2w\sqrt{\omega_{\chi k}^2-M_\chi^2}\sqrt{\omega_{\chi l}^2-M_\chi^2}+M_\eta^{2}-2M_\chi^2}-\omega_{\chi k}-\omega_{\chi l}.
\end{aligned}$$ Since $ \alpha_\chi/M_\chi^2\ll 1/(256\pi^3)\Rightarrow \alpha_\chi/M_\chi^2\ll 10^{-4} $, one infers that $ G $ has a peak (see Fig. \[principle\_plot\] for the plot of the integrand in (eq. \[principle\_gamma3\])) around some points at which $ g(\omega_{\chi k},\omega_{\chi l},w) $ and $ \alpha_\chi(1/\omega_{\chi k}+1/\omega_{\chi l}) $ have the same order of magnitude. Hence, $$\label{order_nu}
\begin{aligned}
& \omega_{\chi k}\omega_{\chi l}=\frac{M_\eta^2}{2(1-w)}+O\left(M_\chi^2\right)+O\left(\alpha_\chi\right),\\
& \omega_{\chi k},\omega_{\chi l}=O(M_\eta)+O\left(\frac{M_\chi^2}{M_\eta}\right) +O\left(\frac{\alpha_\chi}{M_\eta}\right).
\end{aligned}$$
![Plot of the integrand in (eq. \[principle\_gamma3\]). []{data-label="principle_plot"}](principlegamma3){width="15cm"}
Furthermore we note that if we vary $ g(\omega_{\chi k},\omega_{\chi l}, w) $ by an amount of order $ \alpha_\chi/M_\eta $, then $ \alpha_\chi(1/\omega_{\chi k}+1/\omega_{\chi l}) $ would vary by an amount of order $\alpha_\chi^2/M_\eta^3 $, i.e., the latter can be regarded as constant. Hence, to determine the local maximum points of $ G(\omega_{\chi k},\omega_{\chi l},w) $ to an order of $ O(\alpha_\chi/M_\eta) $, we require that $$\label{eqn:max_point}
g(\omega_{\chi k},\omega_{\chi l}, w)=\alpha_\chi\left(\frac{1}{2\omega_{\chi k}}+\frac{1}{2\omega_{\chi l}}\right)~.$$ $ \omega_{\chi k}, \omega_{\chi l} $ are apparently symmetric in these formulae. To simplify the calculations we define a set of new variables: $ \mu=\omega_{\chi k}+\omega_{\chi l}, \nu=\omega_{\chi k}\omega_{\chi l}/M_\eta $. Then (eq. \[eqn:max\_point\]) becomes $$g(\mu,\nu)=\frac{\alpha_\chi\mu}{2M_\eta\nu},$$ and to lowest order in $ \alpha_\chi/M_\eta $ and $ M_\chi^2/M_\eta $, the solution to the above equation is $$\nu_{max}=\frac{M_\eta}{2(1-w)}~.$$
We will expand $ G $ near $ \nu_{max} $ at which $ G $ has a peak and then the integration region becomes a band that is narrow in the $ \nu $ direction and runs along the $ \mu $ direction. From (eq. \[order\_nu\]) we see that the width of the band can be chosen to be $ \alpha_\chi/M_\chi $ since $ \alpha_\chi/M_\chi\gg \alpha_\chi/M_\eta $. First we expand the function $ g $ around $ \nu_{max} $: $$g\left(\mu,\nu_{max}(\mu)+\Delta\right)=g\left(\mu,\nu_{max}(\mu)\right)+\Delta\cdot \left.\frac{\partial g}{\partial \nu}\right|_{\mu,\nu_{max}(\mu)}=\alpha_\chi\frac { \mu } { 2M_\eta \nu_ { \max } ( \mu ) } - \frac { M_\eta } { \mu } ( 1 - w ) \Delta.$$ Inserting it into $ G $ we have $$G \left( \mu , \nu_ { max } ( \mu ) + \Delta \right)=\frac { g \left( \mu , \nu_ { max } ( \mu ) + \Delta \right) ^ { 4 } } { \left[ g \left( \mu , \nu_ { max } ( \mu ) + \Delta \right) ^ { 2 } + \dfrac { \alpha_\chi ^ { 2 } \mu ^ { 2 } } { 4M _\eta^ { 2 } \nu_ { max } ( \mu ) ^ { 2 } } \right] ^ { 4 } },$$ where we have made an approximation: $ \alpha_\chi^2 \mu^2/\left\{M_\eta^2\left[\nu_{max}(\mu)+\Delta\right]^2\right\}\approx \alpha_\chi^2 \mu^2/\left[M_\eta^2\nu_{max}(\mu)^2\right]$. The integrand in (eq. \[principle\_gamma3\]) becomes $$\label{eq: intermediate}
6 \frac { 1 } { ( 2 \pi ) ^ { 4 } }\frac { \sqrt { \omega_{\chi k} ^ { 2 } - M_\chi ^ { 2 } } \sqrt { \omega_{\chi l} ^ { 2 } - M_\chi ^ { 2 } } } { \omega _ { \eta q } } G \left( \mu , \nu _ { max } ( \mu ) + \Delta \right)\left( \frac { \omega _ { \eta q } ^ { 2 } - M_\eta ^ { 2 } } { 2 \omega _ { \eta q } } - \frac { 1 } { 2 } \mu + \frac { M_\chi ^ { 2 } \mu } { 2 M_\eta \nu} \right) \left[ \frac { 1 } { 4 } + \frac { 1 } { \gamma ^ { 2 } M_\eta \nu } \left( 1 - \frac { \mu } { \omega _ { \eta q } } \right) \right].$$ Because in the narrow band we have: $$\nu =\frac { M_\eta } { 2 ( 1 - w ) } + O \left( \frac { M_\chi ^ { 2 } } { M_\eta } \right) + O \left( \frac { \alpha_\chi } { M_\eta } \right) ,$$ $$\omega _ { \eta q } = \mu + g \left( \mu ,\nu _ { max } ( \mu ) + \Delta \right), \quad$$ $$\frac { 1 } { \gamma ^ { 2 } M_\eta\nu} \left( 1 - \frac { \mu } { \omega _ { \eta q } } \right) \ll \frac { 1 } { 4 }.$$ Eq. (\[eq: intermediate\]) can be approximated by $$6 \cdot \frac { 1 } { ( 2 \pi ) ^ { 4 } } \frac { M_\eta ^ { 2 } } { 8 ( 1 - w ) } \frac { 1 } { \mu } \left[ - \frac { M_\eta ^ { 2 } } { 2 \mu } + ( 1 - w ) \frac { M_\chi ^ { 2 } \mu } { M_\eta ^ { 2 } } \right]\frac { g \left( \mu , \nu _ { max } ( \mu ) + \Delta \right) ^ { 4 } } { \left[ g \left( \mu , \nu _ { max } ( \mu ) + \Delta \right) ^ { 2 } + \dfrac { \alpha_\chi ^ { 2 } \mu ^ { 2 } } { 4M_\eta ^ { 2 } \nu _ { max } ( \mu ) ^ { 2 } } \right] ^ { 4 } }.$$ Since outside of the band the integrand is negligible, the integration region of $ \Delta $ can be chosen to be $ (-\infty,\infty) $. We also have restrictions on the range of $ \mu $ and $ w $: in the band we have $$\mu ^ { 2 } = \left( \omega_{\chi k} + \omega_{\chi l} \right) ^ { 2 } \geq 4 \omega_{\chi k} \omega_{\chi l} \approx \frac { 2 M_\eta ^ { 2 } } { 1 - w } \geq M_\eta ^ { 2 }
~~~\Rightarrow~~~ (\mu \geq M_\eta )~~\&~~(w \leq 1 - \frac { 2 M_\eta ^ { 2 } } { \mu ^ { 2 } }).$$ Therefore the measure becomes $$\int _ { - 1 } ^ { 1 } d w \int _ { M_\chi } ^ { 1 / \gamma } d \omega_{\chi k} \int _ { M_\chi } ^ { 1 / \gamma } d \omega_{\chi l} = \int _ { - 1 } ^ { 1 - \frac { 2 M_\eta ^ { 2 } } { \mu ^ { 2 } } } d w \int _ { M_\eta } ^ { 2 / \gamma } d \mu \int _ { -\infty} ^ { \infty } d \Delta \frac { M_\eta } { \sqrt { \mu ^ { 2 } - 4 M_\eta v } }.$$
Hence the integral (eq. \[principle\_gamma3\]) becomes $$\begin{aligned}
\label{J_result}
\begin{aligned}
& 6 \cdot \frac { 1 } { ( 2 \pi ) ^ { 4 } } \int _ { - 1 } ^ { 1 - \frac { 2 M_\eta ^ { 2 } } { \mu ^ { 2 } } } d w \int _ { M_\eta } ^ { 2 / \gamma } d \mu \frac { M_\eta } { \sqrt { \mu ^ { 2 } - \dfrac { 2 M_\eta ^ { 2 } } { 1 - w } } } \frac { M_\eta ^ { 2 } } { 8 ( 1 - w ) } \frac { 1 } { \mu } \left[ - \frac { M_\eta ^ { 2 } } { 2 \mu } + ( 1 - w ) \frac { M_\chi^ { 2 } \mu } { M_\eta ^ { 2 } } \right]\\
&\times \left[ \frac { 2M_\eta \nu _ { max } ( \mu ) } { \alpha_\chi \mu } \right] ^ { 4 } \int _ { - \infty} ^ { \infty } d \Delta\frac { \left[ 1 - \dfrac { 2( 1 - w ) M_\eta ^ { 2 } \nu _ { max } ( \mu ) } { \alpha_\chi \mu ^ { 2 } } \Delta \right] ^ { 4 } } { \left\{ \left[ 1 - \dfrac { 2( 1 - w ) M _\eta^ { 2 } \nu _ { max } ( \mu ) } { \alpha_\chi \mu ^ { 2 } } \Delta \right] ^ { 2 } + 1 \right\} ^ { 4 } }\\
=&6 \cdot \frac { 1 } { ( 2 \pi ) ^ { 4 } } \int _ { - 1 } ^ { 1 - \frac { 2 M_\eta ^ { 2 } } { \mu ^ { 2 } } } d w \int _ { M_\eta} ^ { 2 / \gamma } d \mu \frac { M_\eta } { \sqrt { \mu ^ { 2 } - \dfrac { 2 M_\eta ^ { 2 } } { 1 - w } } } \frac { M_\eta ^ { 2 } } { 8 ( 1 - w ) } \frac { 1 } { \mu } \left[ - \frac { M _\eta^ { 2 } } { 2 \mu } + ( 1 - w ) \frac { M_\chi ^ { 2 } \mu } { M_\eta ^ { 2 } } \right]\\
&{\displaystyle\times \left[ \frac { 2M_\eta \nu _ { max } ( \mu ) } { \alpha_\chi \mu } \right] ^ { 4 } \frac { \pi } { 16 } \frac { \alpha_\chi \mu ^ { 2 } } {2 ( 1 - w ) M_\eta ^ { 2 } \nu _ { max } ( \mu ) }}\\ \\
\approx& {\displaystyle {-\frac{3\cdot 2^{21} \pi^6 \gamma^2 M_\eta^2}{35(\lambda'^2+3h^2)^3}}}~, \\
\end{aligned}\end{aligned}$$ where we have kept only terms of the lowest order in $\alpha_\chi$.
Now let us turn our attention to the other six terms in the expression for $ \partial /\partial \omega J_c[\gamma_3]|_{\omega=0} $. It is easy to check that for any of these terms with signs $\chi_q,\chi_k,\chi_l $, the equation, $$\chi_q\omega_{\eta q}+\chi_k\omega_{\chi k}+\chi_l\omega_{\chi l}=\alpha_\chi\left(\frac{1}{2\omega_{\chi k}}+\frac{1}{2\omega_{\chi l}}\right)~,$$ has no solution. The integrands in these terms do not have a sharp peak and they are much smaller than the two terms we have calculated, and can henceforth be neglected. In addition, $ \partial /\partial \omega J_s[\gamma_i]|_{\omega=0} (i=0,1,2)$ can be neglected as well. The corresponding $ G(\omega_{\eta q},\omega_{\chi k},w) $ in these terms is of order $M_\eta^3/\alpha_\chi^3$ at the peak and is much smaller than (eq. \[fraction\_part\]) which is of order $ M_\eta^4/\alpha_\chi^4 $.
Combining equations (\[I\_result\]) and (\[J\_result\]) we obtain the derivative of the self-energy: $$\label{sunset_result}
\begin{aligned}
&-i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\\
=&-\frac { h ^ { 2 } } { 2 ( 4 \pi ) ^ { 3 } } \frac { 1 } { M_\eta \gamma ^ { 2 } } \left( 1 + \log \frac { M_\eta } { M_\chi } \right)-H\frac{3\cdot 2^{18} \pi^6h^2 \gamma^2 M_\eta^2}{35(\lambda'^2+3h^2)^3}\\
&-h^2Ht_1\frac{1}{16\pi^3M_\eta^2\gamma^3}\biggg[\frac{M_\eta^8}{(M_\eta^4+4M_\chi^2/\gamma^2)^2}-\frac{M_\eta^4}{M_\eta^4+4M_\chi^2/\gamma^2}-\frac{4M_\chi^2M_\eta^4/\gamma^2}{(M_\eta^4+4M_\chi^2/\gamma^2)^2}\\
&+\frac{8M_\chi^4/\gamma^2}{M_\eta^2(M_\eta^4+4M_\chi^2/\gamma^2)}+\log\left(\frac{1}{4}\right)-1+\log\left(\frac{M_\eta^6}{M_\chi^4\sqrt{M_\eta^4+4M_\chi^2/\gamma^2}}\right)\biggg].
\end{aligned}$$ Even if the Hubble parameter is very small, the second term at the right-hand side might be larger than the first term if the coupling constants $ \lambda' $ and $ h $ are small enough. This is due to the resonance effect which amplifies the curved-spacetime effects. The implication of this resonance on matter creation in the early universe shall be explored in a forthcoming article.
Computation of ${
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}$ and ${\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The contribution to ${
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}$ can be determined by the tadpole diagram: $$\begin{aligned}
{
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}&=\left[\frac{\partial^2}{\partial\omega^2}\int_{- \infty}^{\infty}dt\Pi^R(t_1,t_1-t)e^{i\omega t}\right]_{\omega=0}\\
&=-\int_{-\infty}^{\infty}dt\Pi^R(t_1,t_1-t)t^2\\
&\propto\int_{-\infty}^{\infty}dt\delta(t)t^2=0~.
\end{aligned}$$ As for $ {\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi} $, the calculation is similar to that of $ \tilde{\kappa}(t_1,\omega=0) $ and the result is $$\label{kappa_tilde_second_derivative}
{\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}= -\left(1-4 H t_{1}\right) \frac{1}{256 \pi \gamma} \sum_{\xi=\eta,\chi} \frac{g_{\xi}^{2}}{M_{\xi}^{3}}~.$$
#### Summary:
Let us summarise our main results here. The equation of motion for $\varphi$ in an abstract form is given above in (\[eqn: effective\_EoM\]): $$ \begin{aligned}
&\ddot{\varphi}(t_1)+\Bigg\{\left[m_\phi^2+(1+3Ht_1){\ensuremath{\kappa}}(t_1,\omega=0)-3H\left(-i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\right)\right]\varphi(t_1)\\
&\hphantom{\ddot{\varphi}(t_1)+}
+\frac{1}{6}\left[\lambda+(1+3Ht_1){\ensuremath{\widetilde{\kappa}}}(t_1,\omega=0)-3H\left(-i{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right)\right]\varphi(t_1)^3\Bigg\}\\
&+\Bigg\{3H+(1+3Ht_1)\left(-i{
\if 11
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\right)+3H{
\if 12
\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\else
\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\kappa}}(t_1,-\omega)\right|_{\omega=0}}
\fi
}\\
&\hphantom{+}
+\frac{1}{3}\left[(1+3Ht_1)\left(-i{\if 11\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right)+3H{\if 12\ensuremath{\left.\frac{\partial }{\partial \omega}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\else\ensuremath{\left.\frac{\partial^2 }{\partial \omega^2}{\ensuremath{\widetilde{\kappa}}}(t_1,-\omega)\right|_{\omega=0}}\fi}\right]\varphi(t_1)^2\Bigg\}\,\dot{\varphi}(t_1) =0~.
\end{aligned}$$ Using the results obtained in this section we obtain an explicit expression of the equation of motion for $ \varphi $, describing its dynamics in an expanding or a contracting cosmic background: $$\label{eqn: final_EoM}
\begin{aligned}
&\ddot{\varphi}(t_1)+\biggg\{\left[m_\phi^2+\left(1+Ht_1\right)\frac { \lambda + h } { 24 \gamma ^ { 2 } }+H\frac {3} { 2 ( 4 \pi ) ^ { 3 } } \frac { h ^ { 2 } } { M_\eta \gamma ^ { 2 } } \left( 1 + \log \frac { M_\eta } { M_\chi } \right)\right]\varphi(t_1)\\
&\hphantom{\ddot{\varphi}(t_1)+}
+\Bigg[\lambda-\left(1-Ht_1\right)\left(\frac{\lambda^2}{32\pi M_\eta \gamma}+\frac{h^2}{32\pi M_\chi \gamma}\right)\\
&\hphantom{\ddot{\varphi}(t_1)+}-3H\bigg(\frac{32\pi\gamma h^2}{\lambda'^2+3h^2}+\frac{32\pi\gamma \lambda^2}{\lambda^2+3h^2}\bigg)\left(1+\log\frac{M_\chi\gamma}{2}\right)\Bigg]\frac{\varphi(t_1)^3}{6}\biggg\}\\
&+\biggg\{-\frac { h ^ { 2 } } { 2 ( 4 \pi ) ^ { 3 } } \frac { 1 } { M_\eta \gamma ^ { 2 } } \left( 1 + \log \frac { M_\eta } { M_\chi } \right)+3H-H\frac{3\cdot 2^{18} \pi^6h^2 \gamma^2 M_\eta^2}{35(\lambda'^2+3h^2)^3}\\
&-Ht_1\frac{h^2}{16\pi^3M_\eta^2\gamma^3}\biggg[\frac { 3M_\eta \gamma } { 8} \left( 1 + \log \frac { M_\eta } { M_\chi } \right)+\frac{M_\eta^8}{(M_\eta^4+4M_\chi^2/\gamma^2)^2}-\frac{M_\eta^4}{M_\eta^4+4M_\chi^2/\gamma^2}\\
&-\frac{4M_\chi^2M_\eta^4/\gamma^2}{(M_\eta^4+4M_\chi^2/\gamma^2)^2}+\frac{8M_\chi^4/\gamma^2}{M_\eta^2(M_\eta^4+4M_\chi^2/\gamma^2)}+\log\left(\frac{1}{4}\right)-1+\log\left(\frac{M_\eta^6}{M_\chi^4\sqrt{M_\eta^4+4M_\chi^2/\gamma^2}}\right)\biggg]\\
&\hphantom{+}
+\biggg[\left(\frac{32\pi\gamma h^2}{\lambda'^2+3h^2}+\frac{32\pi\gamma \lambda^2}{\lambda^2+3h^2}\right)\left(1+\log\frac{M_\chi\gamma}{2}+\frac{4}{3}Ht_1\right)\\
&+H\left(\frac{2^{13}\pi^4\gamma^2 h^2}{\left(\lambda'^2+3h^2\right)^2}+\frac{2^{13}\pi^4\gamma^2 \lambda^2}{\left(\lambda^2+3h^2\right)^2}-\frac{3\lambda^2}{256\pi\gamma M_\eta^3}-\frac{3h^2}{256\pi\gamma M_\chi^3}\right)\biggg]\frac{\varphi(t_1)^2}{3}\biggg\}\,\dot{\varphi}(t_1) =0~.
\end{aligned}$$
Conclusions and Discussions {#sec:discussion}
===========================
In this paper, we considered a model of two scalar fields $\phi$ and $\chi$ with quartic coupling (eq. \[eqn:action\]). The mass of the $\phi$ field is assumed to be larger than twice the mass of the background field, $\chi$, so that the $\phi$ particles can decay into the $\chi$ particles for the studying of the dissipation effects. In a thermal bath made up of the $\chi$ particles, we studied the dynamics of the thermal average of the scalar field $\phi$ in an expanding or a contracting universe. We have assumed the background to be de Sitter spacetime with Hubble parameter $H\neq 0$. The Hubble parameter is taken to be a small constant in our calculations, our results are applicable to other expanding or contracting cosmic backgrounds to first order in $H$. The effective temperature is taken to be extremely high in our analysis to have the most manageable configuration yet retaining the interesting physics.
From the effective action of $\phi$, we obtained its equation of motion (eq. \[eqn: effective\_EoM\]), which is determined by the thermally and quantum-mechanically corrected self-energy and the self-coupling, and their derivatives. The analytic expressions of these quantities (eq. \[eqn:kappa\]), (eq.\[eqn:kappatilde\]), (eq. \[kappa\_tilde\_first\_derivative\]), (eq. \[sunset\_result\]) and (eq. \[kappa\_tilde\_second\_derivative\]) are the main results of this paper. In the computation of these quantities we expanded the propagator to first order in $H$; and our results match with those in flat space, using Minkowski-space propagators in loop diagrams [@Cheung:2015iqa], if we set $H$ to zero. Also, we developed some mathematical techniques when calculating these relevant Feynman diagrams.
In the equation of motion (eq. \[eqn: effective\_EoM\]), since the derivatives of $ \kappa $ and $ \tilde{\kappa} $ are derived from 1-loop corrected propagators, they appear to be of higher orders than $ \kappa$ and $\tilde{\kappa} $, which are ignored in the effective potential of $ \phi $. However, as shown in (eq. \[kappa\_tilde\_first\_derivative\]) and (eq. \[sunset\_result\]), there are terms that do not tend to zero as the coupling constants go to zero. Therefore they cannot be excluded from the effective potential.
In our calculations we have used the assumption that the Hubble parameter $H$ is small enough such that we can expand all terms of interest to first order in $H$. Thus, strictly speaking, the background universe is not a purely de Sitter spacetime, and the cosmic expansion is not strictly exponential. However, in such a situation we can still obtain valuable information about the effects of cosmic expansion/contraction on the scalar field dynamics. For example, from equation (\[sunset\_result\]), which contributes to the dissipation coefficient, we find that a de Sitter space, which is very close to a flat spacetime, has the chance of showing curved-spacetime features comparable to the flat spacetime features if the coupling constants $ h $ and $\lambda'$ are small enough. This is because in a de Sitter spacetime, we have to integrate over resonances due to the lack of spacetime translation invariance.
Our assumptions require that the temperature be much higher than the masses of the particles and the scale of the Hubble parameter. The effective temperature decreases as the universe expands, the corresponding approximation will fail when the temperature is of the same order as the masses. This happens only near the end of reheating; and thus the working assumptions are valid for the entire analysis if our results are applied to the reheating dynamics. Let us stress that our results also apply to a negative Hubble parameter $H$. We shall be using these results to study the quantum dissipative effects in the process of matter creation in the CST bounce universe [@Li:2011nj].
To discuss the quantum dissipative effects in the extreme conditions in the early universe, a rigorous theoretical framework of first-principle high temperature thermal quantum field theory is needed. The result we have obtained gives us a sense of the difference between the behaviour of a scalar field in a flat spacetime and in a de Sitter spacetime, which is helpful to the study of more realistic and more complicated situations, e.g. the effects of the expansion of the universe on the thermal damping rates of particles in the early universe and the production of matter within some specific inflationary or bounce models.
In this paper, we only calculated the results to the first order in $H$. Theoretically our approach can be extended to arbitrarily high orders, but such attempts are not practical as the integrals get much more complicated. Therefore, when $H$ becomes large enough, for which the approximation we have used would fail, one needs to seek other methods to extract the interesting physics besides matter productions.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research project has been supported in parts by the NSF China under Contract No. 11775110, and No. 11690034. We also acknowledge the European Union’s Horizon 2020 research and innovation programme (RISE) under the Marie Skĺodowska-Curie grant agreement No. 644121, and the Priority Academic Program Development for Jiangsu Higher Education Institutions (PAPD). We would like to express our gratitude to Jin U Kang and Marco Drewes for useful discussions and comments on the manuscript. L.M. and H.X. thank Ella Yang for useful suggestions on improving their draft.
Free Spectral Function of a Scalar Field {#appendix:free-spectral-function}
========================================
In this appendix we calculate the free spectral function of a scalar field $\xi$ in de Sitter spacetime with action $$\begin{aligned}
S&=-\int d^4x\sqrt{-g}\,\,\frac{1}{2}\xi\left[\frac{1}{\sqrt{-g}}\partial _\mu\left(\sqrt{-g}g^{\mu\nu}\partial _\nu\right)+m_{\xi}^{2}\right]\xi\\
&=-\int d^4x\,\,\frac{a^3}{2}\xi\left[\left(3H\partial _t+\partial _{t}^{2}-a^{-2}\nabla^2\right)+m_{\xi}^{2}\right]\xi,
\end{aligned}$$ where $a=e^{Ht} $.
If we set $ \tau=-e^{-Ht}/H $, $ \tilde{\xi}=a(t)\xi(x) $, then the action becomes $$S=-\int d\tau\int d^3x\frac{1}{2}\tilde{\xi}\left[\partial _{\tau}^{2}-\nabla^2+\left(m_\xi^2-2H^2\right)a^2\right]\tilde{\xi}~,$$ from which we obtain the equation of motion for $ \tilde{\xi} $: $$\left[\frac{\partial ^2}{\partial \tau^2}-\nabla^2+(m_\xi^2-2H^2)a^2\right]\tilde{\xi}(\boldsymbol{x},\tau)=0.$$ If we expand $ \tilde{\xi}(\boldsymbol{x},\tau) $ as $$\tilde{\xi}(\boldsymbol{x},\tau)=\int \frac{d^3k}{(2\pi)^3}\tilde{\xi}(\boldsymbol{k},\tau)e^{i\boldsymbol{k}\cdot \boldsymbol{x}}+h.c.~.$$
To solve the equation of motion we employ the WKB method, and obtain, $$\label{xi-expansion}
\tilde{\xi}(\boldsymbol{x},\tau)=\int \frac{d^3k}{(2\pi)^3}\left\{
A_k \frac{\exp\left[\displaystyle -i\int_{\tau_0}^{\tau}\omega(\tau')d\tau'\right]}{\sqrt{2\omega(\tau)}}e^{i\boldsymbol{k}\cdot \boldsymbol{x}}+h.c.~,
\right\},$$ where $ \tau_0 $ is a constant of integration, $ A_k $ a momentum-dependent operator, and $$\omega(\tau)=\sqrt{\boldsymbol{k}^2+(m_\xi^2-2H^2)a^2}~.$$
The conjugate momentum is given by, $$\tilde{\pi}(\boldsymbol{x},\tau)=\int \frac{d^3k}{(2\pi)^3}\frac{1}{\sqrt{2}}\left[-\frac{1}{2}H(m_\xi^2-2H^2)a^3\omega^{-5/2}-i\omega^{1/2}\right]A_k\exp\left(-i\int_{\tau_0}^{\tau}\omega(\tau')d\tau'\right)e^{i\boldsymbol{k}\cdot \boldsymbol{x}}+h.c.~.$$
From the commutation relation $ \left[\tilde{\xi}(\boldsymbol{x},\tau),\tilde{\pi}(\boldsymbol{x}',\tau)\right] =i \delta^3(\boldsymbol{x}-\boldsymbol{x}')$ we obtain $ \left[A_{\boldsymbol{k}},A_{\boldsymbol{k}'}^\dagger\right]=\delta^3(\boldsymbol{k}-\boldsymbol{k}') $. With this commutation relation of $ A_{\boldsymbol{k}}$ and $A_{\boldsymbol{k}'}^\dagger $ we can calculate the free spectral function using (\[xi-expansion\]): $$\begin{aligned}
\Delta^-(\boldsymbol{k},t_1,t_2)&=i\left\langle\left[\frac{\tilde{\xi}(\tau_1)}{a(t_1)},\frac{\tilde{\xi}(\tau_2)}{a(t_2)}\right]\right\rangle_\beta\\
&=\frac{1}{a(t_1)^{3/2}a(t_2)^{3/2}\sqrt{\Omega_\xi(t_1)}\sqrt{\Omega_\xi(t_2)}}\sin\left[\int_{t_2}^{t_1}\Omega_\xi(t')dt'\right],
\end{aligned}$$ where $$\Omega_\xi(t)\equiv \sqrt{\boldsymbol{k}^2/a(t)^2+(m_\xi^2-2H^2)}~.$$
A few useful formulae
=====================
In this appendix we present the relevant formulae that we have used in computing the integrals associated with the Feynman diagrams.
Integrals Involving the Bose Distribution function {#appendix: bose distribution integral}
--------------------------------------------------
When doing the integrals in (eq. \[eqn:tadpole\]), we encounter expressions of the following form, $$\label{eqn:h}
h_n(y)=\frac{1}{\Gamma(n)}\int_{0}^{\infty} dx\frac{x^{n-1}}{\sqrt{x^2+y^2}}\frac{1}{e^{\sqrt{x^2+y^2}}-1}\quad (n\in \mathbb{Z}^{+}).$$ By expanding $ 1/\{[\exp(\sqrt{x^2+y^2})]-1\} $ around $ \sqrt{x^2+y^2}=0 $ into a Laurent series, integrating over $ x $ and then expanding the expression around $ y=0 $, we obtain $$\begin{aligned}
h_1(y)=\frac{\pi}{2y}+\frac{1}{2}\log\frac{y}{4\pi}+\frac{\gamma_E}{2}+\sum_{m=1}^{\infty}\frac{(-1)^{m}}{2^{m+1}}\frac{(2m-1)!!}{m!}\zeta(2m+1)\left(\frac{y}{2\pi}\right)^{2m}~.
\end{aligned}$$ It is easy to check that $ h_n(y) $ satisfies, $$\frac{dh_{n+1}}{dy}=-\frac{yh_{n-1}}{n}~.$$ Setting $ n=2 $ and integrating the above equality we have, $$\label{eqn: h_3}
\begin{aligned}
h_3(y)&=-\frac{1}{2}\int yh_1(y)dy+\frac{\pi^2}{12}\\
&=\frac{\pi^2}{12}-\frac{\pi y}{4}-\left(\frac{\gamma_E}{8}-\frac{1}{16}\right)y^2-\frac{y^2}{8}\log\frac{y}{4\pi}+\sum_{m=1}^{\infty}\frac{(-1)^{m+1}}{2^{m+3}}\frac{(2m-1)!!}{(m+1)!}\frac{\zeta(2m+1)}{(2\pi)^{2m}}y^{2m+2}.
\end{aligned}$$
Fourier Transformation of a General Form {#appendix: Fourier of general form}
----------------------------------------
In this subsection we shall calculate the real and imaginary parts of the following expressions: $$\begin{aligned}
&F^\omega\left[\theta(t)\mathrm{Im}\int \frac{d^3q}{(2\pi)^3}\left(K_0+t K_1+t^2K_2\right)e^{-2i\omega_q t-\alpha t/\omega_q}\right],\\
&F^\omega\left[\theta(t)\mathrm{Im}\int \frac{d^3q}{(2\pi)^3}\left(K_0+t K_1+t^2K_2\right)e^{-\alpha t/\omega_q}\right],
\end{aligned}$$ where $ F^\omega $ denotes a Fourier transformation w.r.t. $ t $ and $ K_0, K_1,K_2 $ are rational functions of $ \boldsymbol{q}^2 $. We use $ K $ to denote any of $ K_0,K_1,K_2 $ and calculate the following integral: $$\begin{aligned}
&F^{\omega}\left[ \theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}} K\left(\omega_{q}\right) e^{-2 i \omega_{q} t- \frac{\alpha}{\omega_q} t}\right]\\
=& \int \frac{d^3 q}{(2\pi)^3}\left[-i\mathrm{Re}K(\omega_q)F^\omega\left(\theta(t)\frac{e^{-2i\omega_q t}-e^{2i\omega_q t}}{2}e^{-\frac{\alpha}{\omega_q}t}\right)+\mathrm{Im}K(\omega_q)F^\omega\left(\theta(t)\frac{e^{-2i\omega_q t}+e^{2i\omega_q t}}{2}e^{-\frac{\alpha}{\omega_q}t}\right)\right].
\end{aligned}$$ Carrying out the Fourier transformation in the integrand and change the variable of integration to $ \omega_q=\sqrt{\boldsymbol{q}^2+m^2} $, the integral becomes, $$\begin{aligned}
&\frac{1}{(2 \pi)^{3}} \int_{m}^{\infty} d \omega_{q} \frac{\omega_{q}}{\sqrt{\omega_{q}^{2}-m^{2}}} 4 \pi\left(\omega_{q}^{2}-m^{2}\right)\\
&\quad\times \bigg\{ \mathrm{Re}K(\omega_q)\left[\frac{1}{2}\left(\dfrac{\omega-2\omega_q}{(\omega-2\omega_q)^2+(\alpha/\omega_q)^2}-\frac{\omega+2\omega_q}{(\omega+2\omega_q)^2+(\alpha/\omega_q)^2}\right)\right.\\
&\qquad\quad \left.-i\left(\frac{\alpha/(2\omega_q)}{(\omega-2\omega_q)^2+(\alpha/\omega_q)^2}-\frac{\alpha/(2\omega_q)}{(\omega+2\omega_q)^2+(\alpha/\omega_q)^2}\right)\right]\\
&\qquad+ \mathrm{Im}K(\omega_q)\left[\frac{i}{2}\left(\dfrac{\omega+2\omega_q}{(\omega+2\omega_q)^2+(\alpha/\omega_q)^2}+\frac{\omega-2\omega_q}{(\omega-2\omega_q)^2+(\alpha/\omega_q)^2}\right)\right.\\
&\qquad\quad \left.+\left(\frac{\alpha/(2\omega_q)}{(\omega+2\omega_q)^2+(\alpha/\omega_q)^2}+\frac{\alpha/(2\omega_q)}{(\omega-2\omega_q)^2+(\alpha/(2\omega_q))^2}\right)\right]\bigg\}.
\end{aligned}$$ Since we only calculate to the lowest order in $ \alpha $, the above expression can be further simplified to, $$\begin{aligned}
&\frac{1}{(2 \pi)^{3}} \int_{m}^{\infty} d \omega_{q} \frac{\omega_{q}}{\sqrt{\omega_{q}^{2}-m^{2}}} 4 \pi\left(\omega_{q}^{2}-m^{2}\right)\\
&\quad \times\left\{\mathrm{Re}K(\omega_q)\left[\frac{2\omega_q}{\omega^2-4\omega_q^2}-i\left(\frac{\alpha}{2\omega_q(\omega-2\omega_q)^2}-\frac{\alpha}{2\omega_q(\omega+2\omega_q)^2}\right)\right]\right.\\
&\qquad\left.+\mathrm{Im}K(\omega_q)\left[i\frac{\omega}{\omega^2-4\omega_q^2}+\left(\frac{\alpha}{2\omega_q(\omega-2\omega_q)^2}+\frac{\alpha}{2\omega_q(\omega+2\omega_q)^2}\right)\right]\right\}\\
&\equiv I_{re}\left[K(\omega_q)\right]+iI_{im}\left[K(\omega_q)\right],
\end{aligned}$$ where $$\left\{
\begin{aligned}
I_{r e}\left[K\left(\omega_{{q}}\right)\right]=&\frac{1}{(2 \pi)^{3}} \int_{m}^{\infty} d \omega_{{q}} 4 \pi \omega_{{q}} \sqrt{\omega_{{q}}^{2}-m^2}\Bigg\{\operatorname{Re} K\left(\omega_{{q}}\right) \frac{2 \omega_{{q}}}{\omega^{2}-4 \omega_{{q}}^{2}}\\
&+\operatorname{Im} K\left(\omega_{{q}}\right)\left[\frac{\alpha}{2\omega_{{q}}\left(\omega+2 \omega_{{q}}\right)^{2}}+\frac{\alpha}{2\omega_{{q}}\left(\omega-2 \omega_{{q}}\right)^{2}}\right]\Bigg\}\\
I_{im}\left[K\left(\omega_{{q}}\right)\right]=&\frac{1}{(2 \pi)^{3}} \int_{m}^{\infty} d \omega_{{q}} 4 \pi \omega_{{q}} \sqrt{\omega_{{q}}^{2}-m^2}\Bigg\{\operatorname{Im} K\left(\omega_{{q}}\right) \frac{\omega}{\omega^{2}-4 \omega_{{q}}^{2}}\\
&-\operatorname{Re} K\left(\omega_{{q}}\right)\left[\frac{\alpha}{2\omega_{{q}}\left(\omega-2 \omega_{{q}}\right)^{2}}-\frac{\alpha}{2\omega_{{q}}\left(\omega+2 \omega_{{q}}\right)^{2}}\right]\Bigg\}
\end{aligned}\right.~.$$ We arrive at, $$\label{eqn: fourier_Im}
\begin{aligned}
& \operatorname{Im} F^{\omega}\left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left(K_{0}+t K_{1}+t^{2} K_{2}\right) e^{-2 i \omega_{q} t- \frac{\alpha}{\omega_{q}} t}\right]\\
=&\operatorname{Im} F^{\omega} \left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}K_{0} e^{-2 i \omega_{q} t- \frac{\alpha}{\omega_{q}} t}\right]+\operatorname{Im}\Bigg\{\left(-i \frac{\partial}{\partial \omega}\right) F^{\omega} \left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}K_{1}e^{-2 i \omega_{q} t-\frac{\alpha}{\omega_{q}}t} \right]\Bigg\}\\
& +\operatorname{Im}\Bigg\{\left(-\frac{\partial^{2}}{\partial \omega^{2}}\right) F^{\omega} \left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}K_{3}e^{-2 i \omega_{q} t-\frac{\alpha}{\omega_{q}} t}\right]\Bigg\}\\
=& I_{i m}\left[K_{0}\right]-\frac{\partial}{\partial \omega} I_{r e}\left[K_{1}\right]-\frac{\partial^{2}}{\partial \omega^{2}} I_{i m}\left[K_{2}\right]~.
\end{aligned}$$ In a similar way we obtain, $$\label{eqn: fourier_Re}
\operatorname{Re} F^{\omega}\left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left(K_{0}+t K_{1}+t^{2} K_{2}\right) e^{-2 i \omega_{q} t- \frac{\alpha}{\omega_{q}} t}\right]=I_{re}\left[K_{0}\right]+\frac{\partial}{\partial \omega} I_{im}\left[K_{1}\right]-\frac{\partial^{2}}{\partial \omega^{2}} I_{re}\left[K_{2}\right]~,$$ and $$\label{eqn: fourier_cross}
\begin{aligned}
&F^{\omega}\left[\theta(t) \operatorname{Im} \int \frac{d^{3} q}{(2 \pi)^{3}}\left(K_{0}+t K_{1}+t^{2} K_{2}\right) e^{- \frac{\alpha}{\omega_{q}} t}\right]\\
=&\int \frac{d^3q}{(2\pi)^3}\left(\mathrm{Im}K_0\frac{\omega_q \alpha}{\alpha^2+\omega^2\omega^2_q}+\mathrm{Im}K_1\frac{\partial}{\partial \omega}\frac{\omega\omega^2_q}{\alpha^2+\omega^2\omega^2_q}-\mathrm{Im}K_2\frac{\partial^2}{\partial \omega^2}\frac{\omega_q \alpha}{\alpha^2+\omega^2\omega^2_q}\right)\\
&+i\int \frac{d^3q}{(2\pi)^3}\left(\mathrm{Im}K_0\frac{\omega\omega^2_q}{\alpha^2+\omega^2\omega^2_q}-\mathrm{Im}K_1\frac{\partial }{\partial \omega}\frac{\omega_q\alpha}{\alpha^2+\omega^2\omega^2_q}-\mathrm{Im}K_2\frac{\partial ^2}{\partial \omega^2}\frac{\omega\omega^2_q}{\alpha^2+\omega^2\omega^2_q}\right)~.
\end{aligned}$$
Calculation of the Tadpole Diagram
==================================
In this appendix we present the detailed calculation of the the tadpole diagram. Inserting the propagator (eq. \[propagator+-\]) into (eq. \[eqn:tadpole\]) we have, $$\begin{aligned}
&\Pi^R(t_1,t_2)\\
=&\frac{1}{2}\delta(t_1-t_2)\sum_{\xi=\eta,\chi}\int \frac{d^3p}{(2\pi)^3}\frac{g_\xi}{a(t_1)^3\sqrt{\boldsymbol{p}^2/a(t_1)^2+M^2_\xi}}\left[\frac{1}{2}+\frac{1}{e^{\sqrt{\boldsymbol{p}^2\beta(t_1)^2/a(t_1)^2+M^2_\xi\beta(t_1)^2}}-1}\right]\\
=&\delta(t_1-t_2)\frac{1}{4\pi^2\beta(t_1)^2}\sum_{\xi=\eta,\chi}\int dx\,g_\xi\left[\frac{1}{2}\frac{x^2}{\sqrt{x^2+M^2_\xi\beta(t_1)^2}}+\frac{x^2}{\sqrt{x^2+M^2_\xi\beta(t_1)^2}}\frac{1}{e^{\sqrt{x^2+M_\xi^2\beta(t_1)^2}}-1}\right]~.
\end{aligned}$$ We replace the divergent integrals that are independent of time by the constants $ C_1 $ and $ C_2 $: $$\begin{aligned}
&C_1=\frac{1}{2}\sum_{\xi=\eta,\chi}\int \frac{d^3p}{(2\pi)^3}\frac{g_\xi}{2\sqrt{\boldsymbol{p}^2+M_\xi^2}}+\text{counter terms},\\
&C_2=\frac{1}{2}\sum_{\xi=\eta,\chi}\int\frac{d^3p}{(2\pi)^3}\left[\frac{g_\xi}{\sqrt{\boldsymbol{p}^2+M_\xi^2}}+\frac{g_\xi M_\xi^2}{2(\boldsymbol{p}^2+M_\xi^2)^{3/2}}\right]+\text{counter terms.}
\end{aligned}$$ yielding $\Pi^R(t_1,t_2)$ in the following form, $$\Pi^R(t_1,t_2)=\delta(t_1-t_2)\left\{C_1-C_2Ht_1+\frac{1}{2\pi^2\beta(t_1)^2}\sum_{\xi=\eta,\chi}g_\xi h_3\left[M_\xi\beta(t_1)\right]\right\}~.$$ Comparing with the case of $ H=0 $, $$\Pi^R(t_1,t_2)\approx\delta(t_1-t_2)\left[C_1+ \sum_{\xi=\eta,\chi}\int\frac{d^3p}{(2\pi)^3}\frac{g_\xi}{2\sqrt{\boldsymbol{p}^2+M_\xi^2}}\cdot\frac{1}{\exp\left(\dfrac{\beta_0\sqrt{\boldsymbol{p}^2+M_\xi^2}}{a_0}\right)-1}\right],$$ we can conclude that $ C_1 =0 $ at zero temperature in flat spacetime. Furthermore the propagator of $ \phi $ has a pole at the physical mass: the first term and the second term in the $ C_2 $ integral are proportional to $ C_1 $ or to the derivative of $ C_1 $ with respect to $ M_\xi $, we deduce that $ C_2=0 $. The renormalised $\Pi^R(t_1,t_2)$ is finally given by, $$\Pi^R(t_1,t_2)=\delta(t_1-t_2)\left\{\frac{1}{2\pi^2\beta(t_1)^2}\sum_{\xi=\eta,\chi}g_\xi h_3\left[M_\xi\beta(t_1)\right]\right\}~.$$
Intermediate Results in the Calculation of the Sunset Diagram {#appendix: C3}
=============================================================
Let us now turn to the contribution of the sunset diagram to the self-energy, $$\begin{aligned}
&-i\left[\Pi^R(t_1,t_2)\right]_{\text{sunset}}\\
=&-ih^2\theta(t_1-t_2)\int \frac{d^3k}{(2\pi)^3}\frac{d^3l}{(2\pi^3)}\mathrm{Im}\left[(\Delta_\chi)_>(\boldsymbol{k},t_1,t_2)(\Delta_\chi)_>(\boldsymbol{l},t_1,t_2)(\Delta_\eta)_>(\boldsymbol{k}+\boldsymbol{l},t_1,t_2)\right]~.
\end{aligned}$$ By expanding the propagators to first order in $ H $ and inserting them into the above equation we obtain the following expression of the imaginary part of $ \kappa $, $$\label{eqn:kappa_H_expansion} \begin{aligned}
&\mathrm{Im}\kappa(t_1,\omega)=\frac{h^2}{8} \mathrm{Im}F^\omega\Bigg\{\theta(t)\int \frac{d^{3}k}{(2\pi)^3}\frac{d^3l}{(2\pi)^3}\\
&\mathrm{Im}\frac { 1 } { \omega _ { \eta q } \omega_{\chi k} \omega_{\chi l} }\left[\left( 1 - 9 H t _ { 1 } + \frac { 9 } { 2 } H t \right) +\left( H t _ { 1 } - \frac { 1 } { 2 } H t \right) \left(3 - \frac { M_\eta ^ { 2 } } { \omega _ { \eta q } ^ { 2 } } - \frac { M_\chi ^ { 2 } } { \omega_{\chi k} ^ { 2 } } - \frac { M_\chi^ { 2 } } { \omega_{\chi l} ^ { 2 } } \right)\right.\\
&\left. - H \left( t _ { 1 } - t \right) \left( f (\omega_{\eta q}) \frac { \gamma M_\eta ^ { 2 } } { \omega _ { \eta q } } + f (\omega_{\chi k}) \frac { \gamma M_\chi^ { 2 } } { \omega_{\chi k} } + f (\omega_{\chi l}) \frac { \gamma M_\chi ^ { 2 } } { \omega_{\chi l} } \right) + i H \left( 2 t _ { 1 } t - t ^ { 2 } \right) \left( \frac { \boldsymbol { q } ^ { 2 } } { 2 \omega _ { \eta q } } + \frac { \boldsymbol { k } ^ { 2 } } { 2 \omega_{\chi k} } + \frac { \boldsymbol { l } ^ { 2 } } { 2 \omega_{\chi l} } \right)\right]\\
&\times \left( 1 + f (\omega_{\eta q})\right) \left( 1 + f (\omega_{\chi k}) \right) \left( 1 + f (\omega_{\chi l}) \right) \exp \left[ - i \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2 \right]\Bigg\} \\
&+ \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
&=\frac{h^2}{8} \mathrm{Im}F^\omega\Bigg\{\theta(t)\int \frac{d^{3}k}{(2\pi)^3}\frac{d^3l}{(2\pi)^3}\frac { 1 } { \omega _ { \eta q } \omega_{\chi k} \omega_{\chi l} }\Bigg\{-\sin \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right) \times\\
&\left[ \left( 1 - 9 H t _ { 1 } \right) \gamma _ { 0 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) + H t _ { 1 } \gamma _ { 1 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) - H t _ { 1 } \gamma _ { 2 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) \right] \\
&+t \cos \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right)\cdot2 H t _ { 1 } \gamma _ { 3 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) \\
&-t \sin \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right)\\
&\times \Bigg[ \frac { 9 } { 2 } H \gamma _ { 0 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right)- \frac { 1 } { 2 } H \gamma _ { 1 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) + H \gamma _ { 2 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) \Bigg] \\
&- t ^ { 2 } \cos \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right) \cdot H \gamma _ { 3 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right)\Bigg\}\\
&\times \left( 1 + f (\omega_{\eta q}) \right) \left( 1 + f(\omega_{\chi k}) \right) \left( 1 + f(\omega_{\chi l}) \right)\Bigg\} \\
&+ \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right),
\end{aligned}$$ where $ \Gamma_\chi(\omega_{\chi k})\approx\frac{\lambda'^2+3h^2}{256\pi^3\gamma^2\omega_{\chi k}}, \Gamma_\chi(\omega_{\chi l})\approx\frac{\lambda'^2+3h^2}{256\pi^3\gamma^2\omega_{\chi l}} $ [@Cheung:2015iqa] and $$\left\{
\begin{aligned}
&\gamma _ { 0 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) = 1~,\\
&\gamma _ { 1 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) = 3 - \frac { M_\eta ^ { 2 } } { \omega _ { \eta q } ^ { 2 } } - \frac { M_\chi ^ { 2 } } { \omega_{\chi k} ^ { 2 } } - \frac { M_\chi^ { 2 } } { \omega_{\chi l} ^ { 2 } }~,\\
&\gamma _ { 2 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) = f (\omega_{\eta q}) \frac { \gamma M_\eta ^ { 2 } } { \omega _ { \eta q } } + f (\omega_{\chi k}) \frac { \gamma M_\chi^ { 2 } } { \omega_{\chi k} } + f (\omega_{\chi l})\frac { \gamma M_\chi ^ { 2 } } { \omega_{\chi l} }~,\\
&\gamma _ { 3 } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right) = \frac { \boldsymbol { q } ^ { 2 } } { 2 \omega _ { \eta q } } + \frac { \boldsymbol { k } ^ { 2 } } { 2 \omega_{\chi k} } + \frac { \boldsymbol { l } ^ { 2 } } { 2 \omega_{\chi l} } ~.
\end{aligned}\right.$$ We note that in (eq. \[eqn:kappa\_H\_expansion\]) there are four kinds of Fourier transformations, which involves $ \sin(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}), $ $t\cos(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}) , $ $ t\sin(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}) , $ and $ t^2\cos(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}) , $ respectively. For the first two we carry out the Fourier transformations and perform the angle integrals and express them in the following form: $$ \begin{aligned}
I_s[\gamma_i] \equiv
&-\mathrm{Im}F^\omega\Bigg\{\theta(t)\int \frac{d^{3}k}{(2\pi)^3}\frac{d^3l}{(2\pi)^3}\\
&\sin \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right) \frac { \gamma_i(\omega_{\eta q},\omega_{\chi k},\omega_{\chi l}) } { \omega _ { \eta q } \omega_{\chi k} \omega_{\chi l} }\\
&\times\left( 1 + f (\omega_{\eta q}) \right) \left( 1 + f(\omega_{\chi k}) \right) \left( 1 + f(\omega_{\chi l}) \right)\Bigg\} \\
+ &\left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
+ & \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
+ & \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
+ &\left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
\approx & \frac { 1 } { 4 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l} \int _ { M_\chi t _ { 1 } - } ^ { M_\chi t _ { 1 + } } d \omega_{\chi k}\big[\gamma _ { i } \left( - \omega + \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
- & \gamma _ { i } \left( \omega - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\big]-\left(\omega\to -\omega\right)~.
\end{aligned}$$ $$ \begin{aligned}
-\frac{\partial}{\partial\omega}I_c[\gamma_i] \equiv
&\mathrm{Im}F^\omega\Bigg\{\theta(t)\int \frac{d^{3}k}{(2\pi)^3}\frac{d^3l}{(2\pi)^3}\\
&t\cos \left[ \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) t \right]\exp \left(-\Gamma_\chi(\omega_{\chi k})t/2-\Gamma_\chi(\omega_{\chi l})t/2\right)\frac { \gamma_i(\omega_{\eta q},\omega_{\chi k},\omega_{\chi l}) } { \omega _ { \eta q } \omega_{\chi k} \omega_{\chi l} }\\
&\times\left( 1 + f (\omega_{\eta q}) \right) \left( 1 + f (\omega_{\chi k})\right) \left( 1 + f (\omega_{\chi l})\right)\Bigg\} \\
&+ \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
\approx& -\frac{\partial}{\partial \omega}\Bigg\{\frac { 1 } { 4 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l} \int _ { M_\chi t _ { 1 } - } ^ { M_\chi t _ { 1 + } } d \omega_{\chi k}\big[\gamma _ { i } \left( - \omega + \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right)\\
&\times \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
& \quad - \gamma _ { i } \left( \omega - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\big]+\left(\omega\to -\omega\right)\Bigg\}
\end{aligned}$$ where $$\left\{
\begin{aligned}
& t _ { 1 \pm } = ( u - 1 ) s \pm \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) } + \frac { \omega } { M_\chi }
\left[ 1 - u + ( 2 u - 1 ) s ^ { 2 } \pm s \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) \left( \frac { 2 u - 3 } { u - 2 } \right) } \right]\\
& u = \frac { M_\eta ^ { 2 } } { 2 M_\chi^ { 2 } } , s = \frac { \omega_{\chi l} } {M_\chi}
\end{aligned}
\right.~.$$
We shall presently compute the derivatives of $ I_s $ and $ I_c $ and setting $\omega=0$: $$\label{I_s} \begin{aligned}
\left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { i } \right] \right|_ { \omega = 0 }
= &2 \cdot \frac { 1 } { 4 ( 2 \pi ) ^ { 3 } } \int _ {M_\chi} ^ { \infty } d \omega_{\chi l}\int_{M_\chi \left[ ( u - 1 ) s - \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) } \right]}^{M_\chi \left[ ( u - 1 ) s + \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) } \right]}d \omega_{\chi k}\\
& ~~\frac { \partial } { \partial \omega } \bigg[\gamma _ { i } \left( - \omega + \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
& ~~- \gamma _ { i } \left( \omega - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\bigg]_{\omega=0}\\
+& 2 \cdot \frac { 1 } { 4 ( 2 \pi ) ^ { 3 } } \int _ {M_\chi} ^ { \infty } d \omega_{\chi l}\bigg\{M_\chi t_{1p}\Big[\gamma _ { i } \left( \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
& - \gamma _ { i } \left( - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\Big]_{\omega_{\chi k}=M_\chi t_{0p}}\\
& - M_\chi t _ { 1 m }\Big[\gamma _ { i } \left( \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
&- \gamma _ { i } \left( - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\Big]_{\omega_{\chi k}=M_\chi t_{0m}}\bigg\}~.
\end{aligned}$$ $$\label{I_c} \begin{aligned}
- \left. \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } } I _ { c } \left[ \gamma _ { i } \right] \right|_ { \omega = 0 }
=& - \frac { 1 } { 2 ( 2 \pi ) ^ { 3 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi l}\Bigg\{\int _ { M_\chi t _ { 0 m } } ^ { M_\chi t _ { 0 p } } d \omega_{\chi k} \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } }\Big[\gamma _ { i } \left( - \omega + \omega_{\chi k} + \omega_{\chi l} , - \omega_{\chi k} , - \omega_{\chi l} \right) \\
&\times\left( 1 + f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \right) f \left( \omega_{\chi k} \right) f \left( \omega_{\chi l} \right)\\
& - \gamma _ { i } \left( \omega - \omega_{\chi k} - \omega_{\chi l} , \omega_{\chi k} , \omega_{\chi l} \right) f \left( \omega_{\chi k} + \omega_{\chi l} - \omega \right) \left( 1 + f \left( \omega_{\chi k} \right) \right) \left( 1 + f \left( \omega_{\chi l} \right) \right)\Big]\bigg|_{\omega=0}\\
& +M_\chi^2(t_{1p})^2\frac{\partial}{\partial \omega_{\chi k}}\Big[\gamma_i(\omega_{\chi k}+\omega_{\chi l},-\omega_{\chi k},-\omega_{\chi l})\left(1+f\left(\omega_{\chi k}+\omega_{\chi l}\right)\right)f\left(\omega_{\chi k}\right)f\left(\omega_{\chi l}\right)\\
&-\gamma_i\left(-\omega_{\chi k}-\omega_{\chi l},\omega_{\chi k},\omega_{\chi l}\right)f\left(\omega_{\chi k}+\omega_{\chi l}\right)\left(1+f\left(\omega_{\chi k}\right)\right)\left(1+f\left(\omega_{\chi l}\right)\right)\Big]\bigg|_{\omega_{\chi k}\to M_\chi t_{0p}}\\
&-M_\chi^2(t_{1m})^2\frac{\partial}{\partial \omega_{\chi k}}\Big[\gamma_i(\omega_{\chi k}+\omega_{\chi l},-\omega_{\chi k},-\omega_{\chi l})\left(1+f\left(\omega_{\chi k}+\omega_{\chi l}\right)\right)f\left(\omega_{\chi k}\right)f\left(\omega_{\chi l}\right)\\
&-\gamma_i\left(-\omega_{\chi k}-\omega_{\chi l},\omega_{\chi k},\omega_{\chi l}\right)f\left(\omega_{\chi k}+\omega_{\chi l}\right)\left(1+f\left(\omega_{\chi k}\right)\right)\left(1+f\left(\omega_{\chi l}\right)\right)\Big]\bigg|_{\omega_{\chi k}\to M_\chi t_{0m}}\Bigg\},
\end{aligned}$$ where $$\left\{
\begin{aligned}
& u=\frac{M_\eta^2}{2M_\chi^2}, s=\frac{\omega_{\chi l}}{M_\chi}\\
& t _ { 0 p } = ( u - 1 ) s + \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) }\\
& t _ { 0 m } = ( u - 1 ) s - \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) }\\
& t _ { 1 p } = \frac { 1 } { M_\chi } \left[ 1 - u + ( 2 u - 1 ) s ^ { 2 } + s \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) } \left( \frac { 2 u - 3 } { u - 2 } \right) \right]\\
& t _ { 1 m } = \frac { 1 } { M_\chi } \left[ 1 - u + ( 2 u - 1 ) s ^ { 2 } - s \sqrt { u ( u - 2 ) \left( s ^ { 2 } - 1 \right) } \left( \frac { 2 u - 3 } { u - 2 } \right) \right]\\
& t _ { 1 + } = t _ { 0 p } + \omega t _ { 1 p } , \quad t _ { 1 - } = t _ { 0 m } + \omega t _ { 1 m }~.
\end{aligned}\right.$$
We also calculate the derivatives of the Fourier transformations of, $ t\sin(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}) $ and $ t^2\cos(\omega_{\eta q}+\omega_{\chi k}+\omega_{\chi l}) $ (which we denote by $ J_s $ and $ J_c $, respectively) and then evaluate them at $ \omega=0 $: $$\label{J_s}
\begin{aligned}
\left.\frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { i } \right]\right| _ { \omega = 0 } \approx & - 2 \frac { 1 } { ( 2 \pi ) ^ { 4 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi k} d \omega_{\chi l} \int _ { - 1 } ^ { 1 } d w \frac { \sqrt { \omega_{\chi k} ^ { 2 } - M_\chi ^ { 2 } } \sqrt { \omega_{\chi l} ^ { 2 } - M_\chi ^ { 2 } } \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) ^ 3} { \omega _ { \eta q } \left[\left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) ^2+\left(\Gamma_\chi(\omega_{\chi k})/2+\Gamma_\chi(\omega_{\chi l})/2\right)^2\right] ^3}\\
&\times \gamma _ { i } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right)\left( 1 + f (\omega_{\eta q}) \right) \left( 1 + f (\omega_{\chi k})\right) \left( 1 + f (\omega_{\chi l}) \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)~,
\end{aligned}$$ $$\label{J_c}
\begin{aligned}
\left.\frac { \partial } { \partial \omega } J _ { c } \left[ \gamma _ { i } \right] \right| _ { \omega = 0 }
& \approx 6 \frac { 1 } { ( 2 \pi ) ^ { 4 } } \int _ { M_\chi } ^ { \infty } d \omega_{\chi k} d \omega_{\chi l} \int _ { - 1 } ^ { 1 } d w \frac { \sqrt { \omega_{\chi k} ^ { 2 } - M_\chi ^ { 2 } } \sqrt { \omega_{\chi l} ^ { 2 } - M_\chi ^ { 2 } } \left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) ^ 4} { \omega _ { \eta q } \left[\left( \omega _ { \eta q } + \omega_{\chi k} + \omega_{\chi l} \right) ^2+\left(\Gamma_\chi(\omega_{\chi k})/2+\Gamma_\chi(\omega_{\chi l})/2\right)^2\right] ^4} \\
&\times \gamma _ { i } \left( \omega _ { \eta q } , \omega_{\chi k} , \omega_{\chi l} \right)\left( 1 + f (\omega_{\eta q}) \right) \left( 1 + f (\omega_{\chi k})\right) \left( 1 + f (\omega_{\chi l}) \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow - \omega _ { \eta q } , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega _ { \eta q } \rightarrow \omega _ { \eta q } , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)\\
& + \left( \omega _ { \eta q } \rightarrow \omega_{\eta q} , \omega_{\chi k} \rightarrow - \omega_{\chi k} , \omega_{\chi l} \rightarrow - \omega_{\chi l} \right) + \left( \omega_{\eta q} \rightarrow - \omega_{\eta q} , \omega_{\chi k} \rightarrow \omega_{\chi k} , \omega_{\chi l} \rightarrow \omega_{\chi l} \right)~.
\end{aligned}$$
The derivative of $ \kappa $ with respect to $ \omega $ can thus be expressed as, $$\label{self_imaginary1}
\begin{aligned}
& -i\left. \frac { \partial } { \partial \omega } \kappa ( t_1,\omega ) \right| _ { \omega = 0 }\\
=&\frac{h^2}{8}\Bigg\{ \left( 1 - 9 H t _ { 1 } \right) \left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 0 } \right] \right| _ { \omega = 0 } + H t _ { 1 }\left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 1 } \right] \right| _ { \omega = 0 } - H t _ { 1 }\left. \frac { \partial } { \partial \omega } I _ { s } \left[ \gamma _ { 2 } \right] \right| _ { \omega = 0 } - 2 H t _ { 1 }\left. \frac { \partial ^ { 2 } } { \partial \omega ^ { 2 } } I _ { c } \left[ \gamma _ { 3 } \right] \right|_ { \omega = 0 }\\
& -\frac { 9 } { 2 } H\left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 0 } \right] \right| _ { \omega = 0 } - \frac { 1 } { 2 } H \left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 1 } \right] \right|_ { \omega = 0 } + H \left. \frac { \partial } { \partial \omega } J _ { s } \left[ \gamma _ { 2 } \right] \right| _ { \omega = 0 } - H\left. \frac { \partial } { \partial \omega } J _ { c } \left[ \gamma _ { 3 } \right] \right|_ { \omega = 0 }\Bigg\}.
\end{aligned}$$
Symbol List
===========
This temporary list is made to ensure that all symbols have been clearly defined:
symbol section illustration
--------------------------------------------------- --------- ------------------------------------------------------------------------------------------------
$ \phi $ 1 the scalar field we study
$ \chi $ 1 scalar field representing the environment
$ \varphi $ 1 expectation value of $ \phi $
$ m_\phi $ 1 mass of $ \phi $
$ m_\chi $ 1 mass of $ \chi $
$ \Delta^{>,<}(x,y) $ 2 two point correlation functions
$ \Delta_{ab}(x,y) \,\,\,(a,b=1,2) $ 2 propagators
$ \Delta^{+,-}(x,y) $ 2 two point correlation functions
$ g_{\mu\nu} $ 2 FLRW metric
$ t_B $ 2 the smaller of $t_1$ and $t_2$
$ \beta_0, a_0 $ 2 parameters
$ \Omega_{\xi}(t) $ 2 energy of $\xi$
$ \lambda, \lambda', h $ 3 coupling constants
$ \Pi(x_1,x_2) $ 3 self-energy of $ \phi $
$ \widetilde{\Pi}(x_1,x_2) $ 3 correction to the quartic coupling constant $ \lambda $
$ \Pi_{ab}(x_1,x_2) $ 3 self-energy component
$ \kappa(t,\omega),\widetilde{\kappa} (t,\omega)$ 3 Fourier transformation of $ \Pi(t,t') $ and $ \widetilde{\Pi}(t,t') $ with respect to $ t-t' $
$\Pi^R(t_1,t_2)$ 3 retarded self-energy
$\widetilde{\Pi}^R(t_1,t_2)$ 3 retared version of correction to the coupling constant
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|
---
abstract: 'We calculate the spatial distributions and the dynamics of a few-body two-component strongly interacting Bose gas confined to an effectively one-dimensional trapping potential. We describe the densities for each component in the trap for different interaction and population imbalances. We calculate the time evolution of the system and show that, for a certain ratio of interactions, the minority population travels through the system as an effective wave packet.'
author:
- 'R. E. Barfknecht'
- 'A. Foerster'
- 'N. T. Zinner'
date: 'Received: date / Accepted: date'
title: 'Effects of interaction imbalance in a strongly repulsive one-dimensional Bose gas'
---
Introduction {#intro}
============
The theoretical study of one-dimensional models has been pushed forward in recent years due to the possibility of constructing and measuring such systems in experiments with ultracold atoms. Examples of successful applications of these experimental techniques are the realization of the Tonks-Girardeau gas of infinitely repulsive bosons [@weiss2; @paredes] and the observation of a cold atomic system that does not exhibit thermalization [@weiss1]. Furthermore, by manipulating the trapping potential, systems with only a few atoms have been constructed, and their properties have been measured [@jochim1; @jochim2]. As common features, these experiments exhibit the control over the interactions between the atoms, often exploring the strongly correlated regime.
To keep up with these advances, one specific theoretical approach has gained attention: a strongly interaction system of atoms with two or more internal degrees of freedom can be mapped to a spin chain, where exchange coefficients can be calculated from the shape of the trapping potential [@deuretz2; @artem1]. Several works have detailed the static and dynamical properties of these models [@pu1; @artem2; @xiaoling2; @deuretz_mdist; @barfk3]. One interesting application of these tools is the investigation of the fundamental properties of quantum magnetism in models with a small number of particles [@amin2]. Recently, the magnetic correlations of a few-body strongly interacting system of fermions has been measured in the lab [@jochim3].
In this work, we focus on a yet unexplored regime of interactions in a one-dimensional strongly interacting Bose mixture. We show that, assuming an imbalance in the interaction between the two species it is possible to obtain radically different spatial correlations in a few-body system. Furthermore, we describe the dynamics of a mixture with imbalanced population in this interaction regime. We observe that the minority species can travel through the system as an effective spin wave packet in a certain interaction regime.
Hamiltonian {#sec:1}
===========
We consider the problem of a two-component Bose gas with population and interaction imbalance in an effective one-dimensional trapping geometry. We assume that the two different components consist of atoms of a same element, but in different hyperfine states. Therefore, the two bosonic species we take into account have the same mass $m$, but are defined by the states $\vert\uparrow\rangle$ or $\vert\downarrow\rangle$; the number of atoms in each species is respectively given by $N_\uparrow$ and $N_\downarrow$, with the total number of atoms written as $N=N_\uparrow+N_\downarrow$. We are interested in the spatial distributions and the dynamics of an imbalanced mixture where a minority species with $N_\downarrow$ atoms interacts with the remaining $N_\uparrow$ majority atoms. The general Hamiltonian for this system, considering contact interactions, is written as
$$\begin{aligned}
\label{ham1}
H=\sum_{i=1}^{N} H_0(x_i)&+&g_{\uparrow\uparrow}\sum_{i<j}^{N_\uparrow}\delta(x_{\uparrow i}-x_{\uparrow j})\nonumber \\+ g_{\downarrow\downarrow} \sum_{i<j}^{N_\downarrow} \delta(x_{\downarrow i}-x_{\downarrow j}) &+& g_{\uparrow\downarrow} \sum_{i}^{N_\uparrow}\sum_{j}^{N_\downarrow} \delta(x_{\uparrow i}-x_{\downarrow j}),\end{aligned}$$
where $H_0(x)=-\frac{\hbar^2}{2m}\frac{\partial}{\partial x}+V(x)$ is the single particle Hamiltonian with a trapping potential given by $V(x)$. In the following we assume that $\hbar=m=1$. We consider the general case of different intraspecies and interspecies interactions, that is $g_{\uparrow\uparrow}\neq g_{\downarrow\downarrow} \neq g_{\uparrow\downarrow}$. The particular case of $g_{\uparrow\uparrow} = g_{\downarrow\downarrow} = g_{\uparrow\downarrow}$ can be solved (for a homogeneous case) with the Bethe ansatz. Generally, the interactions between the different components can be tuned by means of Feshbach or confinement induced [@feshbach; @olshanii] resonances. For a system with very strong interactions ($g\gg 1$), the Hamiltonian can be mapped (up to linear order in $1/g$) to an effective spin chain. In the most general case of different interactions between the different components, this spin chain is described by [@xiaoling] $$\label{spinchain}
H_{S}=E_0 -\sum_{i=1}^{N-1}\alpha_i\left(\frac{1}{g_{\uparrow\uparrow}} P_{1,1}^{i,i+1}+\frac{1}{g_{\downarrow\downarrow}} P_{1,-1}^{i,i+1}+\frac{1}{g_{\uparrow\downarrow}} P_{1,0}^{i,i+1}\right),$$ where $P_{S,M}$ denotes the projection operator on the two-particle eigenstates with total spin $S$ and magnetization $M$, while $E_0$ is the energy in the limit of infinite repulsion (see below). The exchange coefficients $\alpha_i$ are determined by the trapping geometry and can be thought of as being proportional to the spatial overlap of individual atoms in the trap [@deuretz2; @pu1; @artem1]. Different numerical methods to calculate these coefficients up to large values of $N$ are available and allow for the construction of effective hamiltonians with different trapping potentials [@conan; @deuretz_mdist]. In the next sections, we will focus on a system trapped in the infinite square well, where all coefficients $\alpha_i$ have the same constant value. We point out that the results for the spatial distributions shown in the next section also hold for the case of a harmonic trap. Furthermore, we drop the term $E_0$, which only accounts for a constant energy factor. Since we consider the bosonic case where no interaction between atoms is forbidden by the Pauli principle, the two-body scattering happens in the triplet channel $$\label{triplet}
\vert 1,1\rangle=\vert \uparrow\uparrow\rangle;\,\,\,\,
\vert 1,-1\rangle=\vert \downarrow\downarrow\rangle;\,\,\,\,
\vert 1,0\rangle=\frac{\vert\uparrow\downarrow\rangle+\vert\downarrow\uparrow\rangle}{\sqrt{2}},$$ from which the projectors in Eq. can be built as $$\begin{aligned}
P_{1,1}=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix},\,
P_{1,-1}=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix},\
P_{1,0}=\frac{1}{2}
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}.\end{aligned}$$ Following the same reasoning, spin chains for bosons or fermions with a higher number of internal components can be constructed [@decamp]. By expanding the projection operators in terms of the Pauli matrices [@xiaoling], it is possible to reproduce a XXZ Hamiltonian (for $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}\neq g_{\uparrow\downarrow}$) and the Heisenberg Hamiltonian with ferromagnetic correlations (for $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g_{\uparrow\downarrow}$) [@guan2].
Spatial distributions {#sec:1}
=====================
We now focus on calculating the spatial distributions of a two-component bosonic system with population imbalance and arbitrary combinations of the interactions $g_{\uparrow\uparrow},g_{\downarrow\downarrow}$ and $g_{\uparrow\downarrow}$. We point out that, while the interactions may be different, the spin chain approach only remains valid as long as we have $\left(g_{\uparrow\uparrow},g_{\downarrow\downarrow},g_{\uparrow\downarrow}\right)\gg 1$. The spatial distributions of trapped few-body bosonic mixtures over wide interaction ranges has been studied, for instance, in [@miguel2; @miguel; @bruno]. Here, we calculate the densities for individual atoms in the trap and combine those results with the spin orderings obtained from the eigenstates of Eq. . The [*spin densities*]{} are thus defined as [@deuretz1] $$\label{spin densities}
\rho_{\uparrow,\downarrow}(x)=\sum_{i=1}^{N}\rho^{i}_{\uparrow,\downarrow}(x),$$ where $\rho^i_{\uparrow,\downarrow}=m^{i}_{\uparrow,\downarrow}\rho^{i}(x)$ and $m^{i}_{\uparrow,\downarrow}$ is the magnetization probability at site $i$. The quantity $\rho^{i}(x)$ represent the individual distribution of atoms in the trap, and is given by $$\label{onebody}
\rho^i(x)=N!\int_{\gamma} dx_1...dx_N \,\delta(x_i-x)|\Phi_0(x_1,...,x_i,...,x_N)|^2,$$ where the integrals are performed in the region $x_1<...<x_i<...<x_N$. The wave function $\Phi_0(x_1,...,x_i,...,x_N)$ is a spinless fermion wave function, which is constructed as the Slater determinant of $N$ particles in the potential $V(x)$. In our case, those are simply the $N$ lowest eigenstates of the infinite square-well (the energy $E_0$, present in Eq. \[spinchain\], is the sum of the energies of these orbitals). The spatial distribution obtained from $\Phi_0(x_1,...,x_N)$ thus coincides with the expected results from a Tonks-Girardeau gas, where the inifinite repulsion between the atoms mimics the Pauli exclusion principle [@girardeau]. Exploring the determinant form of $\Phi_0(x_1,...,x_N)$, Eq. can be written as [@deuretz1]
$$\label{dets}
\rho^i(x)=\frac{\partial}{\partial x}\left( \sum_{j=0}^{N-1} c_{j}^{i} \frac{\partial^j}{\partial \lambda^j}\det \left[B(x)-1\lambda \right]\vert_{\lambda=0}\right),$$
where $c_{j}^{i}=\frac{(-1)^{N-1}(N-j-1)!}{(i-1)!(N-j-i)!j!}$ and $B(x)$ is a matrix whose elements are given by the superpositions $b_{mn}(x)=\int_{-\infty}^{x}dy\,\varphi_m(y)\varphi_n(y)$ of single-particle orbitals. Eq. then allows us to avoid the multidimensional integration required for evaluating Eq. .
{width="100.00000%"}
In Fig. \[fig:1\] we show the results for the distributions calculated with Eq. for $N=7$ atoms and different population imbalances: $N_\uparrow =4,N_\downarrow=3$, $N_\uparrow=5,N_\downarrow=2$, and $N_\uparrow=6,N_\downarrow=1$. We also consider different interaction regimes, which we define as: the Heisenberg ferromagnetic (HFM) regime ($g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g_{\uparrow\downarrow}$), the Ising ferromagnetic (IFM) regime ($g_{\uparrow\uparrow}=g_{\downarrow\downarrow}<g_{\uparrow\downarrow}$), the antifferomagnetic regime (AFM) ($g_{\uparrow\uparrow}=g_{\downarrow\downarrow}>g_{\uparrow\downarrow}$) and the completely imbalanced (IMB) regime ($g_{\downarrow\downarrow}<g_{\uparrow\downarrow}<g_{\uparrow\uparrow}$). Each of these cases corresponds to a column in Fig. \[fig:1\]: in the column defined by panels $(a),(e)$ and $(i)$, we observe the results for the HFM regime of identical interactions, where the distributions are the same, only scaled to the number of particles in the specific component. The profiles show ferromagnetic correlations of the Heisenberg type, which are expected for isospin bosons [@guan2]. Column $(b),(f),(j)$ show the IFM case where the intraspecies interaction is smaller than the interspecies interactions. This regime is characterized by a phase separated distribution, typical of an Ising ferromagnet. In column $(c),(g),(k)$ we present the AFM case where the intraspecies interactions are larger than the interspecies interactions. Here, we observe that antiferromagnetic correlations arise, being particularly visible in the slightly imbalanced case of panel $(c)$. Finally, we consider in column $(d),(h),(l)$ a completely imbalanced regime (IMB), where $g_{\downarrow\downarrow}<g_{\uparrow\downarrow}<g_{\uparrow\uparrow}$. Here, we see that we again obtain phase-separated profiles, excpet that now the minority atoms sit in the center of the trap. This happens because the minority intraspecies interactions are smaller and the net result is an effectively attractive regime for this component. We note that these distributions resemble the $XY$ phase described in a Bose-Fermi mixture [@deuretz3]. The evident similarity between the last two panels $(k)$ and $(l)$ results from the fact that in both cases there is only one minority particle. Therefore, the intraspecies interaction in this case has no effect on the minority spin density.
Dynamics {#sec:2}
========
We now turn to the dynamics of the system in the case of population and interaction imbalance. We choose to analyze the two cases of $N_\uparrow=5,N_\downarrow=2$ and $N_\uparrow =4,N_\downarrow=3$, with interactions strength ratios set as $g_{\uparrow\downarrow}=3g_{\downarrow\downarrow}$ and $g_{\uparrow\uparrow}=5g_{\downarrow\downarrow}$. The case of a single minority spin in different interaction regimes and trapping potentials has been studied in [@artem2; @loft1]. We will show that, in the imbalanced regime, non-trivial effects for the dynamics of the minority species can arise. The system is initialized in the state where the minority species is placed at the left side, that is $\vert \psi(0)\rangle=\vert \downarrow_1 ... \downarrow_{N_\downarrow}\uparrow_1 ... \uparrow_{N\uparrow}\rangle$. We disregard any external excitations to the system, such that the dynamics can be described completely by Eq. . The time unit $\tau=g_{\downarrow\downarrow}\pi/\alpha$ is set by the energy scale of the trap through the geometrical coefficient $\alpha$. We calculate the time-dependent overlap of the wave function with a state $\vert s \rangle$ defined by a given spin ordering as $$F_{s}(t)=\vert \langle s \vert e^{-i H_S t}\vert \psi(0)\rangle \vert^2.$$ In Fig. \[fig:2\] we show the results for the cases of $(a)$ $N_\uparrow=5,N_\downarrow=2$ and $(b)$ $N_\uparrow =4,N_\downarrow=3$. We focus only in the cases where $\vert s \rangle$ denotes a state where the minority spins are “bunched up" the system, e.g. $\vert \downarrow\downarrow\uparrow\uparrow\uparrow\uparrow\uparrow\rangle$, $\vert \uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow\rangle$, and so on.
{width="75.00000%"}
We clearly observe in both imbalanced cases that the minority spins, initialized in the left, travel through the system together, with higher amplitudes for the projections over the states with the minority at the edges. We also show the sum of the projections over all states $\vert s \rangle$, which nearly amounts to unity at all times, indicating that other spin orderings have little influence on the dynamics. The physical interpretations for this phenomenon comes from the fact that the minority intraspecies interaction is weaker than the remaining two. Additionally, the repulsion between the majority atoms dominates. This leads to a regime where the minority atoms behave as if under the effect of an attractive force. While this combination of strongly imbalanced interactions may be hard to achieve in practice, it is possible to devise a scenario where the majority strongly repulsive bosons are replaced by a Fermi gas. Bose-Fermi mixures have been studied under the same formalism applied here [@deuretz3], and could present similar dynamical effects.
Conclusions {#conc}
===========
We have studied the effects of imbalance in the spatial correlations and dynamics of a strongly interacting two-component Bose gas. The results for spin densities reproduce the expected behavior in the known regimes of ferromagnetic and antiferromagnetic correlations. Furthermore, the completely imbalanced case can present non-trivial spatial distributions that are not found in other regimes. The dynamics of the system is also affected by the population and interaction imbalance. We demonstrated that, given a certain choice of interaction parameters, a minority population initialized at the edge of the system can move around as if bound by an effectively attractive force. Our work suggests that set ups where the interactions can be tuned between different species may be ideal for studying the transfer of spin packets. Such a realization could provide knowledge of quantum transport in spin chains beyond the case of a single impurity. By generalizing the spin chain approach to a higher number of internal components, it is also possible to consider cases of mixed multicomponent systems with different interactions in several scattering channels.
The authors thank Artem G. Volosniev for feedback on the results. The following agencies - Conselho Nacional de Desenvolvimento Cient[í]{}fico e Tecnol[ó]{}gico (CNPq), the Danish Council for Independent Research DFF Natural Sciences and the DFF Sapere Aude program - are gratefully acknowledged for financial support.
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abstract: 'We show explicitly that supersymmetric $E_6$ Grand Unified Theory with a Higgs sector consisting of $\{27+\overline{27}+351''+\overline{351''}+78\}$ fields provides a realistic scenario for symmetry breaking and fermion mass generation. While gauge symmetry breaking can be achieved without the $78$ field, its presence is critical for a successful doublet-triplet mass splitting. The Yukawa sector of the model consists of only two symmetric matrices describing all of quark, lepton and neutrino masses and mixings. The fermion mass matrices are computed at low energy and a fit to the second and third generation masses and mixings is performed. We find a good numerical fit to the low-energy data. Thus, this model, having $11$ superpotential parameters, alongside the two symmetric Yukawa matrices, seems to be the best realistic candidate for a minimal renormalizable supersymmetric $E_6$ unified theory.'
---
OSU-HEP-15-01
[**A Minimal Supersymmetric $\mathrm{E}_6$ Unified Theory**]{}
K.S. Babu$^{a,}$[^1], Borut Bajc$^{b,}$[^2] and Vasja Susič$^{b,}$[^3]
$^{a}$[*Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA* ]{}
$^{b}$ [*J. Stefan Institute, 1000 Ljubljana, Slovenia*]{}
Introduction
============
There are three types of unified groups (simple groups containing the Standard Model gauge group) that admit chiral representations for fermions: $\SU(N)$ for $N\geq 3$ [@Georgi:1974sy], $\SO(4N+2)$ for $N\ge 2$ and $\EE$ [@Gursey:1975ki]. The first two have been thoroughly studied in many versions, renormalizable as well as non-renormalizable, with and without supersymmetry. Strange enough, $\EE$, the only exceptional group which contains chiral representations, has been mainly ignored over the last forty years. Apart from a few exceptions where symmetry breaking was analyzed (via renormalizable potential with $78$ and two $27$s in Ref. [@Kalashnikov:1979dq] and via renormalizable superpotential with $78$, $27$ and $\overline{27}$ in Ref. [@Buccella:1987kc]), only the Yukawa sectors have been touched upon without using explicit constraints arising from symmetry breaking [@Gursey:1978fu; @Achiman:1978vg; @Shafi:1978gg; @Ruegg:1979fr; @Barbieri:1980vc; @Shaw:1980ku].
Recently, two of us have tried to fill this gap by suggesting a possible minimal renormalizable supersymmetric $\EE$ model [@Bajc:2013qra]. The Higgs sector is composed of pairs of fundamental $27+\overline{27}$ and the two-index symmetric $351'+\overline{351'}$ representations. Much to our surprise we found that, although successful in the symmetry breaking pattern to the Standard Model (SM) gauge group, this sector is unable to provide two light Higgs doublets of the minimal supersymmetric Standard Model (MSSM), or technically said, to perform the doublet-triplet (DT) splitting. The impossibility of DT splitting was due to the identical nature of the expressions for a massless doublet and for a massless triplet; this result was obtained by an explicit computation, and we are not aware of a shortcut explanation of this unusual feature.[^4] To put this result in perspective, consider the minimal renormalizable supersymmetric breaking sectors (able to break to the SM group) for other groups. In the minimal renormalizable $\SU(5)$ superpotential (consisting of the adjoint $24$), there are no SM doublets. In the corresponding minimal $\SO(10)$ case (made out of $210$, $126$ and $\overline{126}$) one has in principle enough doublets, but only one (in $\overline{126}$) is coupled to fermions. In a minimal $\EE$ case with $27+\overline{27}+351'+\overline{351'}$, there are many more doublets which couple to fermions, yet the naive DT splitting fails (a completely different approach to DT splitting in $\mathrm{E}_6$ can be found in [@Maekawa:2002bk]). We thus proposed in [@Bajc:2013qra] to add another $27+\overline{27}$ pair. In this way, the Yukawa sector consisted of three Yukawa matrices: the matter fields (3 copies of $27_F$) have one coupling to the $\overline{351'}$ and a coupling to each of the two Higgs-like $27$s. Though we did not perform it explicitly, we strongly believe that in this model a realistic fit of the Yukawa sector is possible to obtain, but the model is essentially not predictive due to the large number of free parameters in the three Yukawa matrices.
The purpose of this paper is to find an $\EE$ theory with only two Yukawa matrices and thus a simpler Yukawa sector than the model in [@Bajc:2013qra], while keeping the symmetry breaking sector as simple as possible. The idea is to add to the minimal breaking sector (instead of the extra $27+\overline{27}$ pair) another multiplet, which preferably has the following properties:
1. It does not couple to the matter bilinears (and so is not a $27$, $\overline{351}$ or $\overline{351'}$).
2. It contributes to the symmetry breaking vacuum.
3. It increases the size of the mass matrices for weak doublets and/or color triplets.
The minimal such multiplet is the adjoint representation $78$, for which all of the above properties hold true. We will see that its inclusion is enough to allow the doublet-triplet splitting in the theory, thus leading to a theory with the correct (SM) vacuum and the correct (MSSM) low energy field theory.
The Yukawa sector in this model consists of two matrices only. Assuming 3 generations, the down quark and charged lepton sectors have 3 extra vector-like fields, while the neutrino sector has 3 extra vector-like lepton doublet-antidoublet pairs, as well as 6 SM singlets, all coming from the extra fields in the decomposition of the fundamental 27 into the SM subgroup. The projection of all these fields into the usual 3 light generations will be performed explicitly and shown, in the simplified case of only two generations, to provide a realistic fit of the masses and mixing angles.
Special attention needs to be paid to the overall neutrino mass scale. In minimal $\SO(10)$ [@Clark:1982ai; @Aulakh:1982sw; @Aulakh:2003kg] this scale poses a serious problem and eventually rules out the low-energy supersymmetry scenario [@Aulakh:2005bd; @Bajc:2005qe; @Aulakh:2005mw; @Bertolini:2006pe]: a too low $\SU(2)_R$ breaking is disallowed by unification constraints and/or $D=6$ proton decay [@Bajc:2008dc], while a too large breaking scale makes the right-handed neutrinos too heavy and thus the seesaw mechanism ineffective to produce a large enough scale for the light neutrinos. In our model this conclusion is avoidable because there are many possible fields that can break $\SU(2)_R$. While the neutrino mass scale can be adjusted by choosing the vacuum expectation value (VEV) of the $(1,3,10)$ of the Pati-Salam (PS) $\SU(2)_L\times\SU(2)_R\times\SU(4)_C$, located in the $\overline{126}$ of $\SO(10)$, which in turn is found in the $\overline{351'}$ of $\EE$, all other fields’ masses depend also on several other VEVs. We can thus arrange a lighter VEV of this $(1,3,10)$ still avoiding dramatic changes in the renormalization group (RG) gauge running.
We arrange the paper in the following way: we define the model and specify the terms in the superpotential in section \[section:define-model\]. We find a suitable vacuum solution, which breaks $\EE$ into the SM group in section \[section:SSB\], perform the doublet-triplet splitting in section \[section:DT-splitting\] and compute the mass matrices in the Yukawa sector and identify the low energy content of the theory in section \[section:Yukawa-sector\]. We then use the obtained results to perform a $2$ generation fit of the masses and mixing angles in section \[section:numeric-fit\]. We finish with a discussion in section \[section:discussion\]. Five different appendices collect various definitions and technical results. All explicit computation in $\EE$ has been performed using methods from [@Kephart:1981gf] and [@Bajc:2013qra]; also, [@Anderson:1999em; @Anderson:2000ni; @Anderson:2001sd] might also prove useful to the reader.
For ease of use we stick to the following color convention in this paper: denotes VEVs coming from the spontaneous symmetry breaking of the unified theory at the scale $M_{GUT}$, while denotes VEVs coming from electroweak (EW) symmetry breaking at the scale $m_{EW}$.
Defining the model\[section:define-model\]
==========================================
The renormalizable $E_6$ SUSY GUT that we consider here is an alternative to the model in [@Bajc:2013qra]. It is motivated by the fact that the minimal Higgs sector in a renormalizable SUSY $\EE$ model, which can break $\EE$ to the Standard Model, is $351'+\overline{351'}+27+\overline{27}$. This minimal breaking sector cannot accommodate doublet-triplet splitting, however, and therefore needs to be extended to get a realistic model. One possible extension is an addition of a $27+\overline{27}$ pair, which was analyzed in [@Bajc:2013qra], while an alternative, taken in this paper, is to extend it by the representation $78$ instead. The Yukawa sector in the present model will consist of only two matrices, unlike that in [@Bajc:2013qra], which has three matrices.
Our renormalizable $\EE$ model thus contains the following:
- The “fermionic sector” of three copies of a chiral supermultiplet $27$, denoted by $27_F^i$, with $i=1,2,3$. Also, we assume a $\mathbb{Z}_2$ *matter parity*, under which the $27_F$ are $-1$, and the remaining chiral superfields are $+1$. With this symmetry, the ansatz $\langle 27_F\rangle=0$ is consistent with the equations of motion, which we shall adopt.
- The “breaking sector” consisting of $351'+\overline{351'}+27+\overline{27}+78$.
The model under consideration is supersymmetric. The problem of SUSY breaking will not be considered, since it is (usually) an orthogonal problem to breaking the gauge group. One can imagine however, that whatever the detailed mechanism of SUSY breaking might be, we get soft SUSY breaking terms at energies not far above $m_{\textrm{EW}}$, such that we get automatic unification of gauge couplings at $M_{\textrm{GUT}}$ assuming no extra particle states up to the unification scale. The effective low energy theory of our model will thus be MSSM. The soft SUSY breaking terms do not give contributions to the fermion masses at tree level. A fit to fermion masses and mixings can then be done without specifying the details of SUSY breaking, although the RG evolution of these parameters does depend on the SUSY scale (which we assume is of order few TeV).
The full superpotential of our model is $$\begin{aligned}
W_{\textrm{full}}&=m_{27}\; 27\;\overline{27}\;+\;m_{351'}\;351'\;\overline{351'}\nonumber \;+\; m_{78}\;78\;78\\
&\quad+\; \lambda_1\; 351'^3 \;+\; \lambda_2\;\overline{351'}^3 \;+\; \lambda_3\;27^2\;\overline{351'} \;+\; \lambda_4\; \overline{27}^2\;351'\nonumber\\
&\quad+\; \lambda_5\;27^3 \;+\; \lambda_6\;\overline{27}^3 \;+\; \lambda_7\;\overline{27}\;78\;27 \;+\; \lambda_8\;\overline{351'}{}\;78\;351'\nonumber \\
&\quad+\; \tfrac{1}{2}\yd^{ij}\;27_F^i\;27^j_F\;27 \;+\; \tfrac{1}{2}\yt^{ij}\;27_F^i\;27^j_F\;\overline{351'}.\label{equation:W-full}\end{aligned}$$
Note that the definitions of $351'$ and $\overline{351'}$ used here are switched compared to Slansky’s defintions [@Slansky:1981yr].
Spontaneous symmetry breaking\[section:SSB\]
============================================
Equations of motion
-------------------
The Higgs sector $351'+\overline{351'}+27+\overline{27}+78$ forms a realistic Higgs sector, which is able to break the gauge group from $E_6$ to $\SU(3)_C\times\SU(2)_L\times\mathrm{U}(1)_Y$. We provide just such a vacuum below.
First, note the decompositions of $27$, $78$ and $351'$ under $\SO(10)\times \mathrm{U}(1)$: $$\begin{aligned}
27&=16(1)+10(-2)+1(4),\\
78&=45(0)+16(-3)+\overline{16}(3)+1(0),\\
351'&=1(8)+10(2)+16(5)+54(-4)+126(2)+\overline{144}(-1).\end{aligned}$$
The representation $351'$ contains $5$ SM singlets, $3$ of which are $\SU(5)$ singlets (in $1$, $16$ and $126$ of $\SO(10)$), and $2$ are part of a $24$ under $\SU(5)$ (in $54$ and $\overline{144}$ of $\SO(10)$). Similarly, the representation $78$ also has $5$ singlets, with one being a $24$ under $\SU(5)$ (the one in $45$ of $\SO(10)$), while the remaining ones are singlets under $\SU(5)$ (the $1$, $16$ and $\overline{16}$ of $\SO(10)$, as well as another one in the $45$). The Higgs sector $351'+\overline{351'}+27+\overline{27}+78$ therefore contains $5+5+2+2+5=19$ singlets in total. We list their VEVs[^5] in Table \[table:singlet-labels\].
0.2cm
With the ansatz $\langle 27_F\rangle=0$, the breaking part of the superpotential is $$\begin{aligned}
W&=m_{351'}\;I_{351'\times\overline{351'}}+m_{27}\;I_{27\times\overline{27}}+m_{78}\;I_{78\times 78}\nonumber\\
&\qquad + \lambda_1 \;I_{351'^3}+\lambda_2 \;I_{\overline{351'}^3}+\lambda_3 \;I_{27^2\times \overline{351'}}+\lambda_4 \;I_{\overline{27}^2\times 351'}\nonumber\\
&\qquad +\lambda_5 \;I_{27^3} + \lambda_6 \;I_{\overline{27}^3}+\lambda_7\; I_{27\times 78\times \overline{27}}+\lambda_8\; I_{351'\times 78\times \overline{351'}}\label{equation:superpotential-breaking-sector}.\end{aligned}$$
Explicit computation yields the following expressions for the superpotential invariants (VEV terms only): $$\begin{aligned}
I_{351'\times\overline{351'}}&=\overline{351'}_{\mu\nu}\;351^{\mu\nu}=\RED{e_1} \RED{f_1}+\RED{e_2} \RED{f_2}+\RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}+\RED{e_5} \RED{f_5},\label{equation:computed-invariant-first}\\
I_{27\times\overline{27}}&=\overline{27}_\mu\; 27^\mu=\RED{c_1} \RED{d_1}+\RED{c_2} \RED{d_2},\\
I_{78\times 78}&=78^{\mu}{}_{\nu}\;78^{\nu}{}_{\mu}=2 \RED{u_1} \RED{u_2} + \RED{w}^2+\RED{v}^2+\RED{y}^2,\\
I_{351'^3}&=351'^{\mu\alpha}\;351'^{\nu\beta}\;351'^{\lambda\gamma}\;d_{\alpha\beta\gamma}d_{\mu\nu\lambda}=3 \left(\RED{e_3} \RED{e_4}^2+\RED{e_1} \RED{e_5}^2 - \sqrt{2} \RED{e_2} \RED{e_4} \RED{e_5} \right),\\
I_{\overline{351'}^3}&=\overline{351'}_{\mu\alpha}\;\overline{351'}_{\nu\beta}\;\overline{351'}_{\lambda\gamma}\;d^{\alpha\beta\gamma}\;d^{\mu\nu\lambda}=3 \left(\RED{f_3} \RED{f_4}^2+\RED{f_1} \RED{f_5}^2 - \sqrt{2}\RED{f_2} \RED{f_4} \RED{f_5}\right),\\
I_{27^2\times\overline{351'}}&=\overline{351'}_{\mu\nu}\;27^\mu\;27^\nu=\RED{c_2}^2 \RED{f_1} + \sqrt{2} \RED{c_1} \RED{c_2} \RED{f_2} + \RED{c_1}^2 \RED{f_3},\\
I_{\overline{27}^2\times 351'}&=351'^{\mu\nu}\;\overline{27}_\mu\;\overline{27}_\nu=\RED{d_2}^2 \RED{e_1} + \sqrt{2} \RED{d_1} \RED{d_2} \RED{e_2} + \RED{d_1}^2 \RED{e_3},\\
I_{27^3}&=27^\mu\;27^\nu\;27^\lambda\;d_{\mu\nu\lambda}=0,\\
I_{\overline{27}^3}&=\overline{27}_\mu\;\overline{27}_\nu\;\overline{27}_\lambda\;d^{\mu\nu\lambda}=0,\\
I_{27\times 78\times \overline{27}}&=\overline{27}_{\mu}\;78^{\mu}{}_{\nu}\;27^\nu\nonumber\\
&=
\tfrac{1}{\sqrt{6}}\RED{u_1}\RED{c_1}\RED{d_2} + \tfrac{1}{\sqrt{6}}\RED{u_2} \RED{c_2} \RED{d_1}-\tfrac{1}{6\sqrt{2}}\RED{w}\left(4 \RED{c_1}\RED{d_1}+\RED{c_2}\RED{d_2}\right)-\tfrac{1}{2} \sqrt{\tfrac{5}{6}} \RED{v} \RED{c_2}\RED{d_2},\\
I_{351'\times 78\times \overline{351'}}&=\overline{351'}_{\mu\nu}\;78^{\nu}{}_{\kappa}\;351'^{\kappa\mu}=\nonumber\\
&=\tfrac{1}{2\sqrt{6}}\RED{u_1}\left(\sqrt{2} \RED{e_2} \RED{f_1}+\sqrt{2} \RED{e_3} \RED{f_2}+\RED{e_5} \RED{f_4}\right)+
\tfrac{1}{2\sqrt{6}}\RED{u_2}\left(\sqrt{2} \RED{e_1} \RED{f_2}+\sqrt{2} \RED{e_2} \RED{f_3}+\RED{e_4} \RED{f_5}\right)\nonumber\\
&\quad +
\tfrac{1}{12\sqrt{2}}\RED{w}\left(-2 \RED{e_1} \RED{f_1}-5 \RED{e_2} \RED{f_2}-8 \RED{e_3} \RED{f_3}+4 \RED{e_4} \RED{f_4}+\RED{e_5} \RED{f_5}\right)
-\tfrac{1}{4} \sqrt{\tfrac{5}{6}} \RED{v}\left(2 \RED{e_1} \RED{f_1}+\RED{e_2} \RED{f_2}-\RED{e_5} \RED{f_5}\right).\label{equation:computed-invariant-last}
\end{aligned}$$
The general formula for $D$-terms in our case yields
$$\begin{aligned}
D^A&= \phantom{+}(27^\dagger)_\mu\;(\hat{t}^A\, 27)^\mu+(\overline{27}^\dagger)^\mu\;(\hat{t}^A\, \overline{27})_\mu+(78^\dagger)^{\nu}{}_{\mu}\;(\hat{t}^A\, 78)^{\mu}{}_{\nu}\nonumber\\
&\quad+(351'^\dagger)_{\mu\nu}\;(\hat{t}^A\, 351')^{\mu\nu}+(\overline{351'}^\dagger)^{\mu\nu}\;(\hat{t}^A\, \overline{351'})_{\mu\nu}.\end{aligned}$$
Of the $78$ $D$-terms, $5$ vanish non-trivially, corresponding to the following generators of the $\SU(3)_C\times\SU(3)_L\times\SU(3)_R$ subgroup of $\EE$: $t_L^8$, $t_R^3$, $t_R^6$, $t_R^7$, $t_R^8$, given explicitly by
$$\begin{aligned}
D_L^8&=\tfrac{1}{\sqrt{3}}\left(|\RED{c_1}|^2+|\RED{c_2}|^2+2 |\RED{e_1}|^2 +2 |\RED{e_2}|^2 +2 |\RED{e_3}|^2 -|\RED{e_4}|^2 -|\RED{e_5}|^2\right.\nonumber\\
&\qquad \left.-|\RED{d_1}|^2 -|\RED{d_2}|^2-2 |\RED{f_1}|^2 -2 |\RED{f_2}|^2 -2 |\RED{f_3}|^2 +|\RED{f_4}|^2 +|\RED{f_5}|^2\right),\\
D_R^3&=\tfrac{1}{6}\left(
-3 |\RED{c_2}|^2 -6 |\RED{e_1}|^2 -3 |\RED{e_2}|^2 +3 |\RED{e_5}|^2 -|\RED{u_1}|^2 \right.\nonumber\\
&\qquad\left.+3 |\RED{d_2}|^2 +6 |\RED{f_1}|^2 +3 |\RED{f_2}|^2 -3 |\RED{f_5}|^2 +|\RED{u_2}|^2\right),\\
D_R^6&=
\tfrac{1}{12} \left(
6 \RED{c_2} \RED{c_1}^\ast+6 \RED{c_1} \RED{c_2}^\ast
-\sqrt{3} \RED{u_1} \RED{w}^\ast-\sqrt{3}\RED{w} \RED{u_1}^\ast
+\sqrt{5} \RED{u_1} \RED{v}^\ast +\sqrt{5} \RED{v} \RED{u_1}^\ast\right.\nonumber\\
&\qquad\left.-6 \RED{d_2} \RED{d_1}^\ast-6 \RED{d_1} \RED{d_2}^\ast
+\sqrt{3} \RED{u_2} \RED{w}^\ast+\sqrt{3}\RED{w} \RED{u_2}^\ast
-\sqrt{5}\RED{u_2} \RED{v}^\ast -\sqrt{5} \RED{v} \RED{u_2}^\ast\right.\nonumber\\
&\qquad\left.+6\sqrt{2} \RED{e_1} \RED{e_2}^\ast+6\sqrt{2}\RED{e_2} \RED{e_1}^\ast
+6\sqrt{2} \RED{e_2} \RED{e_3}^\ast+6\sqrt{2}\RED{e_3} \RED{e_2}^\ast
+6 \RED{e_4} \RED{e_5}^\ast+6 \RED{e_5} \RED{e_4}^\ast\right.\nonumber\\
&\qquad \left. -6\sqrt{2} \RED{f_1}\RED{f_2}^\ast-6\sqrt{2}\RED{f_2}\RED{f_1}^\ast
-6\sqrt{2} \RED{f_2}\RED{f_3}^\ast-6\sqrt{2}\RED{f_3}\RED{f_2}^\ast
-6 \RED{f_4} \RED{f_5}^\ast-6 \RED{f_5} \RED{f_4}^\ast \right),\\
D_R^7&=
\tfrac{i}{12} \left(
6 \RED{c_2} \RED{c_1}^\ast-6 \RED{c_1} \RED{c_2}^\ast
-\sqrt{3} \RED{u_1} \RED{w}^\ast+\sqrt{3}\RED{w} \RED{u_1}^\ast
+\sqrt{5} \RED{u_1} \RED{v}^\ast -\sqrt{5} \RED{v} \RED{u_1}^\ast\right.\nonumber\\
&\qquad\left.+6 \RED{d_2} \RED{d_1}^\ast-6 \RED{d_1} \RED{d_2}^\ast
-\sqrt{3} \RED{u_2} \RED{w}^\ast+\sqrt{3}\RED{w} \RED{u_2}^\ast
+\sqrt{5}\RED{u_2} \RED{v}^\ast -\sqrt{5} \RED{v} \RED{u_2}^\ast\right.\nonumber\\
&\qquad\left.+6\sqrt{2} \RED{e_1} \RED{e_2}^\ast-6\sqrt{2}\RED{e_2} \RED{e_1}^\ast
+6\sqrt{2} \RED{e_2} \RED{e_3}^\ast-6\sqrt{2}\RED{e_3} \RED{e_2}^\ast
+6 \RED{e_4} \RED{e_5}^\ast-6 \RED{e_5} \RED{e_4}^\ast\right.\nonumber\\
&\qquad \left. +6\sqrt{2} \RED{f_1}\RED{f_2}^\ast-6\sqrt{2}\RED{f_2}\RED{f_1}^\ast
+6\sqrt{2} \RED{f_2}\RED{f_3}^\ast-6\sqrt{2}\RED{f_3}\RED{f_2}^\ast
+6 \RED{f_4} \RED{f_5}^\ast-6 \RED{f_5} \RED{f_4}^\ast \right),\\
D_R^8&=\tfrac{1}{2\sqrt{3}}\left(-2 |\RED{c_1}|^2 +|\RED{c_2}|^2 +2 |\RED{e_1}|^2 -|\RED{e_2}|^2 -4 |\RED{e_3}|^2 +2 |\RED{e_4}|^2 -|\RED{e_5}|^2 +|\RED{u_1}|^2\right.\nonumber\\
&\qquad\quad\left. +2 |\RED{d_1}|^2 -|\RED{d_2}|^2 -2 |\RED{f_1}|^2 +|\RED{f_2}|^2 +4 |\RED{f_3}|^2 -2 |\RED{f_4}|^2 +|\RED{f_5}|^2 -|\RED{u_2}|^2
\right).\end{aligned}$$
They can be rewritten into $3$ independent $D$-terms:
$$\begin{aligned}
D^{I}&=|\RED{c_1}|^2 - |\RED{d_1}|^2+ |\RED{e_2}|^2 -|\RED{f_2}|^2 +2 |\RED{e_3}|^2 -2 |\RED{f_3}|^2 -|\RED{e_4}|^2 +|\RED{f_4}|^2 -\tfrac{1}{3}|\RED{u_1}|^2+\tfrac{1}{3}|\RED{u_2}|^2,\label{equation:D-term-summary-first}\\
D^{II}&=|\RED{c_2}|^2-|\RED{d_2}|^2+|\RED{e_2}|^2-|\RED{f_2}|^2+2 |\RED{e_1}|^2-2 |\RED{f_1}|^2-|\RED{e_5}|^2+|\RED{f_5}|^2-\tfrac{1}{3}|\RED{u_2}|^2+\tfrac{1}{3}|\RED{u_1}|^2,\\
D^{III}&=+
\RED{c_1} \RED{c_2}^\ast-\tfrac{\sqrt{3}}{6} \RED{w} \RED{u_1}^\ast+\tfrac{\sqrt{5}}{6} \RED{v} \RED{u_1}^\ast
+\sqrt{2} \RED{e_2}\RED{e_1}^\ast+\sqrt{2} \RED{e_3}\RED{e_2}^\ast+\RED{e_5}\RED{e_4}^\ast\nonumber\\
&\quad - \RED{d_2}\RED{d_1}^\ast+\tfrac{\sqrt{3}}{6} \RED{u_2} \RED{w}^\ast-\tfrac{\sqrt{5}}{6} \RED{u_2} \RED{v}^\ast
-\sqrt{2} \RED{f_1}\RED{f_2}^\ast-\sqrt{2}\RED{f_2}\RED{f_3}^\ast-\RED{f_4}\RED{f_5}^\ast,\label{equation:D-term-summary-last}\end{aligned}$$
via the definitions $D^I:=\sqrt{3}D_L^8+2D_R^3$, $D^{II}:=-2D_R^3$ and $D^{III}:=D_R^6+i D_R^7$, where $D^{III}$ now forms a complex equation. The other independent combination $D^Y=D_L^8+\sqrt{3}D_R^3+D_R^8$ is trivially zero, as it should be, since this $D$-term corresponds to the unbroken generator of $\mathrm{U}(1)_Y$.
A specific vacuum solution
--------------------------
In this section, we obtain a vacuum solution which breaks the gauge group to the Standard Model group $\SU(3)_C\times\SU(2)_L\times\mathrm{U}(1)_Y$. Due to the complexity of the equations we are unable to provide a full classification of vacua; a short discussion on alternative vacua can be found in Appendix \[section:vacuum2\].
The equations of motion in SUSY models are $$\begin{aligned}
D^a&=0,\\
F_\phi&=0,\end{aligned}$$ with the usual definition of the $F$-term: $$\begin{aligned}
F_\phi:=\frac{\partial W}{\partial \phi}.\end{aligned}$$
In our case, there are $19$ Standard Model singlets in the Higgs sector, giving $19$ non-trivial $F$-terms, which can be easily reconstructed from the superpotential in equation and the all-singlet terms of the invariants given by equations –. The non-trivial $D$-terms are given by equations –.
To obtain a vacuum solution, we perform the following steps:
- First, we notice that $F_y$ leads directly to $y=0$ ($y$ is present only in the mass term $78^2$). By taking the self-consistent ansatz[^6] $$\begin{aligned}
c_1=d_1=f_5=e_5&=0,\label{equation:ansatz-first}\\
u_1=u_2=e_2=f_2&=0,\label{equation:ansatz-last}
\end{aligned}$$
the system of equations is greatly simplified: $F_{c_1},F_{d_1},F_{e_2},F_{f_2},F_{e_5},F_{f_5},F_{u_1},F_{u_2},F_y$ and $D^{III}$ are solved automatically.
- Solve $F_{e_3}$ and $F_{f_3}$ for $f_3$ and $e_3$, respectively to get $$\begin{aligned}
e_3&=-\frac{9 \lambda_2 \RED{f_4}^2}{3 m_{351'}-\sqrt{2} \lambda_8 \RED{w}},&
f_3&=-\frac{9 \lambda_1 \RED{e_4}^2}{3 m_{351'}-\sqrt{2} \lambda_8 \RED{w}}.
\end{aligned}$$
- Solve $F_{c_2}$ and $F_{d_2}$ for $f_1$ and $e_1$, respectively to get $$\begin{aligned}
e_1&=\frac{\RED{c_2}}{24 \lambda_4 \RED{d_2}}\left(\sqrt{2} \lambda_7 \left(\sqrt{15} \RED{v}+\RED{w}\right)-12 m_{27} \right),\\
f_1&=\frac{\RED{d_2}}{24 \lambda_3 \RED{c_2}}\left(\sqrt{2} \lambda_7 \left(\sqrt{15} \RED{v}+\RED{w}\right)-12 m_{27} \right).
\end{aligned}$$
- Simultaneously solve $F_{e_4}$ and $F_{f_4}$ for $f_4$: $$\begin{aligned}
f_4&=\frac{(3 \sqrt{2} m_{351'}-2 \lambda_8 \RED{w})(3 \sqrt{2} m_{351'}+\lambda_8\RED{w})}{324 \RED{e_4} \lambda _1 \lambda _2}.
\end{aligned}$$
- Simultaneously solve $F_{e_1}$ and $F_{f_1}$ for $d_2$: $$\begin{aligned}
d_2&=\frac{\big(6\sqrt{2}m_{27}-\lambda_7(\sqrt{15}\RED{v}+\RED{w})\big)\big(6\sqrt{2}m_{351'}-\lambda_8 (\sqrt{15}\RED{v}+\RED{w})\big)}{144 \lambda_3 \lambda_4 \RED{c_2}}.
\end{aligned}$$
- It is now convenient to define a new quantity $A:=\sqrt{15} v+ w$. We can now solve $F_{v}$ for $v$ as a linear equation: $$\begin{aligned}
v&=\frac{\sqrt{10} \RED{A}^2 \lambda_7^2 \lambda_8-8 \sqrt{5} \lambda_7\RED{A} (m_{351'}\lambda_7 +2 m_{27}\lambda_8)+24 \sqrt{10} m_{27} (2 m_{351'} \lambda_7+m_{27} \lambda_8)}{768 \sqrt{3} m_{78} \lambda_3 \lambda_4}.
\end{aligned}$$
- Three variables remain to be determined: $A$, $c_2$ and $e_4$. We are left with only one unsolved $F$-term $F_{w}$, which is a polynomial in $A$: $$\begin{aligned}
0&=P_0+P_1 \RED{A}+P_2 \RED{A}^2+P_3 \RED{A}^3+P_4 \RED{A}^4,\\
P_0&=-576 m_{27} \left(2 m_{351'} \lambda_7+m_{27} \lambda_8\right)\Big(25 m_{27} \left(2 m_{351'} \lambda_7+m_{27} \lambda_8\right) \lambda_8^3\nonumber\\
&\qquad -480 m_{351'} m_{78} \lambda_3
\lambda_4 \lambda_8^2+110592 m_{78}^2 \lambda_1 \lambda_2 \lambda_3 \lambda_4\Big),\\
P_1&=192\sqrt{2}\Big(995328 \lambda_1 \lambda_2 \lambda_3^2 \lambda_4^2 m_{78}^3\nonumber\\
&\qquad +4608 \lambda_3 \lambda_4 \left(24 m_{27} \lambda_1 \lambda_2 \lambda_7 \lambda_8+m_{351'} \left(12 \lambda
_1 \lambda_2 \lambda_7^2-\lambda_3 \lambda_4 \lambda_8^2\right)\right) m_{78}^2\nonumber\\
&\qquad +240 \lambda_3 \lambda_4 \lambda_8^2 \left(-m_{351'}^2 \lambda_7^2+2 m_{351'} m_{27} \lambda
_8 \lambda_7+2 m_{27}^2 \lambda_8^2\right) m_{78}\nonumber\\
&\qquad+25 m_{27} \lambda_7 \lambda_8^3 \left(2 m_{351'}^2 \lambda_7^2+5 m_{351'} m_{27} \lambda_8 \lambda_7+2 m_{27}^2 \lambda_8^2\right)\Big),\\
P_2&=-16\Big(\lambda _8 \left(18432 \lambda _3 \lambda _4 \left(9 \lambda _1 \lambda _2 \lambda _7^2+\lambda _3 \lambda _4 \lambda _8^2\right) m_{78}^2\right.\nonumber\\
&\qquad \left.+240 \lambda _3 \lambda _4 \lambda
_7 \lambda _8^2 \left(5 m_{351'} \lambda _7+16 m_{27} \lambda _8\right) m_{78}\right.\nonumber\\
&\qquad \left.+25 \lambda _7^2 \lambda _8^2 \left(2 m_{351'}^2 \lambda _7^2+14 m_{351'} m_{27} \lambda _8 \lambda
_7+11 m_{27}^2 \lambda _8^2\right)\right)\Big),\\
P_3&=40\sqrt{2}\lambda _7^2 \lambda _8^4 \left(96 m_{78} \lambda _3 \lambda _4+5 \lambda _7 \left(m_{351'} \lambda _7+2 m_{27} \lambda _8\right)\right),\\
P_4&=-25\lambda _7^4 \lambda _8^5.
\end{aligned}$$
Note that the coefficients $P_i$ depend only on the Lagrangian parameters; choosing those, we can determine $A$ numerically.
Finally, we solve the remaining $D$-terms and determine $c_2$ and $e_4$ from $$\begin{aligned}
0&=D^{I}=2 |\RED{e_3}|^2-|\RED{e_4}|^2-2 |\RED{f_3}|^2+|\RED{f_4}|^2,\\
0&=D^{II}=|\RED{c_2}|^2-|\RED{d_2}|^2+2 |\RED{e_1}|^2-2 |\RED{f_1}|^2.\end{aligned}$$ Note that $v$ and $w$ are determined, once $A$ is fixed. We can therefore see that $f_4\propto 1/e_4$, $e_3\propto f_4^2\propto 1/e_4^2$ and $f_3\propto e_4^2$. Therefore $D^{I}$ can be written as a quartic polynomial in $|e_4|^2$; the constant term has a positive coefficient (the $e_3$ term), while the highest order term in $|e_4|^2$ has a negative coefficient, a solution for a real $e_4>0$ will always exist. Similarly, $d_2\propto 1/c_2$, $e_1\propto c_2/d_2\propto c_2^2$ and $f_1\propto d_2/c_2\propto 1/c_2^2$, $D^{II}$ is a quartic polynomial in $c_2$ independent of $e_4$; the constant coefficient will be negative (the $f_1$ term), while the highest order coefficient in $|c_2|^2$ is positive (the $e_1$ term), which again guarantees a real solution $c_2>0$.
The initial Lagrangian parameters constitute of the $3$ masses $m_{27}$, $m_{351'}$, $m_{78}$ and $8$ massless parameters $\lambda_i$, $i=1,\ldots,8$. The simplest order in which to compute the given vacuum solution with these parameters is given below:
1. Take the ansatz for some of the VEVs (symmetric under conjugation symmetry [@Bajc:2013qra]) $$\begin{aligned}
c_1&=0,&d_1&=0,\non\\
e_2&=0,&f_2&=0, \non\\
e_5&=0,&f_5&=0, \non\\
u_1&=0,&u_2&=0, \non\\
y&=0.&&\end{aligned}$$
2. $A$ is determined through the polynomial, and then the VEVs $v$ and $w$ are determined by $$\begin{aligned}
v&=\frac{\sqrt{10} \RED{A}^2 \lambda_7^2 \lambda_8-8 \sqrt{5} \lambda_7\RED{A} (m_{351'}\lambda_7 +2 m_{27}\lambda_8)+24 \sqrt{10} m_{27} (2 m_{351'} \lambda_7+m_{27} \lambda_8)}{768 \sqrt{3} m_{78} \lambda_3 \lambda_4},\\
w&=\RED{A}+\frac{5}{768 m_{78} \lambda_3 \lambda_4}\left(-24 \sqrt{2} m_{27} (2 m_{351'} \lambda_7+m_{27}
\lambda_8)\right.\nonumber\\
&\qquad \left.+\lambda_7\RED{A}(8 m_{351'} \lambda_7+16 m_{27}
\lambda_8-\sqrt{2} \RED{A} \lambda_7 \lambda_8)\right).\end{aligned}$$
3. $e_4$ and $c_2$ are determined through $D^{I}$ and $D^{II}$, respectively, using equations –.
4. The remaining nonvanishing VEVs are $d_2,e_1,f_1,e_3,f_3,f_4$, and they can be computed in terms of $A,w,c_2,e_4$: $$\begin{aligned}
d_2&=\frac{1}{144 \lambda_3 \lambda_4 \RED{c_2}}\left(6\sqrt{2}m_{27}-\lambda_7\RED{A}\right)\left(6\sqrt{2}m_{351'}-\lambda_8 \RED{A}\right),\label{equation:VEVan-1}\\
e_1&=\frac{12 \lambda_3 \RED{c_2}^2}{\sqrt{2} \text{\RED{A}} \lambda_8-12 m_{351'}},\\
f_1&=\frac{\sqrt{2} \RED{A}\lambda_7-12 m_{27}}{3456 \lambda_3^2 \lambda_4
\RED{c_2}^2}\left(6\sqrt{2}m_{27}-\lambda_7\RED{A}\right)\left(6\sqrt{2}m_{351'}-\lambda_8 \RED{A}\right),\\
e_3&=\frac{\left(3 \sqrt{2} m_{351'}-2 \lambda_8 \RED{w}\right) \left(3 \sqrt{2} m_{351'}+\lambda_8\RED{w}\right)^2}{5832 \sqrt{2} \RED{e_4}^2 \lambda_1^2 \lambda_2},\\
f_3&=-\frac{9 \RED{e_4}^2 \lambda _1}{3 m_{351'}-\sqrt{2} \RED{w} \lambda _8},\\
f_4&=\frac{1}{324 \RED{e_4} \lambda _1 \lambda _2}\left(3 \sqrt{2} m_{351'}-2 \lambda_8 \RED{w}\right)\left(3 \sqrt{2} m_{351'}+\lambda_8\RED{w}\right).\label{equation:VEVan-5}\end{aligned}$$
The solution above does indeed break $\EE$ into the Standard Model group. This can be checked by explicitly computing the gauge boson masses, found in Table \[table:gauge-boson-masses\] of Appendix \[Gauge-Boson-masses\]. It is possible to further illuminate this result by considering that under the standard embeddings of the $\EE$ subgroups from Table \[table:singlet-labels\], the $\SU(5)$ breaking in our solution is solely due to non-vanishing VEVs $e_4$ and $f_4$.
Doublet-triplet splitting\[section:DT-splitting\]
=================================================
We tackle now the issue of doublet-triplet splitting. We denote the doublets and antidoublets by $D\sim (1,2,+1/2)$ and $\overline{D}\sim (1,2,-1/2)$, while the triplets and antitriplets are denoted by $T\sim (3,1,-1/3)$ and $\overline{T}\sim (\bar{3},1,+1/3)$. The detailed labels of these states are given in Table \[table:labels-doublets-triplets\] of Appendix \[Particle-identification\].
In the Higgs sector of our model, there are $12$ doublet-antidoublet pairs and $13$ triplet-antitriplet pairs. This is one extra pair of each compared to the renormalizable model with the Higgs sector $351'+\overline{351'}+27+\overline{27}$, in which doublet-triplet splitting surprisingly fails (see [@Bajc:2013qra] for details). The extra states come from the added $78$, and are labeled by the index $0$, i.e. $D_0,\overline{D}_0,T_0,\overline{T}_0$. We shall see that this extra row and column, together with a new vacuum compared to the model without $78$ now enable doublet-triplet splitting in the usual way (by fine-tuning). Note that all the doublets and triplets are located in a $5$ or $45$ (or their conjugates) of $\SU(5)$, except for one extra triplet in the $50$ of $\SU(5)$.
The mass terms for the doublets and triplets are written as
$$\begin{aligned}
\begin{pmatrix}D_0&\cdots& D_{11}\\\end{pmatrix}
\mathcal{M}_{\textrm{doublets}}
\begin{pmatrix}\overline{D}_1\\\vdots \\\overline{D}_{11}\\\end{pmatrix}+
\begin{pmatrix}T_0&\cdots& T_{12}\\\end{pmatrix}
\mathcal{M}_{\textrm{triplets}}
\begin{pmatrix}\overline{T}_1\\\vdots \\\overline{T}_{12}\\\end{pmatrix}.\end{aligned}$$
The matrices $\mathcal{M}_{\textrm{doublets}}$ and $\mathcal{M}_{\textrm{triplets}}$ are similar; we can compactly write a $13\times 13$ matrix $\mathcal{M}$ with block form $$\begin{aligned}
\mathcal{M}&=
\begin{pmatrix}
M_{11}&M_{12}\\
M_{21}&M_{22}\\
\end{pmatrix},\label{equation:DT-matrix}\end{aligned}$$ with the diagonal blocks defined by $$\begin{aligned}
M_{11}&=\scalebox{0.7}{$
\left(\begin{smallmatrix}
m_{78} & 0 & \frac{\lambda_{7} \RED{d_2}}{2 \sqrt{3}} & 0 & 0 & 0 \\
0 & m_{27}+\frac{\lambda_{7} (\sqrt{3}\RED{v}+\sqrt{5}\RED{w})}{3\sqrt{10}}& \frac{\alpha \lambda_{3} \RED{f_4}}{\sqrt{15}} & 6 \lambda_{5} \RED{c_2} & 0 & 0\\
\frac{\lambda_{7} \RED{c_2}}{2 \sqrt{3}} & \frac{\alpha \lambda_{4} \RED{e_4}}{\sqrt{15}} & m_{27}+\frac{\lambda_{7} (\sqrt{5}\RED{w}-\sqrt{3}\RED{v})}{3\sqrt{10}} & 0 & 0 & 0 \\
0 & 6 \lambda_{6} \RED{d_2} & 0 & m_{27}+\frac{\lambda_7 (3\sqrt{3}\RED{v}-\sqrt{5}\RED{w})}{6\sqrt{10}}& 0 & -\frac{\lambda_{4} \RED{d_2}}{\sqrt{10}} \\
0 & 0 & 0 & 0 & m_{351'}+\frac{\lambda_{8} \RED{v}}{2 \sqrt{30}}-\frac{\lambda_{8} \RED{w}}{6 \sqrt{2}} & \sqrt{\frac{3}{5}} \alpha \lambda_{1} \RED{e_4} \\
0 & 0 & 0 & -\frac{\lambda_{3} \RED{c_2}}{\sqrt{10}} & \sqrt{\frac{3}{5}} \alpha \lambda_{2} \RED{f_4} & m_{351'}-\frac{\lambda_{8} \RED{v}}{2 \sqrt{30}}-\frac{\lambda_{8} \RED{w}}{6 \sqrt{2}}\\
\end{smallmatrix}\right)$},\\
M_{22}&=\scalebox{0.65}{$
\left(\begin{smallmatrix}
m_{351'}+\frac{\sqrt{3}}{4\sqrt{10}} \lambda_{8} \RED{v}-\frac{5 \lambda_{8} \RED{w}}{12 \sqrt{2}} & 0 & 0 & \frac{1}{2} \sqrt{3} \alpha \lambda_{2} \RED{f_4} & 0 & \frac{1}{2} \sqrt{15} \beta
\lambda_{2} \RED{f_4} & 0 \\
0 & m_{351'}-\frac{\lambda_{8} \RED{v}}{2 \sqrt{30}}-\frac{\lambda_{8} \RED{w}}{6 \sqrt{2}} & \sqrt{5} \beta \lambda_{2} \RED{f_4} & 0 & 0 & 0 & 0 \\
0 & \sqrt{5} \beta \lambda_{1} \RED{e_4} & m_{351'}+\frac{\lambda_{8} \RED{v}}{2 \sqrt{30}}-\frac{\lambda_{8} \RED{w}}{6 \sqrt{2}} & 0 & 0 & 0 & 2 \sqrt{10} \lambda_{1} \RED{e_4} \\
\frac{1}{2} \sqrt{3} \alpha \lambda_{1} \RED{e_4} & 0 & 0 & m_{351'}-\frac{\sqrt{3}}{4\sqrt{10}}\lambda_{8} \RED{v}+\frac{\lambda_{8} \RED{w}}{12
\sqrt{2}} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & m_{351'}-\frac{7 \lambda_{8} \RED{v}}{4 \sqrt{30}}+\frac{\lambda_{8} \RED{w}}{12 \sqrt{2}} & 0 & 0 \\
\frac{1}{2} \sqrt{15} \beta \lambda_{1} \RED{e_4} & 0 & 0 & 0 & 0 & m_{351'}-\frac{\sqrt{3}}{4\sqrt{10}}\lambda_{8} \RED{v}+\frac{\lambda_{8} \RED{w}}{12 \sqrt{2}} & 0 \\
0 & 0 & 2 \sqrt{10} \lambda_{2} \RED{f_4} & 0 & 0 & 0 & m_{351'}-\frac{\lambda_{8} \RED{v}}{2 \sqrt{30}}-\frac{\lambda_{8} \RED{w}}{6 \sqrt{2}} \\
\end{smallmatrix}\right)$},\end{aligned}$$ and the off-diagonal blocks defined by $$\begin{aligned}
M_{12}&=\scalebox{0.9}{$
\left(\begin{smallmatrix}
-\frac{\lambda_{8} \RED{f_3}}{2 \sqrt{6}} & 0 & 0 & \frac{\alpha \lambda_{8} \RED{e_4}}{24 \sqrt{2}} & -\frac{\lambda_{8} \RED{f_1}}{2 \sqrt{6}} & \frac{\sqrt{5}\beta \lambda_{8} \RED{e_4}}{24\sqrt{2}} & 0 \\
0 & 0 & 0 & 2 \sqrt{\frac{2}{5}} \lambda_{3} \RED{c_2} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -\sqrt{2} \lambda_{4} \RED{d_2} & 0 & 0 \\
0 & -\sqrt{\frac{3}{2}} \lambda_{4} \RED{d_2} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{smallmatrix}\right)$},\\
M_{21}&=\scalebox{0.9}{$
\left(\begin{smallmatrix}
-\frac{\lambda_{8} \RED{e_3}}{2 \sqrt{6}} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -\sqrt{\frac{3}{2}} \lambda_{3} \RED{c_2} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
\frac{\alpha \lambda_{8} \RED{f_4}}{24 \sqrt{2}} & 2 \sqrt{\frac{2}{5}} \lambda_{4} \RED{d_2} & 0 & 0 & 0 & 0\\
-\frac{\lambda_{8} \RED{e_1}}{2 \sqrt{6}} & 0 & -\sqrt{2} \lambda_{3} \RED{c_2} & 0 & 0 & 0\\
\frac{\sqrt{5}\beta \lambda_{8} \RED{f_4}}{24\sqrt{2}} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right) $}.\end{aligned}$$
The separation of the matrix $\mathcal{M}$ into $4$ blocks is arbitrary and is used above merely as a simple way of presenting a large matrix. The matrix $\mathcal{M}_{\textrm{doublets}}$ is obtained out of $\mathcal{M}$ by removing the last row and column, and taking $\alpha=-3$, $\beta=-\sqrt{3}$, while the matrix $\mathcal{M}_{\textrm{triplets}}$ is obtained by taking $\alpha=\beta=2$. Note that the matrices were already simplified by taking the vacuum ansatz of vanishing VEVs $$\begin{aligned}
c_1=d_1=e_2=f_2=e_5=f_5=u_1=u_2=y&=0,\end{aligned}$$ while the rest of the vacuum solution was not plugged-in. Notice also that the coefficients $\alpha$ and $\beta$ are located as factors in front of $e_4$ or $f_4$; this is expected, since $e_4$ and $f_4$ are the $\SU(5)$-breaking VEVs, so they control the difference between the doublets and triplets. Also, the coefficients $\alpha$ are $-3$ and $2$ for the doublets, respectively, which come from the VEV $\langle 24\rangle$ in the terms $5\cdot\langle 24\rangle\cdot\bar{5}$ of $\SU(5)$. The coefficients $\beta$ are $-\sqrt{3}$ and $2$ for the doublets and triplets respectively, which are the Clebsch-Gordan coefficients coming from the terms $5\cdot\langle 24\rangle\cdot \overline{45}$ or $\bar{5}\cdot\langle 24\rangle\cdot 45$ of $\SU(5)$. One can check that this is indeed the case by locating the position of $\alpha$ and $\beta$ coefficients in $\mathcal{M}$ and checking, to which states in Table \[table:labels-doublets-triplets\] these matrix entries correspond to.
The procedure for fine-tuning is now similar to the one attempted in the $351'+\overline{351'}+27+\overline{27}$ case in [@Bajc:2013qra]. Once the vacuum solution is plugged-in, the determinants of the two matrices become zero: $$\begin{aligned}
\det(\mathcal{M}_{\textrm{doublets}})\Big|_{\textrm{vacuum}}&=0,\\
\det(\mathcal{M}_{\textrm{triplets}})\Big|_{\textrm{vacuum}}&=0.\end{aligned}$$ The massless doublet-antidoublet and triplet-antitriplet pairs are simply the unphysical would-be Goldstone bosons from the breaking of $\EE$. They correspond to the broken generators with the same quantum numbers, which can be found in $\SO(10)$ language in the $16$ and $\overline{16}$ parts of the adjoint $78$. The doublet-triplet splitting condition then requires another doublet-antidoublet pair to be massless, which imposes the following condition: $$\begin{aligned}
\mathrm{Cond}(\mathcal{M})&:=\mathrm{pmin}_1 \mathcal{M},\end{aligned}$$ where $\mathrm{pmin}_i$ denotes the $i$-th principal minor of rank $1$ (the subdeterminant when the $i$-th row and column are removed). In our specific case, we chose to remove the first row and column, which correspond to the fields $D_0$ and $\overline{D}_0$. We chose these due to simplicity, but note that this choice is valid only if the Goldstone modes of the given vacuum have nonzero $D_0$ and $\overline{D}_0$ components. This can be checked later on via equations and . The logic behind the whole method is explained in Appendix \[section:simplified-DT\].
Given the notation above, DT-splitting can be performed by a fine-tuning, so that $$\begin{aligned}
\mathrm{Cond}(\mathcal{M}_{\textrm{doublets}})&=0,\label{equation:Cond-D}\\
\mathrm{Cond}(\mathcal{M}_{\textrm{triplets}})&\neq 0.\label{equation:Cond-T}\end{aligned}$$ The above conditions can in principle be computed analytically, but they are too complicated to be of any practical use. The viability of the splitting conditions can be shown numerically though, as well as with some careful considerations. The most convenient way to perform the DT splitting is to make use of the dimensionless parameters $\lambda_5$ and $\lambda_6$ from the superpotential (see ). Since the invariants $27^3$ and $\overline{27}^3$ have no all-singlet terms, $\lambda_5$ and $\lambda_6$ are not involved in the computation of the vacuum. Moreover, the matrix $\mathcal{M}$ contains only a single entry with the parameter $\lambda_5$ and a single entry with the parameter $\lambda_6$. The condition in equations and can be written as $$\begin{aligned}
K_1-K_2\;\lambda_5\,\lambda_6&=0,\label{equation:Cond-D2}\\
K'_1-K'_2\;\lambda_5\,\lambda_6&\neq 0,\label{equation:Cond-T2}
\end{aligned}$$ where $K_1,K_2,K'_1,K'_2$ depend only on the other parameters in the Lagrangian ($m_{351'}$, $m_{78}$, $m_{27}$, $\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_4$, $\lambda_7$, $\lambda_8$) and the vacuum itself, all of which can be computed independently from the parameters $\lambda_5$ and $\lambda_6$. The analytic forms of $K$s are not very illuminating, but they can easily be evaluated numerically for any values of the parameters $m_{351'},m_{78},m_{27},\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_7,\lambda_8$. One can then take $$\begin{aligned}
\lambda_5&=\frac{K_1}{K_2\,\lambda_6},\end{aligned}$$ with $\lambda_6$ arbitrary, and get an extra massless doublet mode. Substituting into condition , we can numerically check that indeed $$\begin{aligned}
K'_1 K_2&\neq K'_2 K_1.\end{aligned}$$
We therefore conclude that the addition of the $78$ in the Higgs sector, both with the new vacuum and enlarged matrices $\mathcal{M}_{\textrm{doublets}}$ and $\mathcal{M}_{\textrm{triplets}}$, now allows for a DT splitting in the model. Without the $78$, a similar attempt at fine-tuning is not possible, since in that case we get $K_2=K'_2=0$ after inserting the vacuum and fine-tuning in $\lambda_5$ or $\lambda_6$ is not possible. Note that this tree-level fine-tuning is stable under quantum corrections due to the non-renormalization theorem for the superpotential.
The linear combinations of $D$s and $\bar{D}$s, which correspond to the MSSM Higgses $H_u$ and $H_d$, can be found by computing the new left- and right- null-eigenvectors of the fine-tuned matrix $\mathcal{M}_{\textrm{doublets}}$. In practice, the physical MSSM Higgses can most easily be extracted by computing the (left and right) null-eigenspace of the fine-tuned matrix $\mathcal{M}_{\textrm{doublets}}$, which is $2$-dimensional. Given any basis of the null-eigenspace, the Higgs will always be orthogonal to the would-be Goldstone boson. The would-be Goldstone itself can be easily identified by the fact that it has no component in the directions $D_i$ (or $\overline{D}_i$) for $i=1,3,4,5,7,8$. This absence of some doublets in the Goldstone can be deduced from the mass matrix in equation , but we also confirmed this by noting that the Goldstone components $\phi_i$ are the ones which have couplings of their derivatives to the the gauge field through the following type of expression (originating from the kinetic terms of scalar fields): $$\begin{aligned}
-ig\,A_{\mu}^{a}\;(\partial^\mu \phi^\dagger_i) \;(\hat{t}^a \langle\phi\rangle)^i.\label{equation:would-be-Goldstone}\end{aligned}$$ Choosing $a$ to be the doublet/antidoublet broken generators and using our vacuum, we identify the doublet components in $\phi^\dagger_i$ to which there is no coupling as indeed those listed above. Explicit computation identifies that the prevailing cause of some components not being present in the would-be Goldstone mode is our ansatz of vanishing VEVs. More precisely, without the vacuum ansatz, the terms in expression can be schematically written as
$$\begin{aligned}
\partial D_{G}^\ast&\propto
+\frac{3 \sqrt{5} \RED{v}+5 \sqrt{3} \RED{w}-\sqrt{30} \RED{y}}{10 \sqrt{2}} \;\partial D_{0}^\ast
-\frac{\RED{d_2}}{2} \;\partial D_{2}^\ast
-\frac{\RED{d_1}}{2} \;\partial D_{3}^\ast
+\frac{\RED{e_5}}{4} \sqrt{\frac{3}{2}} \;\partial D_{4}^\ast\nonumber\\
&\qquad +\frac{\sqrt{5} \RED{f_2}}{4} \;\partial D_{5}^\ast
+\frac{\RED{f_3}}{\sqrt{2}} \;\partial D_{6}^\ast
+\frac{\sqrt{3} \RED{f_2}}{4} \partial D_{7}^\ast
+\frac{\RED{e_5}}{4} \sqrt{\frac{5}{2}} \;\partial D_{8}^\ast\nonumber\\
&\qquad -\frac{\RED{e_4}}{4} \sqrt{\frac{3}{2}} \;\partial D_{9}^\ast
+\frac{\RED{f_1}}{\sqrt{2}} \;\partial D_{10}^\ast
-\frac{\RED{e_4}}{4} \sqrt{\frac{5}{2}} \;\partial D_{11}^\ast,\label{equation:Goldstone1}\\
\partial \overline{D}_{G}{}^\ast&\propto
-\frac{3 \sqrt{5} \RED{v}+5 \sqrt{3} \RED{w}-\sqrt{30} \RED{y}}{10 \sqrt{2}} \;\partial \overline{D}_{0}^\ast
+\frac{\RED{c_2}}{2} \;\partial \overline{D}_{2}^\ast
+\frac{\RED{c_1}}{2} \partial \overline{D}_{3}^\ast
-\frac{\RED{f_5}}{4} \sqrt{\frac{3}{2}} \;\partial \overline{D}_{4}^\ast\nonumber\\
&\qquad -\frac{\sqrt{5} \RED{e_2}}{4} \;\partial \overline{D}_{5}^\ast
-\frac{\RED{e_3}}{\sqrt{2}} \;\partial \overline{D}_{6}^\ast
-\frac{\sqrt{3} \RED{e_2}}{4} \;\partial \overline{D}_{7}^\ast
-\frac{\RED{f_5}}{4} \sqrt{\frac{5}{2}} \;\partial \overline{D}_{8}^\ast\nonumber\\
&\qquad +\frac{\RED{f_4}}{4} \sqrt{\frac{3}{2}} \;\partial \overline{D}_{9}^\ast
-\frac{\RED{e_1}}{\sqrt{2}} \;\partial \overline{D}_{10}^\ast
+ \frac{\RED{f_4}}{4} \sqrt{\frac{5}{2}} \;\partial \overline{D}_{11}^\ast.\label{equation:Goldstone2}\end{aligned}$$
We see that the absence of components $D_i$ and $\overline{D}_i$ for $i=3,4,5,7,8$ are directly related to the vacuum ansatz with vanishing VEVs. $D_1$ and $\overline{D}_1$, however, are always absent.[^7]
The ratios of the various components of the Higgs, and hence the ratios of the EW VEVs of these component, are computed from the null-eigenspaces, while the magnitudes of the VEVs are given by the VEVs $v_u=\langle H_u\rangle $ and $v_d=\langle H_d\rangle$. We have $$\begin{aligned}
v_u^2&=\sum_{i=0}^{11} |\BLUE{v_i}|^2,\\
v_d^2&=\sum_{i=0}^{11} |\BLUE{\bar{v}_i}|^2,\end{aligned}$$ where $v_i=\langle D_i\rangle$ and $\bar{v}_i=\langle \overline{D}_i\rangle$, and additionally the following MSSM relations hold: $$\begin{aligned}
v_u^2+v_d^2&=(246\,\mathrm{GeV})^2,\\
v_u/v_d&=\tan\beta.\end{aligned}$$
Yukawa sector\[section:Yukawa-sector\]
======================================
The Yukawa sector comes from the Yukawa part of the superpotential after inserting the vacuum solution: $$\begin{aligned}
W_{\textrm{Yukawa}}&=27_F^i\;27_F^j\;\big(Y^{ij}_{27}\;\langle 27\rangle + Y_{\overline{351'}}^{ij}\;\langle \overline{351'}\rangle\,\big).\end{aligned}$$
In addition to the GUT scale VEVs in the representations $\overline{351'}$ and $27$, the EW Higgses $H_u$ and $H_d$ also need to be present in both of these representations. The low-energy MSSM Higgses $H_u$ and $H_d$ come from a linear combination of the doublets of type $D\sim(1,2,+1/2)$ and antidoublets of the type $\overline{D}\sim(1,2,-1/2)$, respectively. These states are identified in Table \[table:labels-doublets-triplets\], and their EW VEVs are labeled by $v_i:=\langle D_i\rangle$ and $\bar{v}_i:=\langle \overline{D}_i \rangle$, where $i=0,1,\ldots,11$. As discussed in section \[section:DT-splitting\], we generically expect $v_i\neq 0$ and $\bar{v}_i\neq 0$.
Each generation of fermions is found in a fundamental representation $27$ of $\EE$. The matter content of each generation, subdivided into $\SO(10)$ representations, is the following:
- The $16$ of $\SO(10)$ contains the SM particles[^8] and the right-handed neutrino $\nu^c$.
- The $10$ of $\SO(10)$ contains a vector-like pair of down-type quarks $(3,1,-1/3)+(\bar{3},1,+1/3)$, as well as a vector-like pair of lepton doublets $(1,2,-1/2)+(1,2,1/2)$. These exotic are labeled $d'$, $d'^c$, $L'$ and $L'^c$, respectively.
- The $\SO(10)$ singlet $1$ is denoted by $s$ is sterile, and has a role analogous to the right-handed neutrino.
Suppressing generation indices, the mass terms are computed to be $$\begin{aligned}
&u^{T}\left(-\BLUE{v_1} \yd+\tfrac{1}{2\sqrt{10}} \BLUE{v_5}\yt-\tfrac{1}{2\sqrt{6}} \BLUE{v_7}\yt\right) u^c\nonumber\\
&+
\begin{pmatrix}
d^{cT} & d'^{cT}\\
\end{pmatrix}
\begin{pmatrix}
\phantom{-}\BLUE{\bar{v}_2} \yd+\tfrac{1}{2\sqrt{10}} \BLUE{\bar{v}_4}\yt+\tfrac{1}{2\sqrt{6}} \BLUE{\bar{v}_8}\yt & \phantom{-}\RED{c_2} \yd+\frac{1}{\sqrt{15}}\RED{f_5} \yt \\
-\BLUE{\bar{v}_3} \yd-\tfrac{1}{2\sqrt{10}} \BLUE{\bar{v}_9}\yt-\tfrac{1}{2\sqrt{6}} \BLUE{\bar{v}_{11}}\yt & -\RED{c_1}\yd+\frac{1}{\sqrt{15}}\RED{f_4} \yt \\
\end{pmatrix}\!\!
\begin{pmatrix}
d \\
d'\\
\end{pmatrix} \nonumber\\
&+
\begin{pmatrix}
e^{T} & e'^{T}
\end{pmatrix}
\begin{pmatrix}
-\BLUE{\bar{v}_2} \yd- \frac{1}{2\sqrt{10}} \BLUE{\bar{v}_4}\yt+ \sqrt{\tfrac{3}{8}} \BLUE{\bar{v}_8} \yt & \phantom{-}\RED{c_2} \yd- \tfrac{3}{2}\frac{1}{\sqrt{15}} \RED{f_5} \yt \\
\phantom{-}\BLUE{\bar{v}_3} \yd+\frac{1}{2\sqrt{10}}\BLUE{\bar{v}_9}\yt-\sqrt{\tfrac{3}{8}}\BLUE{\bar{v}_{11}} \yt & -\RED{c_1} \yd- \tfrac{3}{2}\frac{1}{\sqrt{15}} \RED{f_4} \yt \\
\end{pmatrix}\!\!
\begin{pmatrix}
e^c \\
e'^c\\
\end{pmatrix} &&\nonumber\\
&+
\begin{pmatrix}
\nu^{T} & \nu'^{T}\\
\end{pmatrix} \!\!
\begin{pmatrix}
\BLUE{v_1} \yd-\tfrac{1}{2\sqrt{10}}\BLUE{v_5}\yt-\sqrt{\frac{3}{8}} \BLUE{v_7}\yt & -\tfrac{1}{\sqrt{2}} \BLUE{v_6} \yt & \phantom{-}\RED{c_2} \yd-\tfrac{3}{2}\frac{1}{\sqrt{15}} \RED{f_5} \yt \\
-\tfrac{1}{\sqrt{2}} \BLUE{v_{10}} \yt & -\BLUE{v_1} \yd-\sqrt{\frac{2}{5}} \BLUE{v_5} \yt & -\RED{c_1} \yd-\tfrac{3}{2}\frac{1}{\sqrt{15}} \RED{f_4} \yt \\
\end{pmatrix} \!\!
\begin{pmatrix}
\nu^c \\
s \\
\nu'^c
\end{pmatrix} \nonumber\\
&+\frac{1}{2}
\begin{pmatrix}
\nu^{cT} & s^T & \nu'^{cT}\\
\end{pmatrix}\!
\begin{pmatrix}
\phantom{\tfrac{1}{\sqrt{2}}}\RED{f_1} \yt & \tfrac{1}{\sqrt{2}}\RED{f_2} \yt &-\BLUE{\bar{v}_3} \yd+ \sqrt{\frac{2}{5}} \BLUE{\bar{v}_9} \yt \\
\tfrac{1}{\sqrt{2}}\RED{f_2} \yt & \phantom{\tfrac{1}{\sqrt{2}}}\RED{f_3} \yt &\phantom{-}\BLUE{\bar{v}_2} \yd-\sqrt{\frac{2}{5}} \BLUE{\bar{v}_4} \yt \\
-\BLUE{\bar{v}_3} \yd+\sqrt{\frac{2}{5}} \BLUE{\bar{v}_9} \yt & \BLUE{\bar{v}_2} \yd-\sqrt{\frac{2}{5}} \BLUE{\bar{v}_4} \yt & 0\\
\end{pmatrix}\!\!
\begin{pmatrix}
\nu^c \\
s \\
\nu'^c\\
\end{pmatrix}\nonumber\\
&+\frac{1}{2}
\begin{pmatrix}
\nu^T & \nu'^T
\end{pmatrix}
\begin{pmatrix}
\Delta_1 Y_{\overline{351'}} & \tfrac{1}{\sqrt{2}}\Delta_2 Y_{\overline{351'}}\\
\tfrac{1}{\sqrt{2}}\Delta_2 Y_{\overline{351'}}& \Delta_3 Y_{\overline{351'}}\\
\end{pmatrix}
\begin{pmatrix}
\nu \\
\nu' \\
\end{pmatrix}\!.\label{equation:Yukawa-terms-full}\end{aligned}$$
Notice the different Clebsch-Gordan coefficients in front of $\bar v_i$ ($i=2,3,4,9$) that come from the couplings with the $\bar 5$ of $\SU(5)$ and in front of $\bar v_{8,11}$ that come from the $\overline{45}$, while $c_{1,2}$ originate from a $\SU(5)$ singlet and $f_{4,5}$ from a $\SU(5)$ adjoint $24$.
The $\Delta_i$ correspond to VEVs induced in the electrically neutral component of weak triplets of type $(1,3,+1)$, while $\overline{\Delta}_i$ correspond to VEVs induced in weak triplets $(1,3,-1)$. The $\Delta_i$ and $\overline{\Delta}_i$ are found only in the representations $351'$ and $\overline{351'}$, as shown in the definitions of Table \[table:weak-triplets\]. The mass matrix $M_{\Delta}$ has contributions from the terms $351'\cdot\overline{351'}$ and $351'\cdot\langle 78\rangle\cdot\overline{351'}$. Its explicit form is computed to be $$\begin{aligned}
M_\Delta&=\scalebox{0.55}{$
\begin{pmatrix}
m_{351'}-\lambda _8 \left(\frac{\RED{w}}{6 \sqrt{2}}-\frac{1}{2} \sqrt{\frac{3}{10}} \RED{v}+\frac{\RED{y}}{2 \sqrt{5}}\right)
& \frac{\RED{u_1} \lambda _8}{2 \sqrt{3}} & 0 & \phantom{-}6 \lambda _1 \RED{e_1} \\
\frac{\lambda _8}{2 \sqrt{3}}\RED{u_2} & m_{351'}+\lambda_8 \left(\frac{\RED{w}}{12 \sqrt{2}}+\frac{\RED{v}}{4 \sqrt{30}}-\frac{\RED{y}}{2 \sqrt{5}}\right)& \frac{\lambda_8}{2 \sqrt{3}}\RED{u_1} & -6 \lambda _1 \RED{e_2}
\\
0 & \frac{\lambda _8}{2 \sqrt{3}}\RED{u_2} & m_{351'}+ \lambda _8\left(\frac{\RED{w}}{3 \sqrt{2}}-\frac{\RED{v}}{\sqrt{30}}-\frac{\RED{y}}{2 \sqrt{5}}\right) & \phantom{-}6 \lambda _1\RED{e_3} \\
6 \lambda _2\RED{f_1} & -6 \lambda _2\RED{f_2} & 6 \lambda _2 \RED{f_3} & m_{351'}+ \lambda _8 \left(\frac{\RED{w}}{3 \sqrt{2}}+\frac{\RED{v}}{\sqrt{30}}+\frac{\RED{y}}{2 \sqrt{5}}\right)
\end{pmatrix}
$}.
\end{aligned}$$ Integrating out the heavy weak triplets and inserting the $F$-term ansatz of vanishing VEVs, we get $$\begin{aligned}
\begin{pmatrix}
\Delta_1\\ \Delta_2\\\Delta_3\\\Delta_4\\
\end{pmatrix}&=
\scalebox{0.7}{$
\left(\begin{smallmatrix}
m_{351'}-\lambda _8 \left(\frac{\RED{w}}{6 \sqrt{2}}-\frac{1}{2} \sqrt{\frac{3}{10}} \RED{v}\right)
& 0 & 0 & 6 \lambda _1 \RED{e_1} \\
0 & m_{351'}+\lambda_8 \left(\frac{\RED{w}}{12 \sqrt{2}}+\frac{\RED{v}}{4 \sqrt{30}}\right)& 0 & 0
\\
0 & 0 & m_{351'}+ \lambda _8\left(\frac{\RED{w}}{3 \sqrt{2}}-\frac{\RED{v}}{\sqrt{30}}\right) & 6 \lambda _1\RED{e_3} \\
6 \lambda _2\RED{f_1} & 0 & 6 \lambda _2 \RED{f_3} & m_{351'}+ \lambda _8 \left(\frac{\RED{w}}{3 \sqrt{2}}+\frac{\RED{v}}{\sqrt{30}}\right)
\end{smallmatrix}\right)$}^{-1}
\begin{pmatrix}
\lambda_4 \BLUE{v_3}^2\\ \lambda_4 \sqrt{2}\BLUE{v_2} \BLUE{v_3}\\ \lambda_4 \BLUE{v_2}^2\\ \lambda_3 \BLUE{v_1}^2\\
\end{pmatrix}.
\end{aligned}$$
After integrating out the heavy vector-like states from equation , and using the ansatz $c_1=f_5=0$, we get the matrices for the low energy states: $$\begin{aligned}
M_U&=-\BLUE{v_1} \yd+\left(\tfrac{1}{2\sqrt{10}} \BLUE{v_5}-\tfrac{1}{2\sqrt{6}} \BLUE{v_7}\right)\yt,\label{equation:low-energy-masses-first}\\
M_D^T&=\scalebox{0.9}{$ \left(1+(9/4)XX^\dagger\right)^{-1/2} \left(\left(\BLUE{\bar{v}_2}-\tfrac{3}{2}\BLUE{\bar{v}_3}X
\right)\yd+\left(\tfrac{1}{2\sqrt{10}}(\BLUE{\bar{v}_4}-\tfrac{3}{2}\BLUE{\bar{v}_9}X)+\tfrac{1}{2\sqrt{6}}(\BLUE{\bar{v}_8}-\tfrac{3}{2}\BLUE{\bar{v}_{11}}X)\right)\yt\right) $},\\
M_E&=\scalebox{0.9}{$ \left(1+XX^\dagger\right)^{-1/2} \left(\left(-\BLUE{\bar{v}_2}-\BLUE{\bar{v}_3}X
\right)\yd+\left(-\tfrac{1}{2\sqrt{10}}(\BLUE{\bar{v}_4}+\BLUE{\bar{v}_9}X)+\sqrt{\tfrac{3}{8}}(\BLUE{\bar{v}_8}+\BLUE{\bar{v}_{11}}X)\right)\yt\right) $},\label{equation:low-energy-masses-third}\\
M_N&=-\left(1+XX^\dagger\right)^{-1/2}\nonumber\\
&\Bigg(\scalebox{0.95}{$ \bigg(-\frac{1}{\sqrt{10}}\frac{\BLUE{v_1} \BLUE{v_5}}{\RED{f_1}}-\sqrt{\frac{3}{2}}\frac{\BLUE{v_1} \BLUE{v_7}}{\RED{f_1}}+
\frac{1}{\sqrt{3}}\frac{\BLUE{v_5} \BLUE{v_{10}}}{\RED{f_1}}\frac{\RED{c_2}}{\RED{f_4}}+\sqrt{5}\,\frac{\BLUE{v_7} \BLUE{v_{10}}}{\RED{f_1}}\frac{\RED{c_2}}{\RED{f_4}}+
\frac{4}{\sqrt{3}}\frac{\BLUE{v_5} \BLUE{v_6}}{\RED{f_3}}\frac{\RED{c_2}}{\RED{f_4}}-2\sqrt{\frac{10}{3}}\,\Delta_2\,\frac{\RED{c_2}}{\RED{f_4}}\bigg) $}\;\yd\nonumber\\
&\quad +\bigg(\frac{1}{40}\frac{\BLUE{v_5}^2}{\RED{f_1}}+\sqrt{\frac{3}{80}}\frac{\BLUE{v_7} \BLUE{v_5}}{\RED{f_1}}+\frac{3}{8}\frac{\BLUE{v_7}^2}{\RED{f_1}}+ \frac{1}{2}\frac{\BLUE{v_6}^2}{\RED{f_3}}-\Delta_1\bigg)\;\yt\nonumber\\
&\quad + \scalebox{0.95}{ $\bigg(\frac{\BLUE{v_1}^2}{\RED{f_1}}-2 \sqrt{\frac{10}{3}} \frac{\BLUE{v_1} \BLUE{v_{10}}}{\RED{f_1}}\frac{\RED{c_2}}{\RED{f_4}}
+\frac{10}{3}\frac{\BLUE{v_{10}}^2}{\RED{f_1}}\frac{\RED{c_2}^2}{\RED{f_4}^2}
+2 \sqrt{\frac{10}{3}}\frac{\BLUE{v_1} \BLUE{v_6}}{\RED{f_3}}\frac{\RED{c_2}}{\RED{f_4}}
+\frac{8}{3}\frac{\BLUE{v_5}^2}{\RED{f_3}}\frac{\RED{c_2}^2}{\RED{f_4}^2}-\frac{20}{3}\,\Delta_3\,\frac{\RED{c_2}^2}{\RED{f_4}^2}\bigg) $}\;\yz\nonumber\\
&\quad +\bigg(\frac{8\sqrt{10}}{3}\frac{\BLUE{v_1} \BLUE{v_5}}{\RED{f_3}}\frac{\RED{c_2}^2}{\RED{f_4}^2}\bigg)\;\yd\yt^{-1}\yd\yt^{-1}\yd\nonumber\\
&\quad +\bigg(\frac{20}{3}\frac{\BLUE{v_1}^2}{\RED{f_3}}\frac{\RED{c_2}^2}{\RED{f_4}^2}\bigg)\;\yd\yt^{-1}\yd\yt^{-1}\yd\yt^{-1}\yd\Bigg)\nonumber\\
&\left(1+X^\ast X^T\right)^{-1/2}\label{equation:low-energy-masses-last},\end{aligned}$$ where $$\begin{aligned}
X&=-2\sqrt{\tfrac{5}{3}}\,\frac{\RED{c_2}}{\RED{f_4}}\,\yd\,\yt^{-1}.\label{equation:X0-definition}\end{aligned}$$
Notice that the main factor in the expressions is a linear combination of the following matrices: $$\begin{aligned}
\yd,\quad \yt,\quad \yd\yt^{-1}\yd, \quad \yd\yt^{-1}\yd\yt^{-1}\yd,\quad \yd\yt^{-1}\yd\yt^{-1}\yd\yt^{-1}\yd,\end{aligned}$$ which are all symmetric, since $\yd$ and $\yt$ are symmetric, as are $M_U$ and $M_N$. The matrices $M_D$ and $M_E$ are not symmetric, though, due to the projection factor onto the light families.
As usual with vector-like states, the expressions for the low energy masses in equations – are nonlinear, which complicates the analysis of the masses. We comment more on the low-energy part of the Yukawa sector in section \[section:numeric-fit\], in which we also do a numeric fit in the 2-generation case.
Numeric fit of the Yukawa sector\[section:numeric-fit\]
=======================================================
The presented model has $3$ masses and $8$ dimensionless parameters $\lambda_i$ in the breaking sector, as well as two symmetric $3\times 3$ Yukawa matrices in the Yukawa sector, which is easily seen from the superpotential in equation . Note also that the product $\lambda_5\lambda_6$ is fixed by the fine-tuning of the EW Higgs mass. Also, a rotation in family-space can bring one of the Yukawa matrices to be diagonal. We shall limit ourselves to the case where all parameters of the Lagrangian are real. The independent number of real parameters in the breaking sector is thus $3+8-1=10$, while the $3$-family Yukawa sector has $6+3=9$ real parameters. Since the number of independent real parameters is $19$, while there are only $17$ real numbers to be measured ($3$ masses in the up, down, charged lepton sector each, the two differences of masses-squared in the neutrino sector, as well as $3$ angles in each of the CKM and PMNS matrices; we neglect the CP-phases), the general expectation is that a fit is possible to perform. There may exist, however, non-obvious mass relations concealed due to the complexity of the low energy expressions in equations –, which are not respected by the experimental values; these fears can be alleviated by finding points in parameter space, which give a good fit for to the masses and mixing angles.
In this section, we perform a fit in the simplified case of $2$ families. Here, there are again $10$ real parameters in the breaking sector, while the Yukawa sector has $3+2=5$ real parameters. The fit is performed for the results of the quark masses $m_{t}$, $m_{c}$, $m_{b}$, $m_{s}$, the charged lepton masses $m_{\tau}$ and $m_{\mu}$, the difference of the squared neutrino masses $m_{\nu_3}^2-m_{\nu_2}^2$, and the mixing angles $\theta_{cb}$ and $\theta_{23}$ in the CKM and PMNS matrices, respectively.
The most convenient way to perform the fit is to take some of the GUT scale VEVs as the parameters in the fit, instead of the initial parameters in the Lagrangian. This is advantageous since the equations of motion are linear in the Lagrangian parameters. Taking the ansatz $c_1=d_1=e_2=f_2=e_5=f_5=u_1=u_2=y=0$, we need $12$ parameters; only $10$ are independent, while $2$ are determined through the $D$-terms. It is convenient, for example, to take the following quantities as independent parameters: $$\begin{aligned}
m_{351'},\quad c_2,\quad d_2,\quad f_1,\quad f_3,\quad f_4,\quad e_4,\quad v,\quad w,\quad \lambda_5.\label{equation:fit-parameters}\end{aligned}$$
We can then use the equations of motion to determine the remaining initial parameters and VEVs. The $D$-terms are solved by taking $$\begin{aligned}
\label{d2}
e_1&= \pm\sqrt{|\RED{d_2}|^2 - |\RED{c_2}|^2 + 2 |\RED{f_1}|^2}/\sqrt{2},\\
\label{e4}
e_3&= \pm\sqrt{|\RED{e_4}|^2 - |\RED{f_4}|^2 + 2 |\RED{f_3}|^2}/\sqrt{2}.
\end{aligned}$$ Since we want $e_1$ and $e_3$ to be real numbers, the arguments in the square root need to be positive, which limits the space of parameters in equation . Alternatively, we could also take the $6$ independent parameters, for example[^9], to be the VEVs $d_2$, $f_1$, $e_3$, $f_4$ (now with no restriction) and two angles $\theta_{1},\theta_2\in (0,2\pi)$, with the remaining quantities computed as $$\begin{aligned}
f_1&=\sin\theta_1\;\sqrt{|\RED{c_2}|^2 + 2 |\RED{e_1}|^2}/\sqrt{2},\\
d_2&=\cos\theta_1\;\sqrt{|\RED{c_2}|^2 + 2 |\RED{e_1}|^2},\\
f_3&=\sin\theta_2\;\sqrt{|\RED{f_4}|^2 + 2 |\RED{e_3}|^2}/\sqrt{2},\\
\label{ee4}
e_4&=\cos\theta_2\;\sqrt{|\RED{f_4}|^2 + 2 |\RED{e_3}|^2}.\end{aligned}$$
The $F$-terms then yield
$$\begin{aligned}
m_{27}&= \frac{m_{351'}}{2 \RED{w} \left(\sqrt{15} \RED{w}-\RED{v}\right) \RED{c_2} \RED{d_2} \left(4 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)} \left(2 \RED{v}(\RED{w}+\sqrt{15} \RED{v}) \left(\RED{e_4} \RED{f_4}-2 \RED{e_3} \RED{f_3}\right){}^2\right.\nonumber\\
&\left. +\RED{e_1} \RED{f_1}
(\RED{v}-\sqrt{15} \RED{w}) \left(2 \RED{e_3}\RED{f_3}(3 \sqrt{15} \RED{v}-5 \RED{w})-\RED{e_4} \RED{f_4}(3 \sqrt{15} \RED{v}+7 \RED{w})\right)\right),\\
m_{78}&= \frac{\sqrt{15} m_{351'} \left(\RED{e_4} \RED{f_4}-2 \RED{e_3} \RED{f_3}\right){}^2}{2 \RED{w}
\left(\sqrt{15} \RED{w}-\RED{v}\right) \left(4 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)},\\
\lambda _1&=-\frac{\RED{f_3} \RED{f_4} m_{351'}}{\RED{e_4} (\RED{e_4} \RED{f_4}+4 \RED{e_3} \RED{f_3})},\\
\label{lambda2}
\lambda _2&=-\frac{\RED{e_3} \RED{e_4} m_{351'}}{\RED{f_4} (\RED{e_4} \RED{f_4}+4 \RED{e_3} \RED{f_3})},\\
\lambda _3&=-\frac{\RED{e_1} m_{351'} \left(\sqrt{15} \RED{v} \left(\RED{e_4} \RED{f_4}-2 \RED{e_3} \RED{f_3}\right)+3 \RED{w} \left(2 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)\right)}{2 \RED{w} \RED{c_2}^2 \left(4 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)},\\
\lambda _4&=-\frac{\RED{f_1} m_{351'} \left(\sqrt{15} \RED{v} \left(\RED{e_4} \RED{f_4}-2 \RED{e_3} \RED{f_3}\right)+3 \RED{w} \left(2 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)\right)}{2 \RED{w} \RED{d_2}^2 \left(4 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)},\\
\lambda _7&= \frac{3 \sqrt{2} m_{351'} \left(2 \RED{e_3} \RED{f_3}-\RED{e_4} \RED{f_4}\right) \left(\left(\RED{v}-\sqrt{15} \RED{w}\right) \RED{e_1} \RED{f_1}
+2\RED{v}\left(2 \RED{e_3} \RED{f_3}-\RED{e_4} \RED{f_4}\right)\right)}{\RED{w} \left(\sqrt{15} \RED{w}-\RED{v}\right) \RED{c_2} \RED{d_2} \left(4 \RED{e_3} \RED{f_3}+\RED{e_4} \RED{f_4}\right)},\\
\lambda _8&= \frac{3 \sqrt{2} m_{351'} \left(2 \RED{e_3} \RED{f_3}-\RED{e_4} \RED{f_4}\right)}{\RED{w} \left(\RED{e_4} \RED{f_4}+4 \RED{e_3} \RED{f_3}\right)}.
\end{aligned}$$
Using these values, $\lambda_6$ can then be determined by fine-tuning in the doublet mass matrix (see section \[section:DT-splitting\]), from which also the EW VEVs $v_i$ and $\bar{v}_i$ are computed. One can then use these to compute the mass matrices from equations –, and ultimately the masses and the mixing angles. For our numeric fit, we assume:
- the simplified case of second and third generation only;
- all parameters real;
- all errors in measured quantities at the $10\%$ level; we believe that at the present stage such a choice is a good compromise between the realistic case and the simplicity of the analysis;
- the values for the masses and mixing angles at the GUT scale $\sim 10^{16}\,\mathrm{GeV}$ as shown in Table \[table:numeric-masses-and-angles\] [@Xing:2007fb], valid for $\tan{\beta}=10$;
- positive signs in eqs. -.
We performed the fit by minimizing the chi-squared function $$\begin{aligned}
\chi^2&:=\sum_{i}{\frac{(f_i(x)-y_i)}{\sigma_i^2}},\end{aligned}$$ with respect to the initial parameters $x$. The experimental values are denoted by $y_i$, the values computed using our model are $f_i(x)$, and $\sigma_i$ are the $1$-sigma deviations from the values $y_i$. For our fit, we have $9$ different measured quantities, thus $i=1,\ldots,9$. The errors are taken to be $\sigma_i=0.1\,y_i$, giving $$\begin{aligned}
\chi^2=100\sum_i \left(\frac{f_i(x)-y_i}{y_i}\right)^2.\end{aligned}$$ We define the convenient measure $\hat{\chi}^2=\chi^2/9$, which tells us the average $\sigma^2$ deviation per measured value. Also, we define the pulls $\chi_i=(f_i(x)-y_i)/\sigma_i$, which tell us how many sigma a certain quantity deviates from the measured one.
Due to the large number of parameters, the $\chi^2$ function will have many local minima in the parameter space. We give below two such points in the parameter space, corresponding to the best fits that were found and which we deem sufficiently good (with $\hat{\chi}^2 \lesssim 1$). The results are given in Tables \[table:numeric-independent-parameters\], \[table:numeric-Lagrangian-parameters\], \[table:numeric-masses-and-angles\]; the points in the parameter space are given in terms of the independent parameters best suited to a numeric search (as discussed in this section) in Table \[table:numeric-independent-parameters\] and in terms of the original Lagrangian parameters in Table \[table:numeric-Lagrangian-parameters\], while Table \[table:numeric-masses-and-angles\] shows the obtained results for the masses and mixing angles.
Notice from Tables \[table:numeric-independent-parameters\] and \[table:numeric-Lagrangian-parameters\] that of the two Yukawa matrices $\yd$ and $\yt$, $\yt$ was chosen to be the diagonal one. Furthermore, the original Lagrangian parameters in Table \[table:numeric-Lagrangian-parameters\] are given so that it can be checked they roughly fall into the perturbative regime. A possible exception could be the value $\lambda_8$ of the first solution.
Note that the given two points are merely the best ones we found. Due to the high dimensionality of the parameter space, we suspect there are likely many more points which give a comparable or a better fit. There are a number of observations that can be made about these points in general, however, by deducing them from the formulae, comparing the two parameter points given in the tables and through experience obtained by performing the fit:
- *The parameter points are not necessarily close to each other, which also holds true for any specific single parameter.* We can see in Table \[table:numeric-independent-parameters\] that it is not necessary for any parameter to be at a very specific value to obtain a good fit. The suitable areas of parameter space thus form many disconnected regions, and no specific value can be claimed for any parameter. In this sense, the mass formulae of the theory are not very predictive of the original parameters.
- *There is no specific mass or mixing angle, where one would consistently be getting tension.* As observed from Table \[table:numeric-masses-and-angles\], while a specific solution might have most tension with observation coming from a single mass or mixing angle, there are other points, which also give a good fit and where this parameter is predicted better. In this sense we cannot claim any tendencies in tensions of the observables.
- Imagine that in Table \[table:numeric-independent-parameters\] or \[table:numeric-Lagrangian-parameters\] we rescale all the mass parameters with a common factor. Such a rescaling would have no influence on the masses of the quarks and charged leptons, which are controlled only by the EW VEVs, as can be seen from equations – and equation . Rescaling would influence the neutrino masses, however, due to the seesaw mechanism, confirmed by equation . In principle, the rescaling factor can always be adjusted, so that the fit of the mass-square difference of the neutrino masses $m_{\nu_3}^2-m_{\nu_2}^2$ is exact (provided this does not spoil the GUT scale, or the upper bounds on neutrino masses). *Thus only neutrinos are actually sensitive to the GUT scale.*
- The mass parameters are chosen to be at approximately the GUT scale, say at the order of $10^{15-16}\,\mathrm{GeV}$. If this is true for all the mass parameters, there might be a problem with the neutrino masses. It is a well known fact that this GUT scale is a few orders of magnitude too large compared to the seesaw scale for sufficiently large mass differences in the neutrino sector. One can cure this problem by having the spontaneous symmetry breaking occur in multiple stages, with a mass hierarchy between different VEVs and the seesaw scale corresponding to one of the intermediate stages. The gauge coupling unification in such a scenario could then be spoilt by the particles appearing at these intermediate mass scales.
In our case, however, the number of different VEVs is large enough, so that having one or two of the VEVs at a smaller scale does not disturb the breaking pattern. To see this, note that to obtain sufficiently large neutrino masses, it is enough for one of the terms in equation to be of the proper scale, which can be achieved by simply taking $f_1$ or $f_3$ to be several orders of magnitude smaller than the GUT scale (the seesaw type I contributions). In the solutions given in Table \[table:numeric-independent-parameters\], we achieved sufficiently high neutrino masses by taking the parameter$f_1$ (and also $v$) to be a few orders of magnitude below the GUT scale. As seen from the gauge boson masses in Table , small $f_1$ and $v$ do not spoil the one-stage breaking scenario; since the intermediate scales are not associated to an intermediate symmetry breaking at that scale, but are instead purely accidental due to a carefully chosen parameter point, we do not expect too many relevant states (apart from the singlets $\nu^c$ with $f_1$ Majorana mass) to be found at the scale $f_1$. We noticed however the appearance of a lighter color triplet-antitriplet pair, possibly due to the similarities between the doublet and triplet mass matrices. Although this influences the running of the gauge couplings, we will neglect it in view of the (presumably) large threshold uncertainties present anyway.
- A final comment on the neutrino masses: although only the difference of masses-squared needs to be fitted, one still needs to check that the neutrino masses themselves are $< 1\,\mathrm{eV}$ [@Agashe:2014kda]. As seen in Table \[table:numeric-masses-and-angles\], this condition holds true for both of our parameter points.
[lrr]{} &&\
$m_{351'}\;[\mathrm{GeV}]$&$ 1.17\times 10^{16} $&$ -4.17\times 10^{15} $\
$c_2\;[\mathrm{GeV}]$&$ 6.68\times 10^{15} $&$ 3.98\times 10^{15} $\
$d_2\;[\mathrm{GeV}]$&$ -6.78\times 10^{15} $&$ -4.90\times 10^{15} $\
$f_1\;[\mathrm{GeV}]$&$ 4.12\times 10^{11} $&$ -5.52\times 10^{12} $\
$f_3\;[\mathrm{GeV}]$&$ -1.84\times 10^{16} $&$ 1.38\times 10^{16} $\
$f_4\;[\mathrm{GeV}]$&$ 1.61\times 10^{16} $&$ 1.49\times 10^{16} $\
$e_4\;[\mathrm{GeV}]$&$ 5.27\times 10^{15} $&$ -1.69\times 10^{16} $\
$v\;[\mathrm{GeV}]$&$ -7.07\times 10^{13} $&$ 8.44\times 10^{14} $\
$w\;[\mathrm{GeV}]$&$ 6.13\times 10^{15} $&$ -1.78\times 10^{16} $\
$\lambda_5$&$ -1.58\times 10^{-3} $&$ 1.50\times 10^{-1} $\
$(Y_{27})_{11}$&$ -0.723 $&$ 1.93 $\
$(Y_{27})_{12}$&$ 0.703 $&$ -1.19 $\
$(Y_{27})_{22}$&$ -0.676 $&$ 0.730$\
$(Y_{\overline{351'}})_{11}$&$ -0.371 $&$ 0.733$\
$(Y_{\overline{351'}})_{22}$&$ 0.363 $&$ -0.287$\
[lrr]{} &&\
$m_{351'}\;[\mathrm{GeV}]$&$ 1.17\times 10^{16} $&$ -4.17\times 10^{15} $\
$m_{27}\;[\mathrm{GeV}]$&$ -2.92\times 10^{14} $&$ -1.64\times 10^{15} $\
$m_{78}\;[\mathrm{GeV}]$&$ -6.19\times 10^{16} $&$ -5.04\times 10^{15} $\
$\lambda_1$&$ -6.49\times 10^{-1} $&$ -8.91\times 10^{-2} $\
$\lambda_2$&$ 5.66\times 10^{-2} $&$ -1.24\times 10^{-1} $\
$\lambda_3$&$ -1.5\times 10^{-1} $&$ 2.78\times 10^{-1} $\
$\lambda_4$&$ -7.38\times 10^{-5} $&$ -5.04\times 10^{-4} $\
$\lambda_5$&$ -1.58\times 10^{-3} $&$ 1.50\times 10^{-1} $\
$\lambda_6$&$ -7.59\times 10^{-3} $&$ 3.06\times 10^{-2} $\
$\lambda_7$&$ -4.23\times 10^{-1} $&$ 9.55\times 10^{-1} $\
$\lambda_8$&$ 5.08 $&$ 1.16 $\
$(Y_{27})_{11}$&$ -7.23\times 10^{-1} $&$ 1.87 $\
$(Y_{27})_{12}$&$ 7.03\times 10^{-1} $&$ -1.09 $\
$(Y_{27})_{22}$&$ -6.76\times 10^{-1} $&$ 6.30\times 10^{-1} $\
$(Y_{\overline{351'}})_{11}$&$ -3.71\times 10^{-1} $&$ 7.39\times 10^{-1} $\
$(Y_{\overline{351'}})_{12}$&$ 0 $&$ 0 $\
$(Y_{\overline{351'}})_{22}$&$ 3.63\times 10^{-1} $&$ -2.58\times 10^{-1} $\
[rlrlrl]{} &&&\
&&&&&\
$m_c$&$0.236 $&$ 0.226 $&$ -0.432 $&$ 0.205 $&$ -1.30 $\
$m_t$&$ 92.2 $&$ 94.0 $&$ +0.193 $&$ 105 $&$ +1.38 $\
$\theta_{cb}$&$ 0.0409 $&$ 0.0358 $&$ -1.24 $&$ 0.0378 $&$ -0.757 $\
$m_s$&$ 0.013 $&$ 0.0144 $&$ +1.06 $&$ 0.146 $&$ +1.20 $\
$m_b$&$ 0.79 $&$ 0.79 $&$ +0.0021 $&$ 0.781 $&$ -0.112 $\
$m_{\mu}$&$ 0.0599 $&$ 0.0613 $&$ +0.241 $&$ 0.0664 $&$ +1.08 $\
$m_{\tau}$&$ 1.02 $&$ 0.867 $&$ -1.51 $&$ 1.03 $&$ +0.056$\
$m_{\nu_3}$&&$0.135$&&$0.0824$&\
$m_{\nu_2}$&&$0.126$&&$0.0659$&\
$(m_{\nu_3}^2-m_{\nu_2}^2)10^{3}$&$ 2.32 $&$ 2.45 $&$ +0.568 $&$ 2.24 $&$ -0.325$\
$\sin^2 \theta_{23}$&$ 0.386 $&$ 0.343 $&$ -1.12 $&$ 0.327 $&$ -1.52 $\
$\hat{\chi}^2$&&&&&\
Discussion\[section:discussion\]
================================
What we presented here is a model we believe is a good candidate for a minimal supersymmetric renormalizable $E_6$ GUT. Let’s see why this model can be considered as more minimal than our previous candidate [@Bajc:2013qra]. Although the total number of degrees of freedom is now larger ($78>27+27$), what really counts are the number of multiplets and, even more important, the total number of free parameters. The number of parameters in the Higgs sector is now 11 complex minus 5 phases due to field redefinitions. Yukawa sector adds another 3 real and 6 complex parameters. Together with one real gauge coupling we have thus a total of $33$ real parameters. This is 7 more than in the minimal $\SO(10)$ [@Clark:1982ai; @Aulakh:1982sw; @Aulakh:2003kg], but still $16$ less than even in the simplified truncated version of $E_6$ in [@Bajc:2013qra].
Is there any possible low-energy signature of this model? It is often said that $E_6$ could have possible light extra generations, coming from remnants of the three copies of the $27_F$. Light vector-like fermions of the SM group could indeed emerge out of $\EE$ if they were associated with anomaly cancellation in a $\mathrm{TeV}$-scale extra $\mathrm{U}(1)$. In such a setup, the extra $\mathrm{U}(1)$ would be a linear combination of the two $\mathrm{U}(1)$s present in the rank $6$ group $\EE$. Phenomenology of such $\mathrm{TeV}$ scale $\EE$ motivated models has been extensively studied, see for eg. [@rizzo]. In the present setup, however, there are no intermediate $\mathrm{U}(1)$s and any light vector-like states would be accidental. In fact if we try to get such light states from our solution, we find them hard to obtain. The question is, does the matrix for (as an example) down quarks in eq. allow 4 or more zero eigenvalues once we limit all $\bar v$’s to zero? This can in principle be obtained either by putting $c_2,f_4\to0$ (remember that our solution already has $c_1=f_5=0$), or by imposing a vanishing determinant constraint to Yukawa matrices. The first case points towards an SU(5) invariant vacuum[^10], while the second one constrains the Yukawa parameters and so a worse fit to data is expected. So we conclude that such an extreme, albeit interesting situation is unlikely, at least in the given vacuum solution. Although we cannot make the same conclusions in general, any possibility of light states, if available, will occur due to fine-tuning in the superpotential parameters.
Another possibility for having light states could be to have flat directions. We checked by explicit computation that no such states are present in our solution. Although we omit the details of this computation here, the interested reader can reconstruct the mass matrix of the SM singlets via the all-singlet terms in the superpotential, which are given in equations –. If the vacuum solution is then plugged-in, one discovers $4$ massless singlet states, all of which are in fact would-be Goldstone bosons. Note that the adjoint $78$ of $\EE$ contains $5$ SM singlets, while only $1$ remains unbroken among the SM generators, so the $4$ massless singlet states are the would-be Goldstone bosons eaten up by the $4$ broken singlet generators. We thus conclude there are no physical massless singlet states. Other vacua could in principle be possible: one of them is described in Appendix \[section:vacuum2\].
Proton decay is, as usual, quite hidden by details of superpartners’ spectra: it is hard to disentangle the GUT and SUSY breaking information from it. Once however colliders will (hopefully) tell us more about the low energy spectrum, this $E_6$ theory as well as other grand unified theories could be tested better. For the sake of completeness the forms of the low-energy $D=5$ operators are given in Appendix \[Proton-decay\].
The model we presented here is the minimal known $E_6$, although for a more convincing proof we should satisfy three more checks.
- First, there is a possibility to redefine the charges under matter parity, so that the $27+\overline{27}$ parity is now transferred to the fermionic sector. This means that the fields $351'+\overline{351'}+78$ should alone break to the SM gauge group as well as allow for DT splitting. The fermionic sector would now consists of $27^a_F$, $a=1,\ldots,4$ and $\overline{27}_F$. The Yukawa terms could be written schematically as
$$\begin{aligned}
\begin{pmatrix} 27^{a}_F\\ \overline{27}_F\end{pmatrix}^T
\begin{pmatrix}
Y_{\overline{351'}}^{ab}\langle \overline{351'}\rangle &m_{27}^a+\lambda^a\langle 78\rangle\\
m_{27}^b+\lambda^b\langle 78\rangle& y_{351'}\langle 351'\rangle\\
\end{pmatrix}
\begin{pmatrix} 27^{b}_F\\ \overline{27}_F\end{pmatrix}.
\end{aligned}$$
The total number of parameters is now the following: 5 complex parameters come from the Higgs superpotential, 3 phases of which can be rotated away by Higgs field redefinitions; 4 real diagonal components are given by the only Yukawa matrix, and 8 complex and one real parameters are the off-diagonal terms; finally, 1 real gauge coupling sums to a total of 29 real free parameters. This would be 4 real parameters less than the model in this paper. Obviously there is no guarantee that such a model is realistic. We plan to come back to this issue in the future.
- Second, we should study the complete three generation case, not only its two generation subsystem. Although the number of parameters seems naively large enough, it is far from obvious that a successful fit is possible. In fact already in the two generation case considered here we could not find a solution with vanishing $\chi^2$, in spite of enough free parameters. However the (at least partially) successful fit of the the minimal SO(10) analogous case [@Babu:1992ia; @Bajc:2001fe; @Fukuyama:2002ch; @Bajc:2002iw; @Goh:2003sy; @Goh:2003hf; @Bertolini:2004eq; @Babu:2005ia; @Bertolini:2006pe; @Bajc:2008dc; @Joshipura:2011nn; @Altarelli:2013aqa; @Dueck:2013gca] with $10$ and $\overline{126}$ (instead of $27$ and $\overline{351'}$) Yukawa couplings make us feel optimistic. Notice also that the usual obstruction of the neutrino mass either too low or unification violated [@Aulakh:2005bd; @Bajc:2005qe; @Aulakh:2005mw; @Bertolini:2006pe] is here avoided as shown in section \[section:numeric-fit\].
- Third, the theory is not asymptotically free and has a huge gauge coupling beta function, more precisely $159$. This means, similarly as in our previous $E_6$ model, or the minimal renormalizable supersymmetric $\SO(10)$, that a Landau pole is close to the GUT scale and so the theory itself may be already in the non-perturbative regime. An indication of problems being present already at the matching scale $M_{GUT}$ can be found in the calculation of the threshold corrections [@Aulakh:2011at; @Aulakh:2013lxa]. The general problem of having large numbers of degrees of freedom (in our case through large representations) and the associated non-perturbativity is a problem, which is far from easy to solve and well beyond the purpose of this paper, although some progress has been made recently [@Abel:2008tx; @Abel:2009bj] based on previous works on Seiberg dualities. We hope to come back to this very interesting issue soon.
But even in the case the model presented here is the minimal one, other vacua could still be realistic with in principle different predictions.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of K.S.B is supported in part by the US Department of Energy Grant No. de-sc0010108. The work of B.B. and V.S. is supported by the Slovenian Research Agency.
Gauge Boson masses \[Gauge-Boson-masses\]
=========================================
We compute in this appendix the expression for the gauge boson masses, so that we can confirm that solutions really break into the SM group. We write the mass terms in the Lagrangian as $$\begin{aligned}
\mathcal{L}_{mass}&=g^2 A_\mu^{\;a} M^{ab} A^{\mu\,b},\end{aligned}$$ where $g$ is the $\EE$ coupling constant and the matrix $M^{ab}$ is computed via $$\begin{aligned}
M^{ab}&:=\sum_i\mathrm{Tr}\left(\left(\hat{t}^{a} \langle\phi_i\rangle\right)^\dagger \left(\hat{t}^{b} \langle\phi_i\rangle\right)\right).\end{aligned}$$ The sum is over all representations containing VEVs (contributions come from the representations of the breaking sector), while $\hat{t}^a$ denotes the action of the $a$-th generator on the representation $\phi_i$. The mass matrix becomes block diagonal if we choose an appropriate basis (indices $a,b$), so that the basis gauge bosons have well defined transformation properties under the SM group (note that some of these states are complex). We omit the details of this calculation and only give the results, collected in Table \[table:gauge-boson-masses\].
0.2cm
Particle identification \[Particle-identification\]
===================================================
[llll@p[4cm]{}]{} label&[$E_6\supseteq\SO(10)\supseteq\SU(5)$]{}&label&[$E_6\supseteq\SO(10)\supseteq\SU(5)$]{}&doublet
triplet\
$\overline{D}_0,\overline{T}_0$&$\phantom{0}78\phantom{'}\supseteq\phantom{0}16\supseteq\phantom{0}\overline{5}$&$D_0,T_0$&$\phantom{0}78\phantom{'}\supseteq\phantom{0}\overline{16}\supseteq\phantom{0} 5$&$\tfrac{1}{\sqrt{12}}(t_L^6\pm i\, t_L^7)$
$\tfrac{1}{\sqrt{12}}\bar{t}_\alpha{}^{31},\tfrac{1}{\sqrt{12}}t^\alpha{}_{31}$\
$D_1,T_1$&$\phantom{0}27\phantom{'}\supseteq\phantom{0}10\supseteq\phantom{0}5$&$\overline{D}_1,\overline{T}_1$&$\phantom{0}\overline{27}\phantom{'}\supseteq\phantom{0}10\supseteq\phantom{0}\overline{5}$&$L'^c$
$d'$\
$\overline{D}_2,\overline{T}_2$&$\phantom{0}27\phantom{'}\supseteq\phantom{0}10\supseteq\phantom{0}\overline{5}$&$D_2,T_2$&$\phantom{0}\overline{27}\phantom{'}\supseteq\phantom{0}10\supseteq\phantom{0}5$&$L'$
$d'^c$\
$\overline{D}_3,\overline{T}_3$&$\phantom{0}27\phantom{'}\supseteq\phantom{0}16\supseteq\phantom{0}\overline{5}$&$D_3,T_3$&$\phantom{0}\overline{27}\phantom{'}\supseteq\phantom{0}\overline{16}\supseteq\phantom{0}5$&$L$
$d^c$\
$D_4,T_4$&$351'\supseteq\phantom{0}10\supseteq\phantom{0}5$&$\overline{D}_4,\overline{T}_4$&$\overline{351'}\supseteq\phantom{0}10\supseteq\phantom{0}\overline{5}$&$Q d^c-L e^c-4L'^c \nu^c$
$QL-u^c d^c-4d's$\
$\overline{D}_5,\overline{T}_5$&$351'\supseteq\phantom{0}10\supseteq\phantom{0}\overline{5}$&$D_5,T_5$&$\overline{351'}\supseteq\phantom{0}10\supseteq\phantom{0} 5$&$Q u^c-L\nu^c-4L' s$
$u^c e^c-d^c\nu^c+QQ-4d'^c s$\
$\overline{D}_6,\overline{T}_6$&$351'\supseteq\phantom{0}16\supseteq\phantom{0}\overline{5}$&$D_6,T_6$&$\overline{351'}\supseteq\phantom{0}\overline{16}\supseteq\phantom{0}5$&$-L s$
$-d^c s$\
$\overline{D}_7,\overline{T}_7$&$351'\supseteq 126\supseteq\phantom{0}\overline{5}$&$D_7,T_7$&$\overline{351'}\supseteq \overline{126}\supseteq\phantom{0} 5$&$-Qu^c-3L\nu^c$
$-u^c e^c-3 d^c\nu^c-QQ$\
$D_8,T_8$&$351'\supseteq126\supseteq 45$&$\overline{D}_8,\overline{T}_8$&$\overline{351'}\supseteq\overline{126}\supseteq \overline{45}$&$Qd^c+3Le^c$
$QL+u^c d^c$\
$D_9,T_9$&$351'\supseteq\overline{144}\supseteq\phantom{0}5$&$\overline{D}_9,\overline{T}_9$&$\overline{351'}\supseteq 144\supseteq\phantom{0}\overline{5}$&$-Qd'^c+4L'^c\nu^c+L'e^c$
$-QL'+u^c d'^c+4d' \nu^c$\
$\overline{D}_{10},\overline{T}_{10}$&$351'\supseteq\overline{144}\supseteq\phantom{0}\overline{5}$&$D_{10},T_{10}$&$\overline{351'}\supseteq 144\supseteq\phantom{0} 5$&$-L'\nu^c$
$-d'^c\nu^c$\
$D_{11},T_{11}$&$351'\supseteq\overline{144}\supseteq 45$&$\overline{D}_{11},\overline{T}_{11}$&$\overline{351'}\supseteq 144\supseteq\overline{45}$&$-dd'^c-3e'e^c$
$-QL'-u^c d'^c$\
$\phantom{D_{1}}\overline{T}_{12}$&$351'\supseteq 126\supseteq \overline{50}$&$\phantom{D_{1}}T_{12}$&$\overline{351'}\supseteq \overline{126}\supseteq 50$&$/$
$2u^c e^c- QQ$\
In this appendix, we write the definitions of various states needed in the paper. The SM singlet VEVs were already defined in Table \[table:singlet-labels\]. We supplement the list of definitions with Tables \[table:labels-doublets-triplets\] and \[table:weak-triplets\]. In Table \[table:labels-doublets-triplets\] we define the doublets $(1,2,+1/2)$, antidoublets $(1,2,-1/2)$, triplets $(3,1,-1/3)$ and antitriplets $(\bar{3},1,+1/3)$; these definitions are needed for DT splitting. In Table \[table:weak-triplets\], we define the weak triplets $(1,3,\pm 1)$ relevant for type II seesaw. All the states in the two-index $351'$ can be specified by writing the basis states of this representation by using two labels of the fundamental $27$. In this notation, both labels are SM representation in the $27$, assumed to have all the color and weak indices contracted in the correct manner to obtain the desired SM state in the $351'$. More details on this notation can be found in [@Bajc:2013qra].
0.2cm
label [$E_6\supseteq\SO(10)\supseteq\SU(5)$]{} p.n. label [$E_6\supseteq\SO(10)\supseteq\SU(5)$]{} state
----------------------- ---------------------------------------------------------------- ------------------ ----------------------- --------------------------------------------------------------- ------------------------------
$\overline{\Delta}_1$ $351'\supseteq 126\supseteq\phantom{0}\overline{15}$ $L\phantom{'}L$ $\Delta_1$ $\overline{351'}\supseteq\overline{126}\supseteq 15$ $\bar{L}\phantom{'}\bar{L}$
$\overline{\Delta}_2$ $351'\supseteq\overline{144}\supseteq\phantom{0}\overline{15}$ $L\phantom{'}L'$ $\Delta_2$ $\overline{351'}\supseteq 144\supseteq 15$ $\bar{L}\phantom{'}\bar{L}'$
$\overline{\Delta}_3$ $351'\supseteq\phantom{0}54\supseteq\phantom{0}\overline{15}$ $L'L'$ $\Delta_3$ $\overline{351'}\supseteq\phantom{0}54\supseteq 15$ $\bar{L}'\bar{L}'$
$\Delta_4$ $351'\supseteq\phantom{0}54\supseteq\phantom{0}15$ $L'^c L'^c$ $\overline{\Delta}_4$ $\overline{351'}\supseteq\phantom{0}54\supseteq\overline{15}$ $\bar{L}'^c \bar{L}'^c$
: Induced VEVs in weak triplet scalars $(1,3,\pm 1)$ leading to seesaw type II.\[table:weak-triplets\]
DT splitting and Goldstone modes\[section:simplified-DT\]
=========================================================
Analysis of the DT splitting in the group $\EE$ is complicated by the fact that a vacuum breaking to the SM group will automatically cause a doublet and a triplet mode to be massless. These massless modes are would-be Goldstone bosons: $\EE\to\mathrm{SM}$ causes the breaking of $78-12=66$ generators, with a doublet-antidoublet and triplet-antitriplet pair among them. One possible procedure to compute the condition for an extra massless mode in a matrix $M$, for which $\mathrm{det}\,M=0$, is to take $$\begin{aligned}
\frac{\lim_{\varepsilon\to 0}\big(\det(M-\varepsilon I)/\varepsilon\big)}{\langle e|f\rangle}&=0,\end{aligned}$$ where $e$ and $f$ are the already present left and right null-eigenvectors of $M$, respectively. We present below, however, a simplified procedure of computing the conditions of DT splitting in the presence of a Goldstone mode. Its advantages are that it is computationally less intensive and that no issues with singularities, such as $\langle e|f\rangle$=0, arise in the procedure.
Suppose we use a generic label $A$ for an $n\times n$ complex matrix. Although the true scalar mass-squared matrix is in fact the hermitian and positive definite matrix $A^\dagger A$, it is more efficient to work with $A$. $A$ might not necessarily be diagonalizable, but it has a singular value decomposition. The presence of a zero eigenmode in $A^\dagger A$ implies $$\begin{aligned}
\mathrm{det}\,A&=0.\end{aligned}$$ We will rotate this matrix into a basis, where the left and right Goldstone modes correspond to the first basis vector of the rows and columns, respectively. We first write $A$ in block form: $$\begin{aligned}
A&=\begin{pmatrix} m&\mathbf{m_R}^\dagger\\ \mathbf{m_L}&\mathbf{M}\\\end{pmatrix},\end{aligned}$$ where boldface small letters denote $n-1$ column vectors, and boldface capital letters denote matrices. Since $A$ has a massless mode, there exist left and right null-eigenvectors $\vec{e}_0$ and $\vec{f}_0$, respectively: $$\begin{aligned}
A^\dagger\vec{e}_0&=0,&A\vec{f}_0&=0.\label{null-eq}\end{aligned}$$ Providing we choose the phases such that the first components of $\vec{e}_0$ and $\vec{f}_0$ are positive, we define $(n-1)$ column vectors $\mathbf{e}$ and $\mathbf{f}$ via $$\begin{aligned}
\frac{\vec{e}_0}{|\vec{e}_0|}&=:\begin{pmatrix} \sqrt{1-\mathbf{e}^\dagger\mathbf{e}}\\\mathbf{e}\\\end{pmatrix}&
\frac{\vec{f}_0}{|\vec{f}_0|}&=:\begin{pmatrix} \sqrt{1-\mathbf{f}^\dagger\mathbf{f}}\\\mathbf{f}\\\end{pmatrix}.\end{aligned}$$ Writing the null-eigenvector conditions in equations in block form, we get $3$ independent equations (two $(n-1)$-vector, $1$ scalar): $$\begin{aligned}
\sqrt{1-\mathbf{e}^\dagger\mathbf{e}}\;\mathbf{m_R}+\mathbf{M}^\dagger\,\mathbf{e}&=0,\\
\sqrt{1-\mathbf{f}^\dagger\mathbf{f}}\;\mathbf{m_L}+\mathbf{M}^{\phantom{\dagger}}\,\mathbf{f}&=0,\\
m\sqrt{1-\mathbf{e}^\dagger\mathbf{e}}\sqrt{1-\mathbf{f}^\dagger\mathbf{f}}-\mathbf{e}^\dagger\,\mathbf{M}\,\mathbf{f}&=0\end{aligned}$$ These can for example be used to define $m$, $\mathbf{m_L}$ and $\mathbf{m_R}$ in terms of $\mathbf{M}$, $\mathbf{e}$ and $\mathbf{f}$. We now define a $n\times n$ unitary matrix $U(\mathbf{x})$, whose form will be useful for rotating the basis of $A$: in block form, $U$ is written as $$\begin{aligned}
U(\mathbf{x}):=\begin{pmatrix} \sqrt{1-\mathbf{x}^\dagger\mathbf{x}}& \mathbf{x}^\dagger\\ -\mathbf{x}&\Lambda(\mathbf{x})\\\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\Lambda(\mathbf{x}):=I-\frac{\mathbf{x}\,\mathbf{x}^\dagger}{1+\sqrt{1-\mathbf{x}^\dagger \mathbf{x}}}.\end{aligned}$$ It is possible to check explicitly that $U$ is indeed unitary and that the following relations hold: $$\begin{aligned}
U(\mathbf{x})\,U(\mathbf{x})^\dagger&=I,\\
U(\mathbf{x})^{-1}=U(\mathbf{x})^\dagger&=U(-\mathbf{x}),\\
\Lambda(\mathbf{x})^\dagger&=\Lambda(\mathbf{x})=\Lambda(-\mathbf{x}),\\
\Lambda(\mathbf{x})^{-1}&=I+\frac{\mathbf{x}\,\mathbf{x}^\dagger}{\sqrt{1-\mathbf{x}^\dagger\mathbf{x}}\;(1+\sqrt{1-\mathbf{x}^\dagger\mathbf{x}})},\\
\mathrm{det}\Lambda(\mathbf{x})&=1-\frac{\mathbf{x}^\dagger \mathbf{x}}{1+\sqrt{1-\mathbf{x}^\dagger\mathbf{x}}}.\label{equation:lambda-det}\end{aligned}$$ The properly rotated matrix $A$, denoted by $A'$, is then by explicit computation equal to $$\begin{aligned}
A'&:=U(\mathbf{e})\,A\,U(\mathbf{f})^{-1}=\begin{pmatrix} 0&\mathbf{0}^\dagger\\ \mathbf{0}& \mathbf{M_{rot}}\\\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\mathbf{M_{rot}}&=\Lambda(\mathbf{e})^{-1}\mathbf{M}\,\Lambda(\mathbf{f})^{-1}.\end{aligned}$$ We have indeed rotated into a basis, where the first column and row correspond to the zero eigenmodes. Crucially, the formula for the rotated block has only one term, where a simple biunitary rotation is performed on the original block $\mathbf{M}$; another zero-eigenmode can now be simply imposed by taking $\mathrm{det}\mathbf{M_{rot}}=0$. But since we now have $$\begin{aligned}
\mathrm{det}\mathbf{M}&=\mathrm{det}\mathbf{M_{rot}}\;\mathrm{det}\big(\Lambda(\mathbf{e})\Lambda(\mathbf{f})\big).\end{aligned}$$ it is sufficient to impose $\det\mathbf{M}=0$. The only possible caveat is the possibility that either $\mathrm{det}\,\Lambda(\mathbf{e})=0$ or $\mathrm{det}\,\Lambda(\mathbf{f})=0$; considering equation and that $\mathbf{e}$ and $\mathbf{f}$ are parts of normalized vectors, this can happen only if $\mathbf{e}^\dagger\mathbf{e}=1$ or $\mathbf{f}^\dagger\mathbf{f}=1$, which would imply that the zero modes of $A$ have a zero component in the direction of the first vector of the original basis. But since the eigenvalues of $A^\dagger A$ (the determinant of $A^\dagger A-\lambda I$) do not change if we rearrange the rows or the columns of $A$, we can always rearrange the original basis so that we take the $i$-th row and the $j$-th column to be the preferred one for the left and right null-eigenmodes, respectively.
The main result thus states the following: if an $n\times n$ complex matrix $A$ has a zero mode present, an additional zero mode is obtained by demanding the $(i,j)$-th minor of $A$ to vanish (the subdeterminant of $A$, when the $i$-th row and $j$-th column are removed), where the indices $i$ and $j$ can be arbitrarily chosen, as long as the left and right null-eigenvectors of $A$ have respectively a nonzero $i$-th and $j$-th component.
An alternative vacuum\[section:vacuum2\]
========================================
The Yukawa fit presented in this paper is based on the vacuum computed in section \[section:SSB\]. Since we have not been able to obtain (yet) a full classification of all vacua in this $\EE$ model, we cannot conclude much about the broader possibilities of suitable vacua (especially the Yukawa sector), but we were able to find one other alternative Standard Model vacuum. We can obtain it by taking the following ansatz of vanishing VEVs: $$\begin{aligned}
c_2=d_2=e_5=f_5=e_2=f_2=u_1=u_2=y&=0.\label{equation:alt-ansatz}\end{aligned}$$ This ansatz is similar to the ansatz of the original vacuum, but the vanishing of $e_5,f_5$ is now paired up with the vanishing of $c_2,d_2$ instead of $c_1,d_1$. For the purposes of this appendix, we omit the specific form of the solutions, as well as other details, such as the check that the unbroken group is indeed that of the Standard Model.
Looking at the Yukawa terms in equation , we see that the alternative vacuum with $c_2=f_5=0$ decouples the $16_F$ from the $10_F$ of $\SO(10)$ (in leading order of $m_{EW}/M_{GUT}$). The heavy vector-like exotics (in the down-quark sector and charged lepton sector) in the fermionic $27_F$ are thus purely in the $10_F$ part. The low-energy mass matrices are thus simply those for the $16_F$, the analysis of the Yukawa sector becomes linear, and numerically one can make use of the fit for the minimal supersymmetric $\SO(10)$ model [@Babu:1992ia; @Bajc:2001fe; @Fukuyama:2002ch; @Bajc:2002iw; @Goh:2003sy; @Goh:2003hf; @Bertolini:2004eq; @Babu:2005ia; @Bertolini:2006pe; @Bajc:2008dc; @Joshipura:2011nn; @Altarelli:2013aqa; @Dueck:2013gca] with the Higgs in the $10$ and $\overline{126}$ coupling to fermion pairs in $16_F^2$.
The intriguing possibility of this alternative vacuum, which recovers the $\SO(10)$ limit, is somewhat marred by issues in DT splitting. Given the ansatz in equation , the doublet and triplet mass matrices become block diagonal with the following block form (with the basis of the barred states rearranged in the same order as for unbarred): $$\begin{aligned}
\begin{bmatrix}D_0 &D_3 &D_6 &D_9 &D_{10} &D_{11}\\\end{bmatrix},\quad \begin{bmatrix}D_1 &D_2 &D_4 &D_5\\\end{bmatrix},\quad \begin{bmatrix}D_7 &D_8\\\end{bmatrix},\\
\begin{bmatrix}T_0 &T_3 &T_6 &T_9 &T_{10} &T_{11}\\\end{bmatrix},\quad \begin{bmatrix}T_1 &T_2 &T_4 &T_5\\\end{bmatrix},\quad \begin{bmatrix}T_7 &T_8& T_{12}\\\end{bmatrix}.\end{aligned}$$ The Goldstone modes, for example, turn out to be in the first block. The problem now is, however, that a fine-tuning is block specific: the light Higgs lives only in one of the blocks. We see from equation that the low-energy fermionic mass matrices $M_D^T$ and $M_E$ are now controlled solely by the EW VEVs $\bar{v}_2$, $\bar{v}_4$ and $\bar{v}_8$. In the $\bar{v}_8=0$ case, we get the unwanted mass relation $M_D^T=M_E$, but $\overline{D}_{8}$ is in a separate block compared to $\overline{D}_2$ and $\overline{D}_4$. A realistic pattern of fermion masses would thus require a double fine-tuning: one in the second block and one in the third block. Due to this feature, we consider this vacuum to be of less interest: beside the aesthetically unpleasing extra fine-tuning, the additional light Higgs pair $H'_u+H'_d$ pair gives large threshold corrections in the running of the gauge couplings, possibly spoiling unification. The model contains many heavy states though, so the situation regarding the RGE is not clear-cut. For determining the feasibility of this vacuum, further investigation would be necessary.
We conclude this section of the Appendix with a brief elaboration on which kind of ansatz is suitable for a good vacuum, i.e. we motivate equations – and . The considerations will be very similar to the ones in the $\EE$ breaking sector, where the $78$ is omitted [@Bajc:2013qra] and a full classification of vacua is known. Suppose we look for a specific vacuum solution: we want it to be as simple as possible (it has as many vanishing VEVs as possible), yet it needs to be able to break $\EE$ to the SM group. Due to the $D$-terms in SUSY, we assume a conjugate-symmetric ansatz, where a vanishing VEV in $27$ or $351'$ implies that the corresponding (conjugate) VEV in the $\overline{27}$ or $\overline{351'}$ also vanishes, and vice versa. First, we identify the $\SU(5)$ breaking VEVs from Table \[table:singlet-labels\]: $e_4$, $e_5$, $f_4$, $f_5$ and $y$. The $F_y$ equation of motion automatically implies $y=0$. Since $\SU(5)$ needs to be broken, either the pair $e_4$, $f_4$ is non-vanishing, or the pair $e_5$, $f_5$. Considerations of alignment symmetry in [@Bajc:2013qra] imply that the choice is irrelevant, since these pairs are exchanged if one changes the embedding of $\SO(10)$ in $\mathrm{E}_6$, such that the two $\bar{5}$s of $\SU(5)$ in the $27$ are exchanged. Therefore one pair needs to be necessarily non-zero, while we can try a vanishing ansatz for the other pair. In the $27+\overline{27}$ part of the breaking sector, the pairs $c_1$, $d_1$ and $c_2$, $d_2$ also get exchanged under alignment symmetry; we assume one pair to be non-zero (such that the $27+\overline{27}$ part does indeed contribute to the symmetry breaking), but we can again try setting the other pair to vanish in the simple ansatz. The choice of the vanishing pair now points to either the scenario of the main vacuum of this paper, or the the alternative vacuum presented in this appendix. The remaining part of the ansatz, $e_2=f_2=0$ (which transforms into itself under alignment symmetry), is suggested from the solutions in the model without the $78$, while $u_1=u_2=0$ then follows as a consequence of the $F$-terms. We finish the ansatz discussion with the following points:
- The main vacuum solution of the paper follows from the ansatz, which is a direct extension of the solution ansatz in the absence of the $78$ [@Bajc:2013qra]. There, the EOMs were simpler, and this ansatz actually represented the most general SM solution once gauge freedom and the $F$-term equations were accounted for. Note that only the ansatz can be extended, the main solution itself is not merely an extension of the solution when the $78$ is omitted.
- The alternative ansatz leads to a SM vacuum only after the $78$ was included; there is no such option if the $78$ is omitted.
- A complete classification would tell us, whether still other nonequivalent vacua exist, where the VEVs are non-vanishing. If the $78$ is omitted, such vacua did not exist, but we expect this situation to change due to more terms and more VEVs in the EOM.
Proton decay \[Proton-decay\]
=============================
For completeness let’s summarize the analysis of $D=5$ proton decay in this model, similar to the analysis done in [@Bajc:2013qra], obtaining analogous results. The low-energy operators in the superpotential, which are relevant for proton decay, are $$\begin{aligned}
W\big|_{\textrm{proton}}&=-\Big[\big(\overline{C}_1^{inA}-\overline{C}_1'^{imA}(X^T)_{m}{}^{n}\big)\big[(1+X^\ast X^T)^{-1/2}\big]_{n}{}^{j}\;(\hat{\mathcal{M}}_T^{-1})_{AB}\; C_1^{klB}\Big]\;Q_i \hat{L}_j Q_k Q_l\nonumber\\
&\quad -\Big[\big(\overline{C}_2^{njA}+\tfrac{2}{3}\overline{C}_2'^{mjA}(X^T)_{m}{}^{n}\big)\big[(1+\tfrac{4}{9}X^\ast X^T)^{-1/2}\big]_{n}{}^{i}\;(\hat{\mathcal{M}}_T^{-1})_{AB}\; C_2^{klB}\Big]\;\hat{d}^c_i u^c_j u^c_k e^c_l,\end{aligned}$$ with the $X$ defined in equation and the mass matrix of $\mathcal{M}_{\textrm{T}}$ as already defined in equation of section \[section:DT-splitting\]. Note that the $\mathcal{M}_{\textrm{T}}$ has a zero eigenmode corresponding to the would-be Goldstone, so its inverse cannot be directly computed; we instead write $\hat{\mathcal{M}}_{T}^{-1}=\lim_{M\to\infty}(\mathcal{M}_T+M\;e\,f^T)^{-1}$, with $e$ and $f$ being the left and right column-eigenvectors of $\mathcal{M}_T$, respectively.
The $C$ coefficients are computed to be
$$\begin{aligned}
2\;C_1^{ijA}&=-\yd^{ij}\,\delta^{A}{}_{1}+\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{5}
-\tfrac{1}{2\sqrt{6}}\,\yt^{ij}\,\delta^{A}{}_{7}-\tfrac{1}{2\sqrt{3}}\,\yt^{ij}\,\delta^{A}{}_{12},\\
2\;C_2^{ijA}&=-\yd^{ij}\,\delta^{A}{}_{1}+\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{5}
-\tfrac{1}{2\sqrt{6}}\,\yt^{ij}\,\delta^{A}{}_{7}+\tfrac{2}{2\sqrt{3}}\,\yt^{ij}\,\delta^{A}{}_{12},\\
2\;\overline{C}_1^{ijA}&=-\yd^{ij}\,\delta^{A}{}_{2}+
\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{4}+\tfrac{1}{2\sqrt{2}}\,\yt^{ij}\,\delta^{A}{}_{8},\\
2\,\overline{C}_1'^{ijA}&=\phantom{-}\yd^{ij}\,\delta^{A}{}_{3}-
\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{9}-\tfrac{1}{2\sqrt{2}}\,\yt^{ij}\,\delta^{A}{}_{11},\\
2\;\overline{C}_2^{ijA}&=-\yd^{ij}\,\delta^{A}{}_{2}+
\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{4}-\tfrac{1}{2\sqrt{2}}\,\yt^{ij}\,\delta^{A}{}_{8},\\
2\,\overline{C}_2'^{ijA}&=\phantom{-}\yd^{ij}\,\delta^{A}{}_{3}-
\tfrac{1}{2\sqrt{10}}\,\yt^{ij}\,\delta^{A}{}_{9}+\tfrac{1}{2\sqrt{2}}\,\yt^{ij}\,\delta^{A}{}_{11}.\end{aligned}$$
We see that the $C$-coefficients are the same as the coefficients in [@Bajc:2013qra], if we cross out the contributions from the extra Yukawa term in that model. More specifically, notice that there are no $A=0$ contributions from triplets/antitriplets in the new representation $78$, since the $78$ is not present in the Yukawa sector (it does not couple to two $27_F$’s).
Once the model parameters are fit to the experimental values of the fermion masses and mixings, the four-fermion amplitude mediating proton decay is fixed and thus potentially dangerous, but it can always be suppressed by implementing a split supersymmetric scenario without changing any other conclusion.
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[^1]: babu@okstate.edu
[^2]: borut.bajc@ijs.si
[^3]: vasja.susic@ijs.si
[^4]: The sparsity of SU(5) breaking vacuum expectation values in the respective mass matrices is likely to be the fundamental origin of this problem.
[^5]: Notice that the Standard Model singlet VEVs have been denoted by $u_1$, $u_2$, $v$, $w$ and $y$. The notation from [@Buccella:1987kc] is changed due to the states now being those which have well defined transformation properties under the $\SU(5)$ and $\SO(10)$ subgroups of $\EE$. The connection between the two notations is $u_1\propto a_1$, $u_2\propto a_2$, $w\propto a_3-b_3$, $v\propto -\sqrt{3}a_3+2a_4-\sqrt{3}b_3$, $y\propto a_3+\sqrt{3}a_4+b_3$, with the usual normalization $\langle 78^{i}{}_{j} 78^{\ast}_ i{}^{j}\rangle=|u_1|^2+|u_2|^2+|v|^2+|w|^2+|y|^2$.
[^6]: This ansatz is motivated by the ansatz used in the model with the $78$ omitted [@Bajc:2013qra], where a classification of vacua is known. See Appendix \[section:vacuum2\] for further discussion.
[^7]: The $27$ contains $2$ singlets, but $3$ doublets, so the singlets cannot be pushed to all the doublets with a $t^a$ generator, and thus one doublet component is missing. The representation $351'$ has $5$ singlets and $8$ doublets, but the projection relation $d_{ijk}\,351'^{jk}=0$ reshuffles the definitions of singlets and doublets, so there is no loss of doublet components.
[^8]: We use the standard notation $Q$, $L$, $u^c$, $d^c$ and $e^c$ for the representations $(3,2,+1/6)$, $(1,2,-1/2)$, $(\bar{3},1,-2/3)$, $(\bar{3},1,1/3)$ and $(1,1,1)$, respectively. The lepton doublet contains the electron $e$ and the neutrino $\nu$.
[^9]: In both parametrizations of encoding the $D$-terms, we wrote them so that we retained control to make $f_1$ and $f_3$ potentially small; this will be important for the neutrino sector, as described later in this section.
[^10]: This can be easily seen from the expression for $\lambda_2$ in (\[lambda2\]): if $f_4\to0$, we need either $e_3\to0$ or $e_4\to0$; both lead to an SU(5) invariant vacuum with $e_4,f_4\to0$, see .
|
---
author:
- 'E. Spitoni [^1]'
- 'F. Matteucci'
- 'H. Jönsson'
- 'N. Ryde'
- 'D. Romano'
date: 'Received xxxx / Accepted xxxx'
title: 'Fluorine in the Solar Neighborhood: Chemical Evolution Models” '
---
-.2in
Introduction
============
The aim of this paper is to study the evolution of fluorine ($^{19}$F) abundance in the solar neighborhood, by means of a detailed chemical evolution model. The origin of fluorine is still uncertain and widely debated in literature. The production of the only stable isotope, $^{19}$F, is strictly linked to the physical conditions in stars. In particular, we can identify several stellar sites for fluorine production:
i) Asymptotic giant branch (AGB) stars. For solar metallicity the main process of $^{19}$F production is related to nuclear reaction on $^{14}$N including $\alpha$, neutron, and proton captures. On the other hand, at lower metallicity the $^{19}$F production depends on $^{13}$C (Cristallo et al. 2014).
ii) Wolf-Rayet (W-R) stars can be important producers of fluorine which is injected into ISM by their strong stellar winds. Also here, the chain of reactions leading to $^{19}$F production starts from $^{14}$N, which is normally produced during the CNO cycle as a secondary element. However, in massive stars N can be produced as a primary element if they suffer strong rotation, which produces effects similar to those to the dredge-up mechanism in AGB stars. In this case, however, the primary N production is limited only to very metal poor stars (Z$<10^{-5}$) as shown by Meynet & Maeder (2002). The difference secondary/primary N is important, since in the case of secondary N the F behavior would also follow that of a secondary element, namely depending on the original stellar metallicity.
iii) Type II supernovae (SNe II). These SNe can produce $^{19}$F via the neutrino ($\nu$) process. In fact, although neutrinos are characterized by small cross sections, the great amount of them released during the core-collapse turns $^{20}$Ne in the outer envelopes of the collapsing star into $^{19}$F (Woosley & Haxton 1988).
iv) Novae can also in principle be $^{19}$F producers (José & Hernanz 1998), although the yields are still uncertain. In classical novae the reaction chain $^{17}$O$(p, \gamma)^{18}$F$(p, \gamma) ^{19}$Ne is the mechanism involved in the synthesis of $^{19}$F through the production of the short-lived, $\beta^+$-unstable nucleus $^{19}$Ne, which is partially transferred by convection toward the outer cooler layers of the envelope, where it decays into $^{19}$F.
Recently, new observational data (related to 49 K giants with temperatures low enough to show an HF line at 2.3 $\mu$m) by J[ö]{}nsson et al. (2017a) showed that the $\nu$-process is not the main contributor of fluorine production in the solar vicinity, at variance with theoretical predictions. J[ö]{}nsson et al. (2017b) estimated the kinematic probability that stars belong to the thin disk, thick disk, and/or halo. They showed that the majority of the observed stars are part of the Galactic thin disk
The observed \[F/Fe\] vs. \[Fe/H\] and \[F/O\] vs. \[O/H\] ratio both follow and increasing trend, in contrast to what would expect if the $\nu$-process was the dominant fluorine nucleosynthesis source in the solar vicinity, constraining its possible contribution to the cosmic fluorine budget. This tension between data and the $\nu$-process in the solar neighborhood, is confirmed by the observed secondary relation between fluorine and oxygen in J[ö]{}nsson et al. (2017a). This observed secondary behavior can constrain the stellar models of AGB and W-R stars. Hence, AGB stars, W-R stars, SNe Ia, novae, and explosive nucleosynthesis in SNe II without $\nu$ process, are the possible fluorine production sites to be tested in detail chemical evolution models for the solar neighborhood.
In this paper, we present chemical evolution models designed to reproduce the data by J[ö]{}nsson et al. (2017a) in the solar neighborhood testing different nucleosynthesis prescriptions for the fluorine yields. We will present results related both to an update version of the classical “two-infall” model introduced by Chiappini et al. (1997) in which we follow the evolution of the halo-thick disk and thin disk phases, and to a detailed chemical evolution model which follows only the thin disk evolution. The F production sources considered in this paper are: AGB stars (Karakas 2010), massive stars (Kobayashi et al. 2006) and W-R stars (Meynet & Arnould 2000). We will also discuss the case of novae as possible sources for F production adopting the nucleosynthesis yields by José & Hernanz (1998).
Previous papers (Meynet & Arnould 2000, Renda et al. 2004, Kobayashi et al. 2011a) have computed fluorine evolution in the Galaxy. For the first time, Meynet & Arnould (2000) showed that W-R stars could be significant contributors to the solar system abundance of fluorine using a simple model for the chemical evolution. In Renda et al. (2004) the impact of fluorine nucleosynthesis, in both W-R and AGB stars, were considered in chemical evolution model and concluded that the contribution of W-R stars is necessary to reproduce the F abundance in solar vicinity. Kobayashi et al. (2011a) found that the main effect of $\nu$-process of core-collapse supernovae on the evolution of fluorine in the solar neighborhood, is the presence of a plateau at high \[F/O\] values for \[O/H\] $<$ -1.2 in the \[F/O\] vs \[O/H\] relation. As underlined above, this plateau is in contrast with the most recent observational data by J[ö]{}nsson et al. (2017a). Novae have never been included so far in chemical models predicting fluorine.
The paper is organized as follows: in Section 2 we describe the chemical evolution models for the solar vicinity adopted in this paper, in Section 3 the nucleosynthesis prescriptions are described. In Section 4 our results concerning the fluorine abundances in the solar neighborhood predicted by our chemical evolution models are reported. In Section 5, we discuss the effects of the novae nucleosynthesis on fluorine production. In Section 6 we draw our main conclusions.
\[models\]
[c|ccccc]{}
\
Models &AGB stars& SNe Ia& SNe II& Wolf Rayet stars\
&& &&\
&Karakas (2010)& Iwamoto et al. (1999)&Kobayashi et al. (2006) & Meynet & Arnould (2000)\
\
F1& yes & yes &yes &no\
F2 & no & yes &yes &no\
F3& yes $\times$ 1.5 & yes &yes&no\
F4& yes $\times$ 2 & yes &yes &no\
F5& yes $\times$ 3 & yes &yes &no\
F6 &no & no &no &yes\
F7 & yes & yes &yes &yes\
F8& yes & yes &yes &yes $\times$ 2\
F9& yes $\times$ 2 & yes &yes &yes\
\[tab1\]
The chemical evolution models for the solar neighborhood
========================================================
In this Section, we describe the main characteristics of the adopted chemical evolution models for the solar neighborhood in this work:
1. The classical “two-infall” model of Chiappini et al. (1997, 2001). It is assumed that the Galaxy formed by two independent infalls of primordial gas. In the first episode, occurred on short time-scales, the halo-thick disk components have been formed, while in the second one the thin disk was created on longer time-scales.
2. Following Matteucci & Fran[ç]{}ois (1989) approach, we use the “one-infall” chemical evolution model only for the thin disk evolution in the solar neighborhood.
The “two-infall” model
----------------------
As stated in Section 2, J[ö]{}nsson et al. (2017a) data sample is related to thick-disk, thin-disk and halo stars. For this reason, first we want to reproduce those data adopting the classical “two-infall” model capable to trace the chemical evolution of halo-thick disk and thin disk phases.
The Galaxy is assumed to have formed by means of two main infall episodes. The accretion law for a certain element $i$ at the time $t$ in the solar vicinity is defined as: $$\mathcal{A}(t,i)=X_{A_{i}}\left[ c_1 \, e^{-t/ \tau_{H}}+ c_2 \, e^{-(t-t_{max})/ \tau_{D}} \right].
\label{a}$$ The quantity $X_{A_{i}}$ is the abundance by mass of the element $i$ in the infalling gas, while $t_{max}=1$ Gyr is the time for the maximum infall on the thin disk, $\tau_{H}= 0.8$ Gyr is the time-scale for the creation of the halo and thick-disk and $\tau_{D}=7$ Gyr is the timescale to build up the thin disk in the solar neighborhood (as suggested by fitting the G-dwarf metallicity distribution). Here, we assume that the abundances $X_{A_{i}}$ show primordial gas compositions. Finally, the coefficients $c_1$ and $c_2$ are obtained by imposing a fit to the observed current total surface mass density in the solar neighborhood. A threshold gas density of 7 M$_{\odot}$pc$^{-2}$ in the star formation process (Kennicutt 1989, 1998, Martin & Kennicutt 2001) is also adopted for the disk.
The most recent observational data by the Gaia-ESO Survey (Recio-Blanco et al. 2014; Rojas-Arriagada et al. 2017), APOGEE (Hayden et al. 2015) and AMBRE (Mikolaitis et al. 2017) confirmed the existence of two distinct sequences corresponding to thick and thin disk stars. In Grisoni et al. (2017) it was shown that the “two-infall” model is able to perfectly reproduce the AMBRE data in the solar neighborhood, when applied to the thick and thin disks without including the halo.
The “one-infall” model
----------------------
Because of the fact that the majority of the data presented in J[ö]{}nsson et al. (2017a) is supposed to be thin disk stars, we also considered a chemical evolution model which only follows the thin disk.
To reproduce the chemical evolution of the thin disk, we adopt an updated version of the “one-infall” chemical evolution model presented by Matteucci & Fran[ç]{}ois (1989) using the most recent nucleosynthesis yield of Romano et al. (2010).
![The abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood for the “two-infall” chemical evolution model adopting different prescriptions for the channels of the fluorine production. For oxygen we keep Romano et al. (2010, model 15) prescriptions. With the blue line we represent the model F1 of Table \[tab1\] where fluorine is assumed to be produced by both AGB stars, SNe Ia and SNe II. With the green line we have the model F2 (contributions to fluorine production only from SNe Ia and SNe II). The models F3, F4, F5 in which the AGB yields are multiplied by factors of 1.5, 2 and 3, respectively (along with SN Ia SN II channels) are drawn with brown, black, and red lines. Observational data by J[ö]{}nsson et al. (2017a) are indicated with cyan circles.[]{data-label="2IM_FO"}](figure_1.png)
![The abundance ratio \[F/Fe\] as a function of \[Fe/H\] in the solar neighbourhood for the “two-infall” chemical evolution model adopting different prescriptions for the channels of fluorine production. Model lines are the ones described in Fig. \[2IM\_FO\]. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles, whereas the data taken by Pilachowski & Pace (2015) are presented with gray pentagons.[]{data-label="2IM_FFE_AGB"}](figure_2.png)
The infall rate in the thin-disk for a given element $i$ at the time $t$, and in the solar vicinity is defined as: $$\label{infall}
\mathcal{B}(t,i)= X_{A_i} c_2 \, e^{-\frac{t}{\tau_D}}.$$ All model parameters are the same as the thin disk phase of the “two-infall” model described in Section 2.1.
Nucleosynthesis prescriptions for O, Fe, and F
==============================================
In this work our principal aim is to analyze in detail the contribution of the different channels for the production of fluorine in the solar neighborhood, by comparing the predicted abundance ratios \[F/Fe\] vs \[Fe/H\] and \[F/O\] vs \[O/H\] by our chemical evolution models with the data.
We start our study by considering the set of nucleosynthesis yields of the Romano et al. (2010) best model (their model 15) for O, Fe and F (this set of yields has been adopted by Brusadin et al 2013, Micali et al. 2013, Spitoni et al. 2016).
In particular, these yields are:
- For low-and intermediate-mass stars (0.8-8 M$_{\odot}$), we consider the metallicity-dependent stellar yields of Karakas (2010) with thermal pulses. These stars contribute to fluorine, negligibly to O and give no contribution to Fe.
- For SNe Ia, the adopted nucleosynthesis prescriptions are from Iwamoto et al. (1999). These SNe contribute significantly to Fe and negligibly to O and F.
- For massive stars (M$>$8 M$_{\odot}$), which are the progenitors of either SNe II or hypernovae (HNe), depending on the explosion energy, we assume the metallicity-dependent He, C, N and O stellar yields, as computed with the Geneva stellar evolutionary code, which takes into account the combined effect of mass loss and rotation (Meynet & Maeder 2002, Hirschi et al. 2005, Hirschi 2007, Ekström et al. 2008). The Kobayashi et al. (2006) yields not including $\nu$-nucleosynthesis are considered for fluorine. J[ö]{}nsson et al. (2017a) compared their new observational data with chemical evolution models by Kobayashi et al. (2011b) in presence of $\nu$-nucleosynthesis. The model showed a plateau at high \[F/O\] values at low \[O/H\] in contrast with the new data. For this reason we have not taken into account this kind of nucleosynthesis in this work.
In this paper we test also the effects of W-R yields. For W-R stars we assume the F production by the models with mass loss by Meynet & Arnould (2000). These yields do not include oxygen.
Meynet & Arnould (1993) have suggested that W-R stars could significantly contaminate the Galaxy with $^{19}$F. In their scenario, $^{19}$F is synthesized at the beginning of the He-burning phase from the $^{14}$N left over by the previous CNO-burning core, and is ejected in the interstellar medium when the star enters its WC phase. Since the mass loss depends on stellar metallicity, the $^{19}$F yields are metallicity-dependent.
For all the elements heavier than oxygen, namely Fe in this study, we assume the Kobayashi et al. (2006) yields.
![The abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood for the “two-infall” chemical evolution model testing different prescriptions for the channels of fluorine production. With the blue line we represent the model F1 of Table 1. With the red line we consider also the Wolf-Rayet yield by Meynet & Arnould (2000). Model F6 where only Wolf-Rayet stars contribute to the fluorine production is indicated with the brown line in this plot. The black line is the model F8, similar to the F7 model but with the Wolf-Rayet yields multiplied by a factor of 2. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles. []{data-label="WR2_2infall"}](figure_3.png)
![The abundance ratio \[F/Fe\] as a function of \[Fe/H\] in the solar neighbourhood for the “two-infall” chemical evolution model adopting different fluorine yields. Model lines are the ones described in Fig. \[WR2\_2infall\]. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles whereas the data taken by Pilachowski & Pace (2015) are presented with gray pentagons.[]{data-label="WR2_FE_2infall"}](figure_4.png)
Results: testing different nucleosynthesis prescriptions for fluorine
=====================================================================
In this Section, we show the results related to the different prescriptions for the fluorine yields with the aim of reproducing the observational data by J[ö]{}nsson et al. (2017a) both for the “two-infall” model and for a model where it is studied only the evolution of the thin disk (“one-infall” model). In Table 1 we present the list of all models tested here varying the fluorine yield prescriptions. For each model (F1..F9) it is indicated the presence of AGB stars with yields by Karakas (2010) in column 2, of Type Ia SNe with yields by Iwamoto et al. (1999) in column 3, of massive stars with yields by Kobayashi et al. (2006) in column 4, and W-R stars with yields by Meynet & Arnould (2000) in column 5. As stated above we started our work analyzing the effects of the nucleosynthesis prescriptions of the best model by Romano et al. (2010) (their model 15) on the \[F/O\] vs. \[O/H\] and \[F/Fe\] vs. \[Fe/H\] ratios in the light of the new data by J[ö]{}nsson et al. (2017a). We label this model as F1.
The solar values adopted in this work for oxygen and iron are the Asplund (2009) ones, whereas for fluorine the one computed by Maiorca et al. (2014) is used to be coherent with the J[ö]{}nsson et al (2017a) work.
![The abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood for the “two-infall” chemical evolution model. With the blue line we represent the model F1 of Table 1 where fluorine is assumed to be produced by both AGB stars, SNe Ia and SNe II. Model, similar to the F7 model but with the Wolf-Rayet yields multiplied by a factor of 2 is drawn with the black line. With the magenta is represented the model F9 where the contribution of SNe II and SNe Ia are considered along with W-R stars. In this model the AGB yields are multiplied by a factor of 2. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles.[]{data-label="F9_2infall"}](figure_5.png)
We stress that in this work we analyze in detail only fluorine yields; for the other elements shown in this paper (O,Fe), we use the model 15 prescriptions by Romano et al. (2010), which does not include fluorine though.
The “two-infall” model results
------------------------------
J[ö]{}nsson et al. (2017a) compared their new observational data with chemical evolution models by Kobayashi et al. (2011a, 2011b) for thick and thin disk stars where it was considered the contribution of AGB stars and $\nu$ processes to the fluorine production. The main problem of those models is that they are not capable to reproduce the slope of the secondary behavior of the new data presented by J[ö]{}nsson et al. (2017a).
Here, we show the results related to the “two-infall” model of Chiappini et al. (1997, 2001) updated by Romano et al. (2010) for the chemical evolution of the solar neighborhood. In Fig. \[2IM\_FO\] we show the abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood adopting different prescriptions for the channels of fluorine production. First, we consider for F and O the nucleosynthesis yields adopted by the model15 of Romano et al. (2010) (model F1 in Table 1): the ones of Karakas (2010) for AGB stars with thermal pulses, Iwamoto et al. (1999) for Type Ia SNe, and Kobayashi et al. (2006) for massive stars for F and the Geneva stellar evolutionary code for O.
In Fig. \[2IM\_FO\], model F1 clearly shows the transition between the halo-thick disk phase and the thin disk. Because of the presence of a threshold for the surface gas density in the star formation we have a gap in the star formation history (see Spitoni et al. 2016) in the correspondence of the beginning of the second gas infall.
At variance with the models of Kobayashi et al. (2011a,b) shown in Fig. 3 of J[ö]{}nsson et al. (2017a), the F1 model shows a decrease of \[F/O\] abundance values for \[O/H\] larger than 0.2 dex. This behavior is due to the different prescriptions for the yield of oxygen we adopted. In fact, in Romano et al. (2010) the oxygen produced by massive stars is coming from the Geneva tracks with mass loss and rotation (Meynet & Maeder 2002, Hirschi et al. 2005, Hirschi 2007, Ekström et al. 2008). Model F1 is not able to reproduce the observational trend of J[ö]{}nsson et al. (2017a) data for over-solar values of \[O/H\], and the main problem is that we are not capable to reach the high \[F/O\] observed values.
In Fig. \[2IM\_FO\], it is also proven that adopting the Romano et al. (2010) best choice for the yields (their model 15) the main contribution to the fluorine is given by AGB stars. In fact, model F2 (see Table 1), in which the fluorine is created only by means of SN Ia and SN II channels, shows a vary small amount of fluorine production: the \[F/O\] abundance at maximum reaches the value of -0.8 dex.
However, the AGB yields of F are still uncertain. In fact, the recent measurements by Indelicato et al. (2017) and He et al. (2017) of the $^{19}$F$(p, \alpha)^{16}$O reaction, suggest that the computed yields from AGB-stars should be revised and higher values might be expected. Therefore, in Fig. \[2IM\_FO\] the effects of increasing by hand the contribution by AGB stars are presented. Multiplying the yields by factors of 1.5, 2 and 3 (models F3, F4, and F5 in Table 1, respectively), we have a better agreement with the data.
This kind of procedure has been widely used in chemical evolution models in the past. For example, in Fran[ç]{}ois et al. (2004) best model the Mg yields related to the massive stars of Woosley & Weaver (1995) have been multiplied by a factor of two. More recently, in Maas et al. (2017) phosphorus yields by Kobayashi et al. (2006) have been multiplied by a factor of 2.75 with the aim of reproducing the \[P/Fe\] vs \[Fe/H\] abundance ratios in the solar neighborhood as suggested by the new observational stellar data using the chemical evolution model by Cescutti et al. (2012). From Fig. \[2IM\_FO\] we see that models F4 and F5 are able to:
- reach the observed high \[F/O\] ratio values for over-solar \[O/H\] abundances.
- trace the observed secondary slope of the data for sub-solar \[O/H\] values.
The data in Fig. \[2IM\_FO\] present a change in the slope of the \[F/O\] vs \[O/H\] abundance ratios for values larger than \[O/H\]=0. The secondary behavior of the fluorine is not perfectly reproduced for over-solar values of \[O/H\], in fact models F3 and F4 show a decrease of the \[F/O\] abundance ratio for \[O/H\] $>$ 0.
A common problem of models F3, F4, and F5, is that the transition between the halo-thick disk phase and the thin disk has a big impact in the \[F/O\] vs \[O/H\] ratios adopting the Romano et al. (2010) model15 set of yields. We will show in the next subsection related to the “one-infall” results (the chemical evolution only for the thin disk), that a better agreement with the data is achieved. We recall here that the majority of the data presented by J[ö]{}nsson et al. (2017a) are parts of the thin disk system.
However, the predicted fluorine solar values are in agreement with solar ones of Maiorca et al. (2014) as shown in Table 2. In fact, the predicted solar mass fraction for fluorine by the model F4 (AGB yields multiplied by a factor of 2) is 4.91 $\times$ 10$^{-7}$; which compares very well to the value of 4.78 $\times$ 10$^{-7}$ predicted by Maiorca et al. (2014).
![The abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood for the “one-infall” chemical evolution model adopting different nucleosynthesis prescriptions for the fluorine production. Model lines are the ones described in Fig. \[2IM\_FO\]. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles whereas the data taken by Pilachowski & Pace (2015) are presented with gray pentagons. []{data-label="F_O_AGB_VAR_1infall"}](figure_6.png)
![The abundance ratio \[F/Fe\] as a function of \[Fe/H\] in the solar neighbourhood for the “one-infall” chemical evolution model. Model lines are the ones described in Fig. \[F\_O\_AGB\_VAR\_1infall\]. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles whereas the data taken by Pilachowski & Pace (2015) are presented with gray pentagons. []{data-label="AGB_var_1infall"}](figure_7.png)
In Fig. \[2IM\_FFE\_AGB\] the same models of the previous Figure are considered but in the terms of \[F/Fe\] vs \[Fe/H\], and we compare our models with the data by J[ö]{}nsson et al. (2017a) along with Pilachowski & Pace (2015) ones (the stars in Pilachowski & Pace 2015 are thin disk, see their Section 2.1). If we consider both data sets the area spread by the data is well covered by the model predictions. On the other hand, the trend shown by solely the data of J[ö]{}nsson et al. (2017a) - i.e. the abundance ratio \[F/Fe\] always increases with \[Fe/H\] - is not reproduced by any models presented so far in this paper.
In Fig \[WR2\_2infall\] we present the model results for the two-infall model where the W-R star yields by Meynet & Arnould (2000) for the fluorine are taken into account. As suggested by Renda et al. (2004) and J[ö]{}nsson et al. (2017a) a possible source for fluorine production are W-R stars. They might deposit fluorine into the interstellar medium via their strong stellar winds (Meynet & Arnould 2000; Palacios et al. 2005). Just like in the AGB scenario, the fluorine in W-R winds is produced in reactions starting from $^{14}$N, including $\alpha$, neutron, and proton captures. Due to the strong metallicity dependence of the winds of these stars, a possible fluorine production through this channel is expected to start first at slightly sub-solar metallicities and then increase for higher metallicities (Renda et al. 2004).
\[models\]
[c|ccccc]{}
\
Fluorine Solar values &\
& &&\
$(X_{F, \mbox{ }\odot})$ & 2 Infall model &1 Infall model\
\
F1& 3.07 $\times$ 10$^{-7}$ & 3.03 $\times$ 10$^{-7}$\
F2& 9.51 $\times$ 10$^{-8}$ & 9.15 $\times$ 10$^{-8}$\
F3& 3.99 $\times$ 10$^{-7}$ &3.95 $\times$ 10$^{-7}$\
F4& 4.91 $\times$ 10$^{-7}$ &4.87 $\times$ 10$^{-7}$\
F5& 6.74 $\times$ 10$^{-7}$ & 6.71 $\times$ 10$^{-7}$\
F6& 3.23 $\times$ 10$^{-7}$ & 3.10 $\times$ 10$^{-7}$\
F7& 5.79 $\times$ 10$^{-7}$ & 5.64 $\times$ 10$^{-7}$\
F8& 8.58 $\times$ 10$^{-7}$ &8.30 $\times$ 10$^{-7}$\
F9& 7.62 $\times$ 10$^{-7}$ & 7.48$\times$ 10$^{-7}$\
\[tab1\]
![The abundance ratio \[F/O\] as a function of \[O/H\] in the solar neighbourhood for the “one-infall” chemical evolution model adopting different prescriptions for the channels of fluorine production. The model F6, in which only the contribution for fluorine production is given by W-R stars, is indicated with the brown line. Model F7 in which we adopt Romano et al. (2010) yield including W-R stars by Meynet & Arnould (2000) is with the red line. Model F8 (same of model F7, but with the W-R fluorine yields multiplied by 2, see Table 1) is drawn with the black line. Finally, models F9 (same of model F7, but with the AGB fluorine yields multiplied by 2, see Table 1) is labeled with magenta line. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles. []{data-label="F9_O_H_1infall"}](figure_8.png)
In Fig. \[WR2\_2infall\] we note that model F7 which considers the Romano et al. (2010) yield prescriptions for oxygen, coupled with W-R contribution for the fluorine production, is able to increase the \[F/O\] vs \[O/H\] relation for over-solar values of \[O/H\] in comparison with the “reference” model F1. This leads to a better agreement with the data of J[ö]{}nsson et al. (2017a), but the slope of the observed secondary behavior is not well reproduced, and the model shows lower \[F/O\] values at larger \[O/H\]. The predicted solar mass fraction for fluorine by the model F7 is 5.79 $\times$ 10$^{-7}$, therefore still in perfect agreement with Maiorca et al. (2014) value.
In Fig. \[WR2\_2infall\] we also draw the model F8 (see Table 1 for model yield detail) where we tested the metal dependent W-R yield by Meynet & Arnould (2000) for fluorine multiplied by hand by a factor of 2 coupled with the Romano et al. (2010) yield. We note that this model is capable to perfectly reproduce the observed data. The predicted solar mass fraction for fluorine by the model F8 is 8.58 $\times$ 10$^{-7}$ (see Table 2), definitely a larger value than the one predicted by Maiorca et al. (2014), but still within a factor of two.
In summary, from Table 2 we see that the solar fluorine abundance value reached by the model F7 is in agreement with the Maiorca et al. (2014) value, whereas the model F8 (W-R fluorine yields multiplied by a factor of two) shows, as expected, a larger value but within a factor of two.
We also show the effects of considering fluorine as produced only by W-R stars (model F6). We note that W-R stars contribute to increase the fluorine production in the whole range of \[O/H\]. The main feature of model F6 is that it shows the secondary behavior slope observed in the J[ö]{}nsson et al. (2017a) stellar sample for \[O/H\] over-solar values. Concerning the \[F/Fe\] vs. \[Fe/H\] abundance ratios, in Fig \[WR2\_FE\_2infall\] are reported the same models described above and shown in Fig. \[WR2\_2infall\].
We see that models including the W-R contribution for the fluorine provide a better fit to the observed data set of J[ö]{}nsson et al. (2017a) and Pilachowski & Pace (2015) compared to the models presented in Fig. \[2IM\_FFE\_AGB\] without this contribution.
We conclude this Subsection focused on the two-infall model results, presenting in Fig. \[F9\_2infall\] the effects of varying AGB fluorine yields by hands in presence of W-R fluorine contribution. In model F9 (see Table 1) the AGB yields are multiplied by a factor of two along with W-R yields by Meynet & Arnould (2000). In this way we have a better agreement with data compared to reference model F1.
Without any modification of W-R fluorine yields we find a slight decreasing trend in the \[F/O\] vs \[O/H\] relation for over-solar \[O/H\] values but still in agreement with J[ö]{}nsson et al. (2017a) data. Moreover the predicted fluorine solar mass fraction is 7.62 $\times$ 10$^{-7}$, therefore in better agreement with the Maiorca et al. (2014) one.
![The abundance ratio \[F/Fe\] as a function of \[Fe/H\] in the solar neighbourhood for models F6-F7-F8-F9 applied to the “one-infall” chemical evolution model. Model lines are the same of Fig. \[F9\_O\_H\_1infall\]. Observational data of J[ö]{}nsson et al. (2017a) are indicated with cyan circles whereas the data taken by Pilachowski & Pace (2015) are presented with gray pentagons. []{data-label="F9_F_FE_1infall"}](figure_9.png)
The “one-infall” model results
------------------------------
In this Subsection we present the results of chemical evolution models for the thin disk of the Galaxy in the solar neighborhood (the one infall model). In Fig. \[F\_O\_AGB\_VAR\_1infall\] models F1, F2, F3, F4 and F5 are shown (see Table 1 for model detail). In this Figure it is evident a “smoother” chemical evolution compared to the “two-infall” case. This is due to the absence of any gap in the star formation history which is a peculiar feature of the two-infall model (during the transition between the halo-thick disk phase) and the thin disk one.
 
As anticipated in Subsection 6.1, the “one-infall” model results are able to better reproduce the J[ö]{}nsson et al. (2017a) abundance ratios. The same models of Fig. \[F\_O\_AGB\_VAR\_1infall\] are shown in terms of \[F/Fe\] vs \[Fe/H\] in Fig. \[AGB\_var\_1infall\].
In Fig. \[AGB\_var\_1infall\] if we consider both data sets, the area spread by the data is well covered by model predictions. On the other hand, the increase of \[F/Fe\] for larger values of \[Fe/H\] shown by the data of J[ö]{}nsson et al. (2017a) alone, is not reproduced by any model. Moreover, as it can be seen in Table 2, the solar values predicted by the “one-infall” model are not so different from the ones by the two-infall one, although only slightly smaller.
In Fig. \[F9\_O\_H\_1infall\] we show the effects on the relations \[F/O\] vs. \[O/H\] of including the Wolf-Rayet stars contribution to the fluorine production on the “one-infall” chemical evolution model for the thin disk in terms of \[F/O\] vs. \[O/H\] abundance ratios. The model which best fits the data is model F8 which perfectly reproduces the change of slope of the data. In fact, the predicted knee by this model is located around the solar value of \[O/H\], in agreement with the data.
As for the two-infall case, the model F8 shows a higher solar value for the fluorine compared with the Maiorca et al. (2014) value (see Table 2).
Moreover, in Fig. \[F9\_O\_H\_1infall\] we show the model F9 applied to the “one-infall” chemical evolution, in which we multiplied by a factor of two the AGB yields for fluorine along with the W-R yield of Meynet & Arnould (2000). Also in this case, the model F9 applied to the “one-infall” chemical evolution model leads to a better fit of the data compared with the case when we considered it for the two-infall model. Even if this model predicts a decreasing trend for \[F/O\] vs \[O/H\] for over-solar \[O/H\] values, the model line is within the observed error bars. As reported in Table 2 this model predicts a fluorine solar value better in agreement with the Maiorca et al. (2014) value compared to the model F8.
In Fig. \[F9\_F\_FE\_1infall\] models with W-R fluorine contribution are reported for the \[F/Fe\] vs \[Fe/H\] relation. Again, if we consider Pilachowski & Pace (2015) and J[ö]{}nsson et al. (2017a) the best model is the F8. However this model it is not able to reproduce the trend shown by the data by J[ö]{}nsson et al. (2017a). In conclusions, as shown for the two-infall case, we confirm that:
- Increasing by hand the AGB fluorine yields leads to a better fit of \[F/O\] vs \[O/H\] abundance ratios for sub-solar values of \[O/H\], but the secondary behavior at higher \[O/H\] is not reproduced adopting the nucleosynthesis yields of model15 of Romano et al. (2010).
- Models which take into account W-R metallicity dependent yields for fluorine by Meynet & Arnould (2000) are capable to better fit the observed data.
- The best models are the F8 and F9 ones.
Testing the novae as possible sources for F production
======================================================
In the last section we focus on the effects of nova nucleosynthesis on the fluorine production. In principle this source could help to reproduce the secondary behavior of fluorine observed by J[ö]{}nsson et al. (2017a) in the \[F/O\] vs \[O/H\] at high \[O/H\] values.
We tested that the inclusion of the nova yields by José & Hernanz (1998) on the model F7 (see Table 1) has a negligible effect on the chemical evolution of fluorine. In any case, as an exercise, we present in Fig. \[novae\] the model with the “maximum” fluorine yield by novae, showing the model F7 in the \[F/O\] vs \[O/H\] relation, including F produced by novae originated in ONe white dwarf (W-D) of 1.35 M$_\odot$ (model ONe7 of José & Hernanz 1998). Although this W-D model leads the maximum production of F in novae, we are aware that these objects are extremely rare in nature. In the same Figure we present model results in which the fluorine nova yields are multiplied by factors of 3 and 5, respectively.
It is quite interesting to note that in this way we are able to trace perfectly the secondary trend at high \[O/H\] values. However, we remind the reader that nova yields are still very uncertain. It is worth noting that Li synthesized in a nova outburst as inferred from recent observations (Tajitsu et al. 2015, Izzo et al. 2015) exceeds by far the one expected from the theoretical models of José & Hernanz (1998). This fact suggests that there is also room for a significant revision of F yield from novae.
Conclusions
===========
In this article we studied in detail the effects of different nucleosynthesis prescriptions for the fluorine production on the chemical evolution models for the solar neighborhood with the aim of reproducing the observational data by J[ö]{}nsson et al. (2017a). Our main conclusions can be summarized as follows:
- The role played by Wolf-Rayet stars in the fluorine production seems to be essential to reproduce the new observed ratios in the solar neighborhood \[F/O\] vs \[O/H\] by J[ö]{}nsson et al. (2017a). This confirms previous suggestions by Renda et al. (2004).
- We obtain a better agreement with the observed fluorine abundances when we consider the “one-infall” chemical evolution model, relative only to the thin disk.
- The best “one-infall” model reproducing the observed abundance ratios \[F/O\] vs \[O/H\] for the thin disk requires the nucleosynthesis prescriptions of the model 15 of Romano et al. (2010) along with the Wolf-Rayet fluorine yields by Meynet & Arnould (2000) multiplied by a factor of two. On the other hand, this model predict a solar value higher than the one of Maiorca et al. (2014).
- Considering the AGB yields by Karakas et al. (2010) multiplied by a factor of two along with Meynet & Arnould (2000) Wolf-Rayet yields leads to a good fit of the \[F/O\] vs \[O/H\] data and predict a solar value in agreement with Maiorca et al. (2014) value.
- Concerning the \[F/Fe\] vs \[Fe/H\] relation, the models presented here are in agreement with the collection of data composed by J[ö]{}nsson et al. (2017a) and Pilachowski & Pace (2015). The data by J[ö]{}nsson et al. (2017a) show that \[F/Fe\] abundances increase with \[Fe/H\]. This trend is not found by our models for \[Fe/H\] values larger than -0.3 dex. We want to underline that there are still huge uncertainties concerning the nucleosynthesis of F, and this could be the reason of the discrepancy.
- More detailed data for fluorine in the solar neighborhood are required at low metallicities, i.e for \[O/H\] values smaller than -0.4 dex, to confirm the importance of W-R stars in the fluorine production, as we conclude in this paper. In fact, we predict that the inclusion of the fluorine produced by W-R star would affect the \[F/O\] vs \[O/H\] ratio also at small \[O/H\] values leading to a roughly flat \[F/O\] ratio for \[O/H\] smaller that -0.5 dex.
- We also show that the novae, even if their yields are still uncertain could help to better reproduce the secondary behavior of F in the \[F/O\] vs. \[O/H\] relation in presence of W-R stars fluorine contribution.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous referee for the suggestions that improved the paper. E. Spitoni and F. Matteucci thank the financial support by FRA2016 - University of Trieste. N. Ryde acknowledges support from the Swedish Research Council, VR (project number 621-2014-5640), Funds the Royal Physiographic Society of Lund. (Stiftelsen Walter Gyllenbergs fond and Märta och Erik Holmbergs donation), and from the project grant “The New Milky” from the Knut and Alice Wallenberg foundation. H. J[ö]{}nsson acknowledges the support by the Lars Hierta Memorial Foundation, Helge Ax:son Johnsons stiftelse, and Stiftelsen Olle Engkvist Byggmästare.
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[^1]: email to: spitoni@oats.inaf.it
|
---
author:
- '<span style="font-variant:small-caps;">Chitoshi Yasuda</span>[^1], <span style="font-variant:small-caps;">Daisuke Kinouchi</span> and <span style="font-variant:small-caps;">Kenn Kubo</span>'
title: 'Spin-Wave Theory of the Multiple-Spin Exchange Model on a Triangular Lattice in a Magnetic Field : 3-Sublattice Structures'
---
Introduction
============
The multiple-spin exchange (MSE) model on the triangular lattice is one of the two dimensional frustrated quantum spin systems which are presently at the focus of strong interest. The model is believed to describe nuclear magnetism of the solid $^3$He layers adsorbed on graphite [@Franco; @Godfrin; @Siqueira], whose peculiar temperature dependence of the specific heat with a double-peaked structure is quite intriguing [@Ishida]. Several studies were devoted for theoretical understanding of the experimental results in terms of the MSE model [@Kubo; @Kubo2; @Momoi; @Misguich; @Misguich2; @Misguich3; @Momoi2; @Roger2; @Kubo3]. The mean-field theory was applied to the MSE model with two-, three- and four-spin cyclic exchange interactions, which lead to various kinds of the ground state corresponding to different values of parameters [@Kubo; @Kubo3]. Appearance of various mean-field ground states reflects weak stability of the mean-field ground states due to strong frustration. Therefore quantum fluctuations should have strong influence on the ground state of the system. Quantum mechanical ground state was studied by the numerical diagonalization of finite clusters and a spin liquid ground state with a finite spin gap was predicted for the parameters which are thought to be adequate to the $^3$He layer [@Misguich2; @Misguich3]. However recent measurement of the susceptibility down to $\sim $10 $\mu$K did not show any sign which suggests the existence of a spin gap [@Masutomi]. At present, the ground state of both the two-dimensional solid $^3$He on graphite and the MSE model on the triangular lattice are, therefore, hardly understood. The MSE interactions might be relevant as well to the spin liquid state of $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$ where antiferromagnetic interaction is thought to be dominant [@Kurosaki]. Therefore it is meaningful and desired to study the properties of the MSE model on the triangular lattice into detail.
In the present work, we study the spin wave in the $S=1/2$ MSE model with two-, three- and four-spin exchange interactions on the triangular lattice in the magnetic field. The system is described by the Hamiltonian $${\cal H} = J \sum_{<i,j>} \mib{\sigma}_i \cdot \mib{\sigma}_j
+ K \sum_{\rm p} h_{\rm p} + h \sum_i \sigma_i^z \ ,
\label{ham}$$ where $\mib{\sigma}_i$ is the Pauli matrix on the site $i$ and the summations $\sum_{<i,j>}$ and $\sum_{\rm p}$ run over all pairs and minimum diamonds, respectively. Since the three-spin exchange interaction is reduced to the conventional two-spin one, the first term of the Hamiltonian (\[ham\]) describes the sum of two- and three-spin exchange interactions. The Hamiltonian $h_{\rm p}$ is the four-spin exchange interaction for a minimum diamond p. For a diamond of spins 1 $\sim$ 4 with diagonal bonds (1,3) and (2,4), it reads $$\begin{aligned}
h_{\rm p} &=& \sum_{1 \le i < j \le 4} \mib{\sigma}_i \cdot
\mib{\sigma}_j
+ (\mib{\sigma}_1 \cdot \mib{\sigma}_2)
(\mib{\sigma}_3 \cdot \mib{\sigma}_4) \nonumber \\
&+& (\mib{\sigma}_1 \cdot \mib{\sigma}_4)
(\mib{\sigma}_2 \cdot \mib{\sigma}_3)
- (\mib{\sigma}_1 \cdot \mib{\sigma}_3)
(\mib{\sigma}_2 \cdot \mib{\sigma}_4) \ .\end{aligned}$$ The coupling parameters are written in terms of conventional exchange constants as $K=-J_4 /4$ and $J= J_3 - J_2 /2$. Since $J_n$ are known to be negative in solid $^3$He [@Thouless], we take the value of $K$ positive in the following. The third term in Eq. (\[ham\]) is the Zeeman term with a magnetic field $\mib{h}$ ($h = |\mib{h}|$) applied anti-parallel to the $z$-direction as shown in Fig. \[def0\].
Numerous mean-field ground-state phases were found for the Hamiltonian (\[ham\]) by assuming up to 144 sublattices [@Kubo3]. The phase diagram is parametrized by $J/K$ and $h/K$. When $h=0$, the ground state varies according to $J/K$ as follows : I) the ferromagnetic phase for $J/K <-8.61$. II) the intermediate phase for $-8.61 < J/K < -2.26$ where the ground state spin structure varies with the change of $J/K$ and ten small phases with equal to or more than 12 sublattices were identified. These small phases might be artifacts of assumptions of finite number of sublattices. The true ground state in this parameter region is still controversial even at the mean-field level. III) the tetrahedral phase for $-2.26 < J/K <8.22$. The ground state has a 4-sublattice spin structure with zero magnetization, where spin vectors on four sublattices point four vertices of a tetrahedron if their bottoms are put at its center. IV) a phase with six sublattices for $8.22 < J/K < 10$. The ground state has non-coplaner spin structure with uniform vector chiral order and staggered scalar chiral order. Finally, V) the 120$^{\circ}$ phase for $J > 10K$. The spins on three sublattices make the angle $2\pi/3$ to each other.
The mean-field phase diagram in the magnetic field was also studied[@Kubo; @Momoi2; @Kubo3]. One of the interesting behaviors is the appearance of a magnetization plateau with 1/2 of the full polarization in the parameter region adequate to $^3$He layers [@Kubo; @Misguich2; @Momoi2]. This plateau is realized by the 4-sublattice uuud spin structure. Other numerous ground-state phases in the magnetic field have been found, but we introduce only the phases related to the present work. The 120$^{\circ}$ ground state for $J >10K$ is modified by the magnetic field. This state has a 3-sublattice coplanar spin configuration with the spins on one (say A) sublattice antiparallel to the applied magnetic field. The spins on other sublattices (B and C) tilt toward the oblique directions as shown in Fig. \[def0\]. In the present paper, we call it Y-shape state [@y-shape]. This result shows that the four-spin exchange interaction lifts the non-trivial degeneracy of the mean-field ground-state of the antiferromagnetic Heisenberg (AFH) model in the magnetic field [@Kawamura]. A 6-sublattice phase exists for $8\nle J/K \nle 10$ adjacent to the Y-shape state for weak magnetic field. Increase in the magnetic field induces a 12-sublattice structure. Further increase in the magnetic field leads to a magnetization plateau with 1/3 of full polarization due to the 3-sublattice uud state with two sublattices with up spins and the other with down spins. The uud state is stabilized by the four-spin interaction and occupies a large region in the phase diagram. We note that the phase boundary between the Y-shape and the 6-sublattice phases was not determined by the mean-field theory.
Spin waves in this model were studied previously in the absence of the magnetic field [@Momoi; @Kubo2; @Momoi2]. The spin wave in the tetrahedral phase was studied and the quantum corrections to the sublattice magnetization were estimated [@Momoi]. The analysis for $J=0$ showed the stability of the tetrahedral state against the zero-point fluctuations of the spin wave. The spin wave in the 120$^\circ$ phase was studied and it was shown that there are three gapless branches for small $k$ and the spin wave softens at $J=10K$ [@Kubo2]. The softening corresponds to the phase transition to the 6-sublattice phase.
In the present work, we investigate the spin wave in the Y-shape phase and discuss the phase transition under the magnetic field. Although the Y-shape phase does not correspond to the model for the solid $^3$He layer [@Bernu], coplay of the geometrical frustration and the MSE interactions poses a problem of interest in itself. We investigated the dispersions of the spin waves as well as the effects of the zero-point fluctuations of the spin wave on the ground-state energies. As a result, we found that though the Y-shape ground state survives quantum fluctuations for $J/K \nge 12$ and small $h/K$, softening of the spin wave leads to the phase transition to the 6-sublattice phase.
The present paper is organized as follows. In § 2, we present the linear spin-wave theory for the Y-shape state in the magnetic field. In § 3, we simply summarize the magnetic properties in two special cases, i.e., the AFH model in the magnetic field and the MSE model at $h=0$. The dispersion of the spin wave and the phase diagram obtained from the softening of the spin wave are shown in § 4 and 5, respectively. The quantum effects are discussed by studying quantum corrections of the ground-state energy and the sublattice magnetizations in § 6. Finally, § 7 is devoted to summary and discussion.
Spin-Wave Theory for the Y-Shape State
======================================
Assuming the Y-shape state as the ground state, we can perform the Holstein-Primakoff transformation [@Holstein] of the Hamiltonian (\[ham\]) by neglecting higher-order terms as $$\lefbrace{
\sigma_i^z \simeq 1 - 2 a_i^{\dagger} a_i \\
\sigma_i^+ \simeq a_i \\
\sigma_i^- \simeq a_i^{\dagger}
} \ ,
\label{HP_A}$$ for $i \in$ A sublattice, $$\lefbrace{
\sigma_j^z \simeq -\alpha (b_j^{\dagger} + b_j)
-\beta (1-2b_j^{\dagger} b_j) \\
\sigma_j^+ \simeq \frac{1}{2} \{\alpha (1-2b_j^{\dagger} b_j)
-(\beta+1) b_j^{\dagger}-(\beta-1) b_j \} \\
\sigma_j^- \simeq \frac{1}{2} \{\alpha (1-2b_j^{\dagger} b_j)
-(\beta-1) b_j^{\dagger}-(\beta+1) b_j \}
} \ ,
\label{HP_B}$$ for $j \in$ B sublattice, and $$\lefbrace{
\sigma_k^z \simeq \alpha (c_k^{\dagger} + c_k)
-\beta (1-2c_k^{\dagger} c_k) \\
\sigma_k^+ \simeq \frac{1}{2} \{-\alpha (1-2c_k^{\dagger} c_k)
-(\beta+1) c_k^{\dagger}-(\beta-1) c_k \} \\
\sigma_k^- \simeq \frac{1}{2} \{-\alpha (1-2c_k^{\dagger} c_k)
-(\beta-1) c_k^{\dagger}-(\beta+1) c_k \}
} \ ,$$ for $k \in$ C sublattice, where $a_i^{\dagger}$ ($a_i$), $b_j^{\dagger}$ ($b_j$) and $c_k^{\dagger}$ ($c_k$) are the boson creation (annihilation) operators, $\alpha=\sin{\theta}$ and $\beta=\cos{\theta}$ as shown in Fig. \[def0\].
First, we obtain the relation between $h$ and the angle $\theta$ from the condition that the first-order terms of the transformed Hamiltonian should vanish, i.e., $$\cos{\theta} = \frac{1}{12K} \{ 5K+J-\sqrt{(J-K)^2-4Kh}\} \ ,
\label{kakudo}$$ for $h \le h_{\rm c}$ where $$\label{eq:hc}
h_{\rm c}=-12K+3J.$$ The value of $\theta$ monotonously decreases with the increase of $h$ and the uud state ($\theta =0$) is realized for $h \ge h_{\rm c}$. This relation can be also obtained from ${\rm d} E^{\rm cl}/{\rm d} \theta = 0$ where the mean-field ground-state energy $E^{\rm cl}$ per the total number of sites $N$ is given by $$\begin{aligned}
\label{classical_E}
&& E^{\rm cl}/N = (2\beta^2-2\beta-1)J \\
&& \hspace*{1cm} - (8\beta^3-10\beta^2+4\beta+1)K
+ \frac{1}{3}(1-2\beta)h \ . \nonumber \end{aligned}$$
After straightforward calculations the Hamiltonian is rewritten as $${\cal H} = E^{\rm cl}
+ \sum_{\mib{k}}' \mib{v_k}^{\dagger} {\cal D} \mib{v_k}
+ E_0^{\rm q} \ ,
\label{ham2}$$ where $\mib{v_k}^{\dagger}=(a_{\mib{k}}^{\dagger},b_{\mib{k}}^{\dagger},
c_{\mib{k}}^{\dagger},a_{-\mib{k}},b_{-\mib{k}},c_{-\mib{k}})$ and ${\displaystyle \sum_{\mib{k}}' }$ in the second term denotes the summation over a half of the reduced Brillouin zone of the 3-sublattice structure. The matrix ${\cal D}$ reads $${\cal D} = {\left(\begin{array}{@{\,}cc@{\,}} M_{\rm diag} & M_{\rm off} \\
M_{\rm off} & M_{\rm diag} \end{array}\right)} \ ,$$ where $$M_{\rm diag} = {\left(\begin{array}{@{\,}ccc@{\,}}
A_{\mib{k}} & C \Gamma_{\mib{k}} & C \Gamma_{\mib{k}}^* \\
C \Gamma_{\mib{k}}^* & B_{\mib{k}} & D \Gamma_{\mib{k}} \\
C \Gamma_{\mib{k}} & D \Gamma_{\mib{k}}^* & B_{\mib{k}} \end{array}\right)} \ ,$$ and $$M_{\rm off} = {\left(\begin{array}{@{\,}ccc@{\,}}
E_{\mib{k}} & G \Gamma_{\mib{k}} & G \Gamma_{\mib{k}}^* \\
G \Gamma_{\mib{k}}^* & F_{\mib{k}} & H \Gamma_{\mib{k}} \\
G \Gamma_{\mib{k}} & H \Gamma_{\mib{k}}^* & F_{\mib{k}} \end{array}\right)} \ ,$$ with $$\begin{aligned}
&& A_{\mib{k}}=12 \{ -2K +(J+2K)\beta +4K\beta^3 \nonumber \\
&& \hspace*{1cm} + K(1-\beta^2)\Delta_{\mib{k}} \} -2h \ , \nonumber \\
&& B_{\mib{k}}=6 \{ J+2K +(J-4K)\beta -2(J+5K)\beta^2 \nonumber \\
&& \hspace*{1cm} +12K\beta^3 +2K(1-\beta+2\beta^3)\Delta_{\mib{k}} \}
+2h\beta \ , \nonumber \\
&& C=3(1-\beta)(J+4K+4K\beta^2) \ , \nonumber \\
&& D=6 \{ K(4\beta-1) +(J+5K)\beta^2 -8\beta^3K \} \ , \nonumber \\
&& E_{\mib{k}}= -12(1-\beta^2)K\Delta_{\mib{k}} \ , \\
&& F_{\mib{k}}= 2\beta E_{\mib{k}} \ , \nonumber \\
&& G= -3(1+\beta)(J+4\beta^2K) \ , \nonumber \\
&& H= -6(1-\beta^2)(J+5K-8\beta K) \ . \nonumber\end{aligned}$$ Here the wave-number dependence is incorporated by $$\Gamma_{\mib{k}}=\frac{1}{3}({\rm e}^{{\rm i}k_1}
+ {\rm e}^{-{\rm i}k_2} + {\rm e}^{{\rm i}(-k_1+k_2)} ) \ ,$$ $$\Delta_{\mib{k}}=\frac{1}{3} \{ \cos{(k_1+k_2)} +\cos{(2k_1-k_2)}
+\cos{(-k_1+2k_2)} \} \ ,$$ where $k_i = \mib{k}\cdot \mib{\delta}_i $¡Ê$\ i=1, 2$ ¡Ëis an element of the wave vector and the unit vector $\mib{\delta}_1 $ and $\mib{\delta}_2 $ of the triangular lattice are chosen as shown in Fig. \[unit\]. Note that $k_1^2 + k_2^2 -k_1 k_2 = 3k^2 /4$ holds. The third term in Eq. (\[ham2\]) $$E_0^{\rm q}=-\sum_{\mib{k}}' (A_{\mib{k}}+2B_{\mib{k}}) $$ arises from the Bose commutation relation. The Hamiltonian (\[ham2\]) is diagonalized through the Bogoliubov transformation and takes the following form: $${\cal H} = E^{\rm cl}
+ \sum_{\mib{k}}' \mib{u_k}^{\dagger} {\cal D}_{\rm diag} \mib{u_k}
+ E_0^{\rm q} \ ,$$ where $\mib{u_k}^{\dagger}=(\alpha_{\mib{k}}^{\dagger},\beta_{\mib{k}}^{\dagger},
\gamma_{\mib{k}}^{\dagger},\alpha_{-\mib{k}},\beta_{-\mib{k}},
\gamma_{-\mib{k}})$ is a vector of the transformed Bose operators and ${\cal D}_{\rm diag}$ is a diagonal matrix whose $(i, i)$ elements are $\omega_{\mib{k}}^{(1)}, \omega_{\mib{k}}^{(2)}, \omega_{\mib{k}}^{(3)},
\omega_{\mib{k}}^{(1)}, \omega_{\mib{k}}^{(2)}$ and $\omega_{\mib{k}}^{(3)}$ for $i=1\sim 6$. The frequencies of the three branches of the spin wave are given by $\omega_{\mib{k}}^{(1)}$, $\omega_{\mib{k}}^{(2)}$ and $\omega_{\mib{k}}^{(3)}$. We numerically performed the transformation according to the general theory by Colpa [@colpa].
Spin-Wave Spectrum of the AFH Model in the Magnetic Field and the MSE Model at $h=0$
====================================================================================
In this section we reproduce the previously known results for simple cases in order for comparison with the results in the next section.
First we demonstrate the results for $K=0$, i.e., the AFH model. The dispersions of the spin waves assuming the Y-shape state as the ground state for the AFH model with $h/J=0$ and 2 are shown in Figs. \[disp\_K=0\] (a) and (b), respectively. The hexagonal Brillouin zone for the 3-sublattice structure is shown in Fig. \[zone\]. For $h=0$ the frequencies of all spin-wave branches vanish at the $\Gamma$ point ($\mib{k}=0$) and have the linear dispersion for small $k$. In the magnetic field, one of the three branches is lifted to $\omega_{\mib{k}}=2h$ at the $\Gamma$ point corresponding to the uniform precession of the spins about the magnetic field. There are two gapless branches rather than one expected from the SO(2) symmetry. Furthermore one of them has a quadratic dispersion for small $k$. This is caused by a non-trivial continuous degeneracy in the mean-field ground state in the magnetic field [@Kawamura]. Any three-sublattice spin structure fulfilling the relation $\mib{S}_{\rm A}+\mib{S}_{\rm B}+\mib{S}_{\rm C}=\mib{h}/3J$ is a ground state for $h \le 3J$, where $\mib{S}_{\rm A,B,C}$ is the sublattice magnetization on each sublattice. Chubukov and Golosov showed that quantum fluctuations lift this non-trivial degeneracy and select the coplanar spin structure, i.e., the Y-shape state [@Chubukov3]. This is an example of the so-called order from disorder phenomena.
Next we show the result for the MSE model at $\mib{h}=0$. In this case the transformed Hamiltonian (\[ham2\]) can be analytically diagonalized [@Kubo2] as $${\cal H}= E^{\rm cl}+\sum_{\mib{k}}(
\omega_{\mib{k}}^{(1)}\alpha_{\mib{k}}^{\dagger}\alpha_{\mib{k}}
+ \omega_{\mib{k}}^{(2)}\beta_{\mib{k}}^{\dagger}\beta_{\mib{k}}
+ \omega_{\mib{k}}^{(3)}\gamma_{\mib{k}}^{\dagger}\gamma_{\mib{k}})
+ E^{\rm q} \ ,
\label{ham_h=0}$$ where the summation of the second term runs over the Brillouin zone, and $$\begin{aligned}
&& \omega_{\mib{k}}^{(\mu)}=\sqrt{V_{\mib{k}}^{(\mu)}-W_{\mib{k}}^{(\mu)}}
\ , \nonumber \\
&& V_{\mib{k}}^{(1)}=6(J-K)+9K\Delta_{\mib{k}}+3(J+5K)
\Gamma '_{\mib{k}} \ , \nonumber \\
&& W_{\mib{k}}^{(1)}=9\{ K\Delta_{\mib{k}}+(J+K)
\Gamma '_{\mib{k}} \} \ ,
\label{non-field}\end{aligned}$$ and $\Gamma '_{\mib{k}} ={\rm Re}(\Gamma_{\mib{k}})$, $V_{\mib{k}}^{(2)}=V_{\mib{k}+\mib{Q}}^{(1)}$, $W_{\mib{k}}^{(2)}=W_{\mib{k}+\mib{Q}}^{(1)}$, $V_{\mib{k}}^{(3)}=V_{-\mib{k}+\mib{Q}}^{(1)}$, $W_{\mib{k}}^{(3)}=W_{-\mib{k}+\mib{Q}}^{(1)}$ where $\mib{Q}=(-2\pi/3, 2\pi/3)$. The first and third terms of Eq. (\[ham\_h=0\]) are $$E^{\rm cl}=-\frac{3}{2}(J+K)N \ ,$$ $$E^{\rm q}=\frac{1}{2}\sum_{\mu=1}^3 \sum_{\mib{k}}
(\omega_{\mib{k}}^{(\mu)}-V_{\mib{k}}^{(\mu)}) \ .
\label{quantum_E}$$ We show the spin-wave spectrum for $J/K = 12$ and 10 in Figs. \[disp\_h0\] (a) and (b), respectively. The frequencies of branches vanish at the $\Gamma$ point as for the AFH model. They have linear dispersions for small $k$ as $$\begin{aligned}
\omega_{\mib{k}}^{(1)} &\simeq & 3\sqrt{3(J-K)(J+2K)}\ k \ , \\
\omega_{\mib{k}}^{(2)} &\simeq& \omega_{\mib{k}}^{(3)} \\
\simeq && \hspace{-1cm} 3
\sqrt{6(J-K)\{(J-7K)(k_1^2 + k_2^2 ) - (J+5K) k_1 k_2 \} } \ . \nonumber\end{aligned}$$ Figure \[disp\_h0\] shows that the frequency of the second branch $\omega_{\mib{k}}^{(2)}$ vanishes at the M point, which indicates that a second-order phase transition to the 6-sublattice structure occurs due to the softening of the spin wave just at $J=10K$ [@Kubo2]. The softening occurs at six equivalent M points in the Brillouin zone.
Spin-Wave Spectrum of the MSE Model in the Magnetic Field
=========================================================
In Figs. \[disp\_h10\] (a) and (b) we show the dispersion for $J/K=12$ and 10.14 at $h=10K$. In contrast with the spectrum in the AFH model, there is only one gapless branch at the $\Gamma$ point and the gapless branch has a linear dispersion. The result is consistent with the excitations from a stable ground state only with the global SO(2) symmetry and corresponds to that the four-spin exchanges stabilize the coplanar Y-shape state already in the mean-field approximation. The frequency of the lowest branch decreases at the M point with decreasing $J/K$ for a fixed $h$. It vanishes at a critical value of $J/K$ as shown in Fig. \[disp\_h10\] (b). The $J/K$-dependence of the frequency of the spin wave at the point M ($\mib{k}=(2\pi/3,\pi/3)$) is shown in Fig. \[soft\] for various values of $h$. The softening of the spin wave is induced by the competition of the two- and four-spin interactions in the magnetic field. The frequency vanishes apparently as $\omega_k \propto (J-J_{\rm c})^\beta$. Here we assume the critical exponent $\beta$ to be 1/2 which results from the general Landau theory. The softening of the spin wave leads to a phase transition to the 6-sublattice phase in the magnetic field. We determine the phase boundary between the Y-shape and the 6-sublattice phases by determining the critical value in the next section.
The Phase Diagram
=================
In Fig. \[phase\] we show the phase diagram determined from the softening of the spin wave together with the phase boundaries determined by the mean-field calculations. The phase boundary between the Y-shape and the 6-sublattice phases extends from $(J/K, h/K) = (10, 0)$ to the multi-critical point ($(J/K)_{\rm multi}$, $(h/K)_{\rm multi}$) where four phase boundaries appear to merge. Phase transitions from the Y-shape phase to the 6-sublattice and the uud phases are continuous. On the other hand, the transition between the 6-sublattice and the 12-sublattice phases and that between the 12-sublattice and the uud phases are discontinuous. The phase boundary between the Y-shape and the uud phases is given by the relation (\[eq:hc\]) and the multi-critical point is estimated to be ($(J/K)_{\rm multi}$, $(h/K)_{\rm multi}$) $\simeq$ (11.37, 22.11). Though in the AFH model the uud phase occurs only at a critical value $h_{\rm c}=3J$ in the mean-field theory, the phase extends to a finite region of $h$ in the MSE model. The four-spin exchange stabilizes the uud state. It was shown that the thermal or quantum fluctuations stabilize the uud state [@Kawamura; @Chubukov3]. The four-spin exchange has a same effect and as a result magnifies the 1/3 magnetization plateau.
In the phase diagram, we also mark the phase transition points among the Y-shape, 6-, 12-sublattice and uud phases, which are estimated by the vector and scalar chiral order parameters within the mean-field approximation.
Peculiar properties are expected at the multi-critical point. Since the spin-wave analysis is broken up just at the multi-critical point, we investigate systems close to the point. In Fig. \[disp\_uud\_h\] (a) we show the dispersion of the spin wave for $J=12K$ and $h=24K$ just on the phase-transition line between the Y-shape and the uud phases, which is calculated by the spin-wave theory assuming the uud state as the ground state. We found that the lowest gapless branch has a tendency to be flat along the $\Gamma$-M line. Actually, for $J/K=11.37 \simeq (J/K)_{\rm multi}$ the branch along the $\Gamma$-M line is flat as shown in Fig. \[disp\_uud\_h\] (b). As the magnetic field is increased from the value on the phase-transition line, the frequency of the flat lowest branch for $J/K=11.37 \simeq (J/K)_{\rm multi}$ linearly increases with $h/K$ in the range $22.11 < h/K \nle 34$ as demonstrated in Fig. \[gap\]. Since the second branch at the $\Gamma$ point as shown in Fig. \[disp\_uud\_h\] (b) crosses the lowest branch at $h/K \simeq 34$, the $h$-dependence of the frequency of the lowest branch at the $\Gamma$ point has an inflection at $h/K \simeq 34$. When the value of $J/K$ is shifted from the multi-critical point, the flatness due to the balance of the two- and four-spin exchanges is broken up, though the linearity still holds. In the uud phase the frequency of the lowest branch has the relation $\omega_k=2(h-h_1)$ and $2(h_2-h)$ for $h_1 < h < (h_1+h_2)/2$ and $(h_1+h_2)/2 < h < h_2$, respectively, where $h_1$ and $h_2$ are the magnetic fields of the lower and the upper phase boundaries of the uud phase. The linearity of the $h$-dependence of the frequency has been also obtained in the uud phase for the quantum AFH model [@Chubukov3].
Ground-State Energy and Sublattice Magnetization
================================================
The total energy of the ground state is evaluated by $E^{\rm c}+E^{\rm q}$, where $E^{\rm c}$ is the classical energy of Eq. (\[classical\_E\]) and can be analytically calculated, and $E^{\rm q}$ is the quantum energy corresponding to, e.g., Eq. (\[quantum\_E\]) for $h=0$. The magnetic-field dependences of $E^{\rm c}$ and $E^{\rm q}$ per bonds are shown in Figs. (\[enec\]) and (\[eneq\]), respectively, where we normalize them by using $J+K$ to compare with the values of the AFH model. It is found that, as $J/K$ is decreased, the quantum correction of the energy is increased.
The total magnetization is defined by using the expectation value of the spin $z$ component, i.e., $$M=-\frac{1}{N}\sum_{{\rm all}~i} \langle \sigma_i^z \rangle \ .
\label{totalm}$$ Note that we take the direction of the magnetic field anti-parallel to the $z$ direction. The sublattice magnetizations of the A and B sublattices are defined by $$m_{\rm A}=-\frac{3}{N}\sum_{i \in A} \langle \sigma_i^z \rangle
= -1+\Delta m_{\rm A} \ ,
\label{ma}$$ $$m_{\rm B}=-\frac{3}{N}\sum_{j \in B} \langle \sigma_j^z \rangle
= \beta (1-\Delta m_{\rm B}) \ ,
\label{mb}$$ respectively, where from Eqs. (\[HP\_A\]) and (\[HP\_B\]), $$\Delta m_{\rm A} = \frac{6}{N}\sum_{i \in A} \langle
a_i^{\dagger} a_i \rangle \ ,$$ $$\Delta m_{\rm B} = \frac{6}{N}\sum_{j \in B} \langle
b_j^{\dagger} b_j \rangle \ .$$ From a symmetry of the present system, the sublattice magnetization of the C sublattice is same as that of the B sublattice. The total magnetization is written from Eqs. (\[totalm\])-(\[mb\]), i.e., $$M = M^{\rm cl} + \Delta M \ ,$$ where $$M^{\rm cl} = -\frac{1}{3} (1-2\beta) \ ,$$ $$\Delta M = \frac{1}{3} (\Delta m_{\rm A} - 2\beta \Delta
m_{\rm B}) \ .$$
The magnetic-field dependences of $M$ for various values of $J/K$ are shown in Fig \[total\_mag\]. When $h=0$, the total magnetization is to be zero since the ground state has the 120$^\circ$ structure for all systems. For the AFH system, the value of $-M$ is increased with $h$ and saturates to $-M=1/3$ at $h/J=3$, where the phase transition to the uud phase occurs. While we can obtain the same behavior as the AFH system for $J/K=14$, the values of $-M$ become negative for small $h/(J+K)$ for $10.5 \le J/K \le 12$. This result shows that the spin-wave theory on the sublattice magnetization is broken up due to strong quantum fluctuations in the region. We also show the $h$-dependence of the quantum correction $\Delta M$ in Fig. \[total\_qcorr\]. The quantum corrections vanish both at $h=0$ and the phase transition point to the uud phase.
The magnetic-field dependences of the sublattice magnetizations of the A and B sublattices for various values of $J/K$ are shown in Fig. \[sub\_mag\]. The values of $-m_{\rm A}$ and $m_{\rm B}$ are decreased with $J/K$ and $h/(J+K)$ except near the phase transition point to the uud phase. The reduction shows that the quantum effects are enhanced by the addition of the four-spin interaction in the MSE model. We also show the dependence of the quantum corrections of the sublattice magnetizations on the magnetic field in Fig. \[qcorr\]. The quantum corrections are larger for smaller $J/K$ and larger $h/(J+K)$. We can obtain an interesting tendency of the size relation between $\Delta m_{\rm A}$ and $\Delta m_{\rm B}$ within the present spin-wave approximation. In the AFH system with $K=0$, the value of $\Delta m_{\rm A}$ is larger than that of $\Delta m_{\rm B}$ except for $h=0$. The result suggests that the sublattice magnetization along the axis anti-parallel to the direction of the magnetic field is most sensitive to quantum fluctuation. In the MSE system with the large four-spin interaction and small magnetic field, on the other hand, the sublattice magnetization whose quantization axis tends to parallel to the direction of the magnetic field is most sensitive to quantum fluctuation.
Summary and Discussions
=======================
In the present paper, we studied quantum effects on the 3-sublattice structures in the $S=1/2$ multiple-spin exchange model with two-, three- and four-spin exchange interactions on the triangular lattice in the magnetic field. By using the linear spin-wave theory, we found that the coplanar Y-shape state is stable as the ground state of the quantum system, though the four-spin interaction leads to the instability as the softening of the spin wave and the phase transition to the 6-sublattice phase occurs.
We had arguments in the framework of the linear spin-wave theory. Within the approximation, the magnetization becomes negative in the parameter region where the four-spin exchange interaction is dominant. We should take into account higher-order corrections of the spin wave to estimate the magnetization in the region.
The exact diagonalization study of finite clusters predicted that the ground state is a spin-liquid state with a spin gap filled with a large number of singlet states for parameters corresponding to our 6-sublattice phase [@LiMing]. The character of the ground state in this parameter region is, however, still not clarified. In order to numerically examine the possibility of the 6-sublattice phase, we would need the exact diagonalization of clusters with larger sizes. The spin-wave analysis assuming the 6-sublattice structure as the ground state is another interesting problem remained for the future work.
We acknowledge useful discussions with Y. Uchihira. This work is partly supported by Grants-in-Aid for Scientific Research Programs (No. 17071011, No. 17540339, No. 18740239) and 21st Century COE program from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
[99]{}
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[^1]: E-mail address: cyasuda@phys.aoyama.ac.jp
|
---
author:
- Liang He
- Su Yi
title: 'Supplemental Material for “Anomalous thermodynamics of lattice Bose gases in optical cavities”'
---
Hubbard-Stratonovich transformation on the canonical partition function
=======================================================================
In the occupation number representation, the explicit form of the quantum partition function of the system in the canonical ensemble reads $$\begin{aligned}
Z_{\mathrm{C}} & =\sum_{\{n_{i,\sigma}\}}e^{-\beta\left\{ \sum_{i,\sigma}\left[\frac{U_{s}}{2}n_{i,\sigma}(n_{i,\sigma}-1)\right]-\frac{U_{l}}{L}\left[\sum_{i}\left(n_{i,e}-n_{i,o}\right)\right]^{2}\right\} },\label{eq:Partition_function_explicit_form}\end{aligned}$$ with $\beta\equiv(k_{B}T)^{-1}$ being the inverse temperature. The Hubbard-Stratonovich transformation that we use to introduce the CDW order parameter field $\phi$ into the partition function reads $$\begin{aligned}
& \left(\sqrt{\frac{\beta U_{l}L}{\pi}}\right)^{-1}\exp\left(\beta\frac{U_{l}}{L}\left[\sum_{i=1}^{L/2}\left(n_{i,e}-n_{i,o}\right)\right]^{2}\right)\nonumber \\
& =\int_{-\infty}^{+\infty}d\phi\,\exp\left(-\beta U_{l}\left\{ L\phi^{2}+2\left[\sum_{i=1}^{L/2}\left(n_{i,e}-n_{i,o}\right)\right]\phi\right\} \right).\label{eq:HST}\end{aligned}$$ By using Eq. (\[eq:HST\]) we can replace the long range interaction term appearing in Eq. (\[eq:Partition\_function\_explicit\_form\]) by the integral over the $\phi$ field and rewrite the partition function $Z_{\mathrm{C}}$ in terms of $\phi$ as $$Z_{\mathrm{C}}=\sqrt{\frac{U_{l}L}{\pi k_{B}T}}\int_{-\infty}^{+\infty}d\phi\,e^{-(k_{B}T)^{-1}Lf_{\{T,U_{s},U_{l}\}}(\phi)}\label{eq:Z_c_in_sup_mat}$$ with $$\begin{aligned}
& f_{\{T,U_{s},U_{l}\}}(\phi)\equiv U_{l}\phi^{2}-\frac{k_{B}T}{L}\ln\left(\sum_{M=-N}^{N}e^{\frac{2U_{l}\phi M}{k_{B}T}}C_{M}(T)\right),\label{eq:f_function_in_fup_mat}\end{aligned}$$ as also shown in Eqs. (2, 3)in the main text. The summation involved in the $f_{\{T,U_{s},U_{l}\}}(\phi)$ can not be performed analytically, however one can calculated numerically for sufficiently large system. Here, $\phi$ can be regarded as the fluctuating CDW order parameter filed. Its expectation value $\bar{\phi}\equiv\langle\phi\rangle$ equals to CDW order parameter $\langle\sum_{i=1}^{L/2}\hat{n}_{i,e}-\sum_{i=1}^{L/2}\hat{n}_{i,o}\rangle/L$ exactly, which can be shown by introducing a source $J$ that couples to $L^{-1}\sum_{i=1}^{L/2}(n_{i,e}-n_{i,o})$ in the partition function and calculating the derivative $\left.\partial\ln Z_{\mathrm{C}}[J]/\partial J\right|_{J=0}$.
In the thermodynamic limit $L\rightarrow\infty$, the integral with respect to $\phi$ in Eq. (\[eq:Z\_c\_in\_sup\_mat\]) is given exactly by its saddle point integration. Therefore, in the thermodynamic limit, $$Z_{\mathrm{C}}=\left(\sqrt{\frac{U_{l}L}{\pi k_{B}T}}\right)e^{-(k_{B}T)^{-1}L\min\left[f_{\{T,U_{s},U_{l}\}}(\phi)\right]}\label{eq:Z_C_in_THE_limit}$$ and the CDW order parameter $\bar{\phi}$ is given by the value of $\phi$ that minimizes $f_{\{T,U_{s},U_{l}\}}(\phi)$.
Caculation of Free energy and compressibility
=============================================
In order to check the positiveness of the compressibility, we need to calculate second order derivative of free energy $F$with respect to the total particle number $N$, i.e., $\partial^{2}F/\partial N^{2}$. This calculation is facilitated by using the expressioin of canonical partition function shown in Eq. (\[eq:Z\_C\_in\_THE\_limit\]), via which we can get $$F=-k_{B}T\ln Z_{\mathrm{C}}=-k_{B}T\ln\sqrt{\frac{\beta U_{l}L}{\pi}}+L\min\left[f_{\{T,U_{s},U_{l}\}}(\phi)\right].$$ Therefore, by directly calculating $\min\left[f_{\{T,U_{s},U_{l}\}}(\phi)\right]$ at different total particle number $N$ of the system, we can construct the function $F(N)$ and calculate its second order derivative $\partial^{2}F/\partial N^{2}$.
|
---
abstract: 'A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF’s is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.'
---
-12mm -10mm =0
ITEP-TH-30/02\
hep-th/0206039\
[**ON THE FEDOSOV DEFORMATION QUANTIZATION BEYOND THE REGULAR POISSON MANIFOLDS**]{}\
V.A. Dolgushev$^{a,b}$, A.P. Isaev$^{a,c}$, S.L. Lyakhovich$^{d,e}$ and A.A. Sharapov$^e$
[*$^a$ Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna,*]{}\
[*141 980, Moscow Region, Russia*]{}\
[*$^b$ Institute for Theoretical and Experimental Physics,* ]{}\
[*117259, Moscow, Russia*]{}\
[*$^c$ Max-Plank-Institute für Mathematik*]{}\
[*Vivatsgasse 7, D-53111 Bonn, Germany* ]{}\
[*$^d$ Department of Theoretical Physics, Chalmers University of Technology,*]{}\
[*S-412 96 Göteborg, Sweden*]{}\
[*$^e$ Department of Theoretical Physics, Tomsk State University,* ]{}\
[*634050, Lenin av. 36, Tomsk, Russia*]{}
Introduction
============
The Deformation Quantization, as it was originally formulated in [@Berezin; @BFFLS], has undergone an explosive development during the last two decades [@Stern] with a number of important achievements including Kontsevich’s $\ast$-product construction for a general Poisson manifold [@Kontsevich]. Fedosov [@Fedosov; @Fedosovbook] gave a simple and manifestly covariant quantization method working well for any symplectic or regular Poisson manifold and yet giving a simple classification for inequivalent quantizations. Meanwhile, this method can not be directly applied to the irregular Poisson brackets because the main ingredient is lacking for the Fedosov construction: no affine connection can exist respecting an irregular Poisson structure.
In this paper we suggest a simple method extending the Fedosov approach to a broad class of manifolds with irregular Poisson brackets, that gives an explicitly covariant and geometrically transparent algorithm for constructing the $\ast$-products in the irregular case. To make the method working, these Poisson manifolds are assumed to be equipped with certain additional algebraic structures. We also expect that the method will allow for generalizations essentially relaxing conditions imposed on the class of Poisson manifolds. Although the brackets of considered class, being more general than the regular ones, are of interest in their own rights, our approach admits also a purely algebraic reformulation, which allows one to simply derive the Universal Deformation Formula for any triangular Lie bialgebra (see Sec. 5).
To explain the class of the irregular brackets we are going to study, let us give a simple example of this type considered in [@R93; @R98].
Consider a collection of pair-wise commuting vector fields $X_i$, $ i=1,...,n$ defined on a smooth real manifold $M$. These fields may be viewed as derivatives of the commutative algebra of smooth functions $ C^{\infty}(M)$. A constant skew-symmetric matrix $r$ assigns $M$ with a Poisson bracket of the form $$\label{RB}
\{f,g\}=r^{ij}(X_if)(X_jg)\,,\quad\quad \forall\, f,g\in
C^{\infty}(M)\,.$$ In general, this bracket is irregular since the rank of the distribution $
X_i $ can vary from point to point. Nevertheless, it can be easily quantized by the Weyl-Moyal like formula $$f*g=f\cdot g+\sum_{k=1}^{\infty} \left(-\frac{{i\hbar}}{2}\right)^k
\frac{1}{k!}
r^{i_1 j_1}\ldots r^{i_k j_k} (X_{i_1} \ldots X_{i_k} f)(X_{j_1}
\ldots X_{j_k} g)\,, \label{gen-n}$$ ${\hbar}$ being a formal deformation parameter. The associativity of the $\ast$ -product follows trivially from the commutativity of vector fields $X_i$. Note that (\[gen-n\]) may be regarded as the [*universal deformation formula*]{}, in the sense that it works for the action of [*any*]{} commutative Lie algebra.
A natural generalization of the above construction, we are going to study in this paper, consists in allowing the vector fields $X_i$ to form a noncommutative Lie algebra, say $$[X_i,X_j]=f_{ij}^kX_k\,, \label{lalg}$$ $f_{ij}^k$ being structure constants. The Jacobi identity for the Poisson bracket (\[RB\]) implies that $$\label{JI}
\{f,\{g,h\}\}+cycle(f,g,h)=\Lambda ^{ijk}(X_if)(X_jg)(X_kh)=0\,,$$ where $$\Lambda ^{ijk}=f_{mn}^ir^{mj}r^{nk}+cycle(i,j,k)\,.$$ For the commutative algebra we have $\Lambda \equiv 0$, and the Jacobi identity is automatically satisfied. In general case, equation $\Lambda =0$ is rather nontrivial and known in mathematical literature as the [*classical Yang-Baxter equation*]{} (CYBE). For reasons, which we will not explain here, a skew-symmetric matrix $r$ satisfying this equation is called the [*classical triangular $r$-matrix*]{} [@Dr83]. If all the vector fields $X_i$ are linearly independent (at least at one point on $M$), the CYBE provides both necessary and sufficient conditions for the Jacobi identity to hold. In the opposite case, the equation $\Lambda = 0$ gives only sufficient condition. The important observation concerning $r$-bracket (\[RB\]) is that one may always assume the $r$-matrix to be non-degenerate without restricting generality. Indeed, as for any skew-symmetric matrix $r$, one may always find such a basis of the generators $X_i=(X_A,X_\alpha )$ in which $r^{i\alpha }=0$ and ${\rm \det }(r^{AB})\neq 0$. Thus, only the vector fields $X_A$ actually contribute to the Poisson bracket (\[RB\]). From the CYBE it then follows that $$\Lambda ^{\alpha AB}=f_{MN}^\alpha r^{MA}r^{NB}=0\quad \Rightarrow \quad
f_{MN}^\alpha =0$$ and hence, the vector fields $X_A$ form a Lie subalgebra, for which $ r^{AB}$ is a non-degenerate triangular $r$-matrix. For non-degenerate $r$ -matrices the CYBE reduces to the ordinary cocycle condition $$\label{cc}
f^n_{ij}r_{nk}+cycle(i,j,k)=0\,,$$ where $ r_{ik}r^{kj}={\delta}^j_i. $ The solutions to the equation (\[cc\]), forming a linear space of 2-cocycles, are known to be in one-to-one correspondence with the central extensions of the Lie algebra. In other words, equation (\[cc\]) arises as a part of the Jacobi identity for the Lie algebra $$\label{fla}
[y_i,y_j]=f_{ij}^ky_k+r_{ij}c\,,\quad\quad [c,y_i]=0\,$$ The Lie algebra admitting a central extension defined by a non-degenerate cocycle $r_{ij}$ is called [*quasi-Frobenius*]{}, and [*Frobenius*]{} if at least one of such cocycles is trivial (representable in the form $r_{ij}=f^k_{ij}\xi_k$ for some vector $\xi_k$)[@El]. An extended list of examples of (quasi-)Frobenius Lie algebras (from here on QFL algebras, for short) can be found in recent papers [@L1], [@L2], [@L3].
In this paper we give a simple quantization formula for the Poisson bracket (\[RB\]) associated to the action of an arbitrary QFL algebra (\[lalg\]). Our method, being conceptually similar to that of Fedosov, does not require regularity of the bracket (\[RB\], \[lalg\]). The main distinction is a possibility to work with an auxiliary quantum bundle associated to the enveloping of the underlying Lie algebra, instead of the usual quantum Weyl algebra exploited in the original Fedosov construction and its various adaptations and reinterpretations [@BW], [@GL], [@BGL], [@DLS], [@KS]. Since the output $\ast$-product involves only initial algebraic data (much like its commutative version (\[gen-n\])) one may regard it as the Universal Deformation Formula for the respective algebra.
Finally, we should remark that the quantization problem for the $r$-brackets is deeply rooted to the theory of quantum groups [@Takh], dating back to the seminal Drinfeld’s paper [@Dr83], where the existence of quantization was proved for an arbitrary triangular $r$-matrix. Actually, the computations required for obtaining explicit expressions by the Drinfeld method are appeared to be impracticable except for the Abelian case. That is why the most of known examples of the deformation has been constructed by means of [*a twisting*]{} transformation technique rather than by the quantization [@L1], [@L2], [@L3], [@UDF]. In the recent paper [@XU], the Fedosov method has been used to quantize a non-degenerate triangular [*dynamical*]{} $ r$-matrices. In the formal algebraic settings the application of the Fedosov deformation quantization to a certain class of irregular Poisson brackets, including $r$-brackets (\[RB\]), (\[lalg\]) has been also discussed in [@V]. In principle, the method of the work [@XU], as well as it’s predecessor [@Dr83], gives rise to the quantization of the $r$-bracket (\[RB\]), and even more general Poisson structures. At the technical level, however, this method appears to be unduly cumbersome for the application to such a simple class of Poisson brackets as we are going to consider. In this respect, our approach, being elementary in essence, will hopefully be found useful both from theoretical and practical viewpoints.
The structure of the paper is as follows: The deformation quantization of general $r$-brackets is exposed in Section 2. Section 3 is devoted to the classification of such quantizations and some possible modifications of our method. In Section 4, we consider the case of quasi-homogeneous Poisson brackets on two-plane which is shown to provide an interesting class of $r$-brackets associated with the two-dimensional (and therefore quasi-Frobenius) Lie algebras. In Section 5, we show how the construction of the Universal Deformation Formula for a triangular $r$-matrix results in a quantization of the respective Lie bialgebra. In concluding Section, we summarize the results and sketch proposals for further generalizations of our construction within the framework of BRST quantization.
Construction of the $\ast$-product
==================================
Let us start with the action $\rho : L\rightarrow {\rm Vect}(M)$ of an abstract QFL algebra $L$ on a smooth real manifold $M$. Denote by $L_c$ the central extension of $L$ associated with a non-degenerate 2-cocycle $r$. We assume that upon choosing a basis the structure of the algebra $L_c$ is described by the Lie brackets (\[fla\]), so that $L=L_c/ {\Bbb R}c$ and the vector fields $\rho (y_i)=X_i$ commute according to (\[lalg\]).
Introduce the associative algebra ${\cal A}=C^\infty (M)\otimes
U(L_c)$, being a tensor product of the commutative algebra of smooth functions on $M$ and the universal enveloping of the Lie algebra $L_c$ over ${\Bbb C}$ (or more precisely, its formal completion by infinite series). The elements of ${\cal A}$ may be viewed as the sections of the trivial vector bundle $p
:U(L_c)\times M\rightarrow M$, with the fiber-wise associative algebra structure induced by the associative multiplication in $U(L_c)$. Choosing the basis of monomials being symmetric in the Lie algebra generators, we identify the elements of $U(L_c)$ with their Weyl symbols and set $c=1\in{\Bbb C}$. Then the generic element $a\in {\cal A}$ reads $$a(y)=\sum_{k=0}^\infty a^{i_1\cdots i_k}y_{i_1}\cdots y_{i_k}\,,$$ where $y_i$ are formal [*commuting*]{} variables and the coefficients $ a^{i_1\cdots i_k}\in C^\infty (M)$ are symmetric in the indices $ i_1,\ldots ,i_k$. The product of two symbols $a,b\in {\cal A}$, which we will denote by $\circ $, is given by [^1] $$\label{circ}
(a\circ b)(y)= a(\hat{L})b(y)\,, \quad \quad \hat{L}_i=
\left(y_j+\frac12r_{jn}\frac{\partial}{\partial y_n}\right){\cal
R}^j_i\left(\frac{\partial}{\partial y_k}\right) \,,$$ where $$\label{}
{\cal R}(x)= \sum_{m=0}^{\infty}\frac{b_m}{m!}\Lambda ^m(x)
=\left(\frac{e^{\Lambda (x)}- 1}{\Lambda
(x)}\right)^{-1}\,,\qquad \Lambda^i_j(x)=x^kf_{kj}^i\,,$$ $b_m$ being Bernoulli numbers. The formula involves well-known group-theoretical constructions: matrices $\Lambda(x)$ realize the adjoint representation of the Lie algebra $L$; ${\cal R}(x)$ are the matrices of the right shifts on the group $G={\rm
Exp}(L)$, written in the first kind coordinates $x^i$; finally, operators $\hat{L}_i$ define the left regular representation of the algebra $L_c$ corresponding to the Weyl ordering [@KM], $$[\hat{L_i},\hat{L_j}]=f_{ij}^k\hat{L}_k+r_{ij}\,.$$ In what follows the explicit form (\[circ\]) of the $\circ$-product will be inessential for our considerations.
The space of smooth functions $C^{\infty}(M)$ is embedded into ${\cal A}$ as a central subalgebra. Denote by $\pi: {\cal
A}\rightarrow C^{\infty}(M)$ the canonical projection $(\pi
a)(y)=a(0)$. To assign $C^{\infty}(M)$ with a $\ast$-product compatible with the Poisson brackets (\[RB\]) we construct another, less trivial embedding $\sigma: C^{\infty}(M)\rightarrow
{\cal A}$, allowing one to induce the desired $\ast$-product via pull-back of $\circ$-product. To this end, introduce the following Lie algebra of external differentiations: $$\label{der}
\begin{array}{ll}
D_ia=X_ia+[y_i,a]\,,&\\[3mm]
D_i(a\circ b)=(D_ia)\circ b+a\circ (D_i b)\,, &\forall\, a,b \in
{\cal {A }} \,,\\[3mm]
[D_i,D_j]=f_{ij}^kD_{k}\,.&
\end{array}$$ Hereafter we set $[a,b]=a\circ b - b\circ a$. Denote by ${\cal
A}_D$ the subalgebra of $D$-constant elements $$\label{AD}
{\cal A}_D=\{a\in {\cal A}|\,\, D_ia=0\,\}\,.$$ The following theorem establishes isomorphism between the linear spaces $
C^\infty(M)$ and ${\cal A}_D$.
[**Theorem 1.**]{} [*Any $D$-constant element $a\in
{\cal A}_D$ is uniquely determined by its projection $\pi a$ onto subspace $C^{\infty}(M)$ of $y$-independent elements and vice versa.*]{}
[*Proof.*]{} Let $$\label{exp}
a(y)=\sum_{n=0}^\infty a_n\,,\quad\quad a_n=a^{i_1\dots
i_n}y_{i_1}\dots y_{i_n}\,,$$ be an arbitrary element from ${\cal A}$ with a given $a_0=a(0)\in
C^{\infty}(M)$. Expanding the condition $D_ia=0$ via the basis of homogeneous monomials we get a chain of equations $$\partial_i a_{n+1}=V_i a_{n}\,, \label{oni}$$ where we have denoted $$\partial_i=r_{ji}\frac{\partial}{\partial y_j}\,,\quad\quad
V_i=X_i+y_kf^k_{ij}\frac{\partial}{\partial y_j}\,.$$ Besides, the system (\[oni\]) should be supplemented by a set of compatibility conditions resulted from the commutativity of the partial derivatives $\partial_i$, $$\label{ccon}
\partial_i(V_j a_{n})-\partial_j(V_i a_{n})=0\,.$$ To construct a solution to equations (\[oni\]), (\[ccon\]) we use the induction over the degree $n$ in the expansion (\[exp\]). For $n=0$ the equations are obviously consistent and we get $$a_{1}=y_ir^{ji}X_ja_{0}\,,\quad a_{0}\in C^{\infty}(M)\,.$$ Suppose that we have found all $a_k$ with $k=1,2,...,n$. Then equations (\[oni\]) for $a_{n+1}$ are compatible provided the conditions ($\ref{ccon}$) are satisfied. Using identities $$\label{id}
[\partial_i, V_j] -[\partial_j, V_i] =f_{ij}^k\partial_k\,,\quad\quad
[V_i,V_j]=f_{ij}^kV_k\,,$$ we can rewrite (\[ccon\]) as $$\partial_i(V_j a_{n})-\partial_j(V_i a_{n}) =f^k_{ij}\partial_k a_{n}-
(V_i\partial_j-V_j\partial_i)a_{n}\,,$$ and since by induction hypothesis $\partial_ia_{n}=V_ia_{n-1}$ we finally get $$f_{ij}^k\partial_ka_{n}-[V_i,V_j]a_{n-1}=f_{ij}^k( \partial_ka_{n}-V_k
a_{n-1})=0\,.$$ Upon the compatibility conditions have satisfied the straightforward integration of (\[oni\]) leads to the explicit recurrent formula $$\label{iter}
a_{n+1}(y)=\int_0^1y_jr^{ij}\left(X_ia_{n}(ty)+y_kf^k_{im} \frac{
\partial a_{n}(ty)}{\partial y_m}\right)dt$$ for the unique $D$-constant element $a\in {\cal A}_D$ with a prescribed $ a_0=a(0)\in C^{\infty}(M)$. q.e.d.
Up to the second order in $y$ the lift of an arbitrary function $a\in
C^{\infty}(M)$ to a $D$-constant element of ${\cal A}$, denoted by $
\sigma(a) $, reads $$\label{lift}
\sigma(a)=a+y^iX_ia+\frac12(y^iy^jX_iX_ja+y_jy^if_{in}^jX^na)+\cdots$$ $$y^i=r^{ji}y_j\,,\,\,\,\,\,\,X^i=r^{ji}X_j\,.$$
As the next step we introduce the deformation parameter $\hbar$ and extend the space of smooth functions, i.e. the space of classical observables, to that of quantum observables $C^{\infty}(M)[[\hbar]]$. The latter consists of elements $$a(\hbar)=\sum_{k=0}^{\infty}\hbar^k a_k\,, \quad \quad a_k\in
C^{\infty}(M)\,,$$ being formal power series in $\hbar$ with coefficients in smooth functions. The above definitions of $\circ$-product and derivatives (\[der\]) are extended by linearity to the algebra ${\cal A}[[\hbar
]]=C^{\infty}(M)[[\hbar ]]\otimes U(L_c)$; in so doing, we rescale the structure constants of the algebra $L_c$ by multiplying on $i\hbar$, $$\label{rsc}
f^k_{ij}\rightarrow i\hbar f^k_{ij}\,,\quad\quad r_{ij}\rightarrow i\hbar
r_{ij}\,,$$ and redefine the action of the derivatives $$\label{rd}
D_i\rightarrow D_i=X_i+\frac{1}{i\hbar}[\,y_i,\,\cdot\,]\,,$$ so that the lifting map (\[iter\]) remains intact. The relation of the above theorem to the quantization of a general $r$-bracket (\[RB\]) is established by the following
[**Theorem 2**]{}. [*The pull-back of $\circ$-product on ${\cal A}[[\hbar]]$ via ${\Bbb C}[[\hbar]]$-linear isomorphism $\sigma: C^{\infty}(M)[[\hbar]]\rightarrow {\cal A}[[\hbar]] $ induces an associative $\ast$-product on $C^{\infty}(M)[[\hbar]]$*]{}, $$\label{star}
a\ast b=\pi(\sigma (a)\circ \sigma(b))=\sum_{k=o}^{\infty}\hbar^k
D_k(a,b)\,$$ [*satisfying the properties*]{}
i\) [*Locality : $D_{k}(a,b)$ are bi-differential operators of order $k$;*]{}(25.$a$)
[*ii) Correspondence principle:*]{} $a\ast b=a\cdot
b-\displaystyle \frac{i\hbar}{2}\{a,b\}\,\,\,({\rm
mod}\,\hbar^2)$;(25.$b$)
[*iii) Normalization condition:*]{} $1*a=a*1=a$.(25.$c$)
[*Proof*]{}. The associativity of $\ast$-product directly follows from associativity of $\circ$-product and the identity $\sigma \pi |_{{\cal A}_D}={\rm id}$. To prove the locality condition it is convenient to assign the algebra ${\cal
A}[[\hbar ]]$ with a filtration $${\cal A}[[\hbar ]]={\cal
A}_0\supset {\cal A}_1\supset {\cal A}_2\supset \cdots$$ associated to the sequence of subspaces ${\cal A}_k$ consisting of elements $$a=\sum_{2m+n\geq k}\hbar ^ma_m^{i_1\dots
i_n}y_{i_1}\dots y_{i_n}\in {\cal A}_k\,.$$ This filtration defines the topology and convergence in the space of formal power series in $\hbar $ and $y$’s. From the recurrent formula (\[iter\]) we see that the element $a_n$ belongs to ${\cal
A}_n$ and involves [*finite*]{} linear combinations of $a_0$ and its derivatives along $X_i$ up to order $n$. In view of the obvious inclusions (keep in mind redefinitions (\[rsc\])) $${\cal
A}_m\circ {\cal A}_n\subset {\cal A}_{n+m}\,,\quad \quad \pi ({\cal A}
_n)\subset {\cal A}_n$$ we can write $$a*b=\sum_{n+m\leq 2k}\pi (a_n\circ b_m)\quad ({\rm mod}\,\,\hbar ^{k+1})$$ and hence, each power of $\hbar $ in the product expansion is given by a finite order bi-differential operator. More accurate consideration of the above formula shows that $$D_k(a,b)=\frac {(-i)^k}{2^kk!}r^{i_1j_1}\ldots
r^{i_kj_k}(X_{i_1}\ldots X_{i_k}a)(X_{j_1}\ldots X_{j_k}b)+\cdots\,,$$ where dots stand for the bi-differential operators of order less than $k$. For the commutative Lie algebra $L$ the rest terms vanish and we come to the Weyl-Moyal $\ast$-product (\[gen-n\]).
The correspondence principle (25.$b$) is proved by straightforward verification with the help of eq.(\[lift\]). Finally, to prove the normalization condition (25.$c$) it suffices to note that $\sigma(1)=1$. q.e.d.
[*Remark 1*]{}. The constructed $\ast$-product has a rather special structure: the bi-diffrential operators at each power of $\hbar$ are determined by repeated differentiations along the vector fields $X_i$, $$\label{Dn}
D_n(a,b)=\sum_{k,l\leq n}D_n^{i_1\cdots i_kj_1\cdots
j_l}(X_{i_1}\cdots X_{i_k}a)( X_{j_1}\cdots X_{j_l}b)\,,$$ where the coefficients $D_n^{i_1\cdots i_kj_1\cdots j_l}\in {\Bbb
C}$ are universally expressed via the structure constants $f_{ij}^k$ and $r_{ij}$, irrespective of the concrete representation $\rho: X_i\rightarrow {\rm Vect}(M)$. For this reason one may refer to this $\ast$-product as the universal quantization formula. It would be very interesting to compare this result with the Kontsevich quantization formula [@Kontsevich] and/or give its “sigma-model interpretation" [@CF1].
[*Remark 2*]{}. The above quantization admits a natural deformation of the algebraic data entering to the $\ast$-product construction. Namely, the cocycle $r$ may be replaced by any formal series $$\label{def}
r^{\prime}_{ij}=r_{ij}+\hbar r^{(1)}_{ij}+\hbar^2r^{(2)}_{ij}+\cdots\,,$$ where $r^{(k)}$ are arbitrary 2-cocycles of the algebra $L$. Note that the matrix $r^{\prime}$ is formally invertible since $r$ is so. As the result we get another star-product having the same quasi-classical limit (25.$b$). The question arises as to whether the deformation (\[def\]) leads to essentially different quantization or an equivalence transform can be found to establish an isomorphism between two algebras of quantum observables. In the next section we formulate the necessary and sufficient conditions for two star-products, obtained in such a way, to be equivalent.
Two theorems of equivalence
===========================
Given a QFL algebra $L$, consider two star-products $\ast$ and $\ast^{\prime}$ associated to non-degenerate 2-cocycles $r$ and $
r^{\prime}$ of the form (\[def\]). Following [@Dr83], we call these star-products to be [*equivalent*]{} if there exists an invertible operator $$B:C^{\infty}(M)[[{\hbar}]]\mapsto C^{\infty}(M)[[{\hbar}]], \label{eqop}$$ establishing isomorphism of two star-algebras, i.e. $$B(a\ast' b)= (Ba \ast Bb) \label{inter}$$ The most general expression for such an operator $B$ preserving the peculiar structure of $\ast$-product (see Remark 1 of the previous section) looks like $$\label{b}
B=1+\hbar B_1+\hbar^2B_2+\,\cdots\,,$$ where $B_k$ are finite order differential operators of the from $$\label{bk}
B_k=\sum_{n=1}^{N}C^{i_1\cdots i_n}X_{i_1}\cdots
X_{i_n}\,\quad\quad C^{i_1\cdots i_n}\in {\Bbb C}\,.$$
In the paper by Drinfeld [@Dr83] the theorem was stated that the inequivalent Universal Deformation Formulas are classified by the formal series in ${{\hbar}}$ with values in the second cohomologies of the Lie algebra. However the proof of the theorem was given neither in this paper [@Dr83] nor in any of consequent papers. Here we present a simple proof of the classification theorem for the inequivalent Universal Deformation Formulas, obtained by our procedure.
[**Theorem 3.**]{} [*Two star-products $\ast$ and $\ast^{\prime}$ associated to formal 2-cocycles $r$ and $r^{\prime}$ are equivalent if and only if*]{}
$i)$ $r =r^{\prime}\,\, ({\rm mod}\,\, \hbar)$;
$ii)$ $r-r^{\prime}$ [*is a coboundary* ]{}.
[*Proof.*]{} The necessity of the above conditions can be proved by standard deformation quantization arguments based on the consideration of Hochschild cohomologies. So we omit this part of the proof and turn to the sufficiency. By assumption of the theorem we have $r_{ij}-r^{\prime}_{ij}=f_{ij}^k\xi
_k$ , where $\xi _i={\hbar }\xi _i^{(1)}+{\hbar }^2\xi
_i^{(2)}+\,\,\ldots $ is a formal vector. Then the Lie algebra isomorphism $Q: L_c\rightarrow L_c^{\prime }$, defined on the basis elements as $$Q(y_i)=y_i+\xi _ic\,,\quad \quad Q(c)=c\,$$ induces isomorphism of the universal enveloping algebras $U(L_c)$ and $
U(L_c^{\prime })$, which then extends to isomorphism of associative algebras $({\cal A}[[\hbar ]],\circ )$ and $({\cal A}[[\hbar ]],\circ ^{\prime })$. This means that the operator $$Qa(y,{\hbar })=a(y+\xi,\hbar ) \label{Q}$$ intertwines the circle-products $\circ $ and $\circ ^{\prime }$ constructed by $r$ and $r^{\prime }$ respectively. Since $$D_i^{\prime }a=X_ia+\frac 1{i\hbar} [y_i\,,a\,]^{\prime
}=QDQ^{-1}\,, \quad \quad [a,b]^{\prime }=a\circ ^{\prime
}b-b\circ ^{\prime }a\,, \label{DC}$$ $Q$ maps $D$-constant elements to $D^{\prime}$-constant ones. Then it is easy to see that operator $$Ba=\pi Q\sigma (a) \label{eqiop}$$ has the form (\[b\],\[bk\]) and intertwines the star-products $\ast$ and $*^{\prime }$. Indeed, $$(Ba)*^{\prime }(Bb)=(\pi Q\sigma (a))*^{\prime }(\pi Q\sigma
(b))=\pi (Q\sigma (a)\circ ^{\prime }Q\sigma (b))=$$ $$=\pi Q(\sigma (a)\circ \sigma (b))= B\pi (\sigma (a)\circ \sigma
(b))=B(a*b)$$ q.e.d.
So far the role of QFL algebra underlying the quantization was twofold. First, it had appeared as the Lie algebra of vector fields entering to the $r$-bracket construction (\[RB\]). Second, it had defined the fiber-wise $\circ$-product on the auxiliary bundle of the formal universal enveloping algebra $U(L_c)[[\hbar]]$. Now we are going to separate these functions to obtain a more flexible formalism which, as we hope, admits a generalization beyond the case of Lie algebras (see Sec. 6). This may also be viewed as a further deformation of the fiber-wise $\circ$-product giving an equivalent $\ast$-product on the base manifold $M$.
[**Theorem 4**]{}. [*Let* ]{}$(L,r)$[* and* ]{}$(L^{\prime },r^{\prime })$ [*be two QFL algebras with non-degenerate 2-cocycles. If* ]{}$\dim L=\dim L^{\prime }$[* then the associative algebras*]{} $U(L_c)[[\hbar ]]$[* and* ]{}$U(L_c^{\prime })[[\hbar ]]$ [* are isomorphic.* ]{}
[*Remark*]{}. Without loss of generality we may assume that $r=r'$ (if not, choose another basis in the Lie algebra). Then it is sufficient to construct such an isomorphism $\phi:
U(L_c)[[\hbar ]] \rightarrow U(L_c^{\prime })[[\hbar ]]$ for the case when algebra $L$ is fixed. For definiteness, we take $L_c$ to be the Heisenberg-Weyl algebra: $$\label{GW}
\lbrack z_i,z_j]=i\hbar r_{ij}c\,,$$ whose formal universal enveloping algebra, denoted by $W$, consists of formal power series in $z_i$ and $\hbar$ with complex coefficients $$\label{GW1}
a(z,\hbar)=\sum_{k,n\geq 0}\hbar^{k}a_k^{i_1\cdots i_n}z_{i_1}\cdots
z_{i_n}\,.$$ Since the underlying QFL algebra $L$ is commutative the matrix ${\cal R}$ entering to the definition (\[circ\]) of $\circ$-product is equal to unit and we arrive at the usual Weyl-Moyal formula $$a\circ b
=a\left(z_i+\left(\frac{i\hbar}{2}\right)r_{ij}\frac{\partial}{\partial
z_j }\right)b(z)={\rm
exp}\left(\frac{i\hbar}{2}r_{ij}\frac{\partial}{\partial z_i}
\frac{
\partial}{\partial w_j}\right)a(z)b(w)|_{z=w}\,.$$ In what follows we will refer to $W$ as the Weyl algebra.
Now let $L$ and $L^{\prime}$ be arbitrary QFL algebras of equal dimension, then the stated isomorphism $\phi:
U(L_c)[[\hbar]]\rightarrow U(L'_c)[[\hbar]]$ can be written as the composition $\phi=\rho\rho'^{-1}$ of isomorphisms $\rho:
W\rightarrow U(L_c)[[\hbar]]$ and $\rho': W\rightarrow
U(L^{\prime}_c)[[\hbar]]$.
[*Proof*]{}. Prescribing the degrees to the formal variables: $\deg \ y_i=1$ and $\deg \ \hbar =2$, we turn $W$ to a graded associative algebra assigned by the natural filtration $W\supset W_1\supset W_2\ldots $ with respect to the total degree $2k+n$ of the series terms (\[GW1\]). The desired isomorphism $\rho: W\rightarrow U(L_c)[[\hbar ]]$ is then completely determined by its action on the generators $z_i$ of the Heisenberg-Weyl algebra. Having in mind the invertibility of the map $\rho$, we are looking for $\rho$ of the form $$\label{anzats}
y_i=\rho (z_i)=z_i+Y_i(z,\hbar)\,,$$ where $Y_i\in W_2$. Substituting this anzatz to the commutation relation of the algebra $L_c$ we get $$[\rho (z_i),\rho (z_j)]=i\hbar (f_{ij}^k\rho (z_k)+r_{ij})\,,$$ or explicitly $$\label{MC}
\partial _iY_j-\partial _jY_i=f_{ij}^k(z_k+Y_k)-\frac 1{i\hbar}
[Y_i,Y_j]\,,$$ where $$\partial _i =r_{ij}\frac{\partial}{\partial z_j}\,.$$ Let us show that the last equation has a unique solution for $Y_i\in W_2$ subject to additional condition $$\label{con}
z_ir^{ij}Y_j(z,\hbar )=0\,$$ This in particular implies $Y_i(0,\hbar )=0$. Consider an expansion of $Y_i$ in the homogeneous components $$Y_i=\sum\limits_{k\geq 2}Y_i^{(k)}\,, \quad\quad \deg \
Y_i^{(k)}=k\,.$$ Substituting this expansion back to the equation (\[MC\]) we obtain a chain of equations $$\label{Y}
\begin{array}{l}
\partial_iY_j^{(2)}-{\partial}_jY_i^{(2)}=f_{ij}^kz_k,
\\[3mm]\displaystyle
\partial _iY_j^{(n+1)}-\partial _jY_i^{(n+1)}=f_{ij}^kY_k^{(n)}-\frac
1{i\hbar} \sum\limits_{k=2}^n[Y_i^{(n+2-k)},Y_j^{(k)}]\ ,\qquad
n>2\,.
\end{array}$$ Treating now $Y_i^{(n)}$ as 1-forms on a linear space with coordinates $z^i=r^{ij}z_j$ we see that above equations have a structure $dY^{(n)}=F^{(n)}$, where 2-forms $F^{(n)}$ are known. A necessary and sufficient condition for these equations to be solvable is $dF^{(n)}=0$ (the Poincare’ lemma). The last condition can be proved by induction over degree $n$. For $n=2$ the r.h.s. of the first line in (\[Y\]) is obviously closed in view of the cocycle condition $ f_{ij}^lr_{lk}+cycle(i,j,k)=0$. Suppose now that $dF^{(n)}=0$ and $Y^{(n)}$ is a solution to equation $dY^{(n)}=F^{(n)}$. Then it is not hard to check that $$(dF^{(n+1)})_{ijk}=f_{ij}^l(dY^{(n)}-F^{(n)})_{kl}+cycle(i,j,k)=0\,.$$ Thus, using the induction, we obtain that $dF^{(n)}=0$, at any $n$. By the Poincare’ lemma there exists a unique 1-form $Y^{(n)}=d^{-1}F^{(n)}$ vanishing on the linear vector field $v=z^i\partial /\partial z^i$ (condition (\[con\])). The explicit expression reads $$Y_i^{(n)}(z)=\int\limits_0^1dtF_{ij}^{(n)}(tz)z^j\,.$$
The inverse map $\rho^{-1}: U(L_c)[[\hbar]]\rightarrow W$ is obtained by iterating equation $$z_i=y_i-Y_i(z,\hbar)\,$$ with respect to $z_i$. Since ${\rm deg}\,Y_i\geq 2$ these iterations converge to the unique solution $z_i=\rho^{-1}(y_i)\in
W$. q.e.d.
Using this theorem we can reformulate the quantization procedure of Sec. 2 in terms of Weyl algebra bundle for any QFL algebra $L$ of vector fields $X_i\in {\rm Vect}(M)$. Namely, starting with the associative algebra $A[[\hbar]]=C^{\infty}(M)\otimes W$ we introduce a set of its derivatives $D_i:A[[\hbar]]\rightarrow
A[[\hbar]]$, $$D_ia= X_ia+\frac1{i\hbar}[y_i,\,a]\,,$$ In view of the Theorem 4, one can choose the element $y_i=z_i+Y_i(z)$ to satisfy the commutation relations for the Lie algebra $L_c$. Upon this choice for $y$’s the derivatives $D_i$ form the Lie algebra $L$ with respect to the commutator. Now, repeating the proof of the Theorem 1 one can show the existence of an isomorphism between the subspace of $D$-constant elements in ${\cal A}[[\hbar]]$ and the space of quantum observables $C^{\infty}(M)[[\hbar]]$. The pull-back of $\circ$-product on ${\cal A}[[\hbar]]$ via this isomorphism induces $\ast$-product on $C^{\infty}(M)[[\hbar]]$. The latter is obviously equivalent to the $\ast$-product from Sec. 2.
Nonabelian example
==================
The simplest example of the Frobenius Lie algebra is provided by two-dimensional Borel algebra $B$ with the Lie bracket $$\label{borel}
[H,E]=E\,.$$ The coadjoint action of the algebra $B$ on its dual space $B^{\ast}$ is generated by the pair of linear vector fields[^2] $$\label{EandH}
H=y\partial_y\,,\quad\quad E=y\partial_x\,,$$ where $(x,y)$ are coordinates on $B^*\sim {\Bbb R}^2$. This yields a quadratic Poisson brackets on ${\Bbb R}^2$ $$\label{ehv}
\omega =E\wedge H =y^2\partial_x\wedge\partial_y\,.$$ In principle, this bracket can be quantized by means of the general iterative procedure described in the previous sections. In this simple case, however, there is a more direct way to obtain the respective $\ast$-product. Namely, observe that the above Poisson bivector can also be written as the wedge product of the following vector fields: $$X=\partial_x\,, \quad \quad Y=y^2\partial_y\,,$$ so that $$\omega=X\wedge Y\,,\quad \quad [X,Y]=0\,.$$ As the vector fields commute, the bracket can easily be quantized by the Weyl-Moyal like formula $$\label{com}
f\ast
g=\sum_{n=0}^{\infty}\left(-\frac{i\hbar}{2}\right)^n\frac{1}{n!}
(X^nf)(Y^ng)$$ On the other hand, it is easy to prove by induction that $$(y^2\partial)^n=y^n\prod_{k=0}^{n-1}(y\partial_y+k)$$ Using this identity we can rewrite the above $\ast$-product in terms of noncommuting vector fields (\[EandH\]) as $$\label{EH}
f\ast g=\sum_{n=0}^{\infty}\left(-
\frac{i\hbar}{2}\right)^n\frac{1}{n!} (E^nf)(H^{(n)}g)\,,$$ where we put $$H^{(n)}=H(H+1)\cdots(H+n-1)\,.$$ Actually, expression (\[EH\]) is the universal quantization formula as it gives an associative $\ast$-product for any pair of vector fields $E$ and $H$ realizing the Borel algebra (\[borel\]). This may be seen from the following line of reasons. The vector fields (\[EandH\]) induce a special representation $\rho: U(B)\rightarrow D({\Bbb R}^2)$ of the universal enveloping algebra $U(B)$ in the algebra of differential operators on plane $D({\Bbb R}^2)$. Since $H$ and $E$ are linearly independent with functional coefficients this representation is exact, i.e. $\ker \rho = 0$ [^3]. In other words, there are no algebraic relations among the first-order differential operators $E$ and $H$ other than (\[borel\]) and its algebraic consequences. On the other hand, the associativity condition for the $\ast$-product (\[EH\]), being written explicitly, has a form of algebraic equations for the generators $E$ and $H$ and, as we have just argued, they should be satisfied as consequence of the commutation relations of the Borel algebra only. Thus the associativity holds for any representation $\rho :
B\rightarrow {\rm Vect}({\Bbb R }^2)$.
Originally, the Universal Deformation Formula (\[EH\]) was obtained in the paper [@CGG]. The skew-symmetric form of the above quasi-exponential formula (that corresponds to the Weyl ordering) was derived in [@UDF].
The formula (\[EH\]) allows one to quantize an interesting class of irregular Poisson brackets on ${\Bbb R }^2$. Any bracket on plane is given by a single function $$\label{f}
\omega =f(x,y)\partial _x\wedge \partial _y$$ The function $f$ is said to be [*quasi-homogeneous of weight*]{} ${\lambda}$ upon the weights of $x$ and $y$ equal to $\alpha$ and $\beta $ respectively if it is an eigen-vector to the Euler operator $v$ with the eigen-value ${\lambda}$, $$vf={\lambda}f,\qquad v=\alpha x\partial _x+\beta y\partial _y\,.$$ In what follows we assume ${\lambda}\neq 0$. Introduce a Hamiltonian vector field $u$ with respect to the canonical Poisson bracket on ${\Bbb R}^2$, $$u=\partial _xf\partial _y-\partial _yf\partial _x$$ Then it is easy to check that $$\omega =\frac{v\wedge u}{{\lambda}},\qquad [v,u]=({\lambda}-\alpha -\beta )u$$ If $\alpha +\beta ={\lambda}$ the bivector (\[f\]) is representable as exterior product of the commuting vector fields $u$, $v$, otherwise the vector fields form (after an obvious redefinition) the Borel algebra (\[borel\]). Thus, every quasi-homogeneous Poisson bracket on plane can be quantized in a pure algebraic manner by formulas (\[com\]) or (\[EH\]).
For example, let $(m,n,k,l)$ be a set of four integers, then function $$f=x^my^n + x^ky^l\,$$ is quasi-homogeneous of weight ${\lambda}= nk-lm$ with $\alpha =n-l$ and $\beta =k-m$. Conversely, given the integer weights ${\lambda}, \alpha,
\beta$, then the most general quasi-homogeneous, analytical at zero, function has the form $$f(x,y)=\sum_{k} a_kx^{n_k}y^{m_k}\,,$$ $a_k$ being arbitrary constants and the sum goes over all non-negative integers solutions $(n_k, m_k)$ to the linear Diophantus equation $$\alpha n +\beta m ={\lambda}\,.$$ Depending on $\alpha$, $\beta$ and ${\lambda}$ this equation may have finite or infinite number of non-negative solutions, or may have none.
Quantization of triangular Lie bialgebras
=========================================
In this section we briefly discuss how the output algebraic structures entering the quantization formula for $r$-bracket (\[Dn\]) can be interpreted in the formal language of the Lie bialgebra quantization. Here, we do not want to go into detailed definition of all the related mathematical constructions but only recall some basic points, useful for our purposes. More details can be found, for example, in [@Takh], [@UDF].
As it was first shown by Drinfeld [@Dr83] the deformation problem for the triangular Lie bialgebra $(L,r)$ is equivalent to constructing the Universal Deformation Formula. The latter can be immediately read off from the explicit expression for the $\ast$-product (\[star\], \[Dn\]) if one replace the vector fields $X_i$ by the Lie algebra generators $y_i$. This leads to the following *universal twisting element*: $$F=I\otimes I+\sum_{n,m,k=1}^{\infty} {{\hbar}}^n D_n^{i_1\ldots i_m
j_1\ldots j_k} y_{i_1}\ldots y_{i_m} \otimes y _{j_1}\ldots
y_{j_k}\in U(L)\otimes U(L)[[\hbar]]\,,
\label{ON}$$ $I$ being the unit in $U(L)$. The element $F$ possesses special algebraic properties (to be specified further) resulted from the associativity and normalization conditions for the respective $\ast$-product, which, in turn, are based essentially on the Leibnitz rule for the derivatives $X_i(ab)=X_i(a)b+aX_i(b)$ and the “null on constant condition" $X_i(1)=0$. The algebraic formalization for the last two relations is naturally achieved by introducing the co-algebra structure on the universal enveloping algebra $U(L)$, i.e. the co-multiplication $\triangle
: U(L)\rightarrow U(L)\otimes U(L)$ and co-unit $\varepsilon :
U(L)\rightarrow {\Bbb C}$ mappings. Being homomorphisms of the associative algebra $U(L)$ with unit $I$, these operations are completely determined by their action on the generators $y_i$ and $I$: $$\label{coumnoz}
\begin{array}{cc}
\triangle y_i=y_i\otimes I+I\otimes y_i \; ,& \qquad \triangle I
= I \otimes I \; ,\\ [3mm]
\varepsilon ( y_i) =0 \; , &\qquad
\varepsilon ( I ) =1 \; .
\end{array}$$ Now the associativity $(a* b) *c = a * (b * c)$ and normalization condition (25.$c$) are encoded in the following relations for the universal twisting element $F$: $$\label{cocyc}
\begin{array}{c}
\left( (\triangle \otimes I) \, F \right) \, (F\otimes I) =
\left( (I\otimes \triangle) \, F \right) \, (I \otimes F) \; , \\
[3mm]
(\varepsilon \otimes I) F=(I\otimes \varepsilon) F=I \; ,
\end{array}$$ while the correspondence principle (25.$b$) looks like $$\label{qclass}
F=I\otimes I-\frac{i{{\hbar}}}2\, r^{ij}e_i\otimes e_j\,\,\,\, {\rm mod}
\,\,({{\hbar}}^2) \; .$$ Another natural operation $S: U(L)\rightarrow U(L)$, called antipode, is induced by the involutive anti-homomorphism of the Lie algebra $L$: $$\label{counit}
S(y_i)=-y_i\,, \quad S(I)=I\,,\quad S^2={\rm id}\,.$$
The collection of the operations $(\triangle, \varepsilon, S)$ endows $U(L)$ with a structure of the Hopf algebra defined by three axioms:
$i)\,\, (\triangle\otimes I)\triangle
=(I\otimes \triangle)\triangle$ - [*co-associativity*]{};
$
ii)\,\,(\varepsilon \otimes I)\triangle =I=(I\otimes
\varepsilon)\triangle;
$
$iii)\,\, m(S\otimes I)\triangle = u\circ {\varepsilon}=m(I\otimes S)\triangle\,\,\,$ ($\circ$ [*stands for the composition of maps*]{}).
Here $m$ denotes the standard multiplication in $U(L)$ and the map $u$ sends $c\in {\Bbb C}$ to $c\cdot I\in U(L)$. If in addition $\triangle =\tau \triangle$, where automorphism $\tau : a\otimes b \rightarrow b\otimes a$ permutes the multipliers in the tensor square $U(L)\otimes U(L)$, then the Hopf algebra is called co-commutative[^4].
Thus we arrive at the one-to-one correspondence between the variety of $\ast$-products of the form (\[star\]) and solutions to the equations (\[cocyc\]) subject to the “boundary condition" (\[qclass\]); in doing so, the invariance of equations (\[cocyc\]) under the transformation $$F \longrightarrow F'=(\triangle \, B) \, F \, (B^{-1} \otimes
B^{-1}) \, ,\quad B \in U(L)[[\hbar]]\,, \quad \varepsilon B =
I\,,$$ corresponds the to equivalence of two such $\ast$-products [@Dr83]. As another example of invariance we present the following discrete symmetry: $$\label{ON1}
F \rightarrow \widetilde{F} = \left[ (S \otimes S) \,
\tau F\tau \right]^{-1}$$ Using the new solution $\widetilde{F}$ to the equations (\[cocyc\]), having the same quasi-classical limit (\[qclass\]), one can construct another $\ast$-product which is not in general equivalent to the initial one (\[ON\]) in the sense of (\[inter\]). It would be interesting to explicitly compute the characteristic class of the respective $\ast$-product.
In order to illustrate how the universal deformation formula gives rise to the deformation of the Lie bialgebra structure, we remind the construction of the quantum universal enveloping algebra $U_{\hbar}(L)$. As an associative algebra, $U_{\hbar}(L)$ coincides with $U(L)[[\hbar]]$. The respective co-unit $\varepsilon$ remains the same as in (\[coumnoz\]), while the co-multiplication and the antipode for $U_{\hbar}(L)$ are obtained from those of $U(L)$ by the twisting transformation $$\begin{array}{cc}
\triangle_{\hbar} a= F^{-1}(\triangle a) F\,,& \qquad S_{{{\hbar}}}
a=U^{-1} S a U\,,\\ [3mm] U=m((S\otimes I) F)\,, &\qquad a\in
U_{\hbar}(L)\,.
\end{array}$$ The validity of the Hopf algebra axioms for the deformed co-algebra structure can be verified with the help of conditions (\[cocyc\]).
Given the associative bialgebra $U_{\hbar}(L)$, one can twist the multiplication of any $U_{\hbar}(L)$-module algebra or co-algebra, in particular, to write an explicit symbol realization for the deformed coordinate rings of various algebraic varieties [@UDF]. Among the most important examples of such a kind it is worth to mention the Takhtajan approach to the quantum group construction [@Takh]. Let $G$ be the Lie group corresponding to the Lie algebra $L$, and denote by $G_{\hbar}$ corresponding quantum group. The latter is defined in terms of an associative algebra of functions ${\rm Fun} (G_{\hbar})$. As a linear space, the algebra ${\rm Fun}(G_{\hbar})$ is identified with $C^{\infty}(G)[[\hbar]]$ and equipped with the following star-product: $$a\star b = M (F^L (F^{-1})^R (a\otimes b))\,, \label{starr}$$ where $M$ stands for the ordinary commutative multiplication in $C^{\infty}(G)$ and $F^{L}$, $(F^{-1})^{R}$ denote the twisting elements $F$, $F^{-1}$ in the representation of the left $\{
X^L_i \}$ and anti-representation of the right $\{ X^R_i \}$ invariant vector fields on the Lie group $G$, respectively. The associativity of $\star$-product follows from the first equation in (\[cocyc\]) and the fact that $X^R_i ~\mapsto~ X^L_i$ is involutive anti-homomorphism of the Lie algebras of vector fields. Evaluation of the quasi-classical limit for $\star$-product (\[starr\]) leads to the following Poisson bracket: $$\{ a , \, b \} = r^{ij} \, (X_i^L a \,\, X_j^L b - X_i^R a \,\,
X_j^R b )\,.$$ As is seen this bracket is a particular example of $r$-brackets considered in the previous sections.
Note that one can use the transformed twisting element (\[ON1\]) or even a combination of two different elements (\[ON\]) and (\[ON1\]) in the definition (\[starr\]) to obtain a different inequivalent quantizations. However the special “adjoint" structure of the deformation (\[starr\]) (the “right" twisting element is taken to be inverse to the “left" one) implies also that the standard group co-multiplication $\triangle : {\rm Fun}(G_{\hbar})\rightarrow
{\rm Fun}(G_{\hbar})\otimes { \rm Fun}(G_{\hbar})$ defined by the relation $$(\triangle _G f)(g_1,g_2)=f(g_1g_2)\,$$ is indeed a homomorphism of the algebra $(C^{\infty}(G))[[\hbar]], \star )$ to $(C^{\infty}(G)[[\hbar]], \star )\otimes
(C^{\infty}(G)[[\hbar]],\star )$.
Our explicit formulas, being combined with the Takhtajan construction [@Takh], provide a transparent and explicit description for the quantum algebra of functions ${\rm Fun}(G_{\hbar})$, that can be viewed as an alternative to the standard FRT method [@FRT].
Discussion and Conclusion
=========================
To summarize, in this paper we have proposed a simple quantization procedure for the Poisson brackets associated to the classical triangular $r$-matrix and giving, as a byproduct, the explicit quantization formula for Lie bialgebras. Abstracting from the peculiar form of the $r$-brackets (\[RB\]), (\[lalg\]) one can find a noticeable similarity between our quantization algorithm and Fedosov’s construction for symplectic manifolds. Indeed, in both the cases one begins with an auxiliary quantum bundle assigned by a fiber-wise $\circ$-product; in so doing, the space of quantum observables is identified to the center of this algebra. At the next step a new nontrivial embedding is constructed to induce the $\ast$-product on the Poisson manifold as a pull-back of the $\circ$-one. This is achieved by introducing an appropriate Lie algebra of external derivatives and identifying the image of the embedding map with their kernel subspace (in fact $\circ$-subalgebra). Moreover, the Theorem 4 of Sec. 3 suggests that, similar to the Fedosov approach, we can always start with the quantum bundle of Weyl algebras and construct the embedding map in an explicit recurrent manner with any accuracy in the deformation parameter. In our special algebraic situation, however, the use of the universal enveloping algebra bundle seems to be more adequate to the problem, considerably simplifying the whole construction. Another novel point of our approach, looking from the pure algebraic perspective, is the non-commutativity of the aforementioned derivatives for any choice of fiber-wise $\circ$-product. This is a distinction from the Fedosov deformation quantization which is essentially based on the notion of the [*Abelian*]{} connection corresponding, in this language, to a certain pair-wise commuting set of derivatives associated with any symplectic connection.
Let us briefly discuss a possibility to extend our star product construction to a more general class of Poisson manifolds than considered in this paper. It seems useful to seek for the generalizations making use of a natural relationship between the Fedosov deformation quantization and BRST theory [@GL] [^5]. The key idea behind the identification of both the quantization schemes was a realization of the symplectic manifold $M$ as a second class constrained surface embedded into cotangent bundle $T^*M$. Let us explain how this approach, originally found for the symplectic manifold, can be first extended to the case of $r$-brackets (\[RB\]), (\[lalg\]). Consider a trivial vector bundle ${\cal{M}}=M\times V$ over a smooth manifold $M$, assigned by the following Poisson brackets $$\label{11}
\{ x^\mu ,x^\nu \}=0\,,\quad\{p_i,p_j\}=f^k_{ij}p_k + r_{ij}\,,\quad
\{p_i,x^\mu\}=X^\mu_i(x)\,.$$ Here $x^\mu$ are local coordinates on $M$ and $p_i$ are linear coordinates on $V$. It is easy to check that these brackets satisfy the Jacobi identity iff (i) the vector fields $X_i$ generate a Lie algebra (\[lalg\]) and (ii) matrix $r$ satisfies cocycle condition (\[cc\]). If in addition $\det(r_{ij})\neq 0$, then the second class constraints $T_i=p_i\approx 0$ define an embedding of $M$ into $\cal M$ as a zero section. The corresponding Dirac bracket on $\cal M$ induces Poisson bracket on $M$, $$\label{Dirac}
\{x^\mu ,x^\nu\}_{Dirac}=r^{ij}X^\mu_i(x)X^\nu_j(x)\,.$$ This relation identifies the Hamiltonian theory on $M$ to the second class constrained theory on $\cal M$. Note that in general both the brackets (\[11\]) and (\[Dirac\]) are irregular, that does not, however, obstruct the mentioned identification. The common method of BRST treatment of the second class constrained systems implies the conversion of the second class constraints $T_i$ to the first class ones by means of a proper extension of the phase manifold $\cal M$ (see [@BT], and references therein). The minimal possibility to do this is to duplicate the vector bundle $V$, that is to consider the direct product ${\cal
M}'={\cal M}\times V$, where $V$ is a linear symplectic space assigned by canonical Poisson bracket $$\label{111}
\{z_i, z_j\}=r_{ij}\,,$$ $z_i$ being linear coordinates on the second copy $V$. Together Rels. (\[11\]) and (\[111\]) define the Poisson structure on ${\cal M}'$ (we mean that bracket of $z$ are vanishing to other variables). Now it is possible to extend the constraints $T_i$ from $\cal M$ to ${\cal M}^\prime $ getting an equivalent first class constrained theory $T'_i=T_i+z_i+ o(z^2)$, where the higher orders in the formal variables $z_i$ are determined from the requirement of involution of the effective first class constraints $T'_i$ $$\{T'_i, T'_j\}=f_{ij}^kT'_k \,.$$ The Poisson brackets (\[11\]), (\[111\]) on ${\cal M}'$ can be easily quantized if one restricts the space of observables to the functions on ${\cal M}'$ at most linear in $p_i$. The connection with constructions of previous sections is established by the relation $D_ia=[T_i,a]$, where $a=a(x,z,\hbar)$. The construction of nilpotent BRST charge and identification of physical observable algebra with the special BRST cohomology imply a further enlargement of the phase space ${\cal M}'$ by the ghost variables. For the case of $M$ being a symplectic manifold, this BRST quantization programme has been implemented in [@GL], and it seems no special modification is required in the case of the Poisson manifold (\[RB\]), (\[lalg\]).
In this form the above quantization procedure may be generalized to the case when all the structure constants $f^k_{ij}$, $r_{ij}$ entering to the fundamental Poisson brackets (\[11\]), (\[111\]) are allowed to depend on the point $x\in M$ [@LS]. As a particular case this includes the quantization of dynamical $r$-matrix [@XU] with the vector fields $X_i$ generating a Lie bialgebroid.
[**Acknowledgments.**]{} We are thankful to I.A. Batalin, M.A. Grigoriev, O.V. Ogievetsky, M.A. Olshanetsky, I.Yu. Tipunin for useful discussions. We also acknowledge O.V. Ogievetsky for the constructive criticism concerning the preliminary version of this article. The work of SLL is partially supported by RFBR grant 00-02-17-956 and the grant INTAS 00-262. The work of AASh is supported by RFBR grant 02-02-06879 and Russian Ministry of Education under the grant E-00-33-184. The work of VAD is partially supported by the grant INTAS 00-561 and by the Grant for Support of Scientific Schools 00-15-96557. The work of API is partially supported by the RFBR grant 00-01-00299. API is also grateful to Max-Plank-Institute fur Mathematik in Bonn for kind hospitality and support. SLL is thankful to Lars Brink and Robert Marnelius for their warm hospitality at the Chalmers University and to STINT (Swedish Foundation for International Cooperation in Research and Higher Education) for the financial support.
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[^1]: See [@KM] for a detailed survey of the operator calculus, in particular, for the Weyl calculus on Lie algebras.
[^2]: As is seen, the carrier space $B^*$ stratifies on two coadjoint orbits of dimension 0 and 2. The former coincides with the origin $0\in B^*$, while the latter is given by $B^*
-\{0\}\sim S^1\times {\Bbb R }^1$. The existence of a coadjoint orbit of dimension equal to the dimension of the corresponding Lie algebra is a general characteristic property of the Frobenius algebras [@El].
[^3]: Obviously, this property is valid for the coadjoint action of any Frobenius Lie algebra $L$ since there is an open set of points (the orbit of maximal dimension) at which the corresponding vector fields are linearly independent.
[^4]: Omitting the antipode operation $S$ and the related axiom (iii) one arrives at the notion of an associative bialgebra.
[^5]: On the other hand, the BRST language could be useful in attempts to apply the present $\ast$-product construction to the field theory problems, as it provides a uniform description for both deformation quantization of the phase space observable algebra and gauge symmetry or/and the Hamiltonian constraint algebra, see [@BGL].
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---
abstract: 'Maximum likelihood estimation of linear functionals in the inverse problem of deconvolution is considered. Given observations of a random sample from a distribution $P_0\equiv P_{F_0}$ indexed by a (potentially infinite-dimensional) parameter $F_0$, which is the distribution of the latent variable in a standard additive Laplace measurement error model, one wants to estimate a linear functional of $F_0$. Asymptotically efficient maximum likelihood estimation (MLE) of integral linear functionals of the mixing distribution $F_0$ in a convolution model with the Laplace kernel density is investigated. Situations are distinguished in which the functional of interest can be consistently estimated at $n^{-1/2}$-rate by the plug-in MLE, which is asymptotically normal and efficient, in the sense of achieving the variance lower bound, from those in which no integral linear functional can be estimated at parametric rate, which precludes any possibility for asymptotic efficiency. The $\sqrt{n}$-convergence of the MLE, valid in the case of a degenerate mixing distribution at a single location point, fails in general, as does asymptotic normality. It is shown that there exists no regular estimator sequence for integral linear functionals of the mixing distribution that, when recentered about the estimand and $\sqrt{n}$-rescaled, is asymptotically efficient, *viz*., has Gaussian limit distribution with minimum variance. One can thus only expect estimation with some slower rate and, often, with a non-Gaussian limit distribution.'
author:
- 'Catia Scricciolo [^1]'
date:
title: On asymptotically efficient maximum likelihood estimation of linear functionals in Laplace measurement error models
---
Introduction {#intro}
============
The problem of asymptotically efficient estimation of integral linear functionals of the distribution of the latent variable in a standard additive Laplace measurement error model is considered. The focus is on establishing whether asymptotic normality and efficiency hold for the estimator obtained by plugging into the functional of interest the NPMLE of the mixing distribution in a convolution model with the Laplace kernel density. We study the behaviour of the plug-in NPMLE to answer the question of whether there exist integral linear functionals of the mixing distribution that can be consistently estimated by the maximum likelihood method at $n^{-1/2}$-rate, the recentered and $\sqrt{n}$-rescaled version of the plug-in NPMLE being asymptotically normal with zero mean and minimum variance. Situations are distinguished in which the plug-in NPMLE is consistent at parametric rate and asymptotically efficient, albeit the mixing distribution itself can typically be estimated only at slower rates, from those in which there exists no regular sequence of estimators that can be asymptotically efficient. The model is described hereafter and the problem formally stated.
#### Model description
Let $X$ be a real-valued random variable (r.v.) with distribution $P_0$ defined, for every Borel set $B$ on the real line, by the mapping $B\mapsto P_0(B):=\mathrm{P}(X\in B)$. Suppose that $P_0$ is dominated by Lebesgue measure $\lambda$ on $\mathbb R$, with probability density function (p.d.f.) $p_0:={\mathrm{d}}P_0/{\mathrm{d}}\lambda$. Let $X$ satisfy the relationship $$\label{eq:conv}
X=Y+Z,$$ where $Y$ and $Z$ are (stochastically) independent, *unobservable* random variables such that $Y$ has *unknown* cumulative distribution function (c.d.f.) $F_0$ and $Z$ has the *standard classical* Laplace[^2] or double exponential[^3] distribution with scale parameter $s=1$, in symbols, $Z\sim\mathrm{Laplace}\,(0,\,1)$, whose density $k$ has expression $$\label{eq:laplace}
k(z)=\frac{1}{2}e^{-|z|}, \quad z\in\mathbb{R}.$$ The density $p_0$ is therefore the convolution of $F_0$ and $k$ or a location mixture of Laplace densities with mixing distribution $F_0$ supported on a subset $\mathscr{Y}\subseteq\mathbb{R}$, where $\mathscr Y$ stands for *a* support of $F_0$, see, *e.g.*, Billingsley (1995), p. 23, $$p_0(x)\equiv p_{F_0}(x)=\int_{\mathscr{Y}}k(x-y)\,{\mathrm{d}}F_0(y)=\int_{\mathscr{Y}}\frac{1}{2}e^{-|x-y|}\,{\mathrm{d}}F_0(y),\quad x\in\mathbb{R}.$$ For ease of exposition, the density of the standard Laplace distribution is considered as a kernel, but the density of *any* Laplace distribution centered at zero, with *known* scale parameter $s>0$, in symbols, $Z\sim\mathrm{Laplace}\,(0,\,s)$, whose variance is $\sigma^2_Z=2s^2$,[^4] could be employed. Assume that $X_1,\,\ldots,\,X_n$ constitute a random sample from $p_0$. Every $X_i$ then satisfies $$\label{eq:conveq}
X_i=Y_i+Z_i,\quad i=1,\,\ldots,\,n,$$ where $Y_1,\,\ldots,\,Y_n$ and $Z_1,\,\ldots,\,Z_n$ are independent samples from the distributions with c.d.f. $F_0$ and p.d.f. $k$, respectively. The r.v.’s $Y_1,\,\ldots,\,Y_n$ are independent and identically distributed (i.i.d.) as $Y$ and $Z_1,\,\ldots,\,Z_n$ are independent copies of $Z$. Realizations $x_1,\,\ldots,\,x_n$ of the noisy sample data $X_1,\,\ldots,\,X_n$ are observed instead of outcomes of the uncorrupted r.v.’s $Y_1,\,\ldots,\,Y_n$. The r.v.’s $Z_1,\,\ldots,\,Z_n$ represent additive errors and their distribution is called the *error distribution*. In this model, the variable of interest $Y$ cannot be directly observed and empirical access is limited to the sum of $Y$ and the [“noise”]{} $Z$. Therefore, estimating the distribution function $F_0$ of $Y$, or related quantities like the p.d.f. $f_0$ (if it exists), based on a sample from $P_0$, accounts for solving a particular *inverse* problem, called *deconvolution*, which consists in reconstructing (estimating) $F_0$ from indirect noisy observations $X_1,\,\ldots,\,X_n$ drawn from $P_0\equiv P_{F_0}$, the latter being the image of $F_0$ under a known transformation that has to be [“inverted”]{}. As remarked in Groeneboom and Wellner (1992), p. 4, as well as in Bolthausen *et al.* (2002), p. 363, the problem can be viewed as a *missing data problem*: the complete observations would consist of the independent pairs $X^0_i:=(Y_i,\,Z_i)$, with $X^0_i\sim Q:=F_0\times F_Z$,[^5] but part of the data is missing and only outcomes or realizations of the sums $X_i=T(X_i^0):=(Y_i+Z_i)\sim P_0\equiv QT^{-1}$, viewed as transformations of the $X^0_i$’s through the function $T$, are observed.
The statistical model described by relationship , with a zero-mean Laplace measurement error r.v. $Z$ independent of $Y$, is a special case of the *classical error model* $X=Y+Z$, in which $X$ is a measurement of $Y$ in the usual sense, $Z$ has zero mean and is independent of $Y$, see, *e.g.*, Buzas *et al.* (2005), p. 733, and the references therein. Measurement errors with possibly different structures occur in nearly every discipline from medical statistics to astronomy and econometrics, *cf.* the monographs of Fuller (1987) and Buonaccorsi (2010). Furthermore, the Laplace distribution finds applications in a variety of disciplines, from image and speech recognition to ocean engineering, see Kotz *et al.* (2001), chpts. 7–10, pp. 343–397. An application to quality control of the classical Laplace measurement error model is outlined hereafter.
#### Application to steam generator inspection
An application of the Laplace measurement error model to steam generator inspection and testing is described herein, see Easterling (1980) and Sollier (2017) for more details. Steam generators of pressurized water reactors contain many tubes through which heated water flows. For a variety of reasons, such as corrosion-induced wastage, the steam generator tube integrity can be degraded, the walls becoming thinned or cracked. Leaks may occur during normal operating conditions, thus requiring the plant to be shut down. In order to develop an inspection plan, a statistical model for tube degradation is considered. Experimental data evidentiate that measurements are affected by heavy-tailed and biased errors that can be represented by a r.v. $E$ following a Laplace distribution with mean $\mu>0$ and scale parameter $s>0$, in symbols, $E\sim\mathrm{Laplace}\,(\mu,\,s)$, with density $$f_E(e)=\frac{1}{2s}\exp{(-|e-\mu|/s)}, \quad e\in\mathbb R.$$ Denoted by $D$ the *actual* degradation (extent of thinning) of a tube, expressed as a percentage of the initial tube wall thickness, the *measured* degradation $M$ is modeled as $$\label{eq:degradation}
M=D+E,$$ where $E$ is supposed to be independent of $D$. Assuming that the scale parameter $s$ is known and the distribution of $D$ possesses probability density function, say $f_D$, the interest is, in the first place, in estimating the p.d.f. of $D$, based on i.i.d. observations $M_1,\,\ldots,\,M_n$ drawn from the distribution of $M$. An exponential distribution for $D$, with scale parameter $\tau>0$, in symbols, $D\sim\textrm{Exp}\,(1/\tau)$, whose density has expression $$f_D(d)=\frac{1}{\tau}\exp{(-d/\tau)},\quad d>0,$$ provides an exponential-double exponential model for the actual degradation $M$, which has proved to have an adequate fit on experimental data. Statistical procedures for fitness-for-service assessment are described in Carroll (2017).
#### Asymptotic efficiency of the NPMLE for linear functionals of the mixing distribution
For many purposes, interest can lie in only few aspects of the distribution of $Y$, key features of which can be represented as linear functionals of $F_0$. In what follows, symbols $F_0$ and $F$ will be used to indicate probability measures (p.m.’s) on $(\mathscr{Y},\,\mathscr{B}(\mathscr{Y}))$, where $\mathscr{B}(\mathscr{Y})$ denotes the Borel $\sigma$-field on $\mathscr{Y}$, as well as the corresponding cumulative distribution functions, the correct meaning being clear from the context. Letting $$\mathscr P:=\{\mbox{all p.m.'s $F$ on $(\mathscr{Y},\,\mathscr{B}(\mathscr{Y}))$}\}$$ be the collection of all probability measures $F$ on $(\mathscr{Y},\,\mathscr{B}(\mathscr{Y}))$, a functional is a mapping $\psi:\,\mathscr P\rightarrow \mathbb R$ that maps every $F\in\mathscr P$ to a real number $\psi(F)$. The focus is on estimating *integral* linear functionals $$\label{eq:2}
F\mapsto\psi({F}):=\int_{\mathscr{Y}}a(y)\,{\mathrm{d}}F(y)
$$ at the [“point”]{} $F_0$, where the function $a\in L^1(F_0)$ is given. The following examples illustrate choices of $a$ for some common statistical functionals.\
- [[*Distribution function at a point*]{}]{} If, for some fixed $y_1\in\mathbb{R}$, the function $a(\cdot)=1_{(-\infty,\,y_1]}(\cdot)$, then $\psi(F_0)=\int_{\mathscr{Y}}1_{(-\infty,\,y_1]}(y)\,{\mathrm{d}}F_0(y)=F_0(y_1)$ is the c.d.f. of $Y$ at the point $y_1$.\
- [[*Probability of an interval*]{}]{} If, for fixed points $y_1,\,y_2\in\mathbb{R}$, the function $a(\cdot)=1_{(y_1,\,y_2]}(\cdot)$, then $\psi(F_0)=\int_{\mathscr{Y}}1_{(y_1,\,y_2]}(y)\,{\mathrm{d}}F_0(y)=F_0(y_2)-F_0(y_1)=\mathrm P(y_1<Y\leq y_2)$ is the probability of the interval $(y_1,\,y_2]$.\
- [[*Mean*]{}]{} If $a(\cdot)=\textrm{id}_{\mathscr Y}(\cdot)$ is the identity function on $\mathscr Y$, then $\psi(F_0)=\int_{\mathscr{Y}}y\,{\mathrm{d}}F_0(y)
=\mathrm{E}Y$ is the expected value of $Y$ which, for *any* kernel density $k$ (not necessarily the Laplace) with zero mean, $\mathrm{E}Z=0$, is equal to $\mathrm{E}X$: in fact, from the relationship $X=Y+Z$ in , it follows that $\mathrm{E}X=\mathrm{E}Y+\mathrm{E}Z=\mathrm{E}Y$ by linearity of the expected value.\
- [*$r$th moment*]{} If, for any positive integer $r$, the function $y\mapsto a(y)=y^r$, then $\psi(F_0)=\int_{\mathscr{Y}}y^r\,{\mathrm{d}}F_0(y)
=\mathrm{E}Y^r$ is the $r$th moment of $Y$.\
- [[*Moment generating function*]{}]{} If, for some fixed point $t\in\mathbb R$ such that $0<|t|<1$, the mapping $y\mapsto a(y)=e^{ty}$, then $\psi(F_0)$ coincides with the moment generating function (m.g.f.) of $F_0$ at $t$, denoted by $M_{F_0}(t)$ or $M_Y(t)$, that is, $\psi(F_0)=\int_{\mathscr{Y}}e^{ty}\,{\mathrm{d}}F_0(y)=M_{F_0}(t)$. Some features of the mixing distribution $F_0$, like the mean or the variance, can be expressed in terms of the derivatives of the corresponding m.g.f. $M_{F_0}$ evaluated at zero. Therefore, in principle, results for estimating aspects of $F_0$ can be obtained as by-products of the inference on $M_{F_0}$.
A standard and principled method for pointwise estimation of linear functionals consists in plugging *the*[^6] NPMLE $\hat F_n$ of $F_0$ into $\psi(\cdot)$ to obtain the *plug-in estimator* $$\psi({\hat F_n}):=\psi(F)\big|_{F=\hat F_n}.$$ A NPMLE $\hat F_n$ of $F_0$ is a measurable function of the observations $X_1,\,\ldots,\,X_n$ taking values in $\mathscr{P}$, which is not necessarily uniquely defined by the relationship $$\hat F_n\in\mathop{\arg\max}_{\substack{F\in\mathscr P}}
\frac{1}{n}\sum_{i=1}^n\log p_F(X_i),$$ equivalently written as $$\hat F_n\in
\mathop{\arg\max}_{\substack{F\in \mathscr P}}
\mathbb P_n\log p_F,$$ where $$p_F(\cdot):=\int_{\mathscr{Y}}k(\cdot-y)\,{\mathrm{d}}F(y)$$ is the generic location mixture of Laplace densities with mixing distribution $F$ and $${\mathbb P_n}:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$ is the empirical probability measure associated with the random sample $X_1,\,\ldots,\,X_n$, namely, the discrete uniform distribution on the sample values that puts mass $1/n$ on each one of the observations. In the sequel, for a measurable function $f:\,\mathscr X\rightarrow\mathbb R$, where $\mathscr X\subseteq\mathbb R$ is specified at the different occurrences, the notation $\mathbb P_n f$ is used to abbreviate the empirical average $n^{-1}\sum_{i=1}^nf(X_i)$. Analogously, $P_0f$ is used in lieu of $\int f\,{\mathrm{d}}P_0$. Hereafter, unless it is necessary within the context to specify the integral domain, integration is understood to be performed over the entire natural domain of the integrand. Throughout the article, the probability measure $\mathbf{P}$ stands for $P_0^n$, the joint law of the first $n$ coordinate projections of the infinite product probability measure $P_0^{\mathbb{N}}$. Sequences of random variables are meant to convergence (in law or in probability) as the sample size $n$ grows indefinitely large (as $n\rightarrow +\infty$).
#### Historical and conceptual background, overview of the results
The *deconvolution* problem has been intensively studied over the last thirty years. There exists a vast literature on *density* deconvolution, which accounts for reconstructing/estimating the density $f_0$ of $Y$ (if it exists) that satisfies the equation $$p_0(x)=\int_{\mathscr Y} k(x-y)f_0(y)\,{\mathrm{d}}y,\quad x\in\mathbb R,$$ wherein the kernel density $k$ (not necessarily a Laplace) is assumed to be known, based on outcomes $x_1,\,\ldots,\,x_n$ of i.i.d. r.v.’s $X_1,\,\ldots,\,X_n$ as in . We cite key articles of the early 90’s like Carroll and Hall (1988), Stefanski and Carroll (1990), Fan (1991)[^7] that have been ground-breaking and have had a great impact on the area of measurement error, setting the general framework for attacking measurement error/deconvolution problems and developing an approach based on Fourier inversion techniques to construct a deconvolution kernel density estimator for recovering the density of the latent distribution, meanwhile showing how difficult it is to account for measurement errors: in fact, the smoother the error distribution, the stronger its confounding effect on the latent distribution, hence, the slower the optimal attainable rate of convergence for its estimators.
Far less instead seems to be known about *distribution function* deconvolution, keynote contributions, also based on Fourier inversion techniques, being those of Hall and Lahiri (2008), Dattner *et al.* (2011), the former article containing an illuminating critical analysis of the background to the problem of distribution estimation in deconvolution problems. Since the focus of this article is on the behaviour of the NPMLE $\hat F_n$ of the mixing distribution $F_0$, attention is hereafter restricted to review the theory of non-parametric maximum likelihood estimation in deconvolution problems. In general mixture models, the NPMLE $\hat F_n$ of $F_0$ is *discrete*, with at most $l\leq n$ support points, $l$ being the number of distinct values of the data points, *cf.* Lindsay (1983). In deconvolution problems with continuous and symmetric (about the origin) kernels decreasing on $[0,\,+\infty)$, *a* NPMLE $\hat F_n$ always exist (uniqueness is not guaranteed), see Groeneboom and Wellner (1992), Lemma 2.1, pp. 57–58; for kernels that are also strictly convex on $[0,\,+\infty)$, like the (standard) Laplace, the NPMLE $\hat F_n$ is supported on the set of observation points $\{X_1,\,\ldots,\,X_n\}$, so that the corresponding probability measure, still denoted by $\hat F_n$ consistently with the notational convention adopted throughout, is concentrated on the range of the data points $[X_{1:n},\,X_{n:n}]$,[^8] $$\hat F_n([X_{1:n},\,X_{n:n}])=1,$$ see *ibid.*, Corollary 2.2 and Corollary 2.3, p. 59 and p. 60, respectively. Consistency of $\hat F_n$ at a *continuous* distribution function $F_0$ is proved in *ibid.*, §4.2, pp. 79–81. More is known about the *one*-parameter (location only) Laplace model, which can be viewed as a degenerate mixture with point mass mixing distribution at some fixed $\theta\in\mathbb R$, the Dirac measure at $\theta$.[^9] A simple maximization argument to find a MLE of the location parameter $\theta$, denoted by $\hat\theta_n$, is given in Norton (1984): the *sample median* is a MLE which is an *M-estimator*, see Huber (1967), solving the equation $\sum_{i=1}^n\mathrm{sign}(X_i-\theta)=0$.[^10] Therefore, a MLE exists, but may not be unique: if $n$ is odd, that is, $n=2m+1$ for some $m\in\mathbb N$, then the sample median is uniquely defined as the middle observation $X_{m+1:n}$, while, if $n=2m$ is even, then there are two middle observations $X_{m:n}$ and $X_{m+1:n}$ so that, in principle, any value in the interval $[X_{m:n},\, X_{m+1:n}]$ could be chosen, even if the *canonical median* $(X_{m:n}+X_{m+1:n})/2$, the average of the middle observations, is typically used in practice. Therefore, $$\hat\theta_n=
\begin{cases}
X_{m+1:n}, &\,\,\,\text{for } n=2m+1,\\[-0.25cm]
&\qquad\qquad\qquad\qquad\text{} m\in\mathbb N,\\[-0.25cm]
\frac{1}{2}(X_{m:n}+X_{m+1:n}), &\,\,\,\text{for } n=2m.
\end{cases}$$ The sample median $\hat\theta_n$ is a MLE and is *asymptotically efficient*, that is, consistent and, when recentered at $\theta$ and $\sqrt{n}$-rescaled, asymptotically normal with zero mean and variance equal to one, which is the information lower bound corresponding to the amount of information in a single observation, $$\sqrt{n}\big(\hat\theta_n-\theta\big)\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,1\big),$$ where [“$\,\xrightarrow{\mathscr{L}}\,$”]{} denotes convergence in law. This has been established by Daniels (1961) who, motivated by the non-differentiability at zero of the (standard) Laplace density, proved a general theorem on the asymptotic efficiency of the MLE under conditions not involving the second and higher-order derivatives of the likelihood function. Even though, as later on noted by Huber (1967), a crucial step has been overlooked in Daniels’ proof, the assertion remains valid.
In this article, we study the behaviour of the plug-in NPMLE $\psi(\hat F_n)$ to answer the question of whether there exist integral linear functionals $\psi(F_0)$ that can be consistently estimated by the maximum likelihood method at $n^{-1/2}$-rate and for which $$\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)$$ is *asymptotically efficient*, in the sense of Definition 2.8 in Bolthausen *et al*. (2002), p. 349, that is, asymptotically normal with zero mean and minimum variance. In fact, also in non-parametric problems estimation can be performed at $n^{-1/2}$-rate and, in general mixture models $\int_{\mathscr Y} k(\cdot\mid y)\,{\mathrm{d}}F_0(y)$, where $k$ is any (not necessarily the Laplace) kernel density, there may exist linear functionals of $F_0$ that are estimable at parametric rate, even if $F_0$ itself can be pointwise estimated only at slower rates. Fundamental contributions developing the theory of information bounds are van der Vaart (1991), van der Vaart (1998), chpt. 25, pp. 358–432, Bolthausen *et al.* (2002), Part III, pp. 331–457, Groeneboom and Wellner (1992), with emphasis on non-parametric maximum likelihood estimation, and van de Geer (2000), chpt. 11, pp. 211–246, with a focus on asymptotic efficiency of the NPMLE in mixture models. To exemplify the issue, consider estimating the mean functional $\psi(F_0)=\mathrm{E}Y$ which, in a mixture model such that $\mathrm E(X\mid Y)=Y$, is equal to $\mathrm E X$. Then, the sample mean $\bar X_n=n^{-1}\sum_{i=1}^n X_i$ is a $n^{-1/2}$-consistent and, after $\sqrt{n}$-rescaling, asymptotically normal estimator of $\mathrm{E}Y$, but may not be a MLE; furthermore, it does not take into account the information that the sampling density is a mixture. On other side, the MLE may be $n^{-1/2}$-consistent and converge to a normal distribution with smaller variance than that of the sample mean. This is the case for the sample median in the single-parameter (location only) Laplace model, see van der Vaart (1998), Example 7.8 on location models, p. 96. Surprisingly, little is known in general about the asymptotic behaviour of the plug-in NPMLE for linear functionals in Laplace convolution mixtures, even if only for estimating the mean functional. Although the topic is useful, existence of this gap can be partially explained by the fact that the Laplace or double-exponential distribution is not an exponential family model so that standard results may not be valid or immediately available from the theory of exponential families.
In order to investigate whether integral linear functionals of the mixing distribution in a convolution model with the Laplace kernel density are estimable at $n^{-1/2}$-rate, we appeal to van der Vaart’s differentiability theorem, see Theorem 3.1 in van der Vaart (1991), p. 183, a general result that allows for a unified treatment of the information lower bound theory based on the concept of a differentiable functional, see, for a definition of the latter, display (2.2) in van der Vaart (1991), p. 180, or Definition 1.10 in Bolthausen *et al.* (2002), p. 343. The differentiability theorem characterizes differentiable functionals and, by combining the description of the set of differentiable functionals with a result stating that the existence of *regular* estimator sequences for a functional implies its differentiability, provides a way to distinguish situations in which the functional of interest is estimable at $n^{-1/2}$-rate from situations in which this is not the case, see van der Vaart (1998), p. 365, for a definition of a regular estimator sequence. A necessary and sufficient condition for differentiability of a (not necessarily linear) functional is that its gradients are contained in the range of the adjoint of the score operator, where the score operator can be viewed as a derivative (in quadratic mean) of the map $F\mapsto P_F$, see (3.6) in van der Vaart (1991), p. 183, or (25.29) in §25.5 of van der Vaart (1998), p. 372. As previously mentioned, differentiability is necessary for regular estimability of a functional or, equivalently, for the existence of regular estimator sequences, see Theorem 2.1 of van der Vaart (1991), p. 181, so that if the functional is not differentiable, then there exists no regular estimators and estimation at $n^{-1/2}$-rate is impossible. Interestingly, for real-valued functionals, the differentiability condition is equivalent to having positive efficient information, see Theorem 4.1 of van der Vaart (1991), pp. 186–187. We find that the differentiability condition fails for integral linear functionals of the mixing distribution in a convolution model with the Laplace density, this implying that there exists no estimator sequence for $\psi(F_0)$ that is regular at $F_0$ and estimation at $n^{-1/2}$-rate is impossible.
#### Organization
The rest of the article is organized as follows. The main results are presented in Sect. \[section:1\], which is split into two parts. In the first one, asymptotic efficiency of the plug-in NPMLE for integral linear functionals of the mixing distribution in a convolution model with the (one-sided) exponential kernel density is analysed and set in the affirmative. Construction of interval estimators and tests based on a Studentized version of the plug-in NPMLE, when the asymptotic variance is consistently estimated, is revisited. Conditions for extending results to non-linear functionals are discussed as a side-issue. In the second part, the focus is on asymptotically efficient estimation by the plug-in NPMLE for integral linear functionals of the mixing distribution in a convolution model with the double-exponential (Laplace) kernel density. It is shown that, except for the case of a degenerate mixing distribution at a single location point, maximum likelihood estimation completely fails, in the sense that no integral linear functional can be estimated at $n^{-1/2}$-rate, which precludes any possibility for the NPMLE of being asymptotically efficient. Indeed, there exists no regular sequence of estimators for integral linear functionals of the mixing distribution that can be asymptotically efficient, therefore, estimation of linear functionals is impossible at $n^{-1/2}$-rate. Final remarks and comments are exposed in Sect. \[sec:3\]. Proofs of the main results are deferred to the appendices: Appendix A reports the proof of the result for convolution with the exponential density, Appendix B reports the proof of the result for convolution with the Laplace density.
Main results {#section:1}
============
In this section, the main results of the article are presented. First, the case of a convolution model with an exponential kernel density is considered. Since, as previously noted, the Laplace density in can be thought of as two exponential densities spliced together back-to-back, the positive half being a standard exponential density scaled by $1/2$, it is reasonable to begin the analysis from the problem of asymptotically efficient non-parametric maximum likelihood estimation of linear functionals in a convolution model with the *exponential* kernel density. A preliminary study of the one-sided problem, beyond being of interest in itself, is useful to attack the two-sided one by partially reducing it to the previously solved case; it may furthermore provide insight for a better understanding of the reasons why symmetrization leads to a failure of asymptotically efficient estimation of linear functionals in the *double-exponential* (Laplace) case.
#### Convolution with the exponential density
In this paragraph, a standard exponential kernel density on $[0,\,+\infty)$ is considered. This gives rise to a one-sided mixture density generating the data, $$p_0(x)\equiv p_{F_0}(x)=\int_{\mathscr Y} e^{-(x-y)}1\{y\leq x\}\,{\mathrm{d}}F_0(y),\quad x\in\mathscr X,$$ where $\mathscr Y:=\mathrm{support}\,(F_0)
$ is assumed, without loss of generality, to be a proper, left-closed subset of the real line, and $\mathscr X:=[y_{\textrm{min}},\,+\infty)$ is the support of $P_0$, with $y_{\textrm{min}}:=\mathrm{min}\,\mathscr Y>-\infty$. Proposition \[prop:1\] below establishes that, under sufficient conditions, an integral linear functional $\psi(F_0)$ can be consistently estimated at $n^{-1/2}$-rate by the plug-in NPMLE $\psi(\hat F_n)$, which, when recentered about the estimand and $\sqrt{n}$-rescaled, is asymptotically normal and efficient. In stating hereafter the assumptions, Newton’s notation (or the dot notation) for differentiation is adopted, that is, ${\dot a}(y):={\mathrm{d}}a(y)/{\mathrm{d}}y$.
[[**[[Assumptions]{}]{}**]{}]{}
1. $a\in L^1(F_0)$,\
2. $M_{F_0}(1):=\int_{\mathscr Y}e^{y}\,{\mathrm{d}}F_0(y)<+\infty$,\
3. $|\psi(\hat F_n)-\psi(F_0)|=o_{\mathbf{P}}(1)$,\
4. \(i) $a$ is continuous on $\mathscr Y$,\
(ii) either $\mathscr Y$ is compact or $a$ is bounded on $\mathscr Y$,\
(iii) $F_0(y_{\textrm{min}})=0$,\
5. there exists ${\dot a}$ on $\mathscr Y$ and $
\sup_{y\in\mathscr Y}|\dot a(y)|<+\infty$,
6. there exists a constant $0< c_0<+\infty$ such that $$\sup_{y\in\mathscr Y}\bigg|\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_0(y)}\bigg|\leq c_0.$$
Some remarks and comments on the above listed assumptions are in order. Except for Assumption $({\bf{A2}})$, which concerns the plug-in NPMLE $\psi(\hat F_n)$, all other assumptions involve the function $a$ and/or the mixing distribution $F_0$ that jointly define the functional $\psi(F_0)$. Specifically, Assumption $(\bf{A0})$ guarantees that $\psi(F_0)$ is well defined. Assumption $(\bf{A1})$ ensures the existence of the moment generating function of $F_0$ at the point $t=1$. Assumption $(\bf{A2})$ requires consistency of $\psi(\hat F_n)$ at $\psi(F_0)$. If $\hat F_n$ converges weakly to $F_0$ in $P_0^n$-probability, then parts (i) and (ii) of Assumption $(\bf{A3})$ together imply Assumption $(\bf{A2})$ because $a$ is continuous and bounded on $\mathscr Y$. Sufficient conditions for $\hat F_n$ to converge weakly to $F_0$ in the convolution model with the standard exponential kernel density on $[0,\,+\infty)$ are stated in Groeneboom and Wellner (1992), p. 86; see also Theorem 2.3 of Chen (2017), p. 54, for sufficient conditions for strong consistency of $\hat F_n$ in general mixture models. Part (ii) of Assumption $(\bf{A3})$ postulates that either $\mathscr Y$ is a closed and bounded interval $[y_{\mathrm{min}},\,y_{\mathrm{max}}
]$ or $\mathscr Y$ is a right-unbounded interval $[y_{\mathrm{min}},\,+\infty)$ and $a$ is bounded. Assumption $(\bf{A4})$ requires $a$ to be differentiable and bounded on $\mathscr Y$, which, in particular, accounts for $a$ to be right-differentiable at $y_{\mathrm{min}}$, that is, $\dot a_+(y_{\mathrm{min}})<+\infty$,[^11] and, in the case where $\mathscr Y=[y_{\mathrm{min}},\,y_{\mathrm{max}}]$, to be also left-differentiable at $y_{\mathrm{max}}$, that is, $\dot a_-(y_{\mathrm{max}})<+\infty$.[^12] Assumption $(\bf{A5})$ plays its role in the proof of Proposition \[prop:1\] when bounding the worst possible sub-directions, see in the Appendix A.
\[prop:1\] Under Assumptions *(*[**[A0]{}**]{}*)*–*(*[**[A5]{}**]{}*)*, we have $$\label{eq:conv1}
\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)=\sqrt{n}
\mathbb P_n b_{F_0}+o_{\mathbf P}(1)
\,\,\bigg(\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,\|b_{F_0}\|_{2,P_0}^2\big)\bigg),
$$ where the mapping $b_{F_0}:\,\mathscr X\rightarrow \mathbb R$, defined as $$\label{eq:effinfcurv}
x\mapsto b_{F_0}(x):=a(x)-\dot a(x)-\psi(F_0),$$ is the efficient influence function, whose squared $L^2(P_0)$-norm $$\label{eq:var}
\|b_{F_0}\|_{2,P_0}^2:=\int b_{F_0}^2\,{\mathrm{d}}P_0$$ is the efficient asymptotic variance.
Proposition \[prop:1\] establishes that, under sufficient conditions listed as Assumptions ([**[A0]{}**]{})–([**[A5]{}**]{}), *cf.* Lemma 4.6 of van de Geer (2003), p. 461, and van der Geer (2000), p. 231, the plug-in NPMLE $\psi(\hat F_n)$ consistently estimates $\psi(F_0)$ at $n^{-1/2}$-rate; furthermore, when recentered at $\psi(F_0)$ and $\sqrt{n}$-rescaled, it is asymptotically distributed as a zero-mean Gaussian, with variance attaining the lower bound given by the squared $L^2(P_0)$-norm of the efficient influence function, which plays here the same role as the normalized score function for the case of independent sampling from a parametric model $\{p_\theta$, $\theta\in\Theta\subseteq \mathbb{R}^d\}$, $d\geq 1$, $$\label{eq:norminf}
I^{-1}_\theta\dot{\ell}_\theta,$$ where $\dot{\ell}_\theta(\cdot)=\partial[\log p_\theta(\cdot)]/\partial \theta$ is the *score function* of the model and $I_\theta=\mathrm{E}[\dot{\ell}_\theta\dot{\ell}_\theta^{\mathrm T}]$ the Fisher information matrix for $\theta$. In a parametric set-up, the minimum variance lower bound reduces to the Cramér-Rao bound, which states that the inverse of the Fisher information matrix $I^{-1}_\theta$ is a lower bound on the variance of any $\sqrt{n}$-rescaled unbiased estimator $T_n\equiv T_n(X_1,\,\ldots,\,X_n)$ of $\theta$, in symbols, $\textrm{var}(\sqrt{n}T_n)\geq I^{-1}_\theta$. Therefore, the counterpart of $\|b_{F_0}\|_{2,P_0}^2=P_0b_{F_0}^2$ is, by symmetry of ($I_\theta$ hence of) $I^{-1}_\theta$, $$\mathrm E[(I^{-1}_\theta\dot{\ell}_\theta)(I^{-1}_\theta\dot{\ell}_\theta)^{\mathrm T}]=I_\theta^{-1}\mathrm E[\dot \ell_\theta\dot \ell_\theta
^{\mathrm T}]I_\theta^{-1}=I_\theta^{-1}I_\theta I_\theta^{-1}=I_\theta^{-1}.$$ In general, considered a function $\psi$ that maps $\Theta$ into $\mathbb R^m$, $m\geq1$, and denoted by $\dot \psi_\theta$ the derivative of $\theta\mapsto \psi(\theta)$, the matrix $\dot \psi_\theta I^{-1}_\theta\dot\psi_\theta^{\mathrm T}$ is a lower bound on the variance of any $\sqrt{n}$-rescaled unbiased estimator of $\psi(\theta)$.
Even if a statement of the result in Proposition \[prop:1\] appears in Lemma 4.6 of van der Geer (2003), p. 461, as far as we are aware, a complete derivation of the assertion is not available in the literature, *cf.* also van der Geer (2000), p. 231, so the proof reported in the Appendix A might prove helpful. The underlying idea is outlined hereafter. A NPMLE $\hat F_n$ solves the likelihood equation for every path $t\mapsto F_t$, with ${\mathrm{d}}F_t:= (1+th_F)\,{\mathrm{d}}F$, starting at a fixed point $F$ corresponding to $t=0$ and direction $h_F$ such that $\int h_F\,{\mathrm{d}}F=0$, that is, for every parametric sub-model which passes (at $t=0$) through it. In symbols, $$\label{eq:score}
\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathbb P_n \log p_{\hat F_{n,t}}\bigg |_{t=0}=0.$$ For ease of notation, let $$A_{F}h_F(x):=\int_{\mathscr Y} h_F(y)\frac{e^{-(x-y)}}{ p_F(x)}1\{y\leq x\}\,{\mathrm{d}}F(y).$$ Equation reduces to $\mathbb P_n b_{\hat F_n}=0$, where $b_{\hat F_n}(\cdot)=A_{\hat F_n}h_{\hat F_n}(\cdot)$ is the score function (at $t=0$) in an [“information loss model”]{}, see, *e.g.*, §25.5.2 in van der Vaart (1998), pp. 374–375. If $\hat F_n$ dominates $F_0$, which, however, is seldom true, then $-P_0b_{\hat F_n}=\psi(\hat F_n)-\psi(F_0)$ so that $\psi(\hat F_n)-\psi(F_0)=(\mathbb P_n -P_0)b_{\hat F_n}$. Asymptotic equicontinuity arguments then yield that $(\mathbb P_n -P_0)b_{\hat F_n}=(\mathbb P_n -P_0)b_{F_0}+o_{\bf P}(n^{-1/2})=\mathbb P_n b_{F_0}+o_{\bf P}(n^{-1/2})$ because $P_0b_{F_0}=0$, namely, the score has zero mean. So, $\sqrt{n}(\psi(\hat F_n)-\psi(F_0))=\sqrt{n}\mathbb P_n b_{F_0}+o_{\bf P}(1)$. Asymptotic normality follows. The reader is referred to Sect. 11.2 of van de Geer (2000), pp. 211–246, for a more comprehensive treatment of the topic taking into account technical difficulties to which it cannot be here dedicated the necessary space.
For simplicity, a convolution model with an exponential kernel density on $[0,\,+\infty)$ having intensity $\lambda=1$ has been considered (we warn the reader of the clash of notation with the symbol $\lambda$ previously used to denote Lebesgue measure on the real line), but, as revealed by an inspection of the proof of Proposition \[prop:1\], the assertion holds true for every $\lambda>0$.
Part (i) of Assumption ([**[A3]{}**]{}) requires $a$ to be continuous on $\mathscr Y$, which is not true for indicator functions, therefore Proposition \[prop:1\] does not apply to pointwise estimation of the c.d.f. $F_0$ nor to the estimation of the probability of an interval, so that it cannot be concluded that these functionals are estimable at $n^{-1/2}$-rate by the corresponding plug-in NPMLE’s. Indeed, $F_0$ can be *pointwise* estimated only at $n^{-1/3}$-rate, see Groeneboom and Wellner (1992), p. 121. Part (ii) of Assumption ([**[A3]{}**]{}) and Assumption ([**[A4]{}**]{}) require that both $a$ and $\dot a$ are bounded on $\mathscr Y$, which, for example, may not be true for the functions $y^r$ and $e^{ty}$ that define the $r$th moment and the moment generating function of $Y$ at the point $t$, respectively: in fact, both $y^r$ and $e^{ty}$, as well as their first derivatives, are continuous on the half-line $[y_{\mathrm{min}},\,+\infty)$, but not bounded therein. Nonetheless, boundedness can be retrieved by restriction to a compact domain. Therefore, if, besides Assumptions ([**[A0]{}**]{})–([**[A2]{}**]{}) and ([**[A5]{}**]{}), it also holds that $F_0$ has compact support, then, by Proposition \[prop:1\], it can be concluded that $\mathrm E Y^r$ and $M_{F_0}(t)$ are consistently and efficiently estimated at $n^{-1/2}$-rate by their respective plug-in NPMLE’s.
Although Proposition \[prop:1\] asserts that certain integral linear functionals can be consistently estimated at $n^{-1/2}$-rate by the plug-in NPMLE $\psi(\hat F_n)$, which, when recentered at $\psi(F_0)$ and $\sqrt{n}$-rescaled, is asymptotically normal and efficient, two orders of problems may arise that can make it difficult to employ the result for statistical inference:
a)
: computation of the NPMLE $\hat F_n$,
b)
: dependence of the variance $\|b_{F_0}\|_{2,P_0}^2$ on the unknown sampling distribution $P_0$.
As for the former difficulty, although the NPMLE $\hat F_n$ can be found by a one-step procedure computing the slope of the convex minorant of a certain function, *cf.* Groeneboom and Wellner (1992), pp. 62–63 (see also Vardi (1989) for a different approach), as a by-product of Theorem 11.8 of van de Geer (2000), p. 217, which the assertion of Proposition \[prop:1\] relies on, the recentered and $\sqrt{n}$-rescaled plug-in NPMLE $\sqrt{n}(\psi(\hat F_n)-\psi(F_0))$ is equivalent, in the sense of being asymptotically approximable, up to an $o_{\mathbf{P}}(1)$-error term, by the empirical average of the efficient influence function. This is part of a general issue concerning the fact that sequences of efficient estimators for functionals are asymptotically approximable by an empirical average of the efficient influence function, see, *e.g.*, Lemma 2.9 in Bolthausen *et al.* (2002), p. 349. In fact, set the position $$\label{eq:estim}
\tilde\psi_n:=
\frac{1}{n}\sum_{i=1}^n[a(X_i)-\dot a(X_i)]=
\mathbb P_n(a-\dot a)$$ and noted that, from the definition of $b_{F_0}$ in , the term $\mathbb P_nb_{F_0}$ appearing in writes as $\tilde \psi_n-\psi(F_0)$, we have $$\begin{split}
\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)&=\sqrt{n}\mathbb P_nb_{F_0}+o_{\mathbf P}(1)\\
&=\sqrt{n}\big(\tilde \psi_n-\psi(F_0)\big)+o_{\mathbf P}(1)\,\,\bigg(\,\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,\|b_{F_0}\|_{2,P_0}^2\big)\bigg).
\end{split}$$ Thus, both $\sqrt{n}(\psi(\hat F_n)-\psi(F_0))$ and $\sqrt{n}(\tilde \psi_n-\psi(F_0))$ are asymptotically normal and efficient. Moreover, estimators arising from $\tilde\psi_n$ may coincide with simple *naïve* estimators. For example,\
- if $\psi(F_0)=\mathrm{E}Y$, from , for $(a-\dot a)(y)=y-1$, we get the estimator $\tilde\psi_n=\bar X_n-1$, which is the one we would suggest considering that $\mathrm{E}Y=\mathrm{E}X-\mathrm{E}Z
=\mathrm{E}X-1$;\
- if $\psi(F_0)=M_Y(t)$ for any fixed $t<1$ such that $\int_{\mathscr Y} e^{ty}\,{\mathrm{d}}F_0(y)<+\infty$, then the estimator derived from , for $(a-\dot a)(y)=(1-t)e^{ty}$, is $\tilde\psi_n=(1-t)\times{n}^{-1}\sum_{i=1}^n e^{tX_i}$, which is the one we would suggest taking into account that $M_Y(t)=M_X(t)/M_Z(t)=(1-t)M_X(t)$, where $M_{Z}(t)=(1-t)^{-1}$, $t<1$, is the m.g.f. of a standard exponential r.v. $Z$. So, letting $M_n(t):=n^{-1}\sum_{i=1}^n e^{tX_i}$, $t\in\mathbb{R}$, be the empirical m.g.f. for the random sample $X_1,\,\ldots,\,X_n$, it turns out that $\tilde\psi_n={M_n(t)}/{M_Z(t)}$.
As for the difficulty listed in b), by the plug-in approach, replacing the asymptotic variance $\|b_{F_0}\|_{2,P_0}^2$ with a consistent estimator $S_n^2$ leads to the following assertion.
\[cor:1\] Under the conditions of Proposition \[prop:1\], if, in addition, $S_n^2\,\xrightarrow{\mathrm{P}}\,\|b_{F_0}\|_{2,P_0}^2$, where [“$\xrightarrow{\mathrm{P}}$”]{} denotes convergence in $P_0^n$-probability, then $$\sqrt{n}\frac{\tilde\psi_n-\psi(F_0)}{S_n}\,\xrightarrow{\mathscr{L}}\,\mathscr{N}(0,\,1).$$
Replaced the efficient asymptotic variance in with a consistent sequence of estimators, asymptotic normality of the Studentized version of $\tilde\psi_n$ allows to carry out *pointwise* inference on linear functionals by interval estimation or hypotheses testing constructing confidence intervals or tests, respectively. For every $0<\alpha<1$, let $z_{\alpha/2}$ be the $(1-\alpha/2)$-quantile of a standard normal distribution, *i.e.*, $\Phi(z_{\alpha/2})=1-\alpha/2$, where $\Phi(\cdot)$ stands for the c.d.f. of a standard normal. Then, $$C_n:=\big[\tilde\psi_n-z_{\alpha/2} S_n/\sqrt{n},\,\,\,\,\,\,\,\tilde\psi_n+z_{\alpha/2} S_n/\sqrt{n}\big],\quad \mbox{with $\,\mathbf{P}\big(C_n\ni \psi(F_0)\big)=
1-\alpha+o(1)$,}$$ is an approximate $(1-\alpha)$-level confidence interval for $\psi(F_0)$.
Asymptotic normality of the plug-in NPMLE for linear functionals of the mixing distribution can be employed to establish asymptotic normality for *non*-linear functionals. Suppose, for instance, that $F\mapsto\varphi(F)$ is defined as $$\varphi(F):=(g\circ\psi)(F)=g\big(\psi(F)\big),$$ where the function $g:\,\mathbb R\rightarrow\mathbb R$ has non-zero derivative at $\psi(F_0)$ denoted by $\dot g(\psi(F_0))$. Asymptotic normality of $\sqrt{n}(\varphi (\hat F_n)-\varphi(F_0))$ then follows from asymptotic normality of $\sqrt{n}(\psi(\hat F_n)-\psi(F_0))$ by the delta method, see, *e.g.*, chpt. 3 in van der Vaart (1998), pp. 25–34. So, if the convergence in takes place, then $$\sqrt{n}\big(\varphi (\hat F_n)-\varphi(F_0)\big)\,\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,\sigma_{\varphi}^2\big),\quad
\mbox{with $\sigma_\varphi:=\dot g(\psi(F_0))
\|b_{F_0}\|_{2,P_0}$},$$ where efficiency of $\psi(\hat F_n)$ carries over into efficiency of $\varphi (\hat F_n)$, see *ibid.*, p. 386, for details.
Alternatively, set the position $$R_n:=\frac{\varphi (\hat F_n)-\varphi(F_0)}{\psi(\hat F_n)-\psi(F_0)},$$ under the condition $$|R_n-1|=o_{\mathbf{P}}(1),$$ which requires that, in probability, $\varphi(\hat F_n)-\varphi(F_0)$ behaves asymptotically as $\psi(\hat F_n)-\psi(F_0)$, after $\sqrt{n}$-rescaling, the two differences have the same limiting distribution. In fact, if the convergence in takes place, then Slutsky’s lemma implies that $$\begin{split}
\sqrt{n}\big(\varphi(\hat F_n)-\varphi(F_0)\big)&=
[(R_n-1)+1]
\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)\\
&=(o_{\mathbf{P}}(1)+1)\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)
\,\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,
\|b_{F_0}\|_{2,P_0}^2\big),
\end{split}$$ see also the Remark of van de Geer (2000) on p. 223.
#### Convolution with the double-exponential (Laplace) density {#paragraph:Laplace}
In this paragraph, the case of main interest of the article concerning asymptotically efficient maximum likelihood estimation of linear functionals of the mixing distribution in a convolution model with the (standard) Laplace kernel density is considered. It has been recalled in Sect. \[intro\] that, for a *one*-parameter $\theta$ (location only) Laplace model, the sample median $\hat\theta_n$ is a MLE, consistent and asymptotically efficient, even if, for small sample sizes, it may not be the best estimator to use because there exist other unbiased estimators with smaller variances, which are therefore more efficient, see, *e.g.*, Remark 2.6.2 in Kotz *et al.* (2001), p. 82. More precisely, for a sample of odd size $n$ from a general $\mathrm{Laplace}\,(\theta,\,s)$ distribution, the variance of $\hat\theta_n$ is equal to $$\frac{1}{4(n+2)[k(0)/s]^2}=
\frac{s^2}{(n+2)}
,$$ where $k(\cdot)$ is the density of a standard Laplace distribution as defined in , while the asymptotic variance is equal to $$\frac{1}{4n[k(0)/s]^2}=
\frac{s^2}{n}.$$ It is just the case to observe that also the sample mean $\bar X_n$ is asymptotically normal with mean $\theta$, but the asymptotic relative efficiency (ARE) of the median to the mean, namely, the ratio of the variance of the sample mean to the asymptotic variance of the sample median equals $2$: $$\frac{2s^2/n}{s^2/n}=2.$$ On a side note, we recall that, for any function $g$ differentiable at $\theta$, with derivative $\dot g(\theta)$, the plug-in MLE $g(\hat\theta_n)$ is also asymptotically efficient, with $$\sqrt{n}\big(g(\hat\theta_n)-g(\theta)\big)\,\xrightarrow{\mathscr{L}}\,\mathscr{N}\big(0,\,[s\dot g(\theta)]^2\big),$$ see, *e.g.*, Lehmann and Casella (1998), p. 440.
In what follows, we aim at giving results on asymptotically efficient maximum likelihood estimation of linear functionals of the mixing distribution, beyond the case of a degenerate mixing distribution localized at a point $\theta$ on the real line. As recalled in Sect. \[intro\], in the deconvolution problem with the Laplace kernel density, *a* NPMLE $\hat F_n$ always exists and consistency at a *continuous* distribution function $F_0$ holds, but little is known about the asymptotic behaviour of the plug-in NPMLE for linear functionals. The following proposition states that, except for the above recalled degenerate case, estimation of integral linear functionals at $n^{-1/2}$-rate is impossible.
\[prop:2\] Let $F_0$ be a non-degenerate probability measure supported on $\mathscr Y$. Let $\psi(F_0)$ be any integral linear functional evaluated at $F_0$. Then, there exists no estimator sequence for $\psi(F_0)$ that is regular at $F_0$.
Some comments on Proposition \[prop:2\], whose proof is deferred to the Appendix B, are in order. It states that no integral linear functional is estimable at parametric rate, in particular, by the plug-in NPMLE $\psi(\hat F_n)$. One can thus expect estimation, performed by any method, only at slower rates and, possibly, with a non-Gaussian limiting distribution, even if the theorem we invoke to establish Proposition \[prop:2\] does not give any indication about which rates to expect when estimation at $n^{-1/2}$-rate fails, an issue that requires further investigation. A related open question concerns the possible extension of the negative result of Proposition \[prop:2\] to convolution models with general kernel densities that are symmetric about zero, but not differentiable at it, a feature that seems to play a crucial role in causing failure of estimation at parametric rate. To sum-up, only in the case of a degenerate mixing distribution at a point $\theta$, the MLE $\hat \theta_n$ is asymptotically efficient for the location parameter and the plug-in MLE $g(\hat\theta_n)$ is asymptotically efficient for any $g(\theta)$, with $g$ differentiable at $\theta$.
Final remarks {#sec:3}
=============
In this article, we have studied asymptotically efficient maximum likelihood estimation of linear functionals of the mixing distribution in a standard additive measurement error model, when the error has either the exponential or Laplace distribution. In the former case, the plug-in NPMLE of certain linear functionals is $\sqrt{n}$-consistent, asymptotically normal, efficient and equivalent to naïve estimators that are empirical averages of a given transformation of the observations. In the latter case, instead, even if the kernel is generated by symmetrization about the origin of the exponential density, left aside the degenerate case of a single Laplace model in which the MLE, the sample median, is asymptotically efficient for the location parameter, asymptotically efficient estimation of linear functionals completely fails, in the sense that estimation at $n^{-1/2}$-rate is impossible for linear functionals of non-degenerate mixing distributions. An open question then is whether this negative result extends to general kernel densities symmetric about zero, but not differentiable at zero, a feature that seems to play a crucial role in causing the failure.
Appendix A {#appendix-a .unnumbered}
==========
In this section, we present the proof of Proposition \[prop:1\] on the asymptotic efficiency of the plug-in NPMLE for integral linear functionals of the mixing distribution in a convolution model with the *exponential* kernel density on $[0,\,+\infty)$.
#### Proof of Proposition \[prop:1\]
We appeal to Theorem 2.1 of van de Geer (1997), p. 21 (see also Theorem 11.8 of van de Geer (2000), pp. 217–220, for a slightly more general version) and, in showing that Conditions 1–4 are satisfied, we follow the indications exposed in Sect. 3, *ibid.*, pp. 24–27.
[*Verification of Condition 1.*]{} (*Consistency and rates*).\
Under Assumption $(\bf{A1})$ that $M_{F_0}(1)<+\infty$, the MLE $p_{\hat F_n}$ converges in the Hellinger distance $d_{\mathrm H}$, defined as the $L^2$-distance between the square-root densities, at the rate $O_{\mathbf{P}}(n^{-1/3})$. In symbols, for $\delta_n:=n^{-1/3}$, $$d_{\mathrm H}(p_{\hat F_n},\,p_0):=\|p_{\hat F_n}^{1/2}-p_0^{1/2}\|_2=O_{\mathbf{P}}(\delta_n).$$ The result can be obtained by applying Theorem 7.4 in van de Geer (2000), pp. 99–100, see also *ibid.*, p. 124. As a consequence, see, *e.g.*, Corollary 7.5, *ibid.*, p. 100, $$\mathbb{P}_n\log \frac{2p_{\hat F_n}}{p_{\hat F_n}+p_0}
=O_{\mathbf P}(\delta_n^2),$$ where $\delta_n^2=o(n^{-1/2})$. Consistency of $\psi(\hat F_n)$ is guaranteed by Assumption $(\bf{A2})$.
[*Verification of Condition 2.*]{} (*Existence of the worst possible sub-directions and efficient influence functions. Differentiability of $\psi$ in a neighborhood of $F_0$*).\
For real numbers $M>M_{F_0}(1)>0$ and $r>0$, let $$\label{eq:set}
\mathscr P_0:=\{F\in \mathscr P:\,\mathrm{support}\,(F)=\mathscr Y,\,\,\,M_F(1)<M,\,\,\,d_{\mathrm H}(p_F,\,p_0)\leq r\}$$ be a Hellinger-type ball centered at $p_0$ with radius $r>0$. For every $\alpha\in[0,\,1)$ and $F\in\mathscr P_0$, let $F_\alpha:=\alpha F + (1-\alpha)F_0$. We prove
1. *existence of the worst possible sub-directions $h_{F_\alpha}$ such that $h_{F_\alpha}\in L^2(F_\alpha)$ and $\int h_{F_\alpha}\,{\mathrm{d}}F_\alpha=0$*;\
2. *existence of the efficient influence functions $b_{F_\alpha}:=A_{F_\alpha}h_{F_\alpha}$, where $A_{F_\alpha}h_{F_\alpha}(\cdot):=\mathrm E[h_{F_\alpha}(Y)\mid X=\cdot]$*;\
3. *differentiability of $\psi$ at $F_\alpha$*: $$\label{eq:diff}
A^*b_{F_\alpha}(Y)=a(Y)-\psi({F_\alpha})\quad a.s.\,[F_\alpha],$$ *where $A^*b_{F_\alpha}(\cdot):=\mathrm E[b_{F_\alpha}(X)\mid Y=\cdot]$*.
For every $\alpha\in[0,\,1)$, we prove the existence of $h_{F_\alpha}$ such that the corresponding $b_{F_\alpha}=A_{F_\alpha}h_{F_\alpha}$ satisfies $A^*b_{F_\alpha}(y)=a(y)-\psi({F_\alpha})$ for $F_\alpha$-almost all $y$’s. We proceed by first deriving the expression of $b_{F_\alpha}$ as a solution of and then proving the existence of the corresponding worst possible sub-direction $h_{F_\alpha}$ as required in (a) and (b). The function $b_{F_\alpha}$ has to satisfy $$\label{eq:infl}
A^*b_{F_\alpha}(y):=
\mathrm{E}[b_{F_\alpha}(X)\mid Y=y]=
\int_{\mathscr X}
b_{F_\alpha}(x)e^{-(x-y)}1{\{x\geq y\}}\,{\mathrm{d}}x=
a(y)-\psi(F_\alpha),$$ where $\mathscr X=[y_{\textrm{min}},\,+\infty)$, for $F_\alpha$-almost all $y$’s. Differentiating both sides of with respect to $y$, we get $$\label{eq:der}
\int_{\mathscr X}
b_{F_\alpha}(x)e^{-(x-y)}1{\{x\geq y\}}\,{\mathrm{d}}x - b_{F_\alpha}(y)=\dot a(y).$$ Using constraint in , we obtain that $a(y)-\psi(F_\alpha)-b_{F_\alpha}(y)=\dot a(y)$, whence $b_{F_\alpha}(y)=a(y)-\dot a(y)-\psi(F_\alpha)$. The solution is unique up to sets of $F_\alpha$-measure zero. By an extension to $\mathscr X$, $b_{F_\alpha}$ is then defined as in .
\(a) *Existence of $h_{F_\alpha}\in L^2(F_\alpha)$ such that $\int h_{F_\alpha}\,{\mathrm{d}}F_\alpha=0$.*\
Recall that $$p_{F_\alpha}(x):=
\int_{\mathscr Y}e^{-(x-y)}1{\{y\leq x\}}\,{\mathrm{d}}F_\alpha(y),\quad x\in\mathscr X.$$ Defined the function $I_{F_\alpha}:\,\mathscr X\rightarrow\mathbb R$ as $$x\mapsto I_{F_\alpha}(x):=\int_{\mathscr Y}
\dot a(y)\int_{\mathscr Y}
\frac{e^{-(x-u)}}{p_{F_\alpha}(x)}1{\{u\leq y\}}
\,{\mathrm{d}}F_\alpha(u)\,1{\{y\leq x\}}\,{\mathrm{d}}y,
$$ integration by parts yields that $$\begin{split}
&\int_{\mathscr Y}
a(y)\frac{e^{-(x-y)}}{p_{F_\alpha}(x)}1{\{y\leq x\}}\,{\mathrm{d}}F_\alpha(y)\\
&\hspace*{2.2cm}=
\int_{\mathscr Y}
a(y)\frac{{\mathrm{d}}}{{\mathrm{d}}F_\alpha(y)}
{\left(\int_{\mathscr Y}
\frac{e^{-(x-u)}}{p_{F_\alpha}(x)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\right)}1{\{y\leq x\}}\,{\mathrm{d}}F_\alpha(y)
\\&\hspace*{2.2cm}=
a(y)\int_{\mathscr Y}\frac{e^{-(x-u)}}
{p_{F_\alpha}(x)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\bigg|_{y_{\textrm{min}}}^x-
I_{F_\alpha}(x)
\\&\hspace*{2.2cm}=
a(x)\int_{\mathscr Y}\frac{e^{-(x-u)}}
{p_{F_\alpha}(x)}1{\{u\leq x\}}\,{\mathrm{d}}F_\alpha(u)
-I_{F_\alpha}(x)\\&\hspace*{2.2cm}=
a(x)-I_{F_\alpha}(x)
\end{split}$$ because $a(y_{\textrm{min}})<+\infty$ and ${F_\alpha}(y_{\textrm{min}})=0$ by part (iii) of Assumption $(\bf{A3})$ combined with the fact that $F\in\mathscr P_0$. Analogously, since $\dot a_+(y_{\mathrm{min}})<+\infty$, $$\begin{split}
&\hspace*{-0.2cm}
-\int_{\mathscr Y}
\frac{{\mathrm{d}}}{{\mathrm{d}}F_\alpha(y)}{\left(\dot a(y)
\int_{\mathscr Y} e^{-(y-u)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\right)}\frac{e^{-(x-y)}}{p_{F_\alpha}(x)}1{\{y\leq x\}}\,{\mathrm{d}}F_\alpha(y)\\
&\qquad=
-\dot a(y)
\int_{\mathscr Y}e^{-(y-u)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)
\times\frac{e^{-(x-y)}}{p_{F_\alpha}(x)} \bigg|_{y_{\mathrm{min}}}^x\\
&\hspace*{4.5cm}
+\int_{\mathscr Y}
\dot a(y)\int_{\mathscr Y}
\frac{e^{-(x-u)}}{p_{F_\alpha}(x)}1{\{u\leq y\}}
\,{\mathrm{d}}F_\alpha(u)\,1{\{y\leq x\}}\,{\mathrm{d}}y
\\&
\qquad=
-\dot a(x)\int_{\mathscr Y}
\frac{e^{-(x-u)}}{p_{F_\alpha}(x)}1{\{u\leq x\}}
\,{\mathrm{d}}F_\alpha(u)
+I_{F_\alpha}(x)
\\&
\qquad=
-\dot a(x)+I_{F_\alpha}(x).
\end{split}$$ Then, defined the mapping $h_{F_\alpha}:\,\mathscr Y\rightarrow \mathbb{R}$ as $$\label{eq:subdir}
y\mapsto h_{F_\alpha}(y):=
{\left[a(y)-\frac{{\mathrm{d}}}{{\mathrm{d}}F_\alpha(y)}{\left(\dot a(y)\int_{\mathscr Y}e^{-(y-u)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\right)}-\psi({F_\alpha})
\right]},$$ by previous computations, we have that $$\begin{aligned}
\label{eq:ineq}
\hspace*{-0.7cm}
\forall\,x\in\mathscr X,\quad A_{F_\alpha}h_{F_\alpha}(x)&:=&\mathrm E[h_{F_\alpha}(Y)\mid X=x]\nonumber\\
&\,=&
\int_{\mathscr Y}
h_{F_\alpha}(y)
\frac{e^{-(x-y)}
}{p_{F_\alpha}(x)}1{\{y\leq x\}}\,{\mathrm{d}}F_\alpha(y)\nonumber\\
&\,=&a(x)-\dot a(x)-\psi(F_\alpha)=b_{F_\alpha}(x).\end{aligned}$$ In order to check that $h_{F_\alpha}$ has expected value $\int h_{F_\alpha}\,{\mathrm{d}}F_\alpha=0$, it suffices to note that, by applying twice the tower rule and using equalities and , $$\begin{aligned}
\label{eq:null}
\int h_{F_\alpha}\,{\mathrm{d}}F_\alpha=\mathrm E[h_{F_\alpha}(Y)]&=&
\mathrm E[\mathrm E[h_{F_\alpha}(Y)\mid X]]\nonumber\\
&=&\mathrm E[A_{F_\alpha}h_{F_\alpha}(X)]\nonumber\\&
=&\mathrm E[b_{F_\alpha}(X)]\\
&=&\mathrm E[\mathrm E[b_{F_\alpha}(X)\mid Y]]\nonumber\\
&=&\mathrm E[A^*b_{F_\alpha}(Y)]\nonumber\\
&=&\mathrm E[a(Y)-\psi(F_\alpha)]=\int_{\mathscr Y}[a(y)-\psi(F_\alpha)]\, {\mathrm{d}}F_\alpha(y)=0.\nonumber\end{aligned}$$ Next, we show that, for every $\alpha\in[0,\,1)$ and $F\in\mathscr P_0$, $$\label{eq:sup-normdirection}
\sup_{y\in\mathscr Y}|h_{F_\alpha}(y)|<+\infty,$$ which implies that $h_{F_\alpha}\in L^2(F_\alpha)$. Noting that $$\begin{aligned}
\label{eq:expresequiv}
&&\hspace*{-1cm}\frac{{\mathrm{d}}}{{\mathrm{d}}F_\alpha(y)}{\left(\dot a(y)\int_{\mathscr Y} e^{-(y-u)}1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\right)}\nonumber\\
&&\hspace*{0.5cm}=\dot a(y)e^{-y}\times\frac{{\mathrm{d}}}{{\mathrm{d}}F_\alpha(y)}{\left(\int_{\mathscr Y}e^u1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\right)}\nonumber\\
&&\hspace*{4.2cm}
+\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_\alpha(y)}\times \int_{\mathscr Y}e^u1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u)\nonumber
\\
&&\hspace*{0.5cm}=\dot a(y)
+\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_\alpha(y)}\times \int_{\mathscr Y}e^u1{\{u\leq y\}}\,{\mathrm{d}}F_\alpha(u),\end{aligned}$$ we can rewrite $h_{F_\alpha}$ in as $$\begin{aligned}
\label{eq:altern}
\hspace*{-1cm}
h_{F_\alpha}(y)
&=&
\bigg\{a(y)-\dot a(y)-\alpha
\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_0(y)}\times\frac{{\mathrm{d}}F_0(y)}{{\mathrm{d}}F_\alpha(y)}
\int_{\mathscr Y}e^u1{\{u\leq y\}}\,{\mathrm{d}}F(u)\\
&&\qquad\qquad\,\,\,\,\,\, -\,(1-\alpha)
\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_0(y)}\times\frac{{\mathrm{d}}F_0(y)}{{\mathrm{d}}F_\alpha(y)}
\int_{\mathscr Y}e^u
1{\{u\leq y\}}\,{\mathrm{d}}F_0(u)\\
&&\qquad\qquad\,\,\,\,\,\,-\,\alpha\psi(F)-(1-\alpha)\psi(F_0)\bigg\}.\end{aligned}$$ To conclude that $h_{F_\alpha}$ is bounded on $\mathscr Y$, we observe two facts. First, $$\begin{split}
|\alpha\psi(F)+(1-\alpha)\psi(F_0)|&<|\psi(F_0)|+|\psi(F)-\psi(F_0)|\\
&\leq|\psi(F_0)|+\int_{\mathscr{Y}} |a(y)|\,{\mathrm{d}}(F+F_0)(y)\\
&\leq|\psi(F_0)|+2\sup_{y\in\mathscr Y}|a(y)|<+\infty,
\end{split}$$ where $|\psi(F_0)|<+\infty$ by Assumption $(\bf{A0})$ and $\sup_{y\in\mathscr Y}|a(y)|<+\infty$ by parts (i) and (ii) of Assumption $(\bf{A3})$. Second, for every $\alpha\in[0,\,1)$ and $F\in\mathscr P_0$, $$\label{eq:12}
0<\sup_{y\in\mathscr Y}\frac{{\mathrm{d}}F_0}{{\mathrm{d}}F_\alpha}(y)\leq\frac{1}{(1-\alpha)},$$ where $({\mathrm{d}}F_0/{\mathrm{d}}F_\alpha)$ exists because $F_\alpha$ dominates $F_0$. The bound in holds uniformly over $\mathscr P_0$. Therefore, $$\begin{aligned}
\label{eq:suph}
\hspace*{-0.8cm}
\sup_{y\in \mathscr Y}
|h_{F_\alpha}(y)|
&<&\sup_{y\in \mathscr Y}|a(y)|+
\sup_{y\in \mathscr Y}|\dot a(y)|+\frac{1}{1-\alpha}
\sup_{y\in \mathscr Y}\bigg|\frac{{\mathrm{d}}\big(\dot a(y)e^{-y}\big)}{{\mathrm{d}}F_0(y)}\bigg|[M_F(1)+M_{F_0}(1)]\\
&&\qquad\quad\,\,\,\,\,\,+\,|\psi(F_0)|+2\sup_{y\in\mathscr Y}|a(y)|<+\infty\nonumber\end{aligned}$$ by Assumptions $(\bf{A0})$, $(\bf{A1})$, $(\bf{A3})$–$(\bf{A5})$ and the fact that $M_F(1)$ is bounded by a constant $M$ on $\mathscr P_0$.\
(b)–(c) *Definition of $b_{F_\alpha}$ and differentiability of $\psi$ at $F_\alpha$.*\
The function $b_{F_\alpha}$ defined in , which solves equation , is such that $A_{F_\alpha}h_{F_\alpha}(x)=b_{F_\alpha}(x)$ for every $x\in\mathscr X$, in virtue of .
[*Verification of Condition 3.*]{} (*Control on the worst possible sub-directions $h_{F_\alpha}$*).\
Recall that $M_F(1)$ in is bounded by $M$ on $\mathscr P_0$. Besides, the factor $(1-\alpha)^{-1}$, which diverges to $+\infty$ as $\alpha\rightarrow 1$, is counterbalanced by $1-\alpha$. There thus exists a positive constant $B\equiv B(M,\,r)<+\infty$ such that $$\sup_{F\in\mathscr P_0}\,\sup_{0\leq\alpha<1}\,\sup_{y\in\mathscr Y}(1-\alpha)|h_{F_\alpha}(y)|\leq B.$$
[*Verification of Condition 4.*]{} (*Control on the efficient influence functions $b_{F_\alpha}$*).\
The information for estimating $\psi(F_0)$ is positive and finite, $0<\|b_{F_0}\|_{2,P_0}^2<+\infty$. Also, the influence functions are uniformly bounded. In fact, for every $x\in\mathscr X$, we have $|b_{F_\alpha}(x)|<|b_{F_0}(x)|+|\psi(F)-\psi(F_0)|$ so that $$\sup_{F\in\mathscr P_0}\sup_{0\leq \alpha<1}\sup_{x\in\mathscr X}|b_{F_\alpha}(x)|<3\sup_{y\in\mathscr{Y}}|a(y)|+
\sup_{y\in\mathscr{Y}}|\dot a(y)|+ |\psi(F_0)|<+\infty$$ by Assumptions $(\bf{A0})$, $(\bf{A3})$ (parts (i) and (ii)) and $(\bf{A4})$.
Next, to show that relationships (2.10) and (2.11) in van de Geer (1997), p. 21, are satisfied, we follow the reasoning illustrated in Sect. 3.4, *ibid.*, pp. 26–27, and check that, for some positive sequence $r_n\rightarrow0$, $$\label{eq:41}
\lim_{n\rightarrow+\infty}\sup_{F\in\mathscr P_n}\sup_{0\leq\alpha<1}\|b_{F_\alpha}-b_{F_0}\|_{2,P_0}^2=0,$$ where $\mathscr P_n$ is the set obtained from $\mathscr P_0$ in by replacing $r$ with $r_n$. Note that $b_{F_\alpha}-b_{F_0}=\alpha[\psi(F)-\psi(F_0)]=
\alpha\int_{\mathscr{Y}}a(y)\,{\mathrm{d}}(F-F_0)(y)$. Using integration by parts, together with conditions (i) and (ii) of Assumption $(\bf{A3})$, which jointly guarantee that $a$ is bounded on $\mathscr Y$, as well as the fact that every $F\in\mathscr P_n$ has the same support as $F_0$, we find that $\int_{\mathscr{Y}}a(y)\,{\mathrm{d}}(F-F_0)(y)=-\int_{\mathscr{Y}}\dot a(y) (F-F_0)(y)\,{\mathrm{d}}y$. The latter integral can be bounded above by applying inequality (30) in Scricciolo (2018), p. 358, which relates the $L^1$-Wasserstein or Kantorovich distance $W_1(F,\,F_0)=\|F-F_0\|_1$ between distribution functions $F$ and $F_0$ to the Hellinger distance between the corresponding mixtures (of exponential densities) $d_{\mathrm H}\equiv d_{\mathrm H}(p_F,\,p_0)=\|p_F^{1/2}-p_0^{1/2}\|_2$, $$\label{eq:wass}
W_1(F,\,F_0)\lesssim \sqrt{d_{\mathrm H}}\log^{3/4}(1/d_{\mathrm H}),$$ where [“$\lesssim$”]{} indicates inequality valid up to a constant multiple that is universal or fixed within the context, but anyway inessential for our purposes because the bound is uniform over $\mathscr P_n$. The inequality is obtained by setting $p=1$ and $\beta=1$, the latter value being determined by condition (29), *ibid.*, p. 358, on the Fourier transform of a standard exponential density. By Assumption $(\bf{A4})$, which guarantees that $\dot a$ is bounded on $\mathscr Y$, and inequality , we have $$\begin{split}
\|b_{F_\alpha}-b_{F_0}\|_{2,P_0}^2&<\bigg|
\int_{\mathscr{Y}}a(y)\,{\mathrm{d}}(F-F_0)(y)\bigg|^2\\
&=\bigg|
\int_{\mathscr{Y}}\dot a(y) (F-F_0)(y)\,{\mathrm{d}}y\bigg|^2\\
&\leq\bigg(\sup_{y\in\mathscr Y}|\dot a(y)|\bigg)^2W_1^2(F,\,F_0)\lesssim d_{\mathrm H}\log^{3/2}(1/d_{\mathrm H}),
\end{split}$$ where $\lim_{n\rightarrow +\infty}d_{\mathrm H}\log^{3/2}(1/d_{\mathrm H})=0$ because $d_{\mathrm H}\leq r_n$ on $\mathscr P_n$. The limit in follows.
It remains to check that, for the collection of functions $\mathscr I:=\{b_{F_\alpha}:\,d_{\mathrm H}(p_F,\,p_0)\leq r, \,\,\,0\leq \alpha<1\}$, the bracketing integral $$\label{eq:intentropy}
\int_0^1\sqrt{\log N_{[]}\big(\varepsilon,\,\mathscr{I},\,L^2(P_0)\big)}\,{\mathrm{d}}\varepsilon<+\infty,$$ where $N_{[]}(\varepsilon,\,\mathscr{I},\,L^2(P_0))$ is the $\varepsilon$-bracketing number of $\mathscr{I}$ for the $L^2(P_0)$-metric, namely, the smallest number of $\varepsilon$-brackets needed to cover $\mathscr{I}$, see, *e.g.*, 2.1.6 Definition (Bracketing numbers) in van der Vaart and Wellner (1996), p. 83, or Definition 2.2 in van der Geer (2000), p. 16. Under Assumption $(\bf{A3})$ (parts (i) and (ii)) and Assumption $(\bf{A4})$, by the same arguments as before, the $L^2(P_0)$-distance between the lower and upper functions $b_{F^{L}_\alpha}$ and $b_{F^{U}_\alpha}$ of every bracket $[b_{F^{L}_\alpha},\,b_{F^{U}_\alpha}]$ can be bounded above as follows: $$\|b_{F^{U}_\alpha}-b_{F^{L}_\alpha}\|_{2,P_0}\lesssim \|F^{U}-F^{L}\|_1.$$ By 2.7.5 Theorem in van der Vaart and Wellner (1996), pp. 159–162, the bracketing entropy of the class of all uniformly bounded, monotone functions on the real line is of the order $O(1/\varepsilon)$. Therefore, $\log N_{[]}(\varepsilon,\,\mathscr{I},\,L^2(P_0))=O(1/\varepsilon)$ and the integral in is finite. The proof of Condition 4 is thus complete.
The conclusion of Theorem 2.1 follows: $$\psi(\hat F_n)-\psi(F_0)=\int b_{F_0}\,{\mathrm{d}}(\mathbb{P}_n-P_0) +o_{\mathbf{P}}(n^{-1/2})=
\mathbb{P}_n b_{F_0}+o_{\mathbf{P}}(n^{-1/2}),$$ where $\mathbb{P}_n b_{F_0}$ has expected value $P_0 b_{F_0}=0$, as it can be deduced from when $\alpha=0$. Hence, $$\sqrt{n}\big(\psi(\hat F_n)-\psi(F_0)\big)\,\xrightarrow{\mathscr{L}}\,\mathscr{N}(0,\,\|b_{F_0}\|_{2,P_0}^2)$$ and the proof is complete.
Appendix B {#appendix-b .unnumbered}
==========
In this section, we present the proof of Proposition \[prop:2\] which states that no integral linear functional of a non-degenerate mixing distribution in a convolution model with the *Laplace* kernel density is estimable at parametric rate, in particular, by the maximum likelihood method.
#### Proof of Proposition \[prop:2\]
We let, at the outset, $\psi(F_0)$ be any integral linear functional, as defined in , evaluated at the [“point”]{} $F_0$. Arguments are laid down to identify functions $a$ (if any) whose corresponding functionals are estimable at $n^{-1/2}$-rate. To the aim, we appeal to van der Vaart’s differentiability theorem, which provides a necessary and sufficient condition for pathwise differentiability of a (not necessarily linear) functional, see Theorem 3.1, Corollaries 3.2, 3.3 and Lemma 3.4 of van der Vaart (1991), pp. 183–185, or Theorem 3.1, Corollaries 3.1, 3.2 and Proposition 3.1 in Groeneboom and Wellner (1992), pp. 24–28. If differentiability of a functional fails, then, by Theorem 2.1 of van der Vaart (1991), p. 181, the functional is not estimable at $n^{-1/2}$-rate, see also chpt. 25 in van der Vaart (1998), pp. 358–432. A necessary and sufficient condition for differentiability of an integral linear functional $\psi(F_0)$ is that, for $\mathscr X=\mathbb R$, there exists a function $b:\,\mathscr X\rightarrow \mathbb R$, with $b\in L^2(P_0)$, satisfying $$\forall\,y\in\mathscr Y,\quad
\mathrm{E}[b(X)\mid Y=y]
=a(y)-\psi(F_0),$$ explicitly, $$\label{eq:differentiability}
\forall\,y\in\mathscr Y,\quad
\int_{\mathscr X}b(x)\frac{1}{2}e^{-|x-y|}\,{\mathrm{d}}x
=a(y)-\psi(F_0),$$ where the conditional density of $X$, given $Y=y$, is $k(x-y)=e^{-|x-y|}/2$, see §7 in van der Vaart (1991), pp. 189–191, or Example 3.2 in Groeneboom and Wellner (1992), pp. 30–31. If an integral linear functional $\psi(F_0)$ is regularly estimable, then the condition in must be necessarily satisfied and a regular estimator for $\psi(F_0)$ is given by $\mathbb P_nb=n^{-1}\sum_{i=1}^nb(X_i)$. The following arguments are aimed at deriving the expression of $b$. Let $y\in\mathscr Y$ be fixed. For a function $a:\,\mathscr Y\rightarrow \mathbb R$ such that $$\label{eq:456}
\lim_{u\rightarrow-\infty}a(u)e^u=0,$$ where, in the case when $\mathscr Y$ is bounded, $a$ (hence its derivative $\dot a$) is taken to be identically equal to zero on $\mathscr Y^c$ so that the limit is automatically verified, integration by parts yields that $$\begin{aligned}
\label{eq:def}
\hspace*{-0.6cm}
\int_{\mathscr X}
\dot a(x)\frac{1}{2}e^{-(y-x)}1{\{x\leq y\}}\,{\mathrm{d}}x&=&
a(x)\frac{1}{2}e^{-(y-x)}\bigg|_{-\infty}^y-\int_{\mathscr X}
a(x)\frac{1}{2}e^{-(y-x)}1{\{x\leq y\}}\,{\mathrm{d}}x\\
&=&\frac{1}{2}a(y)-\int_{\mathscr X}
a(x)\frac{1}{2}e^{-(y-x)}1{\{x\leq y\}}\,{\mathrm{d}}x,\nonumber\end{aligned}$$ whence $$\label{eq:1integral}
\int_{\mathscr X}[a(x)+\dot a(x)]\frac{1}{2}e^{-(y-x)}1{\{x\leq y\}}\,{\mathrm{d}}x=\frac{1}{2}a(y).$$ The integral analogous to the one on the left-hand side of , but with the right branch of the Laplace density, can be dealt with similarly. For some $a$ satisfying the limit in and also $$\label{eq:378}
\lim_{u\rightarrow +\infty}a(u)e^{-u}=0,$$ where the same proviso on $a$ and $\dot a$ applies for the case when $\mathscr Y$ is bounded, we get $$\begin{split}
\int_{\mathscr X}\dot a(x)\frac{1}{2}e^{-(x-y)}1{\{x>y\}}\,{\mathrm{d}}x&=
a(x)\frac{1}{2}e^{-(x-y)}\bigg|_{y}^{+\infty}+
\int_{\mathscr X}a(x)\frac{1}{2}e^{-(x-y)}1{\{x>y\}}\,{\mathrm{d}}x\\
&=-\frac{1}{2}a(y)+\int_{\mathscr X}
a(x)\frac{1}{2}e^{-(x-y)}1{\{x>y\}}\,{\mathrm{d}}x,
\end{split}$$ whence $$\label{eq:2integral}
\int_{\mathscr X}[a(x)-\dot a(x)]\frac{1}{2}e^{-(x-y)}1{\{x>y\}}\,{\mathrm{d}}x=\frac{1}{2}a(y).$$ Summing side by side and and subtracting $\psi(F_0)$ on both sides of the resulting equation, we obtain $$\int_{\mathscr X}
[a(x)-\mathrm{sgn}(x-y)\dot a(x)-\psi(F_0)]\frac{1}{2}e^{-|x-y|}\,{\mathrm{d}}x=a(y)-\psi(F_0).$$ In order to get rid of the dependence of the function $a(\cdot)-\mathrm{sgn}(\cdot-y)\dot a(\cdot)-\psi(F_0)$ on $y$, the derivative $\dot a(\cdot)$ must be equal to zero, which means that $a(\cdot)$ is identically equal to a constant on ${\mathscr Y}$ and the functional is trivially equal to the constant itself. Conclude that there exists no integral linear functional $\psi(F_0)$ of a non-degenerate mixing distribution $F_0$ that can be estimated at $n^{-1/2}$-rate. This completes the proof.
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[^1]: Catia Scricciolo\
Dipartimento di Scienze Economiche, Università degli Studi di Verona, Polo Universitario Santa Marta, Via Cantarane 24, I-37129 Verona (VR), ITALY,
[^2]: It is also known as the *first law of Laplace* to distinguish it from the *second law of Laplace*, as the normal distribution is sometimes called. It was named after Pierre-Simon Laplace (1749–1827) who, in 1774 (Laplace 1774), obtained $e^{-|z-\theta|}/2$, for $z,\,\theta\in\mathbb{R}$, as the density of the distribution whose likelihood is maximized when the location parameter $\theta$ is equal to the sample median.
[^3]: It is so called because it is formed by reflecting the exponential distribution around its mean.
[^4]: To see that $\sigma^2_Z=2s^2$, one can take into account that, if $V_1$ and $V_2$ are independent r.v.’s, identically distributed as an exponential with parameter $1/s$, in symbols, $V_j\sim \textrm{Exp}\,(1/s)$, $j=1,\,2$, then $V_1-V_2$ has a $\textrm{Laplace}\,(0,\,s)$ distribution. Consequently, $\sigma^2_Z=2\sigma^2_{V_1}=2(1/s)^{-2}=2s^2$.
[^5]: The symbol $F_Z$ denotes the c.d.f. of $Z$, that is, $F_Z(z)=\int_{-\infty}^z k(u)\,{\mathrm{d}}u$, with density $k$ as in .
[^6]: Uniqueness of $\hat F_n$ is not guaranteed, it is therefore with an abuse of language that we refer to *the* NPMLE throughout the article.
[^7]: For a recent reference list, see also Davidian *et al.* (2014).
[^8]: Following a common notational convention, we denote by $X_{r:n}$ the $r$th order statistic.
[^9]: The Dirac measure at $\theta$, denoted by $\delta_\theta(\cdot)$, is defined on the Borel sets $B\in\mathscr B(\mathbb R)$ by $\delta_\theta(B)=1,\,0$ if $B\ni\theta$ or $B\notni \theta$, respectively.
[^10]: The *sign-function* is defined as $\mathrm{sign}(x)=-1,\,0,\,1$ if $x<0$, $x=0$ or $x>0$, respectively.
[^11]: The right-derivative of $a$ at $y_{\mathrm{min}}$, denoted by $\dot a_+(y_{\mathrm{min}})$, is defined as the one-sided limit $\lim_{y\rightarrow y_{\mathrm{min}}+}
[a(y)-a(y_{\mathrm{min}})]/(y-y_{\mathrm{min}})$ if it exists as a real number.
[^12]: The left-derivative of $a$ at $y_{\mathrm{max}}$, denoted by $\dot a_-(y_{\mathrm{max}})$, is defined as the one-sided limit $\lim_{y\rightarrow y_{\mathrm{max}}-
}[a(y_{\mathrm{max}})-a(y)]/(y_{\mathrm{max}}-y)$ if it exists as a real number.
|
---
abstract: |
Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $\Delta_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their [*exponential generating flag function*]{}. These polytopes are simple, include both the simplex $\Delta_{n-1}$ and the permutahedron $\Pi_{n-1}$, and form a Minkowski basis for more general permutahedra.
[**2010 MSC:**]{} 05A15, 52B05, 52B11.
[**Keywords:**]{} polytope, permutahedron, Minkowski sum, flag polynomial, exponential flag function.
author:
- '[*Geir Agnarsson*]{} [^1]'
title: On a special class of general permutahedra
---
Introduction and motivation {#sec:intro}
===========================
The Minkowski sum of simplices yields an important class of polytopes that includes and generalizes many known polytopes. For related references and some of the history of the significance of Minkowski sums of simplices we refer to the introduction in [@Minkowski]. In [@MinkPoly] a closed formula for the $\ell$-flag polynomial (See Definition \[def:flag-poly\]) for an arbitrary Minkowski sum of $k$ simplices is derived. In particular, this yields a closed formula for the $f$-vectors of generalized associahedra from [@Postnikov]. This mentioned formula, however, is in terms of the [*master polytope*]{} $P(k)$, a $(2^k-2)$-dimensional polytope, the structure of which little is known about except when $k\leq 2$ [@Minkowski], [@MinkPoly]. In this paper we focus on the family of Minkowski sum of the simplices of a fixed dimension. These polytopes are interesting for a variety of reasons. We mention a few here without attempting to be exhaustive: (i) The polytopes in this family are all simple, and so they have a nice enumeration of their flags of arbitrary length as we will see shortly (see Lemma \[lmm:simple\]). (ii) This family forms a chain, or an incremental bridge, between the simplex of a given dimension and the standard permutahedron of the same dimension, where each step, or link, is between two such simple polytopes that differ minimally, as we will see in Section \[sec:our-fam\]. (iii) Each polytope in this family is symmetric with respect to permutation of their coordinates, like the simplex and the standard permutahedron. In fact, they make up a subclass of the class of generalized permutohedra studied in [@Postnikov] and [@Post-Vic-Laur], something we will discuss in more detail in Section \[sec:gen-perm\]. (iv) By contracting each face formed by vertices of identical positive support of any polytope of this family, one obtains a hypersimplex; a particular matroid base polytope (or matroid basis polytope) of the uniform matroid formed by all subsets of a fixed cardinality (the rank of the matroid) from a given ground set. In fact, each matroid base polytope of a matroid of a given rank is contained in a “mother”-hypersimplex, that is, its vertices are among the vertices of the “mother”-hypersimplex. The flags of matroid base polytopes have been studied in the literature, in particular in [@Sangwook-conf] and [@Sangwook-JCT], in which a characterization of the faces of the matroid base polytopes is presented. Also, a formula for the $\mathbf{c}\mathbf{d}$-index of rank-two matroid base polytope is presented, describing the number of their flags in the most compact way possible, from a linear relations perspective. (v) Last but not least, this family forms a Minkowski basis for certain generalized permutahedra of the form $P_{n-1}(\tilde{x})$ as defined and discussed in [@Postnikov p. 13] in terms of non-negative integer combination as Minkowski sums. Indeed, matroid base polytopes form a subclass of the family of generalized permutahedra as shown in [@Ard-Bene-Doker] where some of the work from [@Postnikov] is generalized, especially the volume of a general matroid base polytope. This will discussed in Section \[sec:gen-perm\].
The motivation for this paper stems from an observation on the enumeration of the flags of the standard permutahedron, presented in Proposition \[prp:exp-gen-func\] here below, which we now will parse through and discuss.
Recall that the [*permutahedron $\Pi_{n-1}$*]{} is defined to be the convex hull of $\{ (\pi(1), \pi(2),\ldots,\pi(n)) \in {{\mbox{$\mathbb R$}}}^n : \pi \in S_n\}$ where $S_n$ is the symmetric group of degree $n$. The faces of $\Pi_{n-1}$ have a nice combinatorial description as presented in Ziegler [@Ziegler p. 18]: each $i$-dimensional face of $\Pi_{n-1}$ can be presented as an ordered partition of the set $[n] = \{1,\ldots,n\}$ into exactly $n-i$ distinct parts. In particular, $\Pi_{n-1}$ has $\stirling{n}{n-i}(n-i)!$ faces of dimension $i$ for each $i\in\{0,1,\ldots,n\}$, where $\stirling{n}{k}$ denotes the Stirling number of the 2nd kind.
[Conventions:]{} (i) For an $\ell$-tuple $\tilde{x} = (x_1,\ldots,x_{\ell})$ of variables and an $\ell$-tuple of numbers $\tilde{a} = (a_1,\ldots,a_{\ell})$, let ${\tilde{x}}^{\tilde{a}} = x_1^{a_1}x_2^{a_2}\cdots x_{\ell}^{a_{\ell}}$. (ii) For $\tilde{a} = (a_1,\ldots,a_{\ell})$ let $\partial(\tilde{a}) = (a_1, a_2-a_1,a_3-a_2,\ldots,a_{\ell}-a_{\ell - 1})$. The following definition is from [@MinkPoly]:
\[def:flag-poly\] Let $P$ be a polytope with $\dim(P) = d$ and $\ell \in {\mbox{$\mathbb N$}}$. For an $\ell$-tuple of variables $\tilde{x} = (x_1,\ldots,x_{\ell})$ the [*$\ell$-flag polynomial*]{} of $P$ is defined by $${\tilde{f}}^{\ell}_{P}(\tilde{x}) :=
\sum_{\tilde{s}}f_{\tilde{s}}(P){\tilde{x}}^{\partial({\tilde{s}})},$$ where the sum is taken over all chains $\tilde{s} = (s_1,\ldots,s_{\ell})$ with $0\leq s_1\leq s_2\leq \cdots \leq s_{\ell}\leq d$ and $f_{\tilde{s}}(P)$ denotes the number of chains of faces $A_1\subseteq A_2\subseteq \cdots \subseteq A_{\ell}$ of $P$ with $\dim(A_i) = s_i$ for each $i\in\{1,\ldots, \ell\}$.
[Conventions:]{} (i) For a vector $\tilde{c} = (c_1,\ldots,c_n)$ we denote the linear functional $\tilde{x}\mapsto \tilde{c}\cdot\tilde{x}$ by $L_{\tilde{c}}$. (ii) For a given vector $\tilde{c}$ and a polytope $P$, we denote by $F_P(\tilde{c})$ or just $F(\tilde{c})$ the unique face of $P$ determined by $\tilde{c}$ as the points that maximize $L_{\tilde{c}}$ when restricted to $P$. Further, we denote the set of all the faces of $P$ by $\mathbf{F}(P)$. More specifically we denote the set of the $i$-dimensional faces of $P$ by ${\mathbf{F}}_i(P)$, in particular, ${\mathbf{F}}_0(P)$ denotes the set of vertices of $P$. (iii) Finally, for a vector $\tilde{c}$ the set ${{\mathop{\mathrm{supp}}\nolimits}}(\tilde{c}) = \{c_1,\ldots,c_n\}$ is the [*support*]{} of $\tilde{c}$.
Consider now the well-known description of the $i$-dimensional faces of $\Pi_{n-1}$ as the ordered partitions of $[n]$ into $n-i$ parts: more explicitly, each functional $L_{\tilde{c}}$ where the support ${{\mathop{\mathrm{supp}}\nolimits}}(\tilde{c})$ has exactly $n-i$ distinct values $c_1 < c_2 < \cdots < c_{n-i}$, when restricted to $\Pi_{n-1}$, takes its maximum value at exactly one $i$-dimensional face $A$. Here each value $c_i$ of the support corresponds uniquely to one of the ordered parts defining the face $A$. Also, by “merging” two such consecutive values $c_h$ and $c_{h+1}$ (for example, by replacing both $c_h$ and $c_{h+1}$ by their average), we obtain a new functional $L_{\tilde{c}'}$ which is maximized at a face $A'$ of dimension $i+1$ that contains the face $A$. So, by merging two consecutive parts into one part, we obtain a coarser ordered partition of $[n]$. This merging process can clearly be repeated. In this case we informally say that the first partition is a [*refinement*]{} of the last partition, or equivalently that the last partition is a [*coarsening*]{} of the first one.
\[obs:coarse\] For faces $A,B\in\mathbf{F}(\Pi_{n-1})$, then $A\subseteq B$ if, and only if, the ordered partition of $[n]$ corresponding to $A$ is a refinement of the ordered partition of $[n]$ corresponding to $B$.
Given a chain $\tilde{s} = (s_1,\ldots,s_{\ell})$ with $0\leq s_1\leq s_2\leq \cdots \leq s_{\ell}\leq n-1$, the number $f_{\tilde{s}}(\Pi_{n-1})$ of chains of faces $A_1\subseteq A_2\subseteq \cdots \subseteq A_{\ell}$ of $\Pi_{n-1}$ with $\dim(A_i) = s_i$ for each $i\in\{1,\ldots, \ell\}$ can then by Observation \[obs:coarse\] be obtained by first considering any of the $\stirling{n}{n-s_1}(n-s_1)!$ faces $A_1$ of dimension $s_1$, then merging $s_2-s_1$ consecutive parts (in the ordered partition defining the face $A_1$) of the $n-s_1-1$ available consecutive pairs, then merging $s_3-s_2$ consecutive parts of the $n-s_2-1$ available consecutive parts, and so on. Therefore the number $f_{\tilde{s}}(\Pi_{n-1})$ of chains of faces $A_1\subseteq A_2\subseteq \cdots \subseteq A_{\ell}$ where $\dim(A_i) = s_i$ for each $i$ is given by $$\begin{aligned}
f_{\tilde{s}}(\Pi_{n-1}) & = &
\stirling{n}{n-s_1}(n-s_1)!
\binom{n-s_1 -1}{s_2-s_1}\binom{n-s_2-1}{s_3-s_2}
\cdots\binom{n-s_{\ell-1}-1}{s_{\ell}-s_{\ell-1}} \nonumber \\
& = & \stirling{n}{n-s_1}(n-s_1)!
\binom{n-s_1 -1}{s_2-s_1 \ \cdots \ \ s_{\ell}-s_{\ell-1} \ \ n - s_{\ell}- 1}.
\label{eqn:chain-perm}\end{aligned}$$ Such a simple formula for the number of $\tilde{s}$-chains of faces of $\Pi_{n-1}$ as in (\[eqn:chain-perm\]) is not a coincidence, as it is solely the consequence of $\Pi_{n-1}$ being a simple polytope: each vertex of a simple $d$-polytope has $d$ neighboring vertices and is contained in $d$ facets, and so each $k$-face containing a given vertex is uniquely determined by $\binom{d}{k}$ of its neighbors. Hence, for each $h\leq k$ every $h$-face is contained in exactly $\binom{d-h}{k-h}$ $k$-faces, and we obtain in general, as above, the following.
\[lmm:simple\] For any simple $d$-polytope $P$ and a chain $\tilde{s} = (s_1,\ldots,s_{\ell})$ with $0\leq s_1\leq s_2\leq \cdots \leq s_{\ell}\leq d$, we have $$f_{\tilde{s}}(P) =
f_{s_1}(P)
\binom{d-s_1}{s_2-s_1 \ \cdots \ \ s_{\ell}-s_{\ell-1} \ \ d - s_{\ell}},$$ where $f_{s_1}(P)$ is the number of $s_1$-faces of $P$.
Now, assume for a moment that for a $d$-polytope $P$ we have a polynomial ${\tilde{\phi}}^{\ell}_{P}$ of the form $$\label{eqn:phi}
{\tilde{\phi}}^{\ell}_{P}(\tilde{x}) :=
\sum_{\tilde{s}}
\frac{f_{\tilde{s}}(P)}{D(d,s_1)}{\tilde{x}}^{\partial({\tilde{s}})},$$ where $D$ is a bivariate function on non-negative integers. If $D(d,s_1) = 1$ for all $d,s_1$ then ${\tilde{\phi}}^{\ell} = {\tilde{f}}^{\ell}$, the $\ell$-flag polynomial from Definition \[def:flag-poly\]. If $P$ is simple, then by the above Lemma \[lmm:simple\], the multinomial theorem and the definition of the $f$-polynomial ($f = {\tilde{f}}^{1}$ obtained by letting $\ell = 1$ in Definition \[def:flag-poly\]), we obtain the following: $$\begin{aligned}
{\tilde{\phi}}^{\ell}_{P}(\tilde{x})
& = & \sum_{\tilde{s}}\frac{f_{\tilde{s}}(P)}{D(d,s_1)}
{\tilde{x}}^{\partial({\tilde{s}})} \\
& = & \sum_{\tilde{s}}
\frac{f_{s_1}(P)\binom{d-s_1}{s_2-s_1 \ \cdots \ \ s_{\ell}-s_{\ell-1}
\ \ d - s_{\ell}}}{D(d,s_1)}
x_1^{s_1}x_2^{s_2-s_1}\cdots x_{\ell}^{s_{\ell}-s_{\ell-1}} \\
& = & \sum_{s_1}\frac{f_{s_1}(P)}{D(d,s_1)}x_1^{s_1}\sum_{s_2,\ldots,s_{\ell}}
\binom{d-s_1}{s_2-s_1 \ \cdots \ \ s_{\ell}-s_{\ell-1} \ \ d - s_{\ell}}
x_2^{s_2-s_1}\cdots x_{\ell}^{s_{\ell}-s_{\ell-1}} \\
& = & \sum_{s_1}\frac{f_{s_1}(P)}{D(d,s_1)}
x_1^{s_1}(x_2+\cdots+x_{\ell}+1)^{d-s_1} \\
& = & (x_2+\cdots+x_{\ell}+1)^d\sum_{s_1}
\frac{f_{s_1}(P)}{D(d,s_1)}
\left(\frac{x_1}{x_2+\cdots+x_{\ell}+1}\right)^{s_1} \\
& = & (x_2+\cdots+x_{\ell}+1)^d{\tilde{\phi}}^{1}_P
\left(\frac{x_1}{x_2+\cdots+x_{\ell}+1}\right),\end{aligned}$$ showing that ${\tilde{\phi}}^{\ell}_{P}$ is uniquely determined by ${\tilde{\phi}}^{1}_P$ if $P$ is a simple $d$ polytope.
\[cor:simple-flag\] For a simple $d$-polytope $P$ and ${\tilde{\phi}}^{\ell}_{P}$ from (\[eqn:phi\]) we have $${\tilde{\phi}}^{\ell}_{P}(\tilde{x}) =
(x_2 + \cdots + x_{\ell} + 1)^d
{\tilde{\phi}}^1_P\left(\frac{x_1}{x_2 + \cdots + x_{\ell} + 1}\right).$$
In particular, the $\ell$-flag polynomial for any simple polytope is uniquely determined by its $f$-polynomial $${\tilde{f}}^{\ell}_{P}(\tilde{x}) =
(x_2 + \cdots + x_{\ell} + 1)^d
f_P\left(\frac{x_1}{x_2 + \cdots + x_{\ell} + 1}\right).$$ [Remark:]{} Despite this enumerative bonus for simple (and dually for simplicial) polytopes, the number of the flags do not yield much of the actual face lattice structure of simple or simplicial polytopes.
Going back to our motivating permutahedron $\Pi_{n-1}$ and its number $f_i(\Pi_{n-1}) = \stirling{n}{n-i}(n-i)!$ of faces, we see that for $n\geq 1$ $$\sum_{i=0}^{n-1}\frac{f_i(\Pi_{n-1})}{(n-i)!}x^{n-i}
= \sum_{i=0}^{n-1}\stirling{n}{n-i}x^{n-i}
= T_n(x),$$ where $T_n(x)$ is the Touchard polynomial of degree $n$, a.k.a. the Bell polynomial in one variable of degree $n$, as $T_n(x) = B_n(x,\ldots,x)$ where $B_n(x_1,\ldots,x_n)$ is the [*complete Bell polynomial*]{} of degree $n$ in $n$ variables denoted by $\phi_n(x_1,\ldots,x_n)$ in [@Bell p. 263]), and we have the corresponding bivariate exponential generating function [@Bell p. 265] $$\label{eqn:Touch-St2}
T(x,y)
= \sum_{n\geq 0}T_n(x)\frac{y^n}{n!}
= \sum_{n,k \geq 0}\stirling{n}{k}x^k\frac{y^n}{n!} = e^{x(e^y-1)}.$$ This suggests an exponential version of the $\ell$-flag polynomial from Definition \[def:flag-poly\].
\[def:flag-poly-func\] Let $P$ be a $d$-polytope and $\ell\in{\mbox{$\mathbb N$}}$. For an $\ell$-tuple of variables $\tilde{x} = (x_1,\ldots,x_{\ell})$ define the [*exponential ${\ell}$-flag polynomial*]{} of $P$ by $${\tilde{\xi}}^{\ell}_{P}(\tilde{x}) :=
\sum_{\tilde{s}}
\frac{f_{\tilde{s}}(P)}{(d-s_1+1)!}{\tilde{x}}^{\partial({\tilde{s}})},$$ where the sum is taken over all chains $\tilde{s} = (s_1,\ldots,s_{\ell})$ with $0\leq s_1\leq s_2\leq \cdots \leq s_{\ell}\leq d$.
For each $a\geq 0$ define the [*exponential ${\ell}$-generating function*]{} of a given family of polytopes ${\cal{P}} = \{P_d\}_{d\geq 0}$, where each $P_d$ is of dimension $d$, by $${\tilde{\xi}}_{{\cal{P}};a}^{\ell}(\tilde{x},y) :=
\sum_{d\geq a}{\tilde{\xi}}^{\ell}_{P_d}(\tilde{x})\frac{y^{d+1}}{(d+1)!}.$$
In the case of $\ell=1$ we call ${\xi}_{P}(x) := {\tilde{\xi}}_{P}^{1}(x)$ the [*exponential face (or $f$-) polynomial*]{} of $P$ and for a family of polytopes ${\cal{P}} = \{P_d\}_{d\geq 0}$, each $P_d$ a $d$-polytope, we call ${\xi}_{{\cal{P}};a}(x,y) := {\tilde{\xi}}_{{\cal{P}};a}^{1}(x,y)$ the [*exponential face function*]{} of ${\cal{P}}$. When there is not ambiguity and both the family ${\cal{P}}$ and the starting point $a$ are clear, we omit the subscript in ${\xi}_{{\cal{P}};a}$ and simply write ${\xi}$.
For $a=0$ and ${\cal{P}} = \{\Pi_{n-1}\}_{n\geq 1}$ we get by (\[eqn:Touch-St2\]) $$\begin{aligned}
{\xi}(x,y) & = & \sum_{n\geq 1}{\xi}_{\Pi_{n-1}}(x)\frac{y^{n}}{n!} \\
& = & \sum_{n\geq 1}
\left(\sum_{i=0}^{n-1}\frac{f_i(\Pi_{n-1})}{(n-i)!}x^i\right)\frac{y^{n}}{n!} \\
& = & \sum_{n\geq 1}
\left(\sum_{i=0}^{n-1}\frac{f_i(\Pi_{n-1})}{(n-i)!}x^{-(n-i)}\right)
\frac{(xy)^{n}}{n!} \\
& = & \sum_{n\geq 1}T_n(x^{-1})\frac{(xy)^{n}}{n!} \\
& = & T(x^{-1},xy) - 1. \end{aligned}$$ So by Corollary \[cor:simple-flag\] applied to ${\tilde{\xi}}^{\ell}_{\Pi_{n-1}}$ we then get $${\tilde{\xi}}^{\ell}(\tilde{x},y)
= \frac{1}{x_2 + \cdots + x_{\ell} + 1}{\xi}
\left(\frac{x_1}{x_2 + \cdots + x_{\ell} + 1},
(x_2 + \cdots + x_{\ell} + 1)y\right)
= \frac{T\left(\frac{x_2 + \cdots + x_{\ell} + 1}{x_1},
x_1y\right)-1}{x_2 + \cdots + x_{\ell} + 1},$$ and again by (\[eqn:Touch-St2\]) we get the following proposition.
\[prp:exp-gen-func\] The exponential generating function for all the $\ell$-flags of all the permutahedra $\Pi_{n-1}$ for $n\geq 1$ from Definition \[def:flag-poly-func\] is given by $${\tilde{\xi}}^{\ell}(\tilde{x},y) =
\sum_{n\geq 1,\tilde{s}}{\tilde{\xi}}^{\ell}_{\Pi_{n-1}}(\tilde{x})\frac{y^n}{n!} =
\frac{e^{\frac{x_2 + \cdots + x_{\ell} +1}{x_1}(e^{x_1y}-1)} - 1}
{x_2 + \cdots + x_{\ell} +1}.$$
In particular, as the coefficient $[{\tilde{x}}^{\partial({\tilde{s}})}y^n]{\tilde{\xi}}^{\ell}(\tilde{x},y)$ of ${\tilde{x}}^{\partial({\tilde{s}})}y^n$ in the expansion of ${\tilde{\xi}}_e^{\ell}(\tilde{x},y)$ is given by $$[{\tilde{x}}^{\partial({\tilde{s}})}y^n]{\tilde{\xi}}_e^{\ell}(\tilde{x},y)=
\frac{f_{\tilde{s}}(\Pi_{n-1})}{(n-s_1)!n!},$$ then by Proposition \[prp:exp-gen-func\] we have $$f_{\tilde{s}}(\Pi_{n-1}) = (n-s_1)!n![{\tilde{x}}^{\partial({\tilde{s}})}y^n]
\left(
\frac{e^{\frac{x_2 + \cdots + x_{\ell} +1}{x_1}(e^{x_1y}-1)} - 1}
{x_2 + \cdots + x_{\ell} +1}
\right).$$ [Remarks:]{} (i) Needless to say, there are many ways to define [*an*]{} exponential generating function for the ${\ell}$-flags of the permutahedra; we chose one here that would yield nice formulae. (ii) Note that for $\ell= 1$ in Proposition \[prp:exp-gen-func\], the sum $x_2 + \cdots + x_{\ell}$ is empty which yields the exponential face function $${\xi}(x,y) = e^{\left(\frac{e^{xy}-1}{x}\right)} - 1.$$
Having presented our motivating example, a natural question arises whether formulae as in Proposition \[prp:exp-gen-func\] can be generalized to a larger family of polytopes that include the permutahedron $\Pi_{n-1}$. This will be the subject of the rest of the paper, which is organized as follows. In Section \[sec:our-fam\] we formally define the polytopes $\Pi_{n-1}(k-1)$ for each $k\geq 1$ and $n\geq k$ and we present some basic properties. In Section \[sec:gen-perm\] we describe how the [*PI-family*]{} ${\cal{P}}_n = \{\Pi_{n-1}(k-1)\}_{k = 2,\ldots,n}$ fits in with various other families that generalize the standard permutahedron and we demonstrate how ${\cal{P}}_n$ forms a Minkowski basis for one such family of polytopes. The remaining two sections form the meat of this paper. In Section \[sec:flag-poly\] we derive a formula for the $f$-polynomial of $\Pi_{n-1}(k-1)$ and describe its flags in terms of ordered pseudo-partitions of $[n] = \{1,\ldots,n\}$, in a similar way as we did in Observation \[obs:coarse\] for the standard permutahedron $\Pi_{n-1}$. Finally, in Section \[sec:exp-func\] we derive a closed formula for the exponential ${\ell}$-generating function ${\tilde{\xi}}_{{\cal{P}}_k^{\perp};k-1}^{\ell}(\tilde{x},y)$ for an arbitrary but fixed integer $k\geq 1$, where ${\cal{P}}_k^{\perp} = \{\Pi_{n-1}(k-1)\}_{n\geq k}$. Note that both families ${\cal{P}}_n$ and ${\cal{P}}_k^{\perp}$ cover all the polytopes $\Pi_{n-1}(k-1)$ when $n$ and $k$ roam respectively; $\bigcup_{n\geq 2}{\cal{P}}_n = \bigcup_{k\geq 2}{\cal{P}}_k^{\perp}$ are both partitions of the set of all $\Pi_{n-1}(k-1)$.
The PI-family of polytopes and basic properties {#sec:our-fam}
===============================================
In this section we define the PI-family of polytopes we investigate and present some basic properties that naturally generalize those of the permutahedron $\Pi_{n-1}$ and the simplex $\Delta_{n-1}$. First we recall some basic definitions and notations we will be using.
For $n\in{\mbox{$\mathbb N$}}$ and $[n] = \{1,2,\ldots,n\}$, the [*(standard) simplex*]{} $\Delta_{n-1} = \Delta_{[n]}$ of dimension $n-1$ is given by $
\Delta_{n-1} = \Delta_{[n]} =
\{ \tilde{x} = (x_1,\ldots,x_n)\in {{\mbox{$\mathbb R$}}}^n : x_i\geq 0 \mbox{ for all $i$ },
x_1+\cdots+x_n = 1\}.
$ Each subset $F\subseteq [n]$ yields a face $\Delta_F$ of $\Delta_{[n]}$ given by $
\Delta_F = \{ \tilde{x} \in \Delta_{[n]} : x_i = 0 \mbox{ for } i\not\in F\}.
$ Clearly $\Delta_F$ is itself a simplex embedded in ${{\mbox{$\mathbb R$}}}^n$. If ${\cal{F}}$ is a family of subsets of $[n]$, then we can form the [*Minkowski sum*]{} of simplices $$\label{eqn:Minks-basic}
P_{\cal{F}} = \sum_{F\in{\cal{F}}}\Delta_F =
\left\{\sum_{F\in{\cal{F}}}\tilde{x}_F :
\tilde{x}_F \in \Delta_F \mbox{ for each } F\in{\cal{F}}\right\}.$$ In general, every nonempty face of any polytope $P\subseteq {{\mbox{$\mathbb R$}}}^n$ (in particular of $\Delta_{[n]}$) is given by the set of points that maximize a linear functional $L_{\tilde{c}} : \tilde{x}\mapsto\tilde{c}\cdot\tilde{x}$ restricted to $P$. We note that the permutahedron $\Pi_{n-1}$ can be expressed as a [*zonotope*]{}, a Minkowski sum of simplices each of dimension one: $$\Pi_{n-1} = \sum_{F\subseteq [n],\ |F|=2}\Delta_F.$$ In light of this we obtain a natural generalization $$\label{eqn:family}
\Pi_{n-1}(k-1) := \sum_{F\subseteq [n],\ |F|=k}\Delta_F,$$ for each fixed $k\geq 2$, the Minkowski sum of all $(k-1)$-dimensional simplices in ${{\mbox{$\mathbb R$}}}^n$. We will refer to $\Pi_{n-1}(k-1)$ from (\[eqn:family\]) as the [*general permutahedron*]{}. Note that $\Pi_{n-1}(1) = \Pi_{n-1}$, the standard permutahedron, and $\Pi_{n-1}(n-1) = \Delta_{n-1}$, the standard $(n-1)$-dimensional simplex.
We now present some basic facts about Minkowski sums of polytopes in general that we will be using.
\[lmm:sum-faces\] Let $P_1,\ldots, P_k$ be polytopes in ${{\mbox{$\mathbb R$}}}^n$. Then $F\in \mathbf{F}\left(\sum_{i=1}^k P_i\right)$ iff (i) $F = \sum_{i=1}^k A_i$ where each $A_i\in\mathbf{F}(P_i)$ and (ii) there is a linear functional $L$ on ${{\mbox{$\mathbb R$}}}^n$ such that each $L|_{P_i}$ is maximized at $A_i$ and $L|_{\sum_{k=1}^k P_i}$ is maximized at $F$.
Clearly $F\in \mathbf{F}\left(\sum_{i=1}^k P_i\right)$ iff there is a linear functional $L$ on ${{\mbox{$\mathbb R$}}}^n$ such that $L|_{\sum_{i=1}^k P_i}$ is maximized at $F$. Let each $L|_{P_i}$ be maximized at $A_i\in \mathbf{F}(P_i)$. For $\sum_{i=1}^k x_i\in \sum_{i=1}^k P_i$ we then have $$L|_{\sum_{i=1}^k P_i}\left(\sum_{i=1}^k x_i\right)
= \sum_{i=1} L|_{P_i}(x_i)
\leq \sum_{i=1}^k\max(L|_{P_i}),$$ and equality holds iff $L|_{P_i}(x_i) = \max(L|_{P_i})$ for each $i$, which holds iff $x_i\in A_i$ for each $i$. By definition of $F$ and $L$ this implies that $F = \sum_{i=1}^k A_i$. The converse is easier.
Note that the proof implies that $\max(L|_{\sum_{i=1}^k P_i}) = \sum_{i=1}^k\max(L|_{P_i})$.
Assume now that $F = \sum_{i=1}^k A_i = \sum_{i=1}^k A_i'$ where $A_i ,A_i'\in \mathbf{F}(P_i)$ for each $i$, and where both tuples $(A_1,\ldots,A_k)$ and $(A_1',\ldots,A_k')$ are defined by the same functional $L$ and $L'$ respectively as in Lemma \[lmm:sum-faces\]. If $(A_1,\ldots,A_k) \neq (A_1',\ldots,A_k')$, then we may WLOG assume that $A_1'\setminus A_1 \neq \emptyset$. Then for $x_1'\in A_1'\setminus A_1$ and $x_i'\in A_i'$ for each $i\geq 2$ we have $$L|_{\sum_{i=1}^k P_i}\left(\sum_{i=1}^k x_i'\right)
= \sum_{i=1}^k L|_{P_i}(x_i')
< \max(L|_{P_1}) + \sum_{i=2}^k L|_{P_i}(x_i')
\leq \sum_{i=1}^k\max(L|_{P_i}).$$ contradicting the fact that $\sum_{i=1}^kx_i' \in\sum_{i=1}^k A_i$ since $L|_{\sum_{i=1}^k P_i}$ is maximized at $\sum_{i=1}^k A_i$. Hence, we must have $(A_1,\ldots,A_k) = (A_1',\ldots,A_k')$.
\[cor:AB=unique\] The decomposition of $F\in \mathbf{F}\left(\sum_{i=1}^k P_i\right)$ as $F = \sum_{i=1}^k A_i$ from Lemma \[lmm:sum-faces\] is unique.
Consider for a moment a functional $L_{\tilde{c}}$ where $c_1 > \cdots > c_n$. Clearly $L_{\tilde{c}}$ restricted to $\Pi_{n-1}(k-1)$ as defined in (\[eqn:family\]) will yield the vertex $(1,0,\ldots,0)$ of $\Delta_F$ from all the $\binom{n-1}{k-1}$ subsets $F\subseteq [n]$ with $1\in F$. Further, $L_{\tilde{c}}$ will yield the vertex $(0,1,0,\ldots,0)$ of $\Delta_F$ from all the $\binom{n-2}{k-1}$ subsets of $F\subseteq [n]$ with $1\not\in F$ and $2\in F$ and so on. Hence, by (\[eqn:family\]) and Lemma \[lmm:sum-faces\] we see that $L_{\tilde{c}}$ will yield the unique vertex $\left(\binom{n-1}{k-1},\binom{n-2}{k-2},\ldots,
\binom{k-1}{k-1},0,\ldots,0\right)$ of $\Pi_{n-1}(k-1)$. By considering all permutations on $n$ indices, we therefore have the following proposition.
\[prp:vert\] Every vertex of $\Pi_{n-1}(k-1)$ has form $\tilde{u} = (u_1,\ldots,u_n)$ where the support is given by $${{\mathop{\mathrm{supp}}\nolimits}}(\tilde{u}) = \{u_1,\ldots,u_n\}
= \left\{\binom{n-1}{k-1},\binom{n-2}{k-1},\ldots,
\binom{k-1}{k-1},0\right\}.$$ There are exactly $k-1$ copies of $0$ among $u_1,\ldots,u_n$ and hence exactly one copy of each nonzero integer from the above set. In particular $\Pi_{n-1}(k-1)$ has exactly $\frac{n!}{(k-1)!}$ vertices.
From Proposition \[prp:vert\] here above we see that $\Pi_{n-1}(k-1)$ is a degenerate case of the polytope $P_{n-1}(\tilde{v})$ from Postnikov [@Postnikov], defined as the convex hull of $\{(v_{\pi(1)},v_{\pi(2)},\ldots,v_{\pi(n)}) : \pi\in S_n\}$ for a fixed vector $\tilde{v}\in{{\mbox{$\mathbb R$}}}^n$. Namely, $\Pi_{n-1}(k-1) = P_{n-1}(\tilde{v})$ where $\tilde{v} = (v_1,\ldots,v_n) =
(\binom{n-1}{k-1},\binom{n-2}{k-1},\ldots,\binom{k-1}{k-1},0,\ldots,0)$. The combinatorial type of $P_n(\tilde{v})$, for any $\tilde{v}\in{{\mbox{$\mathbb R$}}}^n$ with $v_1 = v_2 = \cdots = v_{k-1} < v_k < \cdots < v_n$, is the same as that of $\Pi_{n-1}(k-1)$ (i.e. they have isomorphic face lattices) so, in particular, the combinatorial type of $P_n(\tilde{v})$ when all the $v_i$ are distinct, is the same as that of the standard permutahedron $\Pi_{n-1}$. In [@Postnikov] the volume $P_n(\tilde{v})$ is studied extensively, and it is shown to be a polynomial in the variables $v_1,\ldots,v_n$.
Note that every functional $L_{\tilde{c}}$ for which the coordinates $c_1,\ldots,c_n$ of $\tilde{c}$ are distinct will always yield a vertex of $\Pi_{n-1}(k-1)$, but not vice versa when $k\geq 3$. For such a $\tilde{c}$ we can, as right before Observation \[obs:coarse\], “merge” two consecutive values the support of $\tilde{c}$ (i.e. replace both values by their average, say) and thereby obtain the unique edge of $\Pi_{n-1}(k-1)$, the endvertices of which form the max-set of this altered $L_{\tilde{c}}$. Note that the edges of $\Pi_{n-1}(k-1)$ are of two types or kinds: 1st kind having $k-1$ zeros among the coordinates of each generic point on the edge (i.e. edge points excluding the endpoints), and the 2nd kind with $k-2$ zeros among the coordinates of each generic point on the edge. The number $e_1$ of edges of the 1st kind is the same as the number of ordered partition of a chosen $(n-k+1)$-subset of $[n]$ into $n-k$ parts, and hence $$e_1 = \binom{n}{n-k+1}\stirling{n-k+1}{n-k}(n-k)!
= \binom{n}{n-k+1}\binom{n-k+1}{2}(n-k)!
= \frac{(n-k)n!}{2(k-1)!}.$$ The number $e_2$ of edges of the 2nd kind is the same as the number ways to choose an $(n-k+2)$-subset of $[n]$, partition it into $n-k+1$ parts and order the $n-k$ singletons of those parts, and so $$e_2 = \binom{n}{n-k+2}\stirling{n-k+2}{n-k+1}(n-k)!
= \binom{n}{n-k+2}\binom{n-k+2}{2}(n-k)!
= \frac{n!}{2(k-2)!}.$$ Hence, the total number of edges is given by $e_1 + e_2 = \frac{(n-1)n!}{2(k-1)!}$. We summarize in the following.
\[prp:v-e-simple\] Every edge of $\Pi_{n-1}(k-1)$ is between a pair of vertices as given in Proposition \[prp:vert\] that differ in exactly two coordinates whose values are consecutive in the support of the vertices. Consequently the edges are of two types: (i) edges between two vertices, both with the same $k-1$ zero coordinates, and (ii) edges between two vertices, both with the same $k-2$ zero coordinates and one with its least nonzero entry where the other vertex has a zero. In particular, the number of edges of $\Pi_{n-1}(k-1)$ is $\frac{(n-1)n!}{2(k-1)!}$ and so $\Pi_{n-1}(k-1)$ is a simple polytope for all $k$ and $n$.
By the above Proposition \[prp:v-e-simple\] every $\Pi_{n-1}(k-1)$ is simple, so the sequence $$\Delta_{n-1} = \Pi_{n-1}(n-1), \Pi_{n-1}(n-2), \ldots, \Pi_{n-1}(2),
\Pi_{n-1}(1) = \Pi_{n-1},$$ can be viewed as discrete transition of simple polytopes from the simplex $\Delta_{n-1}$ to the standard permutahedron $\Pi_{n-1}$, see Figure \[fig:chain\]. This is our first main reason to focus our study on the PI-family consisting of $\Pi_{n-1}(k-1)$ where $k=2,\ldots,n-1$.
(Pi31) at (-4,0) [![A simple transition between $\Delta_3$ and $\Pi_3$ in the case $n=4$.[]{data-label="fig:chain"}](tetrahedron.jpg "fig:")]{}; at (-4,-2) [$\Pi_3(3)$]{}; (Pi32) at (0,0) [![A simple transition between $\Delta_3$ and $\Pi_3$ in the case $n=4$.[]{data-label="fig:chain"}](truncated.jpg "fig:")]{}; at (0,-2) [$\Pi_3(2)$]{}; (Pi33) at (4,0) [![A simple transition between $\Delta_3$ and $\Pi_3$ in the case $n=4$.[]{data-label="fig:chain"}](Pi-3.jpg "fig:")]{}; at (4,-2) [$\Pi_3(1)$]{};
Comparing various types of generalizations of permutahedra {#sec:gen-perm}
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In this section we further promote the importance of the PI-family ${\cal{P}}_n = \{\Pi_{n-1}(k-1)\}_{k = 2,\ldots,n}$ and briefly compare various families of polytopes from the literature, all generalizing the standard permutahedron in one form or another. We present some explicit characterizations of them and show that the PI-family ${\cal{P}}_n$ forms a Minkowski ${{\mbox{$\mathbb Z$}}}^{+}$-basis for a large family of polytopes that generalizes the standard permutahedron.
There are, needless to say, many ways to generalize the standard permutahedron $\Pi_{n-1}$, and we have briefly mentioned two of them (namely, $\Pi_{n-1}(k-1)$ and $P_{n-1}(\tilde{v})$ from above). There are two other classes of important families of polytopes from [@Postnikov] and from [@Post-Vic-Laur] we want to relate $\Pi_{n-1}(k-1)$ to. A good portion of the discussion immediately here below in this section, is, in one form or another, contained in [@Postnikov] and [@Rado], except for some minor observations, propositions, and examples toward the end of this section. We include it all in this short section though as it serves as a second main reason for our investigation, as well as for self-containment of the article.
[First Class:]{} For a collection $\tilde{Y} = \{y_I\}_{I\subseteq [n]}$ of non-negative real numbers $y_I\geq 0$ for each $I\subseteq [n]$, one can define a $P_{n-1}(\tilde{Y})$ as the Minkowski sum of simplices $\Delta_I$ scaled by $y_I$ $$P_{n-1}(\tilde{Y}) := \sum_{I\subseteq [n]}y_I\Delta_I,$$ and is referred to as a [*generalized permutohedron*]{} in [@Postnikov]. Apriori this seems to be more general than the Minkowski sum in (\[eqn:Minks-basic\]). However, if we consider a family ${\cal{F}}$ of subsets of $[n]$ containing (possibly) multiple copies of subsets of $[n]$, then $P_{\cal{F}}$ from (\[eqn:Minks-basic\]) can be written as $$P_{\cal{F}} = \sum_{I\subseteq [n]}n_I\Delta_I,$$ which can have the same combinatorial type as any $P_{n-1}(\tilde{Y})$. The class $P_{n-1}(\tilde{Y})$ (and hence also the class $P_{\cal{F}}$,) includes numerous classes of polytopes with highly interesting combinatorial structures, like the [*associahedron*]{}, the [*cyclohedron*]{}, etc. (see [@Postnikov] for many more examples.)
[Second Class:]{} For a collection $\tilde{Z} = \{z_I\}_{I\subseteq [n]}$ of non-negative real numbers $z_i\geq 0$ for each $I\subseteq [n]$, one can define $P_{n-1}(\tilde{Z})$ by its bounding hyperplanes $$P_{n-1}(\tilde{Z}) := \left\{ \tilde{x}\in{{\mbox{$\mathbb R$}}}^n :
\sum_{i\in[n]} x_i = z_{[n]}, \ \sum_{i\in I} x_i\leq z_I
\mbox{ for }I\subset [n]\right\},$$ which is also refereed to as the generalized permutohedron in [@Postnikov]. The following is a theorem of Rado [@Rado].
\[thm:Rado\] The polytope $P_{n-1}(\tilde{v})$ where the coordinates are ordered $v_1\geq \cdots\geq v_n$, can be presented as those $\tilde{t}\in{{\mbox{$\mathbb R$}}}^n$ satisfying $\sum_{i\in[n]}t_i = \sum_{i\in[n]}v_i$ and $\sum_{i\in I}t_i \leq \sum_{i\in[|I|]}v_i$ for each $I\subseteq [n]$.
From Theorem \[thm:Rado\] we see that if $z_I = z_J$ whenever $|I| = |J|$, then there are uniquely determined $v_1,\ldots,v_n$ such that $P_{n-1}(\tilde{Z}) = P_{n-1}(\tilde{v})$. Therefore, the class of $P_{n-1}(\tilde{Z})$ polytopes strictly includes all polytopes $P_{n-1}(\tilde{v})$.
Further, the following is from [@Postnikov].
\[prp:Postnikov\] For a given collection $\tilde{Y} = \{y_I\}_{I\subseteq [n]}$ of non-negative real numbers $y_I\geq 0$, then $P_{n-1}(\tilde{Y}) = P_{n-1}(\tilde{Z})$ where $z_I = \sum_{J\subseteq I}y_J$ for each $I\subseteq [n]$.
Hence, the the class of $P_{n-1}(\tilde{Z})$ polytopes also includes the class of all $P_{n-1}(\tilde{Y})$ polytopes. By Observation \[obs:Z=Y\] here below, this mentioned inclusion is also strict.
Suppose it is known that $P_{n-1}(\tilde{Z}) = P_{n-1}(\tilde{Y})$ for some $\tilde{Y} = \{y_I\}_{I\subseteq [n]}$. Then we may assume that $z_I = \sum_{J\subseteq I}y_J$ for each $I\subseteq [n]$. By a Möbius inversion we then get $y_I = \sum_{J\subseteq I}(-1)^{|I|-|J|}z_J$ for each $I\subseteq [n]$, so the $y_I$ are uniquely determined in terms of the $z_I$. Hence we have the following.
\[obs:Z=Y\] For $n\in{\mbox{$\mathbb N$}}$ and a collection $\tilde{Z} = \{z_I\}_{I\subseteq [n]}$ of non-negative real numbers, then $P_{n-1}(\tilde{Z}) = P_{n-1}(\tilde{Y})$ if and only if $y_I = \sum_{J\subseteq I}(-1)^{|I|-|J|}z_J\geq 0$ for each $I\subseteq [n]$.
For $n\in{\mbox{$\mathbb N$}}$ we say that a collection $\tilde{T} = \{t_I\}_{I\subseteq[n]}$ is [*symmetric*]{} if $t_I = t_J$ whenever $|I|=|J|$. Hence, from Proposition \[prp:Postnikov\] and the above Observation \[obs:Z=Y\] we have the following.
\[obs:symmetric\] For any $n\in{\mbox{$\mathbb N$}}$ we have that $\tilde{Y} = \{y_I\}_{I\subseteq [n]}$ is symmetric if and only if $\tilde{Z} = \{z_I\}_{I\subseteq [n]}$ where $z_I = \sum_{J\subseteq I}y_J$ is symmetric. Further, if $\tilde{Y}$ is symmetric then $$P_{n-1}(\tilde{Y}) = \sum_{k=1}^ny_k\Pi_{n-1}(k-1)$$ for non-negative real numbers $y_1,\ldots,y_k$, where we interpret $\Pi_{n-1}(0)$ as a singleton point.
We now describe those permutahedra $P_{n-1}(\tilde{v})$ that can be written as $P_{n-1}(\tilde{Y})$ for some $\tilde{y} = \{y_I\}_{I\subseteq [n]}$.
For a sequence $(a_n)_{n\geq 0}$ of real numbers, recall the [*(backward) difference*]{} given by $\Delta(a_n) = a_n - a_{n-1}$ for each $n\geq 1$[^2]. Iteratively we also have the [*$i$-th order difference*]{} by $\Delta^i(a_n) = \Delta(\Delta^{i-1}(a_n))$ for each $i\geq 0$ and where $\Delta^0(a_n) = a_n$ for each $n$. Likewise, for an $n$-tuple $\tilde{a} = (a_1,\ldots, a_n)\in {{\mbox{$\mathbb R$}}}^n$ we let $\Delta(\tilde{a}) = (\Delta(a_2),\ldots,\Delta(a_n)) =
(a_2-a_1, \ldots, a_n-a_{n-1})\in {{\mbox{$\mathbb R$}}}^{n-1}$. Clearly, if $P_{n-1}(\tilde{v})$ can be written of the form $P_{n-1}(\tilde{Y})$ for some $\tilde{Y}$, then by Theorem \[thm:Rado\] and Observation \[obs:Z=Y\] we can assume $\tilde{Y}$ to be symmetric and so $$P_{n-1}(\tilde{v})
= \sum_{I\subseteq [n]}y_{|I|}\Delta_I
= y_1\tilde{1} + \sum_{k=2}^{n}y_k\Pi_{n-1}(k-1),$$ where $\tilde{1} = (1,1,\ldots,1)\in {{\mbox{$\mathbb R$}}}^n$. The following proposition is our main conclusion of this section, the proof of which follows thereafter.
\[prp:Y=X\] For $n\in{\mbox{$\mathbb N$}}$ and $\tilde{v}\in {{\mbox{$\mathbb R$}}}^n$ with $v_1\leq\cdots\leq v_n$, then $P_{n-1}(\tilde{v}) = \sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ for non-negative $y_k\geq 0$ for each $k$, if and only if all the differences of $\tilde{v}$ are non-negative, that is, $\Delta^i(v_k)\geq 0$ for each $i\in\{0,1\ldots,n-1\}$ and $k\in\{i+1,\ldots,n\}$.
[Remark:]{} If $\Delta^i(v_k)\geq 0$ for each $i$ and $k$, then coefficients $y_k$ with $P_{n-1}(\tilde{v}) = \sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ are uniquely determined by $\tilde{v}$. Hence, $\{\Pi_{n-1}(k-1) : k\in \{1,\ldots,n\}\} = {\cal{P}}_n\cup\{\tilde{1}\}$ forms a [*Minkowski basis*]{} for such permutahedra $P_{n-1}(\tilde{v})$.
We will prove Proposition \[prp:Y=X\] in a few small steps. First a lemma in linear algebra.
\[lmm:key\] For $n\in{\mbox{$\mathbb N$}}$ and $k\in\{1,2,\ldots, n\}$ let $$\tilde{v}_{n-1}(k-1)
= \left(0,\ldots,0,\binom{k-1}{k-1},\ldots,\binom{n-1}{k-1}\right).$$ Then $\tilde{u}\in{{\mathop{\mathrm{span}}\nolimits}}_{{{\mbox{$\mathbb R$}}}^{+}}(\{\tilde{v}_{n-1}(k-1) :
k\in\{1,\ldots,n\}\})$ iff all the differences of $\tilde{u}$ are non-negative.
The proof of the above Lemma \[lmm:key\] will use the following trivial fact.
\[clm:trivial\] For any $n\in{\mbox{$\mathbb N$}}$ and $\tilde{v}\in{{\mbox{$\mathbb R$}}}^n$ then $\Delta(\tilde{v}) = \tilde{0}\Leftrightarrow
\tilde{v} = v\tilde{1} = (v,v,\ldots,v)\in {{\mbox{$\mathbb R$}}}^{n-1}$.
(Lemma \[lmm:key\]) Note that $\Delta(\tilde{v}_{n-1}(k-1)) = \tilde{v}_{n-2}(k-2)$, so by induction on $n$, all the differences of $\tilde{v}_{n-1}(k-1)$ are non-negative. Since the difference operator $\Delta$ is linear, then all the differences of $\sum_{k=1}^{n}y_k\tilde{v}_{n-1}(k-1)$ are non-negative if each $y_k\geq 0$. Therefore if $\tilde{u}\in{{\mathop{\mathrm{span}}\nolimits}}_{{{\mbox{$\mathbb R$}}}^{+}}(\{\tilde{v}_{n-1}(k-1) :
k\in\{1,\ldots,n\}\})$, then it is necessary for all the differences of $\tilde{u}$ to be non-negative.
Conversely, let $\tilde{u}\in{{\mbox{$\mathbb R$}}}^n$ have all its differences non-negative. If $n=1$ then clearly $\tilde{u} = u_1 \in{{\mathop{\mathrm{span}}\nolimits}}_{{{\mbox{$\mathbb R$}}}^{+}}(\{\tilde{v}_{0}(0)\})$. Otherwise all the differences of $\Delta(\tilde{u})$ are non-negative, and hence by induction on $n$ we can assume that $\Delta(\tilde{u}) = \sum_{k=2}^{n}y_{k}\tilde{v}_{n-2}(k-1)$ for some non-negative $y_2,\ldots, y_{n}$, and so $$\Delta(\tilde{u})
= \sum_{k=2}^{n}y_{k}\Delta(\tilde{v}_{n-1}(k))
= \Delta\left(\sum_{k=2}^{n}y_k\tilde{v}_{n-1}(k-1)\right).$$ By Claim \[clm:trivial\] we have $\tilde{u} - \sum_{k=2}^{n}y_k\tilde{v}_{n-1}(k) = y_1\tilde{1}$ for some real $y_1$. Since all differences of $\tilde{u}$ are non-negative, in particular $\Delta^0(\tilde{u}) = \tilde{u}$, we have $y_1 = u_1\geq 0$ and hence $\tilde{u}\in{{\mathop{\mathrm{span}}\nolimits}}_{{{\mbox{$\mathbb R$}}}^{+}}(\{\tilde{v}_{n-1}(k-1) : k\in\{1,\ldots,n\}\})$.
We now have what we need to prove Proposition \[prp:Y=X\].
(Proposition \[prp:Y=X\]) We first note that if $\tilde{v}$ is as in Proposition \[prp:Y=X\], that is $P_{n-1}(\tilde{v}) = \sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ where $y_k\geq 0$ for each $k$, then $L_{\tilde{c}}$, where $\tilde{c} = (1,2,\ldots,n)\in{{\mbox{$\mathbb R$}}}^n$, is maximized at $\tilde{v}$, when restricted to $P_{n-1}(\tilde{v})$, and is maximized at $\tilde{v}_{n-1}(k-1)$ when restricted to $\Pi_{n-1}(k-1)$ for each $k$. Hence, when restricted to $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ then $L_{\tilde{c}}$ is maximized at $\sum_{k=1}^{n}y_k\tilde{v}_{n-1}(k-1)$, and so $\tilde{v} = \sum_{k=1}^{n}y_k\tilde{v}_{n-1}(k-1)$. By Lemma \[lmm:key\], all the differences of $\tilde{v}$ must then be non-negative.
For the converse, if all the difference of $\tilde{v}$ are non-negative, then by Lemma \[lmm:key\] there are non-negative real coefficients $y_k\geq 0$ such that $\tilde{v} =\sum_{k=1}^{n}y_k\tilde{v}_{n-1}(k-1)$, at which $L_{\tilde{c}}$ where $\tilde{c} = (1,2,\ldots,n)\in{{\mbox{$\mathbb R$}}}^n$ when restricted to both $P_{n-1}(\tilde{v})$ and $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ is maximized at. Similarly, for any permutation $\pi$ of $\{1,\ldots,n\}$, the linear functional $L_{\pi(\tilde{c})}$ where $\pi(\tilde{c}) = (\pi(1),\ldots,\pi(n))$ when restricted to both $P_{n-1}(\tilde{v})$ and $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ is maximized at $$\pi(\tilde{v}) = (v_{\pi(1)},\ldots,v_{\pi(n)}) =
\sum_{k=1}^{n}y_k\pi(\tilde{v}_{n-1}(k-1)).$$ By definition of $P_{n-1}(\tilde{v})$ we therefore see that every vertex of $P_{n-1}(\tilde{v})$ is also a vertex of $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$. But since every vertex of $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ has by Corollary \[cor:AB=unique\] the unique form $\sum_{k=1}^{n}y_k\tilde{w}_k$ where each $\tilde{w}_k$ is a vertex of $\Pi_{n-1}(k-1)$, and each $\tilde{w}_k$ is the maximum set of the functional $L_{\pi(\tilde{c})}$ when restricted to $\Pi_{n-1}(k-1)$, then [*every*]{} vertex of $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ is indeed the maximum set of some $L_{\pi(\tilde{c})}$. Hence, the the polytopes $P_{n-1}(\tilde{v})$ and $\sum_{k=1}^{n}y_k\Pi_{n-1}(k-1)$ have the same set of vertices, and so must be the same polytope. This completes the proof.
[Example:]{} Consider the polytope $P_3(0,1,2,2)$, and assume it can be written as $P_3(\tilde{Y})$ for some $\tilde{Y} = \{y_I\}_{I\subseteq [4]}$. By Theorem \[thm:Rado\] and Observation \[obs:Z=Y\] we can assume $\tilde{Y}$ to be symmetric and so $P_3(0,1,2,2) = \sum_{k=1}^4 y_k\Pi_3(k-1)$. Looking at the differences of $(0,1,2,2)$ we get $$\left[
\begin{array}{c}
\Delta^0 \\
\Delta^1 \\
\Delta^2 \\
\Delta^3
\end{array}
\right]
=
\left[
\begin{array}{ccccccc}
0 & & 1 & & 2 & & 2 \\
& 1 & & 1 & & 0 & \\
& & 0 & &-1 & & \\
& & &-1 & & &
\end{array}
\right]$$ containing two negative entries in the differences of $\tilde{v} =
(0,1,2,2)$. By Proposition \[prp:Y=X\] $P_3(0,1,2,2)$ cannot be written in the form of $P_3(0,1,2,2) = \sum_{k=1}^4 y_k\Pi_3(k-1)$. However, $P_3(0,1,2,2)$ is still a symmetric polytope and has dimension $3$ by Lemma \[lmm:neq0\] here below.
By Proposition \[prp:Y=X\] we have the following.
\[cor:Mink-basis\] The PI-family ${\cal{P}}_n = \{\Pi_{n-1}(k-1)\}_{k=2,\ldots,n}$ forms a Minkowski ${{\mbox{$\mathbb Z$}}}^{+}$-basis for those polytopes $P_{n-1}(\tilde{v})$ that are of the form $P_{n-1}(\tilde{Y})$ for some family $\tilde{Y} = \{y_I\}_{I\subseteq [n]}$ of non-negative real numbers.
The flag polynomial of the general permutahedron {#sec:flag-poly}
================================================
Having briefly compared three types of polytopes, $P_{n-1}(\tilde{v})$, $P_{n-1}(\tilde{Y})$, and $P_{n-1}(\tilde{Z})$, each of which can be viewed as generalizations of the standard permutahedron, we see that the polytopes $P_{n-1}(k-1)$ for $k\in\{1,\ldots,n\}$ form a Minkowski basis for those polytopes $P_{n-1}(\tilde{v})$ that can be expressed as $P_{n-1}(\tilde{Y})$. Hence, this can be viewed as a further justification for studying them, and so we will in this section focus on the PI-family ${\cal{P}}_n = \{\Pi_{n-1}(k-1)\}_{k=2,\ldots,n}$ for a given $n\in{\mbox{$\mathbb N$}}$. We will discuss the face lattice and its flag polynomial. Since many formal statements are the same for $\Pi_{n-1}(k-1)$ as with the more general $P_{n-1}(\tilde{v})$ and are, in fact, more transparent, we will consider the polytope $P_{n-1}(\tilde{v})$ in many cases, and then derive corollaries about $\Pi_{n-1}(k-1)$.
First, we will derive some facts from linear algebra that will come in handy in this section.
Consider two points $\tilde{a},\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ where neither of them has all its coordinates the same. In this case there is a proper partition $A\cup B = [n]$ such that $c_i > c_j$ for all $(i,j)\in A\times B$. As neither $A$ nor $B$ is empty, we cannot have $a_i= a_j$ for all $(i,j)\in A\times B$, since that would imply $a_i=a_j$ for all $i,j\in[n]$. Hence, there is an $(i,j)\in A\times B$ with $a_i\neq a_j$. If $\tau = (i,j) \in S_n$ then $$L_{\tilde{c}}(\tilde{a}) - L_{\tilde{c}}(\tau(\tilde{a})) =
c_ia_i + c_ja_j - (c_ia_j + c_ja_i) = (c_i-c_j)(a_i -a_j) \neq 0.$$ Hence, we have the following.
\[lmm:neq0\] Let $\tilde{a}, \tilde{c}\in {{\mbox{$\mathbb R$}}}^n$, neither of which have all its coordinates the same. Then there is a transposition $\tau\in S_n$ such that $L_{\tilde{c}}(\tau(\tilde{a}))\neq L_{\tilde{c}}(\tilde{a})$.
In particular, for $\tilde{a}$ and $\tilde{c}$ as in Lemma \[lmm:neq0\], $P_{n-1}(\tilde{a})\not\subseteq\ker(L_{\tilde{c}})$ and so $\dim(P_{n-1}(\tilde{a})) = n-1$. Now, since $\Pi_{n-1}(k-1) = P_{n-1}(\tilde{v}_{n-1}(k-1))$, where $\tilde{v}_{n-1}(k-1))
= \left(0,\ldots,0,\binom{k-1}{k-1},\ldots,\binom{n-1}{k-1}\right)$ is as in Lemma \[lmm:key\], we then have the following.
\[cor:dim\] Let $\tilde{v}\in {{\mbox{$\mathbb R$}}}^n$. Then $$\dim(P_{n-1}(\tilde{v})) = \left\{
\begin{array}{ll}
0 & \mbox{ if } v_1 = \cdots = v_n, \\
n-1 & \mbox{ otherwise. }
\end{array}
\right.$$ In particular $\dim(\Pi_{n-1}(k-1)) = n-1$ for every $k\in\{2,\ldots,n\}$.
We now generalize Corollary \[cor:dim\] slightly.
As the symmetric group $S_n$ denotes the group of bijections $[n]\rightarrow [n]$, we can adopt the notation $S(X)$ for the group of bijections $X\rightarrow X$, where $X$ is a given set. With this convention $S_n = S([n])$ and clearly $S(X)\cong S_{|X|}$ for any finite set $X$. For any collection $X_1,\ldots, X_k$ of disjoint subsets of $[n]$ we then have the internal product $S(X_1)S(X_2)\cdots S(X_k)$, a subgroup of $S([n])$ which is isomorphic to the direct product $S_{n_1}\times S_{n_2}\times\cdots \times S_{n_k}$ where $|X_i| = n_i$. For a vector $\tilde{v}\in{{\mbox{$\mathbb R$}}}^n$ and a subset $X$ of $[n]$ we let ${{\mathop{\mathrm{Proj}}\nolimits}}_{X} : {{\mbox{$\mathbb R$}}}^n \rightarrow {{\mbox{$\mathbb R$}}}^{|X|}$ denote the projection onto all the coordinate in $X$. If $X = \{i\}$ is a singleton set, we let ${{\mathop{\mathrm{Proj}}\nolimits}}_i = {{\mathop{\mathrm{Proj}}\nolimits}}_{\{i\}}$ be the projection onto the $i$-th coordinate. Further we let $\delta_X(\tilde{v})$ denote the indicator function $$\label{eqn:indicator}
\delta_X(\tilde{v}) = \left\{
\begin{array}{ll}
0 & \mbox{ if } |{{\mathop{\mathrm{supp}}\nolimits}}({{\mathop{\mathrm{Proj}}\nolimits}}_X(\tilde{v}))| = 1, \\
1 & \mbox{ otherwise }.
\end{array}
\right.$$ We now have by Corollary \[cor:dim\] the following more general statement.
\[prp:dim-gen\] For disjoint subsets $X_1,\ldots,X_h$ of $[n]$ and $\tilde{v}\in {{\mbox{$\mathbb R$}}}^n$ we have $$\dim({{\mathop{\mathrm{conv}}\nolimits}}(\{\pi(\tilde{v}) : \pi\in S(X_1)\cdots S(X_h)\}))
= \sum_{i=1}^h \delta_{X_i}(\tilde{v})(|X_i| - 1).$$
Note that the above Proposition \[prp:dim-gen\] holds in particular for every partition $X_1\cup\cdots\cup X_h = [n]$ of $[n]$.
We seek to describe the face lattice of the polytope $P_{n-1}(\tilde{v})$ where $\tilde{v}$ has non-negative real entries, in a similar fashion as was done when describing the faces of the standard permutahedron $\Pi_{n-1}$ earlier, namely by considering the max set of a linear functional restricted to the polytope. In [@Post-Vic-Laur] the combinatorial structure of classes of polytopes that include those of $P_{n-1}(\tilde{v})$ is studied in great depth. In particular, the $f$-, $h$- and $\gamma$-vectors of these classes of polytopes are studied. Many explicit formulae for the $h$- and $\gamma$-vectors involving descent statistics of permutations are given. Here we take a different OR-like (operations research) approach, involving linear functionals, that more directly relates to the characterization of the faces as presented in Observation \[obs:coarse\].
We say that two vectors $\tilde{a}$ and $\tilde{c}$ have the same [*order type*]{} if $a_i\leq a_j \Leftrightarrow c_i\leq c_j$ for all $i,j\in\{1,\ldots,n\}$. The order type defines an equivalence relation among vectors $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$, and clearly all vectors of the same type yield the same face of $P_{n-1}(\tilde{v})$, as the set of maximum points of $L_{\tilde{c}}$ when restricted to $P_{n-1}(\tilde{v})$. Denote by $[\tilde{c}]$ the order type equivalence class of the vector $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$. So, if $F(\tilde{c})$ denotes the unique face as the set of maximum points of $L_{\tilde{c}}$ restricted to $P_{n-1}(\tilde{v})$, then, by the above, $F(\tilde{c}) = F(\tilde{c}')$ whenever $[\tilde{c}] = [\tilde{c}']$, and hence the face $F([\tilde{c}])$ is well defined. Also note that $P_{n-1}(\tilde{v})$ and $P_{n-1}(\tilde{v}')$ have the same combinatorial type iff $[\tilde{v}] = [\tilde{v}']$. Finally, if $\delta_X(\tilde{c})$ denote the indicator function from (\[eqn:indicator\]), then clearly $\delta_X([\tilde{c}])$ is well defined.
As real addition is commutative, then for any permutation $\pi\in S_n$ we have $$L_{\pi(\tilde{c})}(\pi(\tilde{x}))
= \pi(\tilde{c})\cdot\pi(\tilde{x})
= \tilde{c}\cdot\tilde{x}
= L_{\tilde{c}}(\tilde{x}).$$ Hence, if let $\pi(F) = \{\pi(\tilde{x}) : \tilde{x}\in F\}$, then clearly we have the following.
\[obs:perm-face\] For any permutation $\pi\in S_n$ we have $\pi(F([\tilde{a}])) = F([\pi(\tilde{a})])$, and, in particular, $P_{n-1}(\tilde{v}) = P_{n-1}(\pi(\tilde{v}))$.
Consider the polytope $P_{n-1}(\tilde{v})$ for a given vector $\tilde{v}$ with non-negative real entries. To describe the face $F(\tilde{c})$ of $P_{n-1}(\tilde{v})$, we first note that $\tilde{c}$ yields a unique ordered partition of $[n]$ $$\label{eqn:part}
[n] = X_1(\tilde{c})\cup\cdots\cup X_h(\tilde{c}),$$ where $c_i=c_j$ for all $i,j\in X_{\ell}(\tilde{c})$ and $c_i < c_j$ if $i\in X_{\ell}(\tilde{c})$ and $j\in X_{\ell'}(\tilde{c})$ where $\ell < \ell'$. Note that $L_{\tilde{c}}$ restricted to the set of vertices ${\mathbf{F}}_0(P_{n-1}(\tilde{v}))$ takes its maximum value on those vertices $\tilde{u}$, the order of whose entries are in agreement with the order of the entries of $\tilde{c}$, that is $c_i < c_j\Rightarrow u_i\leq u_j$. This is clearly a necessary and sufficient condition $\tilde{u}\in {\mathbf{F}}_0(P_{n-1}(\tilde{v}))$ must satisfy in order for $L_{\tilde{c}}(\tilde{u})$ to be a maximum value of $L_{\tilde{c}}$ when restricted to $P_{n-1}(\tilde{v})$. Formally we have a following description.
\[obs:c-face\] For a given $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ the face of $P_{n-1}(\tilde{v})$ determined by $[\tilde{c}]$ is given by $$\begin{aligned}
F([\tilde{c}])
& = &
{{\mathop{\mathrm{conv}}\nolimits}}(\{\tilde{u}\in {\mathbf{F}}_0(P_{n-1}(\tilde{v})) :
c_i < c_j\Rightarrow u_i\leq u_j \}) \\
& = &
{{\mathop{\mathrm{conv}}\nolimits}}(\{\tilde{u}\in {\mathbf{F}}_0(P_{n-1}(\tilde{v})) :
u_i < u_j\Rightarrow c_i\leq c_j \}).\end{aligned}$$
Clearly by Observation \[obs:perm-face\], we can assume $\tilde{v}$ to be ordered in any way convenient for our purposes. In particular, when describing the face $F([\tilde{c}])$ of $P_{n-1}(\tilde{v})$, we can for simplicity assume that the order of $\tilde{v}$ agrees with that of $\tilde{c}$, so $v_i\leq v_j$ whenever $c_i < c_j$, that is we can assume $\tilde{v} \in F(\tilde{c})$ by Observation \[obs:c-face\]. In terms of the partition from (\[eqn:part\]), we then obtain another equivalent form by Proposition \[prp:dim-gen\].
\[prp:face-describe\] For a given $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ the face of $P_{n-1}(\tilde{v})$ determined by $[\tilde{c}]$ that contains $\tilde{v}$ is given by $$F([\tilde{c}]) =
{{\mathop{\mathrm{conv}}\nolimits}}(\{
\mu_1\cdots\mu_h(\tilde{v}) : \mu_i\in S(X_i(\tilde{c}))\subseteq S([n]), \ \
i\in \{1,\ldots,h\}
\}).$$ In particular we have $$\dim(F([\tilde{c}])) =
\sum_{i=1}^h \delta_{X_i(\tilde{c})}(\tilde{v})(|X_i(\tilde{c})| - 1).$$
We note that if $\tilde{c}$ and $\tilde{v}$ are both ordered, $c_1\leq\cdots\leq c_n$ and $v_1\leq\cdots\leq v_n$, and $\delta_{X_{\ell}(\tilde{c})\cup X_{\ell + 1}(\tilde{c})}(\tilde{v}) = 0$ for some $\ell$, then we can replace each $c_i$ where $i\in X_{\ell}(\tilde{c})$ and $c_j$ where $j\in X_{\ell+1}(\tilde{c})$ with a single value between $c_i$ and $c_j$, say $(c_i+c_j)/2$, and thereby obtaining a vector $\tilde{c}'$ with a strictly smaller support than $\tilde{c}$ such that $F([\tilde{c}]) = F([\tilde{c}'])$. In this case we have merged the two consecutive intervals $X_{\ell}(\tilde{c})$ and $X_{\ell +1}(\tilde{c})$ into one interval without altering the corresponding face of $P_{n-1}(\tilde{v})$ that these vectors determine.
\[def:v-reduced\] For ordered vectors $\tilde{c},\tilde{v}\in {{\mbox{$\mathbb R$}}}^n$ we say that $\tilde{c}$ is [*$\tilde{v}$-reduced*]{} if for every $\ell\in\{1,\ldots, h\}$ from (\[eqn:part\]) we have $\delta_{X_{\ell}(\tilde{c})\cup X_{\ell + 1}(\tilde{c})}(\tilde{v}) = 1$.
Turning our attention now back to the more specific PI-family ${\cal{P}}_n = \{\Pi_{n-1}(k-1)\}_{k=2,\ldots,n}$ we note that vectors of distinct order type can yield the same face of $\Pi_{n-1}(k-1)$ when $k\geq 3$, but for $k=2$ (when $\Pi_{n-1}(k-1) = \Pi_{n-1}$, the standard permutahedron) then each face corresponds uniquely to the order type of the vector yielding it.
\[obs:vec-type-face\] For every $k\geq 2$ the map $[\tilde{c}]\mapsto F([\tilde{c}])\in {\mathbf{F}}(\Pi_{n-1}(k-1))$ is always surjective, and it is injective (and hence bijective) iff $k=2$. In particular, the total number of order types $[\tilde{c}]$ where $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ is the same as $|{\mathbf{F}}(\Pi_{n-1})|$, the total number of faces (including the polytope itself) of $\Pi_{n-1}$.
Since $\Pi_{n-1}(k-1) = P_{n-1}(\tilde{v}_{n-1}(k-1))$ where $\tilde{v}_{n-1}(k-1)
= \left(0,\ldots,0,\binom{k-1}{k-1},\ldots,\binom{n-1}{k-1}\right)$ from Lemma \[lmm:key\], then when considering a face $F([\tilde{c}])$ of $\Pi_{n-1}(k-1)$ we can assume $\tilde{c}$ to be $\tilde{v}_{n-1}(k-1)$-reduced. Therefore we can assume the partition (or rather the disjoint union) of $[n]$ induced by $\tilde{c}$ from (\[eqn:part\]) to have the form $[n] = Z(\tilde{c})\cup X_0(\tilde{c})\cup\cdots\cup X_p(\tilde{c})$ where $Z(\tilde{c})$ consists of those indices from $[n]$ whose coordinates of $F(\tilde{c})$ all are zero, which could potentially be empty. In fact, letting $n_{\ell} = |X_{\ell}(\tilde{c})|$, we see that $F(\tilde{c})$ is the unique face that is the convex combination of those vertices of $\Pi_{n-1}(k-1)$ where the $n_p$ largest entries occur in coordinates from $X_p(\tilde{c})$, the next largest entries occur in coordinates from $X_{p-1}(\tilde{c})$, etc., the $n_0$ 2nd smallest entries, not all zero (but where some could be zero), occur in coordinates from $X_0(\tilde{c})$ and lastly the smallest entries, all zero, occur in coordinates from $Z(\tilde{c})$. As noted, with this setup $Z(\tilde{c})$ could be empty. We therefore must relax the notion of partition in order to obtain a description of the face $F(\tilde{c}) = F([\tilde{c}])$.
\[def:OPP\] For $n\in{\mbox{$\mathbb N$}}$ call a tuple $(Z,X_0,\ldots, X_{p})$ an [*ordered pseudo-partition*]{} of $[n]$ (or an [*OPP*]{} for short) if $Z\cup X_0\cup\cdots\cup X_{p} = [n]$ is a disjoint union and either $(X_0,\ldots, X_{p})$, or $(Z,X_0,\ldots, X_{p})$ is an ordered (proper) partition of $[n]$.
[Remark:]{} Although the above Definition \[def:OPP\] is motivated by a vector in $\tilde{c}\in {{\mbox{$\mathbb R$}}}^n$, and its dot-product with a vertex $\tilde{v}\in\Pi_{n-1}(k-1)$, the definition of an OPP does not depend on it.
\[thm:face-gen-perm\] For $n\in{\mbox{$\mathbb N$}}$ and $k\in\{1,\ldots,n\}$, then every $d$-face of $\Pi_{n-1}(k-1)$ is in one-to-one correspondence with an OPP $(Z,X_0,\ldots,X_{p})$ of $[n]$ where (i) $0\leq |Z| \leq k-1$, (ii) $k\leq |Z| + |X_0| \leq n$, and (iii) $n - |Z| - p - 1 = d$.
From an OPP ${\cal{P}} = (Z,X_0,\ldots,X_{p})$ of $[n]$ satisfying the conditions (i) – (iii) in Theorem \[thm:face-gen-perm\] above, we obtain a vector $\tilde{c} = \tilde{c}({\cal{P}})$ with ${{\mathop{\mathrm{Proj}}\nolimits}}_l(\tilde{c}) = 0$ if $l\in Z$ and ${{\mathop{\mathrm{Proj}}\nolimits}}_l(\tilde{c}) = i+1$ if $l\in X_i$. In this case the face $F(\tilde{c})$ is exactly the convex combination of those vertices of $\Pi_{n-1}(k-1)$ where the largest $|X_{p}|$ entries occur in coordinates from $X_{p}$, the largest $|X_{p-1}|$ entries of the remaining $n-|X_{p}|$ ones occur in coordinates from $X_{p-1}$ etc, the largest $|X_0|$ entries of the remaining $n-(|X_1|+\cdots +|X_{p}|)$ ones occur in coordinates from $X_0$, and finally, all the coordinates from $Z$ contain only zeros. Hence, each OPP ${\cal{P}}$ yields a unique face $F(\tilde{c}({\cal{P}}))$.
On the other hand, every (proper) face $F$ of $\Pi_{n-1}(k-1)$ has the form $F([\tilde{c}])$ for some $\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ where ${{\mathop{\mathrm{supp}}\nolimits}}(\tilde{c})=[h]$ for some $h\in[n]$ Viewing $\tilde{c}$ as a function $c : [n]\rightarrow [h]$ with $c(i) = c_i$ for each $i\in [n]$, we obtain a OPP ${\cal{P}} = {\cal{P}}(\tilde{c})$ as in the following way.
Letting $p\geq 0$ be the least integer with $|X_0| + \cdots + |X_{p}| \geq n-k+1$, where $X_i = c^{-1}(h-p+i)$ for each $i=0,\ldots,p$, and $Z = c^{-1}([h-p-1])$, will give us our desired OPP ${\cal{P}}(\tilde{c}) = (Z,X_0,\ldots,X_{p})$.
Clearly we have ${\cal{P}}(F(\tilde{c}({\cal{P}}))) = {\cal{P}}$ and $F(\tilde{c}({\cal{P}}(F(\tilde{c})))) = F(\tilde{c})$. Note that in general $\tilde{c}({\cal{P}}(F(\tilde{c})))\neq\tilde{c}$, but they yield the same face. This proves the one-to-one correspondence between OPPs and (proper) faces of $\Pi_{n-1}(k-1)$.
Finally, by Proposition \[prp:face-describe\] if $F = F(\tilde{c}({\cal{P}}))$ is the unique face obtained from the OPP ${\cal{P}}$, then $$\dim(F) = \sum_{i=0}^{p}(|X_i| - 1) =
|X_0| + \cdots + |X_{p}| - p - 1 =
n - |Z| - p - 1,$$ which completes the proof.
By Theorem \[thm:face-gen-perm\] we can derive the $f$-polynomial of $\Pi_{n-1}(k-1)$ by enumerating all OPP ${\cal{P}}$ satisfying (i) and (ii) in Theorem \[thm:face-gen-perm\] with $d = n - |Z| - p - 1$ being a given fixed number. For disjoint $Z, X_0\subseteq [n]$ there are $\stirling{n-|Z|-|X_0|}{p}p!$ ordered partitions $(X_1,\ldots,X_p)$ of the remaining elements of $[n]\setminus(Z\cup X_0)$.
Letting $i = |Z| \in \{0,\ldots,k-1\}$ and $j = |X_0|$, we get by Theorem \[thm:face-gen-perm\] that $i\in\{0,\ldots,k-1\}$ and $i+j\in \{k,\ldots,n\}$. Hence, each ordered $(X_1,\ldots,X_p)$ of the remaining $n-i-j$ elements from $[n]\setminus(Z\cup X_0)$ will by Theorem \[thm:face-gen-perm\] yield a face of dimension $n - i - p - 1$. As there are $\binom{n}{i}\binom{n-i}{j}$ ways of choosing a legitimate pair $(Z,X_0)$, we have the following Proposition.
\[prp:face-poly\] The $f$-polynomial $f_{\Pi_{n-1}(k-1)}(x) = \sum_{i=0}^{n-1}f_i(\Pi_{n-1}(k-1))x^i$ of $\Pi_{n-1}(k-1)$ is given by $$f_{\Pi_{n-1}(k-1)}(x) = \sum_{\substack{0\leq i\leq k-1 \\ k\leq i+j\leq n}}
\binom{n}{i}\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}{p}!x^{n-i-p-1}.$$
[Remarks:]{} (i) Note that the coefficients $[x^0]f_{\Pi_{n-1}(k-1)}(x)$ and $[x^1]f_{\Pi_{n-1}(k-1)}(x)$ agree with previous Propositions \[prp:vert\] and \[prp:v-e-simple\] on the number of vertices and edges respectively. (ii) When $k=n$ we obtain $$f_{\Pi_{n-1}(k-1)}(x) = f_{\Pi_{n-1}(n-1)}(x) = \frac{(x+1)^n-1}{x},$$ the $f$-polynomial of the $(n-1)$-dimensional simplex.
By Propositions \[prp:face-poly\] and \[prp:v-e-simple\] and Corollary \[cor:simple-flag\] we obtain the $\ell$-flag polynomial of $\Pi_{n-1}(k-1)$ in the following.
\[cor:1-2-ell\] For each $k\in\{2,\ldots,n\}$ the $\ell$-flag polynomial ${\tilde{f}}^{\ell}_{\Pi_{n-1}(k-1)}(\tilde{x})$ of $\Pi_{n-1}(k-1)$ is given by $${\tilde{f}}^{\ell}_{\Pi_{n-1}(k-1)}(\tilde{x}) =
(x_2 + \cdots + x_{\ell} + 1)^{n-1}
f_{\Pi_{n-1}(k-1)}\left(\frac{x_1}{x_2+\cdots+x_{\ell}+1}\right),$$ where $f_{\Pi_{n-1}(k-1)}$ is the $f$-polynomial of $\Pi_{n-1}(k-1)$ given in Proposition \[prp:face-poly\].
We complete this section on the face lattice of $\Pi_{n-1}(k-1)$ by describing the faces of $\Pi_{n-1}(k-1)$ in terms of OPPs of $[n]$ and when one face contains another in a similar fashion as in Observation \[obs:coarse\].
From Observation \[obs:c-face\] we immediately obtain the following.
\[prp:face-chain\] If $\tilde{a},\tilde{c}\in{{\mbox{$\mathbb R$}}}^n$ are such that $a_i\leq a_j\Rightarrow c_i\leq c_j$, then for the corresponding faces of $P_{n-1}(\tilde{v})$ we have $F([\tilde{a}])\subseteq F([\tilde{c}])$.
Proposition \[prp:face-chain\] yields a sufficient condition for the vectors $\tilde{a}$ and $\tilde{c}$ that implies $F([\tilde{a}])\subseteq F([\tilde{c}])$. We will now describe exactly the relationship between $\tilde{a}$ and $\tilde{c}$ such that for faces of $P_{n-1}(\tilde{v})$ we have $F([\tilde{a}])\subseteq F([\tilde{c}])$.
Assume $\tilde{a},\tilde{c}\in {{\mbox{$\mathbb R$}}}^n$ are such for their corresponding faces of $P_{n-1}(\tilde{v})$ we have that $F([\tilde{a}])\subseteq F([\tilde{c}])$. Since there is a permutation $\alpha\in S_n$ with $\alpha(\tilde{a})$ ordered, i.e. $a_{\alpha(1)}\leq\cdots\leq a_{\alpha(n)}$, we can, for simplicity, assume $\tilde{a}$ is ordered $a_1\leq\cdots\leq a_n$. In this case the partition $[n] = X_1(\tilde{a})\cup\cdots\cup X_h(\tilde{a})$ of $[n]$ induced by $\tilde{a}$ as in (\[eqn:part\]) is a union of consecutive intervals. By Observation \[obs:perm-face\] we can assume $\tilde{v}$ is ordered in the same way as $\tilde{a}$ is, so $v_1\leq\cdots\leq v_n$. If $\delta_{X_{\ell}(\tilde{a})}(\tilde{v}) = 1$, then for any transposition $\tau\in S(X_{\ell}(\tilde{a}))\subseteq S_n$ we have $\tau(\tilde{v})\in F([\tilde{a}])\subseteq F([\tilde{c}])$ and hence $\tilde{c}\cdot\tilde{v} = \tilde{c}\cdot\tau(\tilde{v})$. By Lemma \[lmm:neq0\] we must have the following.
\[clm:c=const\] $c_i = c_j$ for all $i,j\in X_{\ell}(\tilde{a})$ with $\delta_{X_{\ell}(\tilde{a})}(\tilde{v}) = 1$.
Assume now $i\in X_{\ell}(\tilde{a})$ and $j\in X_{\ell + 1}(\tilde{a})$ where $\tilde{a}$ is $\tilde{v}$-reduced. In this case one of the following three conditions hold: (i) $v_i < v_j$, (ii) $v_i = v_j$ and $\delta_{X_{\ell}(\tilde{a})}(\tilde{v}) = 1$, and hence there is an $i'\in X_{\ell}(\tilde{a})$ with $v_{i'} < v_i = v_j$, or (iii) $v_i = v_j$ and $\delta_{X_{\ell+1}(\tilde{a})}(\tilde{v}) = 1$, and hence there is an $j'\in X_{\ell+1}(\tilde{a})$ with $v_i = v_j < v_{j'}$.
In case (i) consider the transposition $\tau = (i,j)$. Since $\tilde{v}\in F([\tilde{a}])\subseteq F([\tilde{c}])$ we have $\tilde{c}\cdot\tilde{v} \geq \tilde{c}\cdot\tau(\tilde{v})$, and hence $c_iv_i + c_jv_j \geq c_iv_j + c_jv_i$ or $(c_j - c_i)(v_j - v_i) \geq 0$. Therefore $c_i \leq c_j$ must hold.
In case (ii) consider the transposition $\tau = (i',j)$. As in previous case we have $\tilde{c}\cdot\tilde{v} \geq \tilde{c}\cdot\tau(\tilde{v})$, and hence $c_{i'}v_{i'} + c_jv_j \geq c_{i'}v_j + c_jv_{i'}$ or $(c_j - c_{i'})(v_j - v_{i'}) \geq 0$. Therefore $c_{i'} \leq c_j$ must hold, and so by Claim \[clm:c=const\] $c_i = c_{i'} \leq c_j$ must hold.
Finally, in case (iii) consider the transposition $\tau = (i,j')$. As in previously we have $\tilde{c}\cdot\tilde{v} \geq \tilde{c}\cdot\tau(\tilde{v})$, and hence $c_iv_i + c_{j'}v_{j'} \geq c_iv_{j'} + c_{j'}v_i$ or $(c_{j'} - c_i)(v_{j'} - v_i) \geq 0$. Therefore $c_i \leq c_{j'}$ must hold, and so by Claim \[clm:c=const\] $c_i \leq c_{j'} = c_j$ must hold. Hence, we have the following.
\[clm:c-ordered\] For a $\tilde{v}$-reduced $\tilde{a}$, if $i\in X_{\ell}(\tilde{a})$ and $j\in X_{\ell'}(\tilde{a})$ with $\ell < \ell'$, then $c_i\leq c_j$.
By Observation \[obs:perm-face\] and the previous two Claims \[clm:c=const\] and \[clm:c-ordered\], noting that the ordering of both $\tilde{a}$ and $\tilde{v}$ was assumed for the sake of argument, we have the following summarizing theorem.
\[thm:face-subset\] Let $\tilde{a},\tilde{c}\in {{\mbox{$\mathbb R$}}}^n$ where $\tilde{a}$ is $\tilde{v}$-reduced, and $F([\tilde{a}]), F([\tilde{c}])$ be the corresponding induced faces of $P_{n-1}(\tilde{v})$. Assume $\tilde{v} \in F([\tilde{a}])$ and let $[n] = X_1(\tilde{a})\cup\cdots\cup X_h(\tilde{a})$ be the partition of $[n]$ induced by $\tilde{a}$ as in (\[eqn:part\]). With this setup we have $F([\tilde{a}])\subseteq F([\tilde{c}])$ if and only if we have the following.
1. $a_i < a_j\Rightarrow c_i\leq c_j$.
2. For every part $X_{\ell}(\tilde{a})$ with $\delta_{X_{\ell}(\tilde{a})}(\tilde{v}) = 1$, we have $\delta_{X_{\ell}(\tilde{a})}(\tilde{c}) = 0$.
Finally in this section, we further seek a description of the faces and flags of $\Pi_{n-1}(k-1)$ as described in Observation \[obs:coarse\] for the standard permutahedron $\Pi_{n-1}$. To do so, we apply Theorem \[thm:face-subset\] to describe when exactly one face $F([\tilde{a}])$ of $\Pi_{n-1}(k-1)$ is contained in another $F([\tilde{c}])$ in terms of the characterization given in Theorem \[thm:face-gen-perm\]. We can assume both $\tilde{a}$ and $\tilde{c}$ to be $\tilde{v}_{n-1}(k-1)$-reduced.
Note that, trivially, if $|X_{\ell}(\tilde{a})|=1$, then clearly $\delta_{X_{\ell}(\tilde{a})}(\tilde{c}) = 0$. Assume that $F([\tilde{a}])$ and $F([\tilde{c}])$ correspond to the OPPs $(Z,X_0,\ldots,X_{p})$ and $(Z',X'_0,\ldots,X'_{p'})$ of $[n]$ respectively, and that $\tilde{v}\in F([\tilde{a}])\subseteq F([\tilde{c}])$ is a vertex of $\Pi_{n-1}(k-1)$. By Theorems \[thm:face-gen-perm\] and \[thm:face-subset\] and the above note, we have that $\delta_{X_i}(\tilde{c}) = 0$ for all $i\in\{0,\ldots,p\}$, and hence $X_0\subseteq X_0'$, and further each part from $\{X_1,\ldots,X_{p}\}$ is contained in a part from $\{X'_0,\ldots,X'_{p'}\}$. So, as a direct consequence of Theorem \[thm:face-subset\] We have the following.
\[cor:face-subset-OPP\] For two faces $F([\tilde{a}])$ and $F([\tilde{c}])$ of $\Pi_{n-1}(k-1)$, where both $\tilde{a}$ and $\tilde{c}$ are $\tilde{v}_{n-1}(k-1)$-reduced, corresponding to the OPPs $(Z,X_0,\ldots,X_{p})$ and $(Z',X'_0,\ldots,X'_{p'})$ respectively, we have $F([\tilde{a}])\subseteq F([\tilde{a}])$ if, and only if, the disjoint union $D\cup X_0\cup\cdots\cup X_{p}$ a refinement of $X'_0\cup\cdots\cup X'_{p'}$ where $X_0\subseteq X_0'$ and $D = X_0'\setminus X_0$ is the difference.
[Remark:]{} Note that $D\subseteq [n]\setminus(X_0\cup\cdots\cup X_p)\subseteq Z$.
Note that we have a well defined map $\{0,1,\ldots,p\}\ni i\mapsto i'\in \{0,1,\ldots,p'\}$ where $i'$ is the unique index with $X_i\subseteq X'_{i'}$. In this way we have.
\[obs:inc-surj\] The above map $i\mapsto i'$ is an increasing surjection.
A closed formula for the exponential generating function {#sec:exp-func}
========================================================
In this final section we derive a closed formula for the exponential ${\ell}$-generating function ${\tilde{\xi}}_{{\cal{P}}_k^{\perp};k-1}^{\ell}(\tilde{x},y)$ from Definition \[def:flag-poly-func\] of the family ${\cal{P}}_k^{\perp} = \{\Pi_{n-1}(k-1)\}_{n\geq k}$ of all the general permutahedra, which we will henceforth denote by ${\tilde{\xi}}_k^{\ell}(\tilde{x},y)$, analogous to the result of Proposition \[prp:exp-gen-func\]. Unless otherwise stated $k\geq 1$ is an arbitrary but fixed integer throughout.
If we let $${\tilde{g}}_n^{\ell}(\tilde{x})
:= \sum_{\tilde{s}}\frac{f_{\tilde{s}}(\Pi_{n-1}(k-1))}{(n-s_1)!}
x_1^{n-s_1}x_2^{s_2-s_1}\cdots x_{\ell}^{s_{\ell}-s_{\ell-1}},$$ then by Definition \[def:flag-poly-func\] we have $$\label{eqn:gn}
x_1^n{\tilde{g}}_n^{\ell}(x_1^{-1},x_2,\ldots,x_{\ell})
= {\tilde{\xi}}_{\Pi_{n-1}(k-1)}^{\ell}(\tilde{x})$$ and so for $${\tilde{g}}^{\ell}(\tilde{x},y)
:= \sum_{n\geq 1}{\tilde{g}}_n^{\ell}(\tilde{x})\frac{y^n}{n!}
= \sum_{n\geq k}{\tilde{g}}_n^{\ell}(\tilde{x})\frac{y^n}{n!}$$ we have $$\label{eqn:g-2-g-tilde}
{\tilde{g}}^{\ell}(x_1^{-1},x_2,\ldots, x_{\ell},x_1y) =
\sum_{n\geq k}x_1^n{\tilde{g}}_n^{\ell}(x_1^{-1},x_2,\ldots,x_{\ell})\frac{y^n}{n!}
= {\tilde{\xi}}_k^{\ell}(\tilde{x},y).$$ Hence, it suffices to obtain a closed formula for ${\tilde{g}}^{\ell}(\tilde{x},y)$. Further we note that for $\ell = 1$ we obtain by (\[eqn:gn\]) that $x_1^n{\tilde{g}}_n^1(x_1^{-1}) = {\tilde{\xi}}^1(x_1)$ and hence ${\tilde{g}}_n^1(x_1) = x_1^n{\tilde{\xi}}^1(x_1^{-1})$ and so by Proposition \[prp:face-poly\] that $$\label{eqn:g}
g_n(x) := {\tilde{g}}_n^{1}(x) =
\sum_{\substack{0\leq i\leq k-1 \\ k\leq i+j\leq n}}
\binom{n}{i}\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}{p}!
\frac{x^{i+p+1}}{(i+p+1)!}.$$ Further, by (\[eqn:gn\]) for general $\ell$ and $\ell=1$, and Corollary \[cor:simple-flag\] applied to ${\tilde{\xi}}^{\ell}$ and (\[eqn:g\]) we obtain $$\begin{aligned}
\lefteqn{{\tilde{g}}_n^{\ell}(\tilde{x}) =
\frac{1}{x_2 + \cdots + x_{\ell} +1}g_n(x_1(x_2 + \cdots + x_{\ell}+1))} \\
& & = \frac{1}{x_2 + \cdots + x_{\ell} +1}\left[
\sum_{\substack{0\leq i\leq k-1 \\ k\leq i+j\leq n}}
\binom{n}{i}\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}{p}!
\frac{(x_1(x_2 + \cdots + x_{\ell}+1))^{i+p+1}}{(i+p+1)!}
\right].\end{aligned}$$ Hence, it suffices to obtain a closed formula for $$\label{eqn:gxy}
g(x,y) := {\tilde{g}}^1(x,y)
= \sum_{n\geq 1}g_n(x)\frac{y^n}{n!} = \sum_{n\geq k}g_n(x)\frac{y^n}{n!},$$ since then $$\label{eqn:g-tilde-y}
{\tilde{g}}_n^{\ell}(\tilde{x},y) =
\sum_{n\geq k}{\tilde{g}}_n^{\ell}(\tilde{x})\frac{y^n}{n!}
= \frac{g(x_1(x_2 + \cdots + x_{\ell}+1),y)}{x_2 + \cdots + x_{\ell} +1}.$$
By (\[eqn:g\]) we have $$\label{eqn:g=sumgi}
g_n(x) = \sum_{i=0}^{k-1}g_{n;i}(x)$$ where for each $i\in\{0,1,\ldots,k-1\}$ $$\begin{aligned}
g_{n;i}(x) & = & \sum_{j=k-i}^{n-i}
\binom{n}{i}\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}{p}!
\frac{x^{i+p+1}}{(i+p+1)!} \nonumber \\
& = &
\binom{n}{i}
\sum_{j=k-i}^{n-i}
\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}{p}!
\frac{x^{i+p+1}}{(i+p+1)!}, \label{eqn:gix}\end{aligned}$$ and so its $(i+1)$-th derivative w.r.t. $x$ is $$\label{eqn:i+1-th-diff}
g_{n;i}^{(i+1)}(x) =
\binom{n}{i}
\sum_{j=k-i}^{n-i}
\binom{n-i}{j}\sum_{p=0}^{n-i-j}\stirling{n-i-j}{p}x^p
:= \binom{n}{i}\gamma_{n;i}(x).$$ From this we deduce that for $$\label{eqn:gixy}
g_i(x,y) := \sum_{n\geq 1}g_{n;i}(x)\frac{y^n}{n!}$$ where $g_{n;i}(x)$ is given in (\[eqn:gix\]), we have $$\begin{aligned}
g_i^{(i+1)}(x,y) & := & \frac{\partial^{i+1}}{{\partial x}^{i+1}}g_i(x,y) \\
& = & \sum_{n\geq 1}g_{n;i}^{(i+1)}(x)\frac{y^n}{n!} \\
& = & \sum_{n\geq 1}\binom{n}{i}\gamma_{n;i}(x)\frac{y^n}{n!} \\
& = & \sum_{n\geq i}\binom{n}{i}\gamma_{n;i}(x)\frac{y^n}{n!} \\
& = & \frac{y^i}{i!}\sum_{n\geq i}\gamma_{n;i}(x)\frac{y^{n-i}}{(n-i)!}.\end{aligned}$$
Before we continue, we need the following.
\[lmm:root-ord-parts\] For $m, N\in{\mbox{$\mathbb N$}}$ we have $$\sum_{i=1}^{N-m+1}\binom{N}{i}\stirling{N-i}{m-1} = m\stirling{N}{m}.$$
Call an unordered partition of $[N]$ [*rooted*]{} if it has one distinguished part, the [*root*]{} $r$. For each of the unordered partitions of $[N]$ into $m$ parts, we have $m$ possible roots, and so the number of rooted partitions of $[N]$ into $m$ parts is, on one hand, given by $m\stirling{N}{m}$.
On the other hand, we can start by choosing a root $r$ of valid cardinality $i\in\{1,\ldots,N-m+1\}$, and then consider the $\stirling{N-i}{m-1}$ unordered partitions of the remaining subset of $[N]\subseteq r$. For each $i$ this can be done in $\binom{N}{i}\stirling{N-i}{m-1}$ possible ways. Adding these ways for all possible $i$ will give us the expression on the left in the stated equation.
The coefficient $[x^h](\gamma_{n;i}(x))$ of $x^h$ in the polynomial $\gamma_{n;i}(x)$ defined in (\[eqn:i+1-th-diff\]) is by direct tallying and the above Lemma \[lmm:root-ord-parts\] given by $$\begin{aligned}
[x^h](\gamma_{n;i}(x)) & = &
\sum_{q=k-i}^{n-i-h}\binom{n-i}{q}\stirling{n-i-q}{h} \\
& = & \sum_{q=1}^{n-i-h}\binom{n-i}{q}\stirling{n-i-q}{h} -
\sum_{q=1}^{k-i-1}\binom{n-i}{q}\stirling{n-i-q}{h} \\
& = & \stirling{n-i}{h+1}(h+1) -
\sum_{q=1}^{k-i-1}\binom{n-i}{q}\stirling{n-i-q}{h}.\end{aligned}$$ In light of this, we can further write $\gamma_{n;i}(x)$ as $\gamma_{n;i}(x) = \alpha_{n;i}(x) - \sum_{q=1}^{k-i-1}\alpha_{n;i,q}(x)$, where $$\alpha_{n;i}(x) := \sum_{h\geq 0}\stirling{n-i}{h+1}(h+1)x^h, \ \
\alpha_{n;i,q}(x) := \sum_{h\geq 0}\binom{n-i}{q}\stirling{n-i-q}{h}x^h.$$ Defining the corresponding exponential series $$\alpha_i(x,y) := \sum_{n\geq i}\alpha_{n;i}(x)\frac{y^{n-i}}{(n-i)!}, \ \
\alpha_{i,q}(x,y) := \sum_{n\geq i}\alpha_{n;i,q}(x)\frac{y^{n-i}}{(n-i)!},$$ we then get $$\begin{aligned}
g_i^{(i+1)}(x,y) & = &
\frac{y^i}{i!}\sum_{n\geq i}\gamma_{n;i}(x)\frac{y^{n-i}}{(n-i)!} \\
& = & \frac{y^i}{i!}\sum_{n\geq i}\left(
\alpha_{n;i}(x)
- \sum_{q=1}^{k-i-1}\alpha_{n;i,q}(x)\right)\frac{y^{n-i}}{(n-i)!} \\
& = &
\frac{y^i}{i!}\left(\alpha_i(x,y) - \sum_{q=1}^{k-i-1}\alpha_{i;q}(x,y)\right).\end{aligned}$$ Up to a constant we obtain $\int \alpha_{n;i}(x)\,dx = \sum_{h\geq 0}\stirling{n-i}{h+1}x^{h+1}$, and so by (\[eqn:Touch-St2\]) we get $$\int\alpha_i(x,y)\,dx
= \sum_{h,n\geq i}\stirling{n-i}{h+1}x^{h+1}\frac{y^{n-i}}{(n-i)!}
= \sum_{h,m\geq 0}\stirling{m}{h+1}x^{h+1}\frac{y^{m}}{m!}
= e^{x(e^y-1)}-1,$$ and so by differentiating $$\label{eqn:alphai}
\alpha_i(x,y) = e^{x(e^y-1)}(e^y-1).$$ Similarly, but with neither integration nor differentiation, we obtain by direct manipulation and by again (\[eqn:Touch-St2\]) $$\begin{aligned}
\alpha_{i;q}(x,y) & = & \sum_{n\geq i}\alpha_{n;i,q}(x)\frac{y^{n-i}}{(n-i)!} \\
& = & \sum_{n\geq i}\left(\sum_{h\geq 0}\binom{n-i}{q}\stirling{n-i-q}{h}x^h
\right)\frac{y^{n-i}}{(n-i)!} \\
& = & \sum_{m\geq 0}\left(\sum_{h\geq 0}\binom{m}{q}\stirling{m-q}{h}x^h
\right)\frac{y^m}{m!} \\
& = & \sum_{m,h\geq 0}\binom{m}{q}\stirling{m-q}{h}\frac{y^m}{m!}x^h \\
& = &
\frac{y^q}{q!}\sum_{m,h\geq 0}\stirling{m-q}{h}\frac{y^{m-q}}{(m-q)!}x^h \\
& = & \frac{y^q}{q!}e^{x(e^y-1)}.\end{aligned}$$ Consequentially, by the above and (\[eqn:alphai\]) we then get $$\begin{aligned}
g_i^{(i+1)}(x,y)
& = &
\frac{y^i}{i!}\left(\alpha_i(x,y) - \sum_{q=1}^{k-i-1}\alpha_{i;q}(x,y)\right) \\
& = & \frac{y^i}{i!}\left(e^y - E_{k-i-1}(y)\right)e^{x(e^y-1)},\end{aligned}$$ where $E_m(x) = 1 + x + \cdots + \frac{x^m}{m!}$ is the $m$-th degree polynomial approximation of $e^y$. By the the above and the defining sum of $g_i(x,y)$ in (\[eqn:gixy\]) we have $$\label{eqn:gixy-cond}
g_i^{(i+1)}(x,y) = \frac{y^i}{i!}\left(e^y - E_{k-i-1}(y)\right)e^{x(e^y-1)}, \ \
g_i(0,y) = g_i'(0,y) = \cdots = g_i^{(i)}(0,y) = 0 \mbox{ for all }y.$$ The closed expression for $g_i(x,y)$ is uniquely determined by (\[eqn:gixy-cond\]) and by integrating $i+1$ times we get $$\label{eqn:gixy-form}
g_i(x,y) =
\frac{y^i(e^y - E_{k-i-1}(y))}{i!(e^y-1)^{i+1}}\left(
e^{x(e^y-1)} - E_i(x(e^y-1))\right).$$ By (\[eqn:gxy\]), (\[eqn:g=sumgi\]), (\[eqn:gixy\]) and (\[eqn:gixy-form\]) we then get $$g(x,y)
= \sum_{n\geq 1}g_n(x)\frac{y^n}{n!}
= \sum_{i=0}^{k-1}g_i(x,y)
= \sum_{i=0}^{k-1}\frac{y^i(e^y - E_{k-i-1}(y))}{i!(e^y-1)^{i+1}}\left(
e^{x(e^y-1)} - E_i(x(e^y-1))\right),$$ and so by the above expression, (\[eqn:g-tilde-y\]) and (\[eqn:g-2-g-tilde\]) we have the following main theorem of this section.
\[thm:main-flag-func\] The exponential $\ell$-generating function for all the $\ell$-flags of all the general permutahedra ${\cal{P}}_k^{\perp} = \{\Pi_{n-1}(k-1)\}_{n\geq k}$ from Definition \[def:flag-poly-func\] is given by $$\begin{aligned}
{\tilde{\xi}}_k^{\ell}(\tilde{x},y)
& = & \sum_{n\geq 1,\tilde{s}}
{\tilde{\xi}}^{\ell}_{\Pi_{n-1}(k-1)}(\tilde{x})\frac{y^n}{n!} \\
& = & \frac{1}{S}\sum_{i=0}^{k-1}
\frac{(x_1y)^i(e^{x_1y} - E_{k-i-1}({x_1y}))}{i!(e^{x_1y}-1)^{i+1}}
\left(e^{\frac{S}{x_1}\left(e^{x_1y}-1\right)}
- E_i\left(\frac{S}{x_1}(e^{x_1y}-1)\right)
\right).\end{aligned}$$ where $S = x_2 + \cdots + x_{\ell} + 1$ and $E_m(x) = 1 + x + \cdots + \frac{x^m}{m!}$ is the $m$-th polynomial approximation of $e^x$.
[Remarks:]{} (i) The question remains, whether one could possible further simplify the expression given in Theorem \[thm:main-flag-func\]. Since, however, we are dealing with enumeration of OPP as described in Theorem \[thm:face-gen-perm\], in which the cardinalities of the parts depend both on $n$ and $k$, it seems unlikely to the author that a substantial simplification exists. (ii) Letting $k=2$ we obtain the exponential $\ell$-generating function for all the $\ell$-flags of all the general permutahedra $\{\Pi_{n-1}\}_{n\geq 2}$ as the following $${\tilde{\xi}}_2^{\ell}(\tilde{x},y) =
\frac{1}{x_2 + \cdots + x_{\ell} + 1}
\left(e^{\frac{x_2 + \cdots + x_{\ell} + 1}{x_1}\left(e^{x_1y}-1\right)}
- 1 - y(x_2 + \cdots + x_{\ell} + 1)\right),$$ which is consistent with Proposition \[prp:exp-gen-func\], when we note that the family $\{\Pi_{n-1}\}_{n\geq 1}$ there contains $\Pi_0$ for $n=1$, whereas in Theorem \[thm:main-flag-func\] the family $\{\Pi_{n-1}(k-1)\}_{n\geq k}$ becomes $\{\Pi_{n-1}\}_{n\geq 2}$ omitting the singleton $\Pi_0$.
When $\ell = 1$ in Theorem \[thm:main-flag-func\] the sum $x_2 + \cdots + x_{\ell}$ is empty and so we obtain the face function of all the general permutahedra in the following.
\[cor:l=1\] The exponential generating function for all the faces of all the general permutahedra $\{\Pi_{n-1}(k-1)\}_{n\geq k}$ from Definition \[def:flag-poly-func\] is given by $${\xi}_k(x,y) = {\tilde{\xi}}_k^{1}(x,y) =
\sum_{i=0}^{k-1}\frac{(xy)^i(e^{xy} - E_{k-i-1}({xy}))}{i!(e^{xy}-1)^{i+1}}
\left(
e^{\left(\frac{e^{xy}-1}{x}\right)} - E_i\left(\frac{e^{xy}-1}{x}\right)
\right).$$ where $E_m(x) = 1 + x + \cdots + \frac{x^m}{m!}$ is the $m$-th polynomial approximation of $e^x$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author would like to thank James F. Lawrence for many helpful discussions regarding the theory of polytopes in general.
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[^1]: Department of Mathematical Sciences, George Mason University, MS 3F2, 4400 University Drive, Fairfax, VA – 22030, USA, [geir@math.gmu.edu]{}
[^2]: The [*forward difference*]{} is defined as $\Delta(a_n) = a_{n+1}- a_n$
|
---
abstract: 'Double Dirac fermions have recently been identified as possible quasiparticles hosted by three-dimensional crystals with particular non-symmorphic point group symmetries. Applying a combined approach of ab initio methods and dynamical mean field theory, we investigate how interactions and double Dirac band topology conspire to form the electronic quantum state of [Bi$_2$CuO$_4$]{}. We derive a downfolded eight-band model of the pristine material at low energies around the Fermi level. By tuning the model parameters from the free band structure to the realistic strongly correlated regime, we find a persistence of the double Dirac dispersion until its constituting time reveral symmetry is broken due to the onset of magnetic ordering at the Mott transition. We analyze pressure as a promising route to realize a double-Dirac metal in [[Bi$_2$CuO$_4$]{}]{}.'
author:
- Domenico Di Sante
- Andreas Hausoel
- Paolo Barone
- 'Jan M. Tomczak'
- Giorgio Sangiovanni
- Ronny Thomale
bibliography:
- 'CuBBiO-lib.bib'
title: Strongly correlated double Dirac fermions
---
[*Introduction.*]{} Electrons in solids witness a reduced spatial symmetry in comparison to the spacetime continuum. While the high-energy perspective constrains us to Majorana, Weyl, and Dirac fermions in accordance with the inhomogeneous Lorentz (or Poincaré) group, electronic quasiparticles in solids at low energies can display emergent fermionic behaviour within and even beyond this classification [@charly; @berni]. Graphene constitutes one of the most prominent material discoveries where Dirac-type quasiparticles have been realized [@neto]. The current rise of Weyl semimetals [@wan], along with Majorana quasiparticles in superconducting heterostructures [@nick; @fu], complements this evolution in contemporary condensed matter physics.
Recently, Wieder [*et al.*]{} [@charly] have brought up the possibility to realize double Dirac quasiparticles in certain 3D crystals with specific non-symmorphic point group symmetry. This was followed up on by a systematic analysis of all double space groups (SGs) accounting for $S=1/2$ electrons in spin-orbit coupled crystals with time reversal symmetry [@berni]. In particular, SG 130 and 135 were found to establish eminently suited ground for generic double Dirac fermions protected by point group symmetry. Among all material candidates for SG 130, it is was already realized in Ref. that [[Bi$_2$CuO$_4$]{}]{} might be a prime candidate due to its filling-enforced semimetallicity [@michi], nurturing the hope to observe double Dirac fermions close to the Fermi level. All such band structure classifications, however, always need to be extended to account for the role of interactions in the material, which turn out to be of vital relevance in [Bi$_2$CuO$_4$]{}. Most of the topological band properties, even the metallic ones, display a certain degree of persistence against weak interactions as long as those do not break any protecting symmetry. Interaction-induced instabilities, however, do change the symmetry class of the quantum state, possibly affecting the whole range of constituting symmetries including charge conservation, time reversal, and point group operations.
In this Letter, we study strong interaction effects of double Dirac fermions, analyzing the band structure properties and correlation effects in [[Bi$_2$CuO$_4$]{}]{}. From density functional theory, we distill an effective eight-band tight-binding Hubbard model which is dominated by the $d_{x^2-y^2}$ orbital of the four Cu atoms in the unit cell. Spin-orbit coupling (SOC) is found to be weak because the heavy atoms of the compound do not significantly contribute to the low-energy density of states. As expected, the double Dirac dispersion is located close to the Fermi energy, as identified by an 8-fold band degeneracy at the A point. We quantify the strength of electron-electron interactions by means of the constrained random phase approximation (cRPA). By employing a combined approach of density functional theory (DFT) and dynamical mean field theory (DMFT), we determine the critical value for the Mott transition at ambient pressure to be $U_{\text{Mott}}\approx 0.85$eV. This value is approximately equal to the $d_{x^2-y^2}$ bandwidth, and substantially smaller than $U_{\text{cRPA}}\approx 1.58$eV, suggesting that [[Bi$_2$CuO$_4$]{}]{} at ambient pressure parametrically locates itself deep in the Heisenberg limit of the Mott regime. Below the Néel temperature T$_N$ the magnetic state displays an AFM-C type collinear order of the four local Cu magnetic moments. In the stochiometric compound at pristine conditions, magnetism thus dominates the electronic state, rendering the double Dirac fermions inaccessible. We explore the effect of hydrostatic pressure to drive [[Bi$_2$CuO$_4$]{}]{} into a metallic phase where the double Dirac fermions emerge. We find the bandstructure formation and correlation effects of [[Bi$_2$CuO$_4$]{}]{} to be highly sensitive to pressure. Starting from ambient pressure, we find an initial change of the magnetic ordering from AFM-C to AFM-G type, along with a reduced on-site Coulomb repulsion. This could pave the way towards a high-pressure double Dirac metal.
![ (Color online) (a) Tetragonal Brillouin zone of SG 130 P4/ncc with corresponding high-symmetry points. (b) Crystal structure of [[Bi$_2$CuO$_4$]{}]{}. Atom colors: Bi = blue, Cu = yellow, O = red. Blue plaquettes highlight the square-planar CuO$_4$ coordination. (c) DFT bandstructure in the metallic phase with (red solid line) and without (black dashed line) SOC. The left inset is a schematic representation of the square-planar crystal field and orbital filling for Cu-type d$^9$. The right inset shows a zoom around the eight-fold degenerate A point, which comes along a characteristic four-fold degneracy along the R-Z line.[]{data-label="fig1"}](./Fig1_v2.png){width="\columnwidth"}
---- --------------- --------------- --------------- ---------------- ----------- --------------- -------------------
P t$^{AB}_{1u}$ t$^{AB}_{1d}$ t$^{AB}_{2u}$ t$^{AB}_{2d}$ t$^A_1$ t$^A_2$ $U_{\text{cRPA}}$
t$_{1,2}$ t$_{4,2}$ t$_{1,3}$ t$_{1,2^{-1}}$ t$_{1,4}$ t$_{1,1^{1}}$
0 66 40 5 -24 36 -20 1.58
(74) (36) - (-40) (21) (-18)
30 80 21 6 -20 73 -16 1.45
50 88 13 2 -13 65 -7 1.36
---- --------------- --------------- --------------- ---------------- ----------- --------------- -------------------
: Hopping integrals in meV between the d$_{x^2-y^2}$-type Wannier functions on different Cu atoms for different pressure P \[GPa\] with ambient pressure P $=0$ GPa. The first row takes over the notation introduced in Ref. [@Janson], while t$_{i,j}$ in the second row is transfer integral between Cu-$i$ and Cu-$j$ site ($i,j = 1...4$) as in Fig. \[fig2\]. When a $j$ index has a superscript $\pm 1$ it refers to a Cu-$j$ atom in the \[00$\pm 1$\] unit cell. $U_{\text{cRPA}}$ (eV) is the cRPA value for the effective onsite Coulomb interaction. The $P=0$ hopping values in parenthesis refer to Ref. [@Janson].[]{data-label="tab1"}
![ (Color online) Side view of the d$_{x^2-y^2}$-type MLWFs belonging to two neighbouring Cu-chains, responsible for the transfer integral t$^{AB}_{1u}$ (Tab. \[tab1\]). Brown (green) color refers to positive (negative) values of the MLWF. The individual MLWFs are not symmetric with respect to the CuO$_4$ plaquette plane. This is due to a small off-centering of the Cu atoms, in opposite directions for Cu-1 and Cu-2. The inset shows a top view on the Cu-1 d$_{x^2-y^2}$-type MLWF.[]{data-label="fig2"}](./Fig2.png){width="\columnwidth"}
{width="\textwidth"}
[*Band structure and effective tight-binding model.*]{} [[Bi$_2$CuO$_4$]{}]{} crystallizes in the tetragonal spage group P4/ncc (SG 130) (Fig. \[fig1\](a)) [@GarciaMunoz]. Four inequivalent Cu atoms in the unit cell share a square-planar CuO$_4$ coordination, stacked along the out-of-plane direction and intercalated by Bi atoms, as depicted in Fig. \[fig1\](b). The Cu-O hybridization is quite strong, leading to a mixed $d$-$p$ character of the band structure of the occupied states [@Supp]. The Bi-$p$ manifold only starts $\approx 1.5$ eV above the Fermi level, suggesting that the low energy model is given by Cu and O electronic states. The square-planar crystal field is characterized by a higher-in-energy $d_{x^2-y^2}$ orbital experiencing a head-to-head interaction with the O-$p$. The level separation is schematically drawn in the left inset of Fig. \[fig1\](c). Cu is nominally in the oxidation state 2+ (i.e. $d^9$ configuration). For this reason, a half-filled $d_{x^2-y^2}$-derived single spin-degenerate band per Cu atom in the unit cell is expected to dominate at low energies. Fig. \[fig1\](c) displays the band dispersion of the electronic states around the Fermi level. Due to the dominant Cu orbital, the effect of SOC is almost negligible, only inducing small spin splittings. We estimate the energy scale of SOC to be $\approx
20$ meV. Despite its small scale, however, SOC is a crucial ingredient for the symmetries in the double SG representation that ensure the existence of a double Dirac fermion with linear dispersion at the Brillouin zone corner $\text{A}=(\pi,\pi,\pi)$. As the band filling is given by an odd multiple of 4 (180 = 8$\times$22+4), the double Dirac fermion in [[Bi$_2$CuO$_4$]{}]{} is located almost at the Fermi level, as visible in the right inset of Fig. \[fig1\](c).
While the aforementioned electronic structure details, as well as the magnetic properties at ambient pressure, have already been analyzed previously [@Janson; @Parmigiani], a closer inspection of the hybridization profile and the Coulomb matrix elements in terms of ab-initio derived maximally localized Wannier functions (MLWFs) [@Marzari] is indispensible to further analyze the fate of double Dirac fermions in the presence of interactions. The relatively large extension of the MLWFs is a consequence of the strong Cu-O hybridization. Their lobes indeed stretch out over the O-atoms with a clockwise winding shape, as shown in the top view of Fig. \[fig2\]. Our analysis elucidates the origin of the strong three-dimensional character of [[Bi$_2$CuO$_4$]{}]{}, as the MLWFs extend along the out-of-plane direction in an asymmetric manner, which is a consequence of buckled CuO$_4$ plaquettes. Moreover, two $d_{x^2-y^2}$-type MLWFs localized on different chains have a strong overlap, even though they do not belong to the same $ab-$plane. This is illustrated by the t$^{AB}_{1u}$ transfer integral shown in Fig. \[fig2\] and Tab. \[tab1\]. Coherent electron hopping is also enabled between neighbouring plaquettes along the normal $c-$direction. This process is accounted for by the transfer integral t$^{A}_1$, which is indeed comparable with the hoppings t$^{AB}_{1u}$ and t$^{AB}_{1d}$ that involve a significant in-plane component. As further discussed below, the hoppings turn out to be rather sensitive to pressure, and even impose a change of the magnetic ordering pattern, which is relevant to classifying the possible topological character of symmetry-broken phases in [[Bi$_2$CuO$_4$]{}]{} [@Schnyder].
[*Mott transition and spectral function.*]{} Having obtained an effective band structure description of [[Bi$_2$CuO$_4$]{}]{} close to $E_{\text{F}}$ in terms of a half-filled single band per Cu atom, we focus on the double Dirac point at the A point. In particular, we analyze its evolution as a function of interaction strength, starting from the free electron limit up to the realistic cRPA value for the electron-electron interaction $U$ [@note_UcRPA]. [[Bi$_2$CuO$_4$]{}]{} is parametrically located within the applicability bounds of DFT+DMFT, partly because of its three-dimensional character. Fig. \[fig3\](a) displays the quasiparticle weight $Z\!=\!(1\!-\!\partial\text{Im} \Sigma(\omega)/\partial\omega |_{\omega \rightarrow 0}\!)^{-1}$. Before reaching $U_{\text{cRPA}}$, we encounter a Mott-type metal-to-insulator transition (MIT) at $U_{\text{Mott}} \approx 0.85$ eV. At the Mott transition, the quasiparticle weight is suppressed by interactions, and the fraction of doubly-occupied Cu-sites $\langle n_{\uparrow}n_{\downarrow}\rangle$ reduces from the free-particle value of $1/4$ towards zero (inset in Fig. \[fig3\](a)). At intermediate $U$, the spectral function, as depicted in Fig. \[fig3\](b), is well described by the DFT single particle band structure renormalized in terms of bandwidth by the quasiparticle weight $Z$, along with the appearance of incoherent lower and upper Hubbard bands. Since the double Dirac fermion is located close to the Fermi level, it contributes to the quasiparticle peak for interaction strengths below the Mott transition. Even though the double Dirac velocities are strongly reduced by electron-electron interactions, the eightfold-degeneracy of the double Dirac point is preserved up to $U_{\text{Mott}}$. Within DMFT, as long as we consider the paramagnetic phase which does not allow the system to undergo a magnetic phase transition, the double Dirac point does not get destroyed for larger $U$. Yet, its weight gets damped by electron-electron scattering, and transferred towards the high-energy (Hubbard) bands, alongside with the whole low-energy spectral weight.
P (GPa) $U$ (eV) NM FM AFM-C AFM-G AFM-A
--------- ---------- ----- ---- ------- ------- -------
0
0.0 442 88 0 2.4 54
0.5 555 77 0 2.2 49
1.58 812 60 0 1.8 41
2.1 940 54 0 1.6 37
30
1.45 636 97 28 0 65
: DFT+U total energy differences (meV/unit cell) for the magnetic patterns at ambient pressure and at 30 GPa. The lowest-lying AFM type state is chosen as reference energy. Non-magnetic (NM) and ferromagnetic (FM) configurations are higher up in energy than the AFM states.[]{data-label="tab2"}
![(Color online) (a-c) Evolution of the DFT band structure for $P=0$ (ambient pressure), 6, and 30 GPa. Red symbols refer to the $d_{x^2-y^2}$ character. (d) $U_{\text{cRPA}}$ and $U_{\text{Mott}}$ as a function of pressure. It is suggestive to expect an insultar to metal transition around $P=50-60$ GPa.[]{data-label="fig4"}](./Fig4.pdf){width="\columnwidth"}
[*Magnetic order and hydrostatic pressure.*]{} At $U_{\text{cRPA}}$, allowing for magnetic ordering, the DFT+DMFT ground state of [[Bi$_2$CuO$_4$]{}]{} is not paramagnetic, but displays intra unit cell magnetic order. As a consequence, the double Dirac cone is absent due to the breaking of time-reversal symmetry. This finding holds for finite temperature $T$, below the Néel temperature $T_{\text{N}}$. For $T>T_{\text{N}}$, while there is no long-range magnetic order, the broadening is too strong to resolve detailed dispersive features. In Tab. \[tab2\], we report the DFT+U energies at $T=0$ for different magnetic configurations. Mapping these energies on a Heisenberg model and performing classical Monte Carlo simulations, we estimate a Néel temperature of 56 K for U = 1.58 eV, which compares reasonably well with the experimental value of 50 K [@Supp; @GarciaMunoz; @Janson]. The AFM-C type configuration, characterized by a ferromagnetic alignment of the chains along the $c$-direction, is found to be preferred. Independent of the considered interaction strength, however, it is always energetically close to an AFM-G type phase, where the magnetic moments are antiferromagnetically aligned along all three directions. This is the consequence of a competitive balance between $t^A_1$ and $t^{AB}_{1d}$ transfer integrals (Tab. \[tab1\]). This tuned frustration is relieved by pressure, favoring $t^A_1$ over $t^{AB}_{1d}$. It explains why we observe a change to AFM-G type magnetic order as a function of pressure (see Tab. \[tab2\]).
Along with the change of the magnetic order, pressure has a profound impact on the electronic structure. As illustrated in Fig. \[fig4\](a-c), the bands above and below the $d_{x^2-y^2}$ manifold get closer to the Fermi level. This results in a more efficient screening of the Coulomb interaction for the target $d_{x^2-y^2}$ states. Fig. \[fig4\](d) and TABLE \[tab1\] report the corresponding reduction of $U_{\text{cRPA}}$. In addition, the slight increase of the $d_{x^2-y^2}$ bandwidth ($\sim 10$%) leads to a larger critical $U_{\text{Mott}}$ for the Mott transition. Even though $U_{\text{cRPA}}$ remains larger than $U_{\text{Mott}}$ all the way up to 50 GPa [@note_pressure], the trend towards the restoration of the metallic phase is evident [@noteU]. Assuming the absence of a structural phase transition [@note_pressure], this suggests a transition into a high-pressure double Dirac metal for $\simeq 60$ GPa.
[*Conclusion.*]{} Our analysis identifies [[Bi$_2$CuO$_4$]{}]{} as a prototypical material where crystal structure and orbital character conspire to give rise to correlated double Dirac fermions close to the Fermi level. At ambient pressure, the interactions turn out to drive the system into Mott state along with magnetic intra-cell ordering where the double Dirac cone is absent. As a function of hydrostatic pressure, we find that the material could be driven into a metallic state where the double Dirac features would emerge. For this, our ab initio calculations indicate a pressure regime of $\approx 60$ GPa, which is still within the range of experimental pressure cell transport setups. Our study suggests several routes to realize a double-Dirac metal in [[Bi$_2$CuO$_4$]{}]{} at low temperatures. For instance, a combined pressure and doping approach could establish a convenient perturbation of the pristine material in order to render correlated double Dirac fermions accessible to experimental investigation.
We are grateful to B. A. Bernevig for drawing our attention to double Dirac fermions, and to C. Felser for detailed information on [[Bi$_2$CuO$_4$]{}]{} crystals. We thank M. Baldini, J. Cano, A. Schnyder, and A. Toschi for helpful discussions. This work was supported by DFG-SFB 1170 Tocotronics, ERC-StG-336012-Thomale-TOPOLECTRICS, and NSF PHY-1125915. The authors acknowledge computational resources from the Leibniz Supercomputing Centre under the Project-ID pr94vu.
DFT+DMFT Computational Details
==============================
Density functional theory calculations were performed by using the VASP ab-initio simulation package [@VASP1] within the projector-augmented-plane-wave (PAW) method [@PAW; @VASP2]. The generalized gradient approximation as parametrized by the PBE GGA functional for the exchange-correlation potential was used [@PBE], by expanding the Kohn-Sham wavefunctions into plane-waves up to an energy cut-off of 600 eV. The Brillouin zone has been sampled on a 8$\times$8$\times$12 regular mesh, and when considered, spin-orbit coupling (SOC) was self-consistently included [@SOC_VASP]. The low energy model has been extracted by projecting onto Cu d$_{x^2-y^2}$-like maximally localized Wannier functions (MLWF) using the WANNIER90 package [@WANNIER90]. Electron-electron interaction was included within the framework of dynamical mean field theory (DMFT), by mapping the lattice problem into an impurity model subject to a self-consistency condition [@DMFT]. We solved the impurity model by continuous-time quantum Monte Carlo, as implemented in the w2dynamics package [@CTQMC; @w2dynamics].
Setting up the $d_{x^2-y^2}$ low-energy Hamiltonian
===================================================
[*Ab initio*]{} computation of the screened Coulomb interaction
---------------------------------------------------------------
Our aim is to set up a Hamiltonian that faithfully represents the low-energy electronic degrees of freedom of our material. In particular, we need to compute the strength of the Coulomb repulsion experienced by charge carriers that reside in states close to the Fermi level. In a solid, the polarizability of charges screens the bare Coulomb repulsion $v({ \mathbf}{r},{ \mathbf}{r}{ ^\prime})=\frac{e^2}{4\pi\epsilon_0}1/\left| { \mathbf}{r}-{ \mathbf}{r}{ ^\prime}\right|$. When setting up a Hamiltonian for low-energy excitations, its Coulomb interaction thus needs to be screened by all excitations that are not included in that Hamiltonian.
A successful approach to disentangle and determine screening effects for the Coulomb interaction is the constrained random phase approximation (cRPA)[@PhysRevB.70.195104]. To be on par with the many-body Hamiltonian used in the main manuscript, we here (i) include all contributions to the charge polarization except for those that are confined (constrained) to the subspace of Cu-$d_{x^2-y^2}$ bands that constitute the low-energy spectrum, and (ii) express the partially screened Coulomb interaction in a basis of maximally localized Wannier functions[@miyake:085122].
The underlying DFT calculations employ a full potential LMTO method [@fplmto], a Brillouin zone discretized into 8x8x8 $k$-points, local orbitals for the Bi-$5d$ states, and neglects the spin-orbit coupling which was shown to be weak (see main manuscript). The cRPA is used as implemented in Ref. [@miyake:085122] with a 4x4x4 $k$-mesh.
At ambient conditions, we find that the bare, i.e. unscreened, Coulomb interaction $v({ \mathbf}{r},{ \mathbf}{r}{ ^\prime})$ in the Wannier basis $\chi^{\hbox{\tiny W}}_{{ \mathbf}{R}\alpha}({ \mathbf}{r})$ $$\begin{aligned}
&&V^{\alpha\beta\alpha{ ^\prime}\beta{ ^\prime}}_{{ \mathbf}{R},{ \mathbf}{R}{ ^\prime}}=\frac{e^2}{4\pi\epsilon_0} \,\times\\
&&\quad\int d^3r d^3r{ ^\prime}\chi^{\hbox{\tiny
W}*}_{{ \mathbf}{R}\alpha}({ \mathbf}{r}) \chi^{\hbox{\tiny W}}_{{ \mathbf}{R}\beta}({ \mathbf}{r})\frac{1}{\left|{ \hbox{\textbf r}}-{ \hbox{\textbf r}}{ ^\prime}\right|}
\chi^{\hbox{\tiny W}*}_{{{ \mathbf}{R}{ ^\prime}}\alpha{ ^\prime}}({ \mathbf}{r}{ ^\prime}) \chi^{\hbox{\tiny W}}_{{ \mathbf}{R}{ ^\prime}\beta{ ^\prime}}({ \mathbf}{r}{ ^\prime})\nonumber
\label{eqV}\end{aligned}$$ amounts to $V^{\alpha\alpha\alpha\alpha}_{{ \mathbf}{R},{ \mathbf}{R}}=12.0$eV for the on-site intra-orbital component of [[Bi$_2$CuO$_4$]{}]{}. Using the static cRPA polarizability, this interaction is screened down to $U^{\alpha\alpha\alpha\alpha}_{{ \mathbf}{R},{ \mathbf}{R}}=2.1$eV. Figure \[figS1\](a) displays the static ($\omega=0$) values of the density-density components $U^{\alpha\alpha\beta\beta}_{{ \mathbf}{R},{ \mathbf}{R}}$ within the unit-cell ${ \mathbf}{R}=0$. In our case orbital-offdiagonal ($\alpha\ne\beta$) components refer to the interaction between $d_{x^2-y^2}$-derived orbitals of the four equivalent Cu-atoms within the unit-cell. We see that while the decay of the screened interaction with distance is still algebraic (in contrast to a Yukawa-type of potential from e.g., the Thomas-Fermi theory of screening), the nearest neighbor interaction is already reduced to only 0.52eV, i.e. a factor of four smaller than the on-site repulsion. This finding justifies the use of a Hubbard Hamiltonian in which only intra-atomic interactions are retained. Note, however, that the on-site Hubbard $U$ is the energetic price for two electrons to be simultaneously on the same atomic site [*relative*]{} to occupying two different sites. It is apparent that any non-local interaction will reduce the [*effective*]{} on-site interaction in a model that takes into account only local interactions. Indeed, in can be shown, that to first approximation, the effective on-site interaction is simply given by $U^{eff}=U^{on-site}-U^{nearest-neighbor}$[@2015arXiv150807466S]. Therefore, in our case $U^{eff}=1.58eV$. For the one-band/Cu Hubbard model, the gap in the Mott phase will be roughly given by $U^{eff}$. Given that the experimental (optical) gap has been estimated to be 1.3-1.8eV[@Abdulkarem20111443; @doi:10.1021/jp0725533; @doi:10.1021/jp210130v], our Coulomb interaction thus seems very reasonable.
Many-body perturbation theory: renormalization of ligand states and exchange self-energies
------------------------------------------------------------------------------------------
In the preceding section we computed the screened Coulomb interaction on the basis of the DFT eigenvalues and eigenfunctions and set up the effective Hubbard $U$ to be used in dynamical mean-field (DMFT) calculations for the Cu-$d_{x^2-y^2}$ orbitals.
While this procedure gave a reasonable estimate of the Hubbard $U$, let us point out possible short-comings of this approach. To this end, we perform electronic structure calculations beyond DFT, using the quasi-particle self-consistent (QS)[*GW*]{}[@PhysRevLett.93.126406] approach. The results are shown Figure \[figS1\](b-d) and discussed in the following.
#### (1) exchange self-energies of the Cu-$d_{x^2-y^2}$
We first focus on the $d_{x^2-y^2}$ orbitals that are contained in our low-energy many-body Hamiltonian. The latter is solved using DMFT, which provides a local, on-site, self-energy $\Sigma$. As discussed in the main manuscript the system is Mott insulating: The quasiparticle weight $Z=[\left. 1-\partial_\omega\Re\Sigma(\omega)]^{-1}\right|_{\omega=0}$ vanishes, as the self-energy diverges.
While the perturbative [*GW*]{} approximation is incapable to describe the Mott insulating nature of [[Bi$_2$CuO$_4$]{}]{}, we nonetheless witness a considerable correlation-induced narrowing of the Cu-$d_{x^2-y^2}$ dispersion. Within QS[*GW*]{} we find a local quasi-particle weight $Z=0.29$. From this one would, however, expect a bandwidth-narrowing from $\epsilon^{LDA}({ \mathbf}{k})$ to $Z\times \epsilon^{LDA}({ \mathbf}{k})$. Yet, as is apparent in Figure \[figS1\](d), the QS[*GW*]{} bandwidth is only about 50% smaller than in DFT. The reason for this is exchange contributions in the [*GW*]{} self-energy that favour the delocalization of charges and hence an [*increase*]{} in the bandwidth[@jmt_pnict]. This non-local exchange self-energy is included neither in DFT nor DFT+DMFT, and is a major argument for more advanced electronic structure theories, that e.g., combine many-body perturbation theory with DMFT. One example for such a theory is QS[*GW*]{}+DMFT[@jmt_sces14].
In our case, where there is only one orbital per copper site, we can easily incorporate the effects of the non-local self-energies into an effective Hamiltonian beyond DFT (for details, see Ref. [@jmt_sces14]): The aim is to use the QS[*GW*]{} self-energy to construct an improved—[*non-local*]{} self-energy-containing—one-particle Hamiltonian for usage with DMFT, that will add a non-perturbative [*local*]{} self-energy. To achieve this we have to assure that local renormalizations are not taken into account twice (“double-counted”). Hence we need to remove all local contributions from the [*GW*]{} self-energy. Here, this simply amounts to rescaling the QS[*GW*]{} bands with the inverse of the local [*GW*]{} quasi-particle weight $1/Z$. The resulting band-structure is shown in Figure \[figS1\](d); it has a bandwidth that is [*larger*]{} than the DFT one by 27%, as it includes the non-local self-energy that tends to delocalize charge carriers. Again, owing to the one-band nature of our setup, we can alternatively stick to the DFT Hamiltonian and instead [*reduce*]{} the value of the Hubbard $U^{eff}$ by 27%, yielding $1.15$eV.
#### (2) ligand states.
Given the known underestimation of gap values in $sp$-semiconductors and problems of ligand states in e.g., oxides, it stands to reason to question the DFT as viable starting point for screening processes that involve ligand states. So now we focus in our QS[*GW*]{} calculation on bands outside the Cu-$d_{x^2-y^2}$ subspace. We note that there is a finite (upward) downward shift of (conduction) valence states. As a consequence the $d_{x^2-y^2}$ dispersion gets further isolated and screening effects are likely to become less effective. It can thus be expected that performing the cRPA on top of QS[*GW*]{} will result in an [*increased*]{} Hubbard $U$ for the $d_{x^2-y^2}$ orbitals. See also the discussion on CuBi$_2$O$_4$ under pressure below. This effect thus competes with the reduction of the Hubbard $U$ discussed in the preceding paragraph. Performing cRPA calculation on top of QSGW, we find $U^{on-site}=2.9$eV, $U^{nearest-neighbor}=0.63$eV. Together with the band-width widening from the preceding section, we thus find an effective interaction $U^{eff} = (U-Unn)\times(1-0.27) = (2.9-0.6)\times 0.73 = 1.68$eV which is almost the same as the LDA result of 1.58eV used in the main text.
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
[[Bi$_2$CuO$_4$]{}]{} under pressure: changes in the Hubbard $U$
----------------------------------------------------------------
Applying pressure to the system will modify both the one-particle dispersion (see Figure 4 in the main text) and the screened Coulomb interaction (Hubbard $U$). In order to assess whether pressure could drive the system through an insulator-to-metal transition, we here compute the Hubbard $U$ at the experimentally accessible pressure of 50GPa. Quite in general, the impact of pressure onto the Coulomb interaction is two-fold[@jmt_wannier; @jmt_mno]:
#### (1) basis functions.
The Wannier basis in which hopping and interaction parameters are expressed is modified. Quite, counter-intuitively, the increased extent of maximally localized Wannier functions under compression [*increases*]{} the interaction values[@jmt_wannier]. In our case, however, this effect is moderate. Indeed the real-space extend (“spread”) of the Wannier functions increases from 5.42$\AA^2$ at ambient conditions to 6.52$\AA^2$ at 50GPa. In fact we find that the bare, i.e., unscreened Coulomb interaction is basically unaffected by pressure with the on-site matrix element changing from $V=12$eV to $V^{50GPa}=11.7$eV.
#### (2) screening.
In our case the change in the Hubbard $U$ is caused by changes in the particle-hole transitions that screen the bare interaction. These transitions are, within RPA, determined by the one-particle spectrum. As seen in Fig. 4 of the main text, Bi-p orbitals strongly move down in energy and eventually merge into the d$_{x^2-y^2}$ manifold, causing a sizeable increase of the Coulomb interaction screening.
Using this electronic structure of [[Bi$_2$CuO$_4$]{}]{} under pressure, we find a value of $U=1.68$eV for the on-site matrix element, and an effective $U$ that incorporates the nearest neighbor interaction of $U^{eff}=1.36$eV. This value brings us much closer to the interaction-driven insulator-to-metal transition, which happens at $U_c=1.2$eV for the band-structure corresponding to 50GPa. These findings advocate that the double Dirac dispersion can be realized in [[Bi$_2$CuO$_4$]{}]{} under realistic pressures.
pressure 0GPa 30GPa 50GPa
---------------------------- ------ ------- -------
on-site $U$ 2.1 1.80 1.68
nearest-neighbour $U^{nn}$ 0.51 0.36 0.32
$U^{eff}$ 1.58 1.45 1.36
: cRPA values for the interaction in the basis of maximally localized Wannier functions using the LDA band-structure.[]{data-label="tabS1"}
Many-body perturbation theory: momentum-dependence of the self-energy
----------------------------------------------------------------------
![Real parts of the QSGW intra-d$_{x^2-y^2}$ self-energy for all k-points on a 4x4x4 Brillouin-zone mesh (blue), and the local (k-summed) element (red). All curves have been arbitrarily shifted by Re$\Sigma_{loc}(\omega=0)$.[]{data-label="figS2"}](QSGW_sig_re.png){width="\columnwidth"}
![(a) Energy vs. volume curve and (b) pressure vs. volume curve.[]{data-label="figPressure"}](./Fig_pressure_atomic_relaxation.pdf){width="\columnwidth"}
Fig. \[figS2\] shows the intra-d$_{x^2-y^2}$ element of the QSGW self-energy in the maximally localized Wannier basis. As apparent from the graph, the self-energies for different k-points (different blue curves) are all parallel to each other, i.e., they differ only by a constant shift. This means that the dynamics of the self-energy is independent of the momentum. Conversely, this implies that the non-locality in the self-energy is purely static: $\Sigma(k,\omega)=\Sigma(k)+\Sigma(\omega)$. This “space-time separation” of the self-energy is in-line with recent findings using the non-perturbative dynamical vertex approximation for the 3D Hubbard model[@jmt_dga3d].
Hydrostatic Pressure
====================
To estimate the values of pressure we computed the energy versus volume curve, as shown in Fig. \[figPressure\](a). Subsequently this curve was fitted using the following Murnaghan equation of state:
$$\begin{aligned}
E(V) = E_0 + \frac{B_0V}{B_0'}\left[\left(\frac{V_0}{V}\right)^{B_0'}\frac{1}{B_0'-1}+1\right]-\frac{B_0V_0}{B_0'-1}\end{aligned}$$
where $E_0$ is the energy minimum, $V_0$ the volume at the minimum, $B_0$ is the bulk modulus and $B_0'$ is its derivative with respect to pressure. The obtained parameters were used to produce the $P(V)$ curve
$$\begin{aligned}
P(V) = \frac{B_0}{B_0'}\left[\left(\frac{V_0}{V}\right)^{B_0'}-1\right]\end{aligned}$$
shown in Fig. \[figPressure\](b).
![Specific heat $C_V$, AFM-C susceptibility $\chi_{AFM-C}$ and order parameter $M_{AFM-C}$ as a function of temperature evaluated from exchange interactions at $U=1.58$eV.[]{data-label="fig_mc"}](montecarlo.eps){width="\columnwidth"}
U (eV) $J_1$ $J_2$ $J_3$ $J_4$
-------- ------- ------- ------- -------
0.0 4.25 5.5 0.07 4.48
0.5 3.5 4.81 0.06 3.72
1.58 2.38 3.75 0.04 2.56
2.1 2.12 3.38 0.03 2.29
: Estimated exchange interactions (in meV) for different values of the $U$ parameter[]{data-label="tab_J"}
Néel temperature
================
To evaluate the magnetic ordering temperature we map the total energy of different magnetic phases on a Heisenberg model $$H=\frac{1}{2}\sum_{ij} J_{ij}\bm S_i \bm S_j.$$ We consider four exchange interactions $J_1,J_2,J_3,J_4$, denoting the intra-chain interaction between Cu-1(2) and Cu-4(3) along the $c$ axis ($J_1$) and three inter-chain interactions between Cu-1 and Cu-2 ($J_2$), Cu-1 and Cu-3 ($J_3$) and between Cu-2 and Cu-4 ($J_4$). Only the four considered magnetic configurations FM, AFM-C, AFM-G and AFM-A can be accomodated in the unit cell, which allows to estimate three independent exchange interactions. Setting the AFM-C energy as the reference, the Heisenberg parameters can be expressed in terms of total energy differences as $$\begin{aligned}
J_1 &=& \frac{1}{8 S^2}\,\left(\Delta E_{FM}-\Delta E_{AFM-A}\right) \nonumber\\
J_2 &=&\frac{1}{16 S^2}\,\Delta E_{FM} \\
J_3+J_4 &=& \frac{1}{8 S^2}\,\left(\Delta E_{FM}+\Delta E_{AFM-G}-\Delta E_{AFM-A}\right) \nonumber\end{aligned}$$ The ratio $J_3/J_4$ can be then estimated assuming that it is equal to the ratio $(t_{1,3}/t_{4,2})^2$ between the corresponding transfer integrals. The estimated exchange interactions for $S=1$ and different values of the Coulomb parameter $U$ are given in Table \[tab\_J\].
The Heisenberg model is then used to estimate the Néel temperature from classical Monte Carlo simulations, in order to take spatial non-local fluctuations beyond mean field, the latter being the only one included in single-site DMFT. We used a Metropolis algorithm for 14$\times$14$\times$14 supercells and 10$^7$ Monte Carlo steps for averages. The diverging beehaviour of the specific heat shown in Fig. \[fig\_mc\] as a function of temperature allows one to identify the critical temperature at which the magnetic ordering sets in. In Fig. \[fig\_mc\] we also show the AFM-C order parameter and its associated susceptibility, also displaying a divergence at a critical temperature $T_{N}\simeq 56~$ K.
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---
abstract: |
We study the thermal evolution of neutron stars described within the equation of state with induced surface tension (IST) that reproduces properties of normal nuclear matter, fulfills the proton flow constraint, provides a high-quality description of hadron multiplicities created during the nuclear-nuclear collision experiments, and is equally compatible with the constraints from astrophysical observations and the GW170817 event. The model features strong direct Urca processes for the stars above $1.91\, M_{\odot}$. The IST equation of state shows a very good agreement with the available cooling data, even without introducing nuclear pairing. We also analysed an effect of the singlet proton/neutron and triplet neutron pairing on the cooling of neutron stars of different mass. We demonstrate a full agreement of the predicted cooling curves with the experimental data. Moreover, the IST EoS provides a description of Cas A with both paired and unpaired matter.
[Keywords: neutron stars, equation of state, cooling]{}
author:
- 'S. Tsiopelas, V. Sagun'
title: Neutron star cooling within the equation of state with induced surface tension
---
Introduction {#Intro}
============
\
\
\
\
Equation of state {#EOS}
=================
The IST EoS is formulated to include neutrons, protons and electrons. It accounts for strong short range repulsion and relatively weak long-range attraction between nucleons, while electrons are treated as an ideal Fermi gas. The former part of the nucleon-nucleon interaction is modelled with the hard core radius, similarly to the famous Van der Waals EoS. The hard core repulsion of nucleons leads to an appearance of excluded volume. However, contrary to the Van der Waals approximation, the IST EoS instead of the constant excluded volume has a density-dependent one. The key element of the model is the IST coefficient. It accounts not only for the correct values of four virial coefficients of hard spheres, but also extends the causality range of the model to the density range typical for the NSs [@LS2; @violetta1]. Moreover, the long-range attraction and asymmetry between neutrons and protons are accounted via the mean-field potentials, parameters of which were fitted to the properties of matter at saturation density. More detailed information about the model and its application to the NSs can be found in Refs. [@LS2; @Sagun-Lopes22019; @violetta1].
We adopt the softer parameterisation of the IST EoS that corresponds to the set B of Ref. [@Sagun2020] (for simplicity we keep referring to this parameterisation of the model as set B). The set B was chosen due to the better agreement with the astrophysical constraints and results coming from the GW170817 [@LIGO]. It gives the values of the symmetry energy $E_{sym}= 30.0$ MeV, symmetry energy slope $L= 93.2$ MeV and nuclear incompressibility factor $K_{0}= 201.0$ MeV at normal nuclear density that is in full agreement with the present nuclear matter results [@LS2; @Sagun-Lopes22019]. As it is shown on Fig.\[fig:1\] the maximum mass $M_{max}=2.08 M_{\odot}$ is consistent with the recent measurements of the most massive NSs, i.e. PSR J0348+0432 [@2013Sci...340..448A] and PSR J0740+6620 [@2020NatAs...4...72C].
![Mass-radius relation for non-rotating NSs calculated for set B of the IST EoS [@Sagun2020]. Horizontal bands correspond to the two most massive NSs, e.g. PSR J0348+0432 [@2013Sci...340..448A] (magenta band) and PSR J0740+6620 [@2020NatAs...4...72C] (blue band). The shaded grey area represents the M-R constraint taken from Refs. [@2010ApJ...722...33S; @2013ApJ...765L...5S], while the constraint depicted as a cyan area was taken from Ref. . The red line represents the allowed range of NS radius, according to GW170817 event.[]{data-label="fig:1"}](M-R_new "fig:")\
Cooling processes {#cool}
=================
\
\
\
$$\begin{aligned}
n \rightarrow p + e + \bar{\nu}_{e} ,~~~~
p + e \rightarrow n + \nu_{e} \;.\end{aligned}$$
[As it was mentioned above, the realization of these reactions depends on the proton fraction in the star interior. For the IST EoS adopted in this work, these fast processes can only proceed in the core of stars with central densities $ n_{c} $ higher than $n_{DU}=0.862\, fm^{-3}$. For NSs with lower $ n_{c} $, where the Fermi momenta of the involved particles fail to satisfy the kinematic restriction of $p_{F,n} \leq p_{F,p} + p_{F,e} $, neutron $\beta$-decay and its inverse can still proceed with the help of a spectator proton or neutron that provides the extra $ p_{F,i} $ needed for the conservation of momentum. In this case, though, the efficiency of these so-called modified Urca (MU) processes in removing heat from the star is lower than that of the DU ones, resulting in an overall slower cooling of the star. Nevertheless, neutrino emission from MU processes surpasses the rest of the emissivities in the core, unless n and p are in a paired state, which in turn results in a further suppression of the neutrino emission rates. However, pairing partially compensates for the delay in the cooling of the NS core as it introduces PBF, an additional efficient channel for carrying the heat away in the form of neutrino-antineutrino pairs [@Potekhin2015].]{}
![Density dependence of the critical temperature for the considered singlet and triplet neutron gaps (upper panel) and proton singlet pairing gaps (lower panel).[]{data-label="fig:2"}](Tcall_log2 "fig:") ![Density dependence of the critical temperature for the considered singlet and triplet neutron gaps (upper panel) and proton singlet pairing gaps (lower panel).[]{data-label="fig:2"}](p1s0pairing "fig:")
Results {#res}
=======
![Cooling curves for stars of different mass $M/M_{\odot}=1.2, 1.4, 1.6, 1.8, 2.0$ for the case of unpaired matter. $T^{\infty}_{S}$ denotes the surface temperature at infinity. The solid curves correspond to the light-element envelope ($\eta=10^{-7}$), while the dashed curves were obtained for the Fe envelope. The data points are taken from [@2015MNRAS.447.1598B]. []{data-label="fig:3"}](Teff_IST_noSF_FeH "fig:")\
At the beginning we focused on the thermal evolution of NSs without any sort of pairing between the nucleons. As you can see on Fig.\[fig:3\] all the cooling curves depicted in color exhibit a slow cooling due to the domination of the MU processes, since the DU ones are not kinematically allowed. Finally, at central densities of over $n_{DU}=0.862\, fm^{-3}$ of beta-stable and charge neutral matter, which correspond to NSs with masses $M \geq 1.91\, M_{\odot}$, the DU processes are switched on in the core of the star so that it undergoes enhanced cooling (see the black curves on Fig.\[fig:2\]). In order to model the uncertainties of the heat-blanketing effect of the envelope, we compare the thermal evolution of NSs with a non-accreted envelope containing heavy elements (dashed curves on Fig.\[fig:3\]) with the envelope containing light elements (solid curves on Fig.\[fig:3\]).\
![The same as Figure \[fig:3\], but considering the effect of neutron superfluidity in the $^{1}S_{0}$ channel via the SFB model [@Schwenk2003] and proton superconductivity in the $^{1}S_{0}$ channel with the AO model [@Amundsen1985a] (upper panel) and CCDK model [@Chen1993] (lower panel).[]{data-label="fig:4"}](Teff_IST_SF9_FeHexp "fig:") ![The same as Figure \[fig:3\], but considering the effect of neutron superfluidity in the $^{1}S_{0}$ channel via the SFB model [@Schwenk2003] and proton superconductivity in the $^{1}S_{0}$ channel with the AO model [@Amundsen1985a] (upper panel) and CCDK model [@Chen1993] (lower panel).[]{data-label="fig:4"}](Teff_IST_SF10_FeHexp "fig:")
![The same as lower panel of Figure 4, but supplemented with the triplet neutron pairing in the core of the star described by the T72 model [@Takatsuka1972] (upper panel) and the AO model [@Amundsen1985b] (lower panel). []{data-label="fig:5"}](Teff_IST_SF6_FeH "fig:") ![The same as lower panel of Figure 4, but supplemented with the triplet neutron pairing in the core of the star described by the T72 model [@Takatsuka1972] (upper panel) and the AO model [@Amundsen1985b] (lower panel). []{data-label="fig:5"}](Teff_IST_SF3_FeH "fig:")
\
\
Conclusions {#Concl}
===========
We studied the cooling of NSs with the novel IST EoS that was previously applied to the analysis of nuclear matter properties, heavy-ion collision experimental data, and was recently generalized for the description of matter inside NSs. The considered model parameterisation is in good agreement with the GW170817 and the recent measurements of the most massive NSs. For the stars with $M \geqslant 1.91\, M_{\odot}$ the model allows for the occurrence of DU processes that leads to much faster cooling. An effect of neutron pairing in $^{1}S_{0}$ and $^{3}P_{2}$ channels, as well as proton pairing in $^{1}S_{0}$ channel is thoroughly analysed with different superfluidity/superconductivity models. In addition, effects of different types of envelopes on the thermal evolution of NSs were also compared. We show that for unpaired matter the obtained cooling curves are fully consistent with the currently existing observational data. Remarkably, the IST EoS with unpaired matter predicts the surface temperature of Cas A both with the fast cooling curve for $M_{DU}=2.0\, M_{\odot}$ with the accreted envelope of light-elements and the low-mass star with the heavy-elements envelope.
The presence of the proton(neutron) superconductivity(superfluidity) in the singlet channel slightly slows down the cooling and makes the agreement with the data even better. The two considered scenarios of the triplet neutron pairing in the core of the star (a shallow and an extended neutron superfluidity) lead to too rapid cooling of all NSs, such that the old stars ($t > 10^{5} $ yr) cannot be reproduced. Thus, we can conclude that the calculations favours the vanishing neutron triplet pairing gap.
Acknowledgements
================
We are thankful for the fruitful discussions and suggestions to M. Fortin and O. Ivanytskyi. V.S. acknowledges the support from the Fundação para a Ciência e Tecnologia (FCT) within the project UID/04564/2020.
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|
---
abstract: 'The astonishing spread of Android OS, not only in smart phones and tablets but also in IoT devices, makes this operating system a very tempting target for malware threats. Indeed, the latter are expanding at a similar rate. In this respect, malware fingerprints, whether based on cryptographic or fuzzy-hashing, are the first defense line against such attacks. Fuzzy-hashing fingerprints are suitable for capturing malware static features. Moreover, they are more resilient to small changes in the actual static content of malware files. On the other hand, *dynamic analysis* is another technique for malware detection that uses emulation environments to extract behavioral features of Android malware. However, to the best of our knowledge, there is no such fingerprinting technique that leverages dynamic analysis and would act as the first defense against Android malware attacks. In this paper, we address the following question: could we generate effective fingerprints for Android malware through *dynamic analysis*? To this end, we propose [DySign]{}, a novel technique for fingerprinting Android malware’s dynamic behaviors. This is achieved through the generation of a digest from the *dynamic analysis* of a malware sample with respect to existing known malware. It is important to mention that: (i) [DySign]{} fingerprints are approximates of the observed behaviors during dynamic analysis so as to achieve resiliency to small changes in the behaviors of future malware variants; (ii) Fingerprint computation is agnostic to the analyzed malware sample or family. [DySign]{} leverages state-of-the-art *Natural Language Processing* (NLP) techniques to generate the aforementioned fingerprints, which are then leveraged to build an enhanced Android malware detection system with family attribution. The evaluation of the proposed system on both real-life malware and benign apps demonstrates a good detection performance with high scalability.'
author:
-
-
-
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bibliography:
- 'reference.bib'
title: 'DySign: Dynamic Fingerprinting for the Automatic Detection of Android Malware'
---
Introduction
============
The rapid growth in technologies triggers the development and evolvement of mobile devices to enhance both economic and social interactions. Hence, mobile applications (referred to as *apps* henceforth) running on smart devices are gaining ubiquity due to their convenience. For instance, nowadays users purchase products online and in retail stores at their fingertips using such *apps* like Apple pay app. However, the growth of the mobile market *apps* has increased the concerns about apps’ security. Android[@android_os] is one of the most adopted mobile OS in smart devices, especially in the emerging Internet of Things (*IoT*) world through Brillo [@brillokey], an Android-based IoT system. However, this *IoT* mega-trend makes Android security more crucial than ever before. This is due to the fact that *IoT* devices are everywhere and control important services in cars[@android_auto], TVs[@android_tv], watches[@android_wear], etc. Consequently, this has motivated malware writers to launch attacks against mobile *apps*. These attacks may cause direct financial losses or sensitive data leakages since some *apps* perform monetary transactions using sensitive information such as credit card numbers and passwords. Malware attacks targeting smart devices may also harm IoT devices. This deployment coverage made Android more tempting for cyber-attackers. For example, according to G DATA [@GData_2015_stats], $1,548,129$ and $2,333,777$ new Android malware were discovered in 2014 and 2015, which represents approximately an average of $4,250$ and $6,400$ new malware per day respectively. Furthermore, about $53$% of malware are SMS Trojans designed to steal funds and personal information from Android-based mobile devices [@Canfora:2015].
In this context, it is a desideratum to develop a scalable, efficient, and accurate framework that tackles two distinct problems: (i) Malware detection - distinguishing malicious from benign applications, and (ii) malware family attribution - assigning malware samples to known families.
**Problem Statement:** In the literature, malware analysis may be categorized into static and dynamic analyses. In static analysis, fingerprints are the first defense line against malware attacks. Two common static analysis techniques are used for fingerprinting Android malware, namely, cryptographic and fuzzy/approximate hashing techniques. Using cryptographic hashes may be easily defeated by the tiniest change in malware Android packaging (APK). The *Fuzzy/approximate hashing* technique is more resilient to small changes. Moreover, it has the possibility of detecting malware variants produced by APK repacking. On the other hand, dynamic analysis misses a fuzzy fingerprint, similarly to APK file fuzzy fingerprint, that could effectively capture Android malware run-time behaviors instead of APK static content. Dynamic analysis is commonly used to obtain dynamic features that are fed to a classifier to detect Android malware or cluster them according to their families. However, this dynamic analysis process suffers from the following main drawbacks: i) The detection needs an intermediate, which is the learning model, between the dynamic analysis of malicious Android APK and the new app to check its similarity; ii) in most cases, the extracted dynamic features are driven by the malware dataset. Accordingly, we choose features that give the most accurate model. Although these features could fingerprint the malware family of the dataset, it is hard to predict if extracting the same features from other Android malicious apps could fingerprint them. As such, malware or family-agnostic features are needed in order to have a resilient fingerprinting technique; iii) directly using dynamic analysis output to compare between malware dynamic behaviors lacks portability due to the inconsistent sizes of the output. Moreover, there is no defined approach for similarity computation.
The aforementioned drawbacks induce the need for a dynamic analysis based fingerprint with a fixed size to achieve portability and compute similarity between malware dynamic behaviors. Such fingerprint should be agnostic to the malware sample or family. Hence, the fingerprint extraction approach needs to be general enough to cover most of the essential information of the dynamic analysis output in most Android malware. In addition, it needs to be scalable to compute a digest relatively to the known malware and achieve a fast detection decision. To the best of our knowledge, there is no such a fingerprinting technique that abstracts the dynamic analysis in one digest for the purpose of Android malware detection.
**Approach Overview:** In this paper, we propose a novel fingerprinting approach, namely , which aims at generating a signature that is based on the dynamic analysis of Android malware apps. In particular, the proposed approach aims at achieving the following properties: (i) fingerprints are approximates of the observed behaviors during dynamic analysis so as to achieve resiliency to small changes in the behaviors of future malware variants; (ii) fingerprint computation is agnostic to the analyzed malware sample or family. We choose these properties since they allow our proposed approach to efficiently detect malware variants and other samples of the same family efficiently. The key idea of lies in the fact that Android malicious apps, such as SMS Trojans, tend to have similar overall dynamic behaviors, which are distinguishable from the behaviors of benign apps. In addition, apps targeted by a given malware tend to share similar behaviors than apps that are targeted by different malware families. In a sandboxing environment, malware runtime behaviors are translated into an analysis report. Therefore, malicious apps with similar behaviors would produce similar analysis reports. In the context of , we leverage the output of Android malware dynamic analysis using *sandboxing* environments to generate *relative* fingerprints from the known Android malware apps analysis reports. More precisely, leverages state-of-the-art Natural Language Processing (NLP) techniques to produce the aforementioned fingerprints using the *bag of words* model in the generation from the analysis reports. Considering the latter as a word makes completely agnostic to the malware sample or family. Furthermore, we leverage to build an enhanced Android malware detection system with family attribution. is evaluated on both real-life malware and benign apps and the obtained results demonstrate high detection and attribution performances.
**Contributions:** In summary, this paper makes the following contributions:
1. We introduce , a novel fingerprinting system for automatically generating dynamic fingerprints for dynamic analysis of Android malware apps.
2. We leverage state-of-the-art of Natural Language Processing (NLP) techniques in order to propose an approach that is resilient to change in the dynamic behaviors of Android malicious apps.
3. We conduct a large-scale evaluation of using $8,639$ malicious and benign apps. Our evaluation demonstrates that achieves a good detection performance with high scalability.
The remainder of this paper is organized as follows: Section \[sec:scenarios\] presents some usage scenarios of . Section \[sec:background\] gives a light background on Android OS. Section \[sec:methodology\] details our methodology. We evaluate in Section \[sec:evaluation\]. In Section \[sec:related\_work\], we discuss the related work. In Section \[sec:conclusion\], we provide some concluding remarks on this research together with a discussion of future research.
Usage Scenarios {#sec:scenarios}
===============
The main aim of is to generate an approximate fingerprint from the dynamic analysis of malicious apps. The fingerprint is generated with respect to a database of known apps analysis. Our main concern after accuracy is scalability of the fingerprinting since is intended to be the first fingerprint’s defense line, along with static file fuzzy fingerprint, to tackle the overwhelming volume of malicious apps on a daily basis. has two main usage scenarios: i) *Mobile OS monitoring*: In this scenario, we have a set of installed apps that run in a given smart device (the number of apps could be fixed in the case of an IoT device since it is mono-task with a deterministic goal). Having a runtime report database of the these apps would help to periodically fingerprint the behaviors of these apps in order to check for the existence of abnormal behaviors. In this scenario, could raise an exception of behavior change after a suspicious update or a hack; ii) *Cloud service analyzer*: In this scenario, is used as a core of cloud checking service of the received analysis reports, either automatically or by user submission, from the Android device of suspicious apps. The goal is to match the runtime analysis against malicious apps. These scenarios are general applications of . However, we believe that it can be extended to many other usages due to the simplicity and scalability of .
Background {#sec:background}
==========
Android Architecture
--------------------
Android has been settled by Android Open Source Project (AOSP) team, maintained by Google and supported by the Open Handset Alliance (OHA) [@oha_handset]. It encompasses the Original Equipment Manufacturers (OEMs), chip-makers, carriers and application developers. Android apps are written in Java. However, the native code and shared libraries are generally developed in C/C++ [@android_ndk]. The typical Android architecture consists of Linux kernel, which is designed for an embedded environment consisting of limited resources. On top of Linux kernel, the native libraries developed in C/C++ support high-performance third-party reusable shared libraries. Moreover, Android apps written in Java are translated into Dalvik bytecode. It is specifically optimized for resource-constrained mobile OS platforms.
Android Threats
---------------
Once the app is installed, it may create undesirable consequences for the device security. Following are some examples of malicious activities that have been reported: i) Personal-information leakage occurs when users give dangerous permissions to malicious apps and unknowingly allow access to sensitive data and its exfiltration without user knowledge or consent; ii) malicious apps can also spy on the users by monitoring their voice calls, SMS/MMS, recording audio/video without user knowledge or consent; iii) compromising the device to act as a bot and remotely control it through a server by sending various commands to perform malicious activities.
DYSIGN Methodology {#sec:methodology}
==================
{width="70.00000%"}
Fingerprint Computation
-----------------------
Our ultimate goal is to automatically fingerprint Android malware based on dynamic analysis. To this end, we use natural language processing techniques, where we consider the output of the dynamic analysis as a plaintext file and model it as a *bag of words*. The latter treats the text document as a set of words separated by predefined delimiters such as *spaces* and *curly-brackets*. Given a set of analysis report *bag of words*, we compute a *relative fingerprint* for each report based on the word frequency in one document and the rest. In other words, we distinguish between the reports by giving a high weight to the words with a high frequency in the given report and low frequency in the others. The result is a vector of words’ weights for each analysis report. To compute ’s vector, we leverage the so-called *Term Frequency-Inverse Document Frequency* *tf-idf* [@itidf_wiki], a well-known technique adopted in the fields of *information retrieval* and *natural language processing*. The latter computes vectors of inputted text documents by considering both the frequency in the individual documents and in the whole set. Let $D=\{d_1, d_2,\ldots, d_n\}$ be a set of text documents, where $n$ is the number of documents, and let $d=\{w_1, w_2,\ldots, w_m\}$ be a document, where $m$ is the number of words in $d$. The *tf-idf* of a word $w$ and document $d$ is the product of *term frequency* of $w$ in $d$ and the *inverse document frequency* of $w$, as shown in Formula \[equ:tfidf\]. The *term frequency* (Formula \[equ:tf\]) is the occurrence number of $w$ in $d$. Finally, the *inverse document frequency* of $w$ (Formula \[equ:idf\]) represents the number of documents $n$ divided by the number of documents that contain $w$ in the logarithmic form. The computation of *tf-idf* is very scalable, which suites our needs (Section \[sec:evaluation\]).
$$\mbox{\it tf-idf}(w,d) = \mbox{\it tf}(w,d) \times \mbox{\it idf}(w)
\label{equ:tfidf}$$
$$\mbox{\it tf}(w,d) = |w \in d, ~d=\{w_1, w_2, ... w_n\}: w = w_i|
\label{equ:tf}$$
$$\mbox{\it idf}(w) = log{\frac{|D|}{1 + |d: w \in d|}}
\label{equ:idf}$$
The result of *tf-idf* is a set of vectors $V=\{v_1, v_2, \ldots, v_n\}$ ( fingerprints) of word weights for each document $d \in D$. Computing the similarity using *DySign* is straightforward using the *cosine similarity* as shown in Formula \[equ:cos\].
$$\mbox{cosine-similarity}(v_1, v_2) = \mbox{cos}(\theta) = \frac{v_1 \cdot v_2}{||v_1||||v_2||}
\label{equ:cos}$$
*How could be used for Android malware detection and family attribution?* We answer this question through an illustrating example, in which we compute fingerprints from the analysis reports of malware samples from Drebin malware dataset [@arp2014drebin; @spreitzenbarth2013mobile] along with benign apps downloaded from Google Play [@google_play]. The example is summarized in Table \[tab:insight\_detection\] and Table \[tab:insight\_attribution\]. How is used for malware detection is illustrated in Table \[tab:insight\_detection\], where we compute the similarity between malware analysis reports and benign ones. This example shows the potential of in distinguishing between malware and benign apps.
\# *App1* *App2* *TFIDF Cosine*
---- --------------------- ---------------------------- ----------------
1 00453ca8 (FakeInst) com.BigBawb.coin.apk 0.19
2 00453ca8 (FakeInst) com.interestcalculator.apk 0.21
3 21262a59 (FakeInst) com.sleggi.MiFreetime.apk 0.16
4 00453ca8 (FakeInst) 21262a59 (FakeInst) 0.42
5 00453ca8 (FakeInst) com.sleggi.MiFreetime.apk 0.27
: Insight of Android Malware Detection[]{data-label="tab:insight_detection"}
First 8 characters from malware hash and its malware family.
As shown in Table \[tab:insight\_attribution\], could be used to segregate between Android malware families by requiring a higher similarity between the fingerprints of the same malware family. Based on these insights, we generalize and build a system on top of for Android malware detection and family attribution.
\# *Malware1* *Malware2* *TFIDF Cosine*
---- --------------------- ------------------------ ----------------
1 090b5be2 (Plankton) bedf51a5 (DroidKungFu) 0.56
2 149bde78 (Plankton) bedf51a5 (DroidKungFu) 0.46
3 090b5be2 (Plankton) 149bde78 (Plankton) 0.71
: Insight of Android Malware Family Attribution []{data-label="tab:insight_attribution"}
First 8 characters from malware hash and its malware family.
*How is agnostic to malware samples and families?* is agnostic by design since no features are extracted specifically for a given malware family or sample. In other words, considers the analysis report as a bag of words. It only considers the frequency of the word in a document relatively to the other ones. This ensures that the extracted information is broad enough to cover most malware samples and without relying on specific features.
Architecture Overview
---------------------
In this section, we present the architecture of framework for Android malware detection, built on top of ’s fingerprint. There are two main processes in framework, as depicted in Section \[fig:approach\_overview\]. i) The first process is building the analysis report database. The initial phase of this process consists of a bulk sandboxing and reports insertion into a database of known Android malware (Algorithm \[alg:init\_db\]). Afterwards, the process proceeds as a continuous task of updating the report’s database with new apps (Algorithm. \[alg:update\_db\]).
ii\) The second process is the detection process, in which we check the runtime behaviors of newly received apps against known malware behaviors. First, the new app is executed in a sandboxing environment during a time $T$ to get the analysis report. The latter will be used along with the database reports to compute the fingerprint using *tf-idf*. Finally, we compute the similarity between the fingerprint of the new app and the existing fingerprints to identify whether it is malicious or not and its family in case it is malicious. The complete process is presented in Algorithm \[alg:dysign\_detection\_process\]. Using ’s fingerprint, we do not only detect malware but also attribute the unknown samples to their Android malware families. Further, we can also ascribe a family to the unknown samples if we already have samples of this family in ’s dynamic analysis database. Algorithm \[alg:dysign\_detection\_process\] describes the process of generating a dynamic fingerprint.
A cornerstone in framework is the sandboxing system, which heavily influences the produced analysis reports. We use *DroidBox* [@droidbox_github], a well-established sandboxing environment based on the Android software emulator [@android_emulator] provided by Google Android SDK [@android_sdk]. Running the app may not lead to a sufficient coverage of the executed app. As such, to simulate the user interaction with the apps, we leverage *MonkeyRunner* [@monkeyrunner], which produces random UI actions aiming for a broader execution coverage. However, this makes the app execution non-deterministic since *MonkeyRunner* generates random actions. Therefore, this yields different analysis reports for every execution, where the accuracy of the results may vary. To tackle this issue, we run the app in a sandboxing environment for a long time $T$ in order to assure the maximum of information in the resulting report. However, a long time $T$ could lead to execution bottleneck since *DroidBox* can only handle one app at a time. In this context, executing the dataset apps in a sandboxing environment during the initial setup of a *reports database* is a computation bottleneck in . This is because of the defined time $T$, during which the app needs to run in order to get the analysis report. Hence, the initialization phase could take a very long time (may reach few days). To overcome this challenge, we develop a multi-worker sandboxing environment to exploit the maximum available resources and boost the initialization setup.Another problem is the similarity computation, which could be a bottleneck for the framework and could lead to inefficient matching against new unknown apps. To address this issue, we resort to LSH K-Nearest Neighbor (KNN) [@lshforest05bawa]. Similarity computation needs to be conducted in an efficient way that is much faster than the brute-force computation. To this end, we leverage *Locality Sensitive Hashing* (LSH) techniques, and more precisely *LSH Forest* [@lshforest05bawa], a tunable high-performance algorithm for similarity computation. The key idea behind *LSH Forest* is that similar items hashed using LSH are most likely to be in the same bucket (collide) and dissimilar items in different ones as we will explain it in the next Section.
Locality-Sensitive Hashing
--------------------------
Our system employs Locality-Sensitive Hashing (LSH) for feature reduction [@dasgupta2011fast; @lshforest05bawa]. The main idea in LSH is to define a hash function [h]{} such that [h(s1) = h(s2)]{} if the two sets of chains $s_{1}$ and $s_{2}$ are similar [@andoni2006near]. The hash is calculated over all sets of traces, and only those with similar hash values are clustered (hashed) to the same bucket. In the case of similarity, similar traces will be hashed to the same bucket. In our case, we assume that most dissimilar pairs will never hash to the same bucket, and therefore will never be checked. Once all traces have been hashed to a corresponding bucket, any bucket containing more than one hash value is identified and a list of candidate traces is extracted. Finally, similarity analysis is performed to rank the candidate pairs obtained from the previous steps. To create the signature from traces, we must use one of the hash function pairs. We choose $minhash$ due to its efficiency. When using $minhash$ (with $N$ unique hash functions) as signatures to represent the register chains, LSH can be used by splitting the minhash values into a signature matrix with $b$ bands consisting of $r$ rows each. Depending on the number of used bands, the number of minhash values for a given band will be the number of minhashes divided by the number of bands $( N / \# bands)$. The number of rows will be equal to the number of register chain minhash signatures. Finally, for each band $b$, the minhash values (the portion of one column within that band) are hashed to one bucket of a larger number of buckets.
Experimental Results {#sec:evaluation}
====================
In this section, we present the evaluation results of our proposed system. The implementation subsection shows the setup of our experiments. To evaluate the performance of malware detection using , we use a mixed dataset, i.e., malware and benign apps. As for the evaluation of the attribution performance, we use a malware-only dataset.
Implementation {#sec:implementation}
--------------
For modularity purposes, is implemented using separate Python scripts, which altogether form our analytical system. The scripts are used for parsing, cleaning, and tf-idf computation. We develop a multi-sandboxing system on top of *DroidBox* to be able to execute multiple Android apps simultaneously by leveraging the multicore CPUs to have numerous instances of *DroidBox*. We also use *MonkeyRunner* to simulate UI interaction with the user. SQLite has been used to store the features due to its efficiency and ease of use.
DataSet
-------
The first step towards the evaluation of is to select appropriate datasets that can be utilized for Android malware fingerprinting. Obtaining representative datasets is a fundamental challenge, and there is certainly a strong need for standard ones. Hence, the utilized dataset consists of: i) *malware-only* dataset using the well-known Drebin dataset [@arp2014drebin; @spreitzenbarth2013mobile], and ii) *mixed* dataset using Drebin dataset along with benign apps downloaded from Google Play [@google_play]. Statistics about the dataset are presented in Table \[tab:dataset\_describe\] and Table \[tab:dataset\_family\_number\]. In Table\[tab:dataset\_describe\], we use a subset of $3,414$ Android malware samples, from Drebin dataset, distributed on $8$ families \[tab:dataset\_family\_number\]. From this dataset, we exclude all malware families with only few samples due to the high skewness of the dataset. This would prevent having, for instance, a family with $800$ samples and other families with only 1, 2, or even 20 samples.
*Drebin Dataset* *Drebin Mixed With Benign*
------------ ------------------ ----------------------------
Total Size 3414 8639
Malware 3414 3414
Benign / 5225
: Android Dataset Description[]{data-label="tab:dataset_describe"}
Malawre Family Number of Samples
--- ---------------- -------------------
0 FakeInstaller 866
1 DroidKungFu 611
2 Opfake 566
3 Plankton 515
4 GinMaster 314
5 BaseBridge 295
6 Iconosys 127
7 FakeDoc 120
8 Benign Apps 5225
: Dataset Description By Malware Family[]{data-label="tab:dataset_family_number"}
Results {#sec:results}
-------
To evaluate our approach using the previous datasets, we split the training data into ten sets, reserving one set as a testing set and using nine sets as training sets. We repeat this process numerous times. We use *precision* (P) and *recall* (R):
$$P =\frac{TP}{ {TP + FP}}, ~R =\frac{TP}{ {TP + FN}}, ~F1 = 2 \times \frac{P \times R}{P + R}
\label{equ:precision_recall_fscore}$$
**Detection Performance:** Since the application domain targeted by *DySign* is much more sensitive to false-positives than false-negatives, we employ the F-measure, where the results of $F_{1}$ measure are summarized in Table \[tab:accuracy\_results\]. We use two types of datasets: (i) The mixed dataset, used for detection performance assessment, and (ii) the malware-only dataset, used to assess ’s family attribution, as shown in Table \[tab:accuracy\_results\]. The obtained results show that our approach achieves good detection and attribution performance in short time.
*F1-Score* *Precision* *Recall* *Time*
-------------------------- ------------ ------------- ---------- --------------
**Mixed (Detection)** **85%** **94%** **78%** **4min 45s**
**Drebin (Attribution)** **80%** **82%** **79%** **2min 20s**
: Detection and Attribution Performance of []{data-label="tab:accuracy_results"}
**Attribution Performance:** Figure \[fig:attribution\_confmatrix\] presents the confusion matrix for a more granular view of ’s family attribution. The darker, in the matrix, is the diagonal, the more accurate is the attribution. However, due to the unbalanced malware families (Table \[tab:dataset\_family\_number\]), there are some cells in the diagonal that are more darker because of the high number of samples in that family in the testing set. For this reason, we apply the log function on the original confusion matrix to have clearer results. Notice that all the produced results are based on the sandboxing reports of only $T=15s$ for each app whether it is a malware sample or a benign app. Therefore, the accuracy could be significantly improved by having a longer time $T$.
**Reports Size Analysis:** Figure \[fig:size\_ditribution\_comparision\] shows the size distribution of the analysis reports. Figures \[fig:ben\_size\_byte\] and \[fig:mal\_size\_byte\] show the size distribution in *bytes* for benign and malware reports respectively. To enhance the readability of the results, we apply the log function on the *byte* distributions. The results are shown in Figures \[fig:ben\_size\_log\] and \[fig:mal\_size\_log\] for benign and malware reports. The most noticeable is the size of the malware comparing with benign reports. Malware reports tend to be bigger than benign ones. This difference happens in a very short time since we execute the apps for only $T=15s$. Our observations show that: i) Malicious apps tend to have similar behaviors and are generally eager to access the resources to perform their malicious tasks as soon as they are executed. ii) Malware apps tend to be self-driving, i.e., in most cases, they do not need UI interaction emulator. Instead, for example, they try to connect to a given IP address with a specific payload .
**Accuracy Performance and Dataset Size:** Figure \[fig:accuracy\_scalability\] shows the effect of the dataset size on the detection and family attribution. It also shows the direct relation between the number of samples in the dataset and the accuracy. The bigger is the dataset, the more accurate are the results. However, we could not test for higher scalability since we are limited by the size of Drebin dataset after excluding small families. According to the obtained results with our limited dataset, we conclude that by having a bigger dataset, framework could achieve more accurate results. We let the validation of such conclusion as future work with much larger datasets.
**Scalability Analysis:** shows high scalability, as summarized in Figure \[fig:scalability\_analysis\]. First, computation time is very fast and linearly scalable with the number of reports. Our system could compute *tf-idf* from $100,000$ analysis reports in about $200s$, as shown in Figure \[fig:tfidf\_scalability\]. Notice that we over-sample from our dataset in order to get $100,000$ analysis reports used in the scalability evaluation. Figure \[fig:lsh\_scalability\] shows the linearity of *LSH matching time* with the number of reports. Notice that for a $100,000$-report dataset, we match $10,000$ testing reports against $90,000$ reports in the training dataset.
Related Work {#sec:related_work}
============
In this section, we briefly introduce the existing works on Android malware analysis. They are categorized into static [@arp2014drebin; @feng2014apposcopy], dynamic [@ali2016aspectdroid; @canfora2016acquiring], and hybrid [@bhandari2015draco; @zhang2014semantics].
**Static Analysis Approaches:** Static analysis techniques perform code disassembling and decompilation without actually running it. This approach is undermined by the use of various code transformation techniques [@faruki2015android]. We may divide static analysis based techniques into the following categories: i) **Signature-based analysis:** This analysis deals with extracted syntactic pattern features [@feng2014apposcopy; @karbab2016apkdna; @Karbab:2016:CBC:2991079.2991124], and create a unique signature matching for a particular malware. However, such signature cannot handle new variants of existing known malware. Moreover, the signature database should be updated to handle new variants. AndroSimilar [@faruki2013androsimilar] has been proposed to detect zero-day variants of the known malware. It is an automated statistical feature signature-based method. However, it is sensitive due to code transformation methods. ii) **Resource-based analysis:** The Manifest file contains important meta-data about the components (i.e., activities, services, receivers, etc.) and required permissions. There are some methods that have been proposed to extract such information and subject it to analysis [@chin2011analyzing; @fuchs2009scandroid]. iii) **Permission-based analysis:** Discovering the dangerous permission request is not adequate to proclaim a malware app, but nevertheless, permissions mapping requested and used permissions are an important risk identification technique [@sarma2012android; @barrera2010methodology]. **Dynamic Analysis Approaches.** Dynamic analysis techniques allow us to learn malicious activities. Android app execution is event-based with asynchronous multiple entry points. It is important to trigger those events. Dynamic techniques are divided into the following two categories: i) **Resources usage based:** Some malicious apps may cause Denial of Service (DoS) attacks by over-utilizing the constrained hardware resources. Range of parameters such as CPU usage, memory utilization statistics, network traffic pattern, battery usage and system-calls for benign and malware apps are gathered from the Android subsystem. Automatic analysis techniques along with machine learning techniques are used [@shabtai2012andromaly; @reina2013system; @damopoulos2014best]. ii) **Malicious behavior based:** It is related to abnormal behaviors such as sensitive data leakage and sending SMS/emails [@enck2014taintdroid; @burguera2011crowdroid; @elish2012user; @huang2014asdroid].
Limitations and Concluding Remarks {#sec:conclusion}
==================================
We have reported, in this paper, the first investigation of the possibility of using dynamic features for Android malware fingerprinting. [DySign]{} leveraged state-of-the-art *machine learning* and *Natural Language Processing* (NLP) techniques to produce agnostic fingerprints. The evaluation of [DySign]{} on both real-life malware and benign apps demonstrated a good detection and attribution performances with high scalability. Our work has a few limitations though. First, fingerprinting approach is not deterministic, i.e., multiple executions could lead to slightly different fingerprints. However, the core information captured by such fingerprints is the same. Second, *DySign* detection is limited by the Android malware families in the analysis database, and therefore, it cannot detect malware belonging to new families. We plan to address these limitations in future work. In addition, we suggest exploring the applicability of a hybrid model in our detection system.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the anonymous reviewers for their insightful comments that allowed us to improve this paper.
|
---
abstract: 'Conventional molecular dynamics simulations macromolecules require long computational times because the most interesting motions are very slow compared with the fast oscillations of bond lengths and bond angles that limit the integration time step. Simulation of dynamics in the space of internal coordinates, that is with bond lengths, bond angles and torsions as independent variables, gives a theoretical possibility to eliminate all uninteresting fast degrees of freedom from the system. This paper presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived which are applicable to branched chain molecules with any number of internal degrees of freedom. Equations use the canonical variables and they are much simpler than existing analogs. In the numerical tests the internal coordinate dynamics are compared with the traditional Cartesian coordinate molecular dynamics in simulations a 56 residue globular protein. For the first time it has been possible to compare the two alternative methods on identical molecular models in conventional quality tests. It is shown that the traditional and internal coordinate dynamics require the same time step size for the same accuracy and that in the standard geometry approximation of aminoacids, that is with fixed bond lengths, bond angles and rigid aromatic groups, the characteristic step size is 4 fsec, that is two times higher than with fixed bond lengths only. The step size can be increased up to 11 fsec when rotation of hydrogen atoms is suppressed.'
address:
- 'Laboratoire de Biochimie Théorique, CNRS UPR9080 Institue de Biologie Physico-Chimique 13, rue Pierre et Marie Curie, Paris,75005, France$^1$'
- 'Pacific Institute of Bioorganic Chemistry, Russian Academy of Sciences Vladivostok, 690022, Russia'
author:
- 'Alexey K. Mazur'
title: 'Quasi-Hamiltonian Equations of Motion for Internal Coordinate Molecular Dynamics of Polymers'
---
$^1$Present address. Email: alexey@ibpc.fr
Introduction
============
Computer simulations of molecular dynamics (MD) have become an invaluable tool applied to numerous problems in chemical physics [@Allen:87] and biophysics [@Brooks:88]. Normally molecular movements are simulated by the method of point particles [@Hockney:81], i.e. motion of each atom is computed by Newton’s equations while molecular structures are maintained by harmonic potentials that keep bond lengths and bond angles close to their standard values. The maximum time step with which Newton’s equations can be numerically integrated is limited to rather small values (1 fsec for systems containing explicit hydrogen atoms), in order to reproduce bond-length oscillations accurately. This time step is annoyingly small for simulations of macromolecules, notably biopolymers. The unique molecular properties of biopolymers result from specific concerted atom movements, conformational transitions, which can occur over a very broad time range: from picoseconds to seconds. Because of the necessarily small time step, interesting motions, for the most part, remain beyond the reach of even the fastest computers. This presents the central problem in methodology in the field.
Twenty years ago Ryckaert et al [@SHAKE:] proposed an approach to this problem which is now called constraint dynamics. They found a simple and efficient way to impose holonomic constraints upon atom-atom distances in a system of point particles governed by Newton’s equations. Their well-known algorithm, SHAKE, and a few other techniques based upon the same idea [@Rattle:; @EEM:] are now often used to fix bond lengths, increasing the time step limit to 2 fsec for systems with explicit hydrogens. Constraint dynamics has been extensively reviewed [@Allen:87; @Ciccotti:86; @Tildesley:93]. These simple algorithms can be applied to any molecules regardless of their size, chemical structure etc. However, although it seemed initially that the method could fix not only bond lengths but also, by triangulation, could constrain bond angles, dihedral angles, rigid planar groups etc. [@SHAKE:], this was found to be true only for very small molecules. In large complex polymers like proteins even bond angles cannot be fixed in this way [@vanGunsteren:82].
The intrinsic limitation of constraint dynamics can be qualitatively understood from the underlying physical model. Imposing a distance constraint implies that a reaction force is introduced which is applied along a line joining two atoms. Reactions must be calculated at each time step to balance all other forces in the system, which for large molecules presents a difficult problem because they are all coupled and form a system of algebraic equations solved by iterations. With only bond lengths to hydrogens fixed reactions are coupled in small groups and iterations converge rapidly. With all bond lengths fixed reactions are coupled globally and looping becomes possible. In general, in this case, the convergence is not guaranteed [@Barth:95], but, in practice, it remains acceptable. When bond angles are triangulated coupling is much stronger and convergence becomes too slow. Efforts to overcome these limitations continue [@Durup:91]; in particular, it was shown that an exact non-iterative calculation of reactions is possible, in principle [@Turner:95], but, so far, this is practical only for water molecules [@Miyamoto:92] and for linear unbranched chains in which case the problem can be reduced to inversion of a banded matrix [@EEM:; @Gronbech:94].
Despite these difficulties it remains desirable to model chain molecules within the so-called standard geometry approximation [@Go:69], that is with standard fixed conformations of individual chemical groups and with rotations around single bonds as the only motion considered. Rotatable torsion angles completely define molecular conformations and present a convenient minimum set of generalized coordinates which are directly used in static conformational analysis and Monte-Carlo simulations of polymers. In principle, MD simulations in the space of generalized, rather than Cartesian, coordinates are also possible by using one of the appropriate general formalisms of the classical mechanics. In this way all limitations imposed by the fast vibrations of bond lengths, bond angles etc. would be eliminated automatically. This approach was first applied to liquid butane with one internal degree of freedom [@Ryckaert:75] and soon after Pear and Weiner pioneered simulations of linear chains of up to 15 bonds based upon the Newton-Euler equations of rigid body dynamics [@Pear:79]. Interest in this line of research next decreased because of its complexity and because it initially appeared that constraint dynamics could solve all problems, but the difficulties encountered have led to a revival of work.
To date several research groups have reported attempts to find a suitable technique for MD simulations in generalized coordinates . Both methods of rigid body dynamics and the Lagrange-Hamilton formalism [@BKS0:; @BKS1:; @BKS2:; @Dor-int:; @Gibson:90] were employed. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as [*internal coordinate molecular dynamics*]{} (ICMD). The full set of internal coordinates includes not only torsions, but also bond lengths and bond angles, and in some approaches these variables may also be free, if desired. The term [*internal coordinate molecular dynamics*]{} underlines the main distinction of these approaches from conventional MD, i.e. they all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space.
None of the ICMD techniques has yet been sufficiently well developed to reach its main goal, that is, to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than the classical MD with bond length constraints. Compared with the pure Newtonian dynamics there are two obvious problems with ICMD. The first is the derivation of equations of motion for large molecules. This task seemed too complicated at the beginning [@SHAKE:], but several acceptable solutions have been proposed, which represent the main achievement of the early reports on the subject [@Pear:79; @Perram:88; @BKS0:; @BKS2:; @Rice:94]. The second problem is the cost of additional calculations to be performed in order to obtain generalized accelerations. This must be low enough to be compensated by an increase in the step size. Among these calculations the necessity to invert the mass matrix of the system is an evident obstacle. This matrix is diagonal for the Newton’s equations but for internal coordinates it is a full positive definite matrix, therefore, a direct inversion scales as $O(n^{3})$ with the number of degrees of freedom and this quickly becomes impractical when n exceeds 100. Fortunately, a solution of this problem was found some time ago in robot mechanics [@Vereshchagin:74; @Featherstone:87; @Rodriguez:87; @Rodriguez:92], namely, there are several recursive techniques that allow one to compute exact generalized accelerations for $O(n)$ calculations if the system can be treated as a tree of articulated rigid bodies. Two such algorithms have been employed recently [@Mathiowetz:94; @Rice:94] for ICMD and it was shown that, as in Newtonian dynamics, the cost of the time step can be made close to that of the evaluation of forces [@Mathiowetz:94].
In spite of this progress the capabilities of ICMD still are not obvious because it is unclear how much the time step can be increased in practically interesting cases. The uncertainty is caused by two different reasons. First, biopolymers with only torsion degrees of freedom are very complex, heterogeneous and essentially unharmonic systems in which it is usually impossible to distinguish the fastest motions [*a priori*]{} by intuition. For instance, in simulations of small peptides time steps can be increased up to 15-20 fsec [@Dor-int:; @Mathiowetz:94; @Dor-enk:; @Pet-La:] and one might expect that a considerable increase of the step size should be possible for larger molecules as well. Surprisingly, the first simulations of torsion dynamics of proteins reported recently [@Mathiowetz:94; @Rice:94] encountered a step size barrier of only 2 fsec, the same as in the constraint dynamics with fixed bond lengths. This unusual effect of molecular size still have no clear explanation. Most of the uncertainty, however, results from unresolved numerical problems involved in integration of ICMD equations of motion. Their analytical form does not give a possibility to employ common symplectic MD integrators and that is why various general-purpose predictor-corrector schemes were usually employed. It is well-known, however, that in MD applications such algorithms tend to loose stability with lower step sizes. In addition, it was shown that they poorly conserve momenta when applied to equations of motion in generalized coodrinates [@Dor-int:].
This study presents an attempt to overcome the numerical difficulties in ICMD reported in ref. . These difficulties originate from the general form of the equations of motion which depends only upon the choice of coordinates, and so they could not be overcome by simple remedies. That is why an essentially new analytical formulation of ICMD has been developed with new equations which are essentially Hamiltonian, but have a slightly different form. Surprisingly, they turned out to be much simpler than in other analytical ICMD formulations. Numerical integration in the space of conjugate Hamiltonian variables eliminates all earlier problems associated with poor conservation of momenta [@Dor-int:] and results in trajectories as stable as those obtained with Newtonian MD.
This paper principally presents an analytical background, but includes representative tests in order to obtain reliable estimates of the step size in a few important simulation modes and to make sure that there are no hidden numerical problems that can undermine the method. It turns out that for protein torsion dynamics with all explicit hydrogens the optimal time step is 4 fsec, two times larger than with only bond lengths fixed and the same as that for rigid water simulations [@Fincham:92]. This value is probably determined by fast collisions and rotations of hydrogen atoms because without explicit hydrogens or with weighted inertia of hydrogen-only rigid bodies the possible time step is beyond 10 fsec. The computer cost of a time step for the torsion dynamics is similar to that of usual constraint dynamics with fixed bond lengths. It is, therefore, a method of choice for long simulations in which the accuracy of the standard geometry approximation is acceptable. It should be also stressed that ICMD qualitatively differs from the constraint dynamics in its efficiency with respect to the number of constraints: additional constraints in ICMD result in an increased speed of calculations.
Derivation of Equations
=======================
Let ${\bf r}_\alpha $ denote the position vector of atom $\alpha $. The conformation of the molecule is specified by the set of $N$ such vectors {${\bf r}_\alpha \}$. Kinetic energy $$K=\sum_{\alpha =1}^N\frac{m_\alpha{\bf\dot r_\alpha}^2}2$$ where overdot denotes time derivative; the potential energy $$U=U(\{{\bf r}_\alpha \})$$ and the Lagrangian $$L=K-U$$ Assuming that the set of generalized coordinates is defined $\{\theta_i\}\ i=1,...n$, so that $${\bf r}_\alpha ={\bf r}_\alpha \left( \{\theta _i\}\right)$$ the Lagrangian equations of motion are $$\frac d{dt}\frac{\partial K}{\partial \dot\theta _i}-\frac{\partial K}{\partial
\theta _i}=-\frac{\partial U}{\partial \theta _i}$$ Now consider how molecular conformation is specified by bond lengths, bond angles and dihedrals. If in Fig.1 the Cartesian coordinates are given for atoms $\alpha$, $\beta$ and $\gamma$, ${\bf r}_\delta$ is computed from ${\bf r}_\alpha$, ${\bf r}_\beta$ and ${\bf r}_\gamma$ given the values of dihedral $\omega$, planar angle $\varphi$ and bond length $l$ $${\bf r}_\delta ={\bf r}_\delta({\bf r}_\alpha,{\bf r}_\beta,{\bf r}
_\gamma,\omega,\varphi,l)$$ Similarly, ${\bf r}_\varepsilon$ is computed from ${\bf r}_\beta$, ${\bf r}
_\gamma$ and ${\bf r}_\delta$, and so on. This procedure needs global coordinates of the first three atoms for initialization, but for the moment we skip this point to consider it later. The above process is not just an algorithm for computing {${\bf r}_\alpha\}$ from $\{\theta_i\}$, but also a definition of a specific ordering of atoms and internal coordinates in the molecule. This ordering is indispensable for transformation of $\{\theta
_i\}$ into {${\bf r}_\alpha\}$; it is implicit in Eq.(4) but is revealed when derivatives $\partial{\bf r}_\alpha/\partial\theta_i$ need to be computed. For example, when dihedral $\omega$ in Fig.1 is varied then by definition ${\bf r}_\alpha$, ${\bf r}_\beta$ and ${\bf r}_\gamma$ are not affected while ${\bf r}_\delta$ moves, as well as the following atoms. The infinitesimal displacements of atoms in response to variation of internal coordinates correspond to the motion of a tree: the very first three atoms form the base, internal coordinates close to the base move almost the whole molecule while variables in a specific branch move only the upper part of that branch. Of course, there is no distinction between the base and the tip of the tree in the real motion: when the base gets its six degrees of freedom it becomes equivalent to any of the branches.
Let $D_i$ denote the set of atoms in the molecule that move when internal coordinate $\theta _i$ is varied. Than we have $$\frac{\partial K}{\partial\theta_i}=\sum_{\alpha\in D_i}m_\alpha {\bf\dot r}
_\alpha\frac{\partial {\bf\dot r}_\alpha}{\partial\theta_i}$$ $$\frac{\partial K}{\partial\dot\theta_i}=\sum_{\alpha\in D_i}m_\alpha
{\bf\dot r}_\alpha\frac{\partial{\bf\dot r}_\alpha}{\partial\dot\theta_i}=
\sum_{\alpha\in D_i}m_\alpha{\bf\dot r}_\alpha\frac{\partial{\bf r}
_\alpha}{\partial \theta_i}$$ and Eq.(5) gives $$\sum_{\alpha\in D_i}m_\alpha{\bf\ddot r}_\alpha\frac{\partial{\bf r}_\alpha}
{\partial \theta_i}=-\frac{\partial U}{\partial \theta_i}$$
In order to make the following step it is necessary to assume that there are no loops in the molecule. Internal coordinate description of flexible loops is a separate problem which needs special treatment, but without loops the whole set $D_i$ moves as a single rigid body if only $\theta_i$ is varied. If $\theta_i$ is a bond lengths $D_i$ is translated along the bond so that for any atom $$\delta{\bf r}_\alpha={\bf e}_i\delta\theta_i$$ where ${\bf e}_i$ is the unit vector of the bond. If on the other hand $\theta_i$ is an angle or a dihedral, rotation of $D_i$ occurs: $$\delta{\bf r}_\alpha={\bf e}_i\times\left({\bf r}_\alpha-{\bf r}
_i\right)\delta\theta_i$$ where the cross denotes a vector product, while the unit vector ${\bf e}_i$ and the position vector ${\bf r}_i$ specify the axis of rotation.
For a translational variable Eq.(9) gives $$\sum_{\alpha\in D_i}m_\alpha{\bf\ddot r}_\alpha\frac{\partial{\bf r}_\alpha}
{\partial\theta_i}={\bf e}_i\sum_{\alpha\in D_i}m_\alpha{\bf\ddot r}_\alpha=
{\bf e}_i{\bf\dot P}_i=-\frac{\partial U}{\partial\theta_i}$$ where ${\bf P}_i$ is the total momentum of the set of atoms $D_i$. It is straightforward to show that, in the case of a translational variable, $\partial U/\partial\theta_i$ equals to the sum of the forces applied to atoms of set $D_i$ projected upon vector ${\bf e}_i$, and so Eq. (12) is nothing but the corresponding projection of the Newton equation. For a rotational variable we have $$\sum_{\alpha\in D_i}m_\alpha{\bf\ddot r}_\alpha\frac{\partial{\bf r}_\alpha }
{\partial\theta_i}={\bf e}_i\sum_{\alpha\in D_i}\left({\bf r}_\alpha-
{\bf r}_i\right)\times\left(m_\alpha{\bf\ddot r}_\alpha\right)={\bf e}_i
{\bf\dot M}_i=-\frac{\partial U}{\partial\theta_i}$$ where ${\bf M}_i$ is the total angular momentum of atoms of set $D_i$ around the fixed point in space given by the current value of vector ${\bf r}_i$. Again it is easy to show that, for a rotational variable, $\partial U/
\partial\theta_i$ equals the projection upon ${\bf e}_i$ of the sum of the torques around point ${\bf r}_i$ applied to atoms of set $D_i$, and therefore Eq.(13) is a projection of the Newton equations for the torques.
In order to shorten the notation let us introduce indicator $s_i$ such that
$$s_i=\left\lbrace\begin{array}{l@{\ \ if\ \theta_i\ -\ }l}
1 & rotational\\
0 & translational\\
\end{array}
\right.$$
and unite Eqs.(12) and (13) $$s_i{\bf e}_i{\bf\dot M}_i+\left(1-s_i\right){\bf e}_i{\bf\dot P}_i=
-\frac{\partial U}{\partial\theta_i}$$ Now we need to transform Eq.(15) so that a full time derivative appears on the left. We have $${\bf e}_i{\bf\dot P}_i=\frac d{dt}\left({\bf e}_i{\bf P}_i\right)
-{\bf\dot e}_i{\bf P}_i$$ and $${\bf e}_i{\bf\dot M}_i=\frac d{dt}\left({\bf e}_i{\bf Q}_i\right)
-{\bf\dot e}_i{\bf Q}_i+{\bf P}_i\left({\bf e}_i\times{\bf\dot r}_i\right)$$ where ${\bf Q}_i$ is the total angular momentum of set $D_i$ around moving point given by vector ${\bf r}_i$. Although ${\bf M}_i$ and ${\bf Q}_i$ refer to the same physical quantity distinction between them must be made because these are two different time functions. ${\bf M}_i$ is angular momentum around a fixed node; it equals to ${\bf Q}_i$ at the given moment of time but it has a different time derivative: $${\bf\dot M}_i=\frac\partial{\partial t}{\bf Q}_i
\begin{array}{|l}\\{\bf r}_i=const\\\end{array}$$ and, accordingly, the two values diverge immediately afterwards. The Newton equation (13) is valid for ${\bf M}_i$ but not for ${\bf Q}_i$.
Substitution of Eqs.(16) and (17) into Eq.(15) gives $$\frac\partial{\partial t}\left[s_i{\bf e}_i{\bf Q}_i+\left(1-s_i\right)
{\bf e}_i{\bf P}_i\right]=-\frac{\partial U}{\partial\theta_i}+s_i{\bf\dot e}
_i{\bf Q}_i-s_i{\bf P}_i\left({\bf e}_i\times{\bf\dot r}_i\right)+\left(
1-s_i\right){\bf\dot e}_i{\bf P}_i$$ Now consider more carefully the term in the brackets on the left. By a simple transformation we get $$s_i{\bf e}_i{\bf Q}_i+\left(1-s_i\right){\bf e}_i{\bf P}_i=\sum_{\alpha
\in D_i}m_\alpha{\bf\dot r}_\alpha\left[s_i{\bf e}_i\times\left({\bf r}
_\alpha-{\bf r}_i\right)+\left(1-s_i\right){\bf e}_i\right]=\frac
{\partial K}{\partial\dot\theta_i}=\frac{\partial L}{\partial\dot\theta _i}$$ i.e. this is nothing but the conjugate momentum corresponding to variable $\theta_i$ and Eq.(19) is just one of the Hamiltonian equations. We have, therefore $$\sum_{k=1}^na_{ik}\dot\theta_i=s_i{\bf e}_i{\bf Q}_i+\left(1-s_i\right){\bf e}
_i{\bf P}_i$$ where $\left\{a_{ik}\right\}$ are the coefficients of the mass matrix of the system. Eqs.(19) and (21) present the resultant equations of motion to be integrated by a computer. This is done in two steps. At first the r.h.s. of Eq.(19) is evaluated and the time step is made for momenta. At the same time momenta propagated from the previous step are substituted into the r.h.s. of Eq.(21) and the linear system is solved by inverting the mass matrix, which is done by using the recursive algorithm by Rodriguez et al [@Rodriguez:87; @Rodriguez:92]. This algorithm has been developed within the context of the Newton-Euler approach to rigid body dynamics, where each its step has a physical interpretation and corresponds to a special decomposition of reactions in hinges between rigid bodies. Here these interpretations are lost, but due to the same structure of matrix $\left\{
a_{ik}\right\}$ it still can be applied just as a formal mathematical procedure. The generalized velocities thus obtained are used to make the time step in generalized coordinates. A specific example of an integrator for Eqs. (19) and (20) employed in the numerical tests reported below is detailed in the Appendix.
Eq.(19) is rather simple and presents no computational problems. All terms on the right are familiar functions of atom coordinates and atom velocities. It is clear that in an unbranched chain molecule sets $D_i$ can be ordered so that $$D_1\supset D_2\supset ...\supset D_{n-1}\supset D_n$$ therefore all ${\bf P}_i$ can be computed starting from the tip of the chain, moving to the base and at the ith variable adding only the contribution from subset $D_i/D_{i+1}$. In order to compute ${\bf Q}_i$ angular momenta of sets $D_i$ are first computed with respect to the zero of global coordinates in the same way as momenta. After that ${\bf Q}_i$ are computed by moving the ith node of rotation to ${\bf r}_i$ The total force applied to atoms of set $D_i$ is computed similarly to ${\bf P}_i$ and the total torque - similarly to ${\bf Q}_i$. It can be noted that the recurrent calculations of derivatives of the conformational energy used in this field for a long time [@BKS0:; @Noguti:83; @Abe:84] are nothing but the above summation of forces and torques. Due to these algorithms computations of forces in ICMD is in fact similar to the classical MD, contrary to the common oppinion [@Jain:93; @Mathiowetz:94; @Rice:94]. Generalization of the above procedure for a branched chain is straightforward. These computations present only technical difficulties and there is no need to detail them here. The computer time necessary to evaluate the non-potential terms in Eq.(19) is negligible.
It seems reasonable to call Eqs.(19) and (21) Quasi-Hamiltonian because of the meaning of the parameters involved. One should note that there is already some confusion in the classification of equations of motion in internal coordinates. It certainly has little sense to base such classification upon the method used for the derivation of equations because their final form depends only upon the choice of coordinates. For example, by substituting Eq. (21) into Eq. (19) we can get the earlier equations for generalized accelerations [@BKS0:] which are equivalent to any other equations for accelerations of internal coordinates [@Jain:93]. Inversely, Eq. (19) may be obtained by separating a full time derivative in the corresponding equations for generalized accelerations derived by any appropriate technique.
Conservation of Momenta
=======================
Let us consider now the generalized coordinates that position in space the first three atoms in the molecule. These coordinates may be called external and they affect positions of all other atoms, therefore ${\bf P}_1$ and $
{\bf Q}_1$ are the total momentum and angular momentum around a node which is yet to be specified. In the absence of external field $\partial U/
\partial\theta_i=0$ therefore for external variables the r.h.s. of Eq.(19) includes only inertial terms. All these terms become zero, however, if variable $\theta _i$ can be defined so that the corresponding ${\bf e}_i$ and ${\bf r}_i$ are fixed in space. For instance, if the Cartesian coordinates of the first atom are used as the first three translational coordinates Eq.(19) gives $$\frac d{dt}{\bf P}_x=\frac d{dt}{\bf P}_y=\frac d{dt}{\bf P}_z=0$$ where ${\bf P}$ is the total momentum of the molecule. Thus, the conservation of the total momentum appears to be explicitly present in equations of motion and, more importantly, in their finite-difference approximations. The same property is implicit in the Newton equations. On the face of it, this might seem rather minor, but it appears to be somehow related with the numerical stability with large step sizes. In the Newton’s case, among the principal first integrals of mechanics only the total energy is affected by the numerical approximation errors, while the conservation of momenta commonly is “ideal”, that is it depends only upon the length of the floating point numbers in the computer. In contrast, in the numerical integration of standard ICMD equations for generalized accelerations all first integrals are similarly affected by approximation errors. The importance of this qualitative difference has been recognized in our earlier study [@Dor-int:] where it was possible to significantly improve the numerical stability with large step sizes by enforcing conservation of momenta. By using Eqs. (19) and (20) ideal numerical conservation can be ensured with an appropriate choice of the external coordinates, and we will see below that in this way one manages to obtain the quality of numerical trajectories comparable to that of classical MD.
The above considerations dictate the following procedure for positioning the first three atoms in a molecule. Fig.1 shows the global coordinate frame $xyz$ and the local frame $x^{\prime}y^{\prime}z^{\prime}$ that is bound with the molecule. The origin of the local frame coincides with the first atom (atom $\alpha$ in Fig.1). The second atom ($\beta$) always rests on axis $O^{\prime}x^{\prime}$ and the third atom ($\gamma$) always rests in plane $x^{\prime}O^{\prime}y^{\prime}$. The second atom moves along $O^{\prime
}x^{\prime}$ axis if its bond lengths (bond $\alpha$-$\beta$ in Fig.1) is not fixed. Similarly, the third atom moves in $x^{\prime}O^{\prime
}y^{\prime}$ plane if bond $\beta$-$\gamma$ and/or bond angle $\alpha$-$
\beta$-$\gamma$ are variable. With these conventions, the six rigid body degrees of freedom of frame $x^{\prime}y^{\prime}z^{\prime}$ with respect to the global coordinates $xyz$ complement the internal degrees of freedom in the molecule to the full set required.
Frame $x^{\prime}y^{\prime}z^{\prime}$ is considered to rotate around the zero of the global coordinates, with variations of the first three generalized coordinates, $\delta\theta_1$, $\delta\theta_2$ and $\delta
\theta_3$, corresponding to rotations around axes $x$, $y$ and $z$, respectively. In this case $\dot\theta_1$, $\dot\theta_2$ and $\dot\theta_3$ obtained from Eq.(21) are the components $\omega_x$ $\omega_y$ and $\omega_z$ of the angular velocity of frame $x^{\prime}y^{\prime}z^{\prime}$. They are used to integrate the kinematic equations for the quaternion that controls orientation of frame $x^{\prime}y^{\prime}z^{\prime}$ [@Branets:73; @Evans:77]. Vector ${\bf Q}={\bf Q}_1={\bf Q}_2={\bf
Q}_3$ is the total angular momentum around the global coordinate origin therefore Eq.(19) guarantees conservation of the angular momentum. One can note, however, that since the first atom is moved by variations $\delta
\theta_1$, $\delta\theta_2$ and $\delta\theta_3$ the Cartesian coordinates of the first atom can no longer be taken as independent. However, we can still select the three global translational variables so that variations $\delta\theta_4$, $\delta\theta_5$ and $\delta\theta_6$ correspond to translations of all atoms along global axes $x$, $y$ and $z$. In this case $\dot\theta_4$, $\dot\theta_5$ and $\dot\theta_6$ obtained from Eq.(21) refer to $\upsilon_x$, $\upsilon_y$ and $\upsilon_z$ components of velocity of the point that is rigidly connected with $x^{\prime}y^{\prime
}z^{\prime}$ frame, but instantaneously coincides with the zero of coordinates as the center of rotation. The velocity of the first atom is then computed from its position vector ${\bf r}_1$ and vectors ${\bf\upsilon}$ and ${\bf\omega}$. It is clear that in this case vector ${\bf P}={\bf
P}_4={\bf P}_5={\bf P}_6$ still is the total momentum of the molecule.
Numerical Tests
===============
The formal validity of the analytical formulation of ICMD method described in the sections above has been checked in numerous tests with biopolymers of different size and chemical structure. These calculations clearly confirmed the expected superiority of this new method over our previous formulation [@BKS0:; @BKS1:; @BKS2:; @Dor-int:]. In particular, results of the tests obtained with enforced conservation of momenta [@Dor-int:] have been considerably improved, especially as regards the magnitude of the total energy drift (data not shown). The results of a few representative calculations are shown in Fig. 2. Following many previous methodological MD studies a single protein molecule in vacuum was considered here, and the time step comparisons are made by using the conservation of the total energy to access the accuracy of the numerical trajectory. The protein molecule chosen is the immunoglobulin binding domain of streptococcal protein G [@pgb:] (PGB, file 1pgb in the Protein Database [@PDB:]) which is a 56 residue $\alpha $/$\beta $ protein subunit. PGB is somewhat larger than the commonly used test proteins, but it does not have neither S-S bridges nor proline residues, so that no complications connected with the treatment of the flexible rings are involved. Two indicators of the conservation of the total energy are used: the module of the drift and the relative fluctuation which is computed as $$\delta =\frac{\sqrt{\left\langle \left( E-\left\langle
E\right\rangle \right) ^2\right\rangle }}{\sqrt{\left\langle \left(
K-\left\langle K\right\rangle \right) ^2\right\rangle }}$$ where $\left\langle E\right\rangle $ and $\left\langle K\right\rangle $ are the average total and kinetic energies, respectively. The value of the drift is calculated by a linear regression during 1 psec intervals along the trajectory and averaged.
In Figs. 2(a-d) the ICMD simulations are compared with the classical MD. For the usual MD the popular package AMBER [@AMBER:] with force field AMBER94 [@AMBER94:] was used, with distance dependent dielectric constant, $\varepsilon =r$, and no truncation of non-bonded interactions. In calculations with fixed bond lengths the SHAKE algorithm [@SHAKE:] was employed with a tolerance of 10$^{-6}$. ICMD trajectories were computed by a new program (DY) with the same force filed and all internal coordinates as variables. Special care has been taken to ensure identity of the force filed implementations in the two programs and identity of the starting states. The equilibration was more or less standard and included minimization of the crystal structure followed by a 12.5 psec run starting from Maxwell distribution at 300 K with periodic temperature control and a step size of 0.5 fsec. After that the step size was reduced to 0.25 fsec and a short trajectory of 150 fsec was calculated, with the final part stored and used for generating initial data. The final point was used as the starting state for a test trajectory of 10 psec, with initial half-step velocities taken at appropriate time intervals from coordinates. These last preparations were necessary to provide smooth starts of leapfrog integrators used in AMBER and DY with different time step sizes. The same starting states were imported from AMBER to DY where internal coordinates and generalized velocities were computed from atomic coordinates and velocities. The test trajectory was repeatedly computed with gradually growing step size first in Cartesian and next in internal coordinates.
Figures 2(a,b) present results obtained in calculations with completely free molecular models, i.e. with no constraints. The agreement between the two sets of simulations is close to ideal. Note that the commonly used value of the time step (1 fsec) corresponds to a relative fluctuation of $\delta $ =0.1, which can be reasonably used as a reference level of acceptable accuracy. Results of analogous test with bond lengths constraints applied to hydrogen atoms are shown in Figs. 2(c,d). Again one can note a remarkable agreement between the two different methods. The relative fluctuation of 0.1 corresponds to the well-known time step size level of 2 fsec. A similar agreement was observed in simulations with all bond lengths fixed, and it should be added that absolute values of all energy components were very close in all comparative tests (results not shown). These results demonstrate that ICMD simulations are as accurate as traditional MD and that both techniques suffer from similar time step limitations. Figures 2(a- d), therefore, give an appropriate reference to which standard geometry ICMD simulations may be compared.
The results of ICMD tests performed with the set of variables corresponding to the standard geometry approximation are shown in Figs. 2(e,f). Compared to Figs. 2(c,d) both the relative fluctuation and the drift of the total energy are considerably reduced. The time step size corresponding to the relative fluctuation of 0.1 is 4 fsec, that is two times larger than for all- atom models with fixed bond lengths. This value is close to the well-known step size limit for dynamics of rigid water molecules [@Fincham:92], suggesting that the limitation is probably due to the fast rotation of hydorgen atoms in hydrogen-only rigid bodies like hydroxyl or methyl groups. One can note that the observed increase in the step size is rather moderate compared to earlier estimates obtained in simulations with smaller molecules [@Dor-int:]. The apparent reason is that these fast rotations should mainly occur in protein cores, and so the test molecule must be sufficiently large and must include a large number of hydrogen-only rigid bodies for these limitations to be detected.
Figures 2 (g,h) present an additional proof of the fact that, in the above ICMD tests, the time steps are limited by rotation of hydrogens and illustrates one simple possibility to get rid of this limitation. It is clear that rigid body rotation can be slowed down by artificially increasing its inertia. Imagine, for instance, that in ethane (CH$_3$-CH$_3$) both carbon atoms are no longer point masses, but rather spheres that rotate together with the neighboring hydrogens. Rotation of hydrogens will be slowed down but the overall motion of the molecule will be perturbed only slightly because atom masses do not need to be changed. This rather general trick can be employed at all levels for balancing the time scales of different motions in any hamiltonian system. Note, for instance, that a similar approach is applied to electron degrees of freedom in the Car-Parrinello method [@Car:85] and that the so called weighted mass MD [@Pomes:90; @Mao:91] is also based upon this idea. Results presented in Fig. 5 (g,h) were obtained with point masses of atoms in the rotation nodes of hydrogen-only rigid bodies replaced by spheres with the same mass and fixed moments of inertia equal to 15 (atom mass unit)$\cdot$ Å$^2$. As a result, both the drift and the relative fluctuation are significantly reduced compared to Fig. 2(e,f). The time step size that corresponds to the relative fluctuation of 0.1 is about 11-12 fsec.
Some comparative data on the computer cost of simulations in AMBER and in DY in different modes are given in the Appendix. Overall Table 1A shows that ICMD is complementary to the traditional MD. As expected, for unconstrained simulations and for the two usual modes of bond length constraints AMBER is somewhat faster than DY. On the other hand, in the standard geometry simulations the cost of a time step in ICMD is roughly the same as in constraint dynamics with fixed bond length to hydrogens. Thus, the observed increase in the step size is obtained essentially for no cost, and in terms of computer time per picosecond ICMD gives the best score.
Conclusions
===========
In recent years there has been a slow but steady progress in the development of techniques for ICMD. As shown here, improved ICMD methodology now makes possible simulations of torsion dynamics of biopolymers that fulfill the same strict criteria upon accuracy and stability as classical MD. Overall ICMD simulations within the standard geometry approximation are at least two times faster than constraint dynamics.
It should be kept in mind, however, that the numerical tests presented here mainly serve to confirm the validity of the proposed equations. They do not prove that the method can be freely used in calculations of the physical properties of molecules. It is known that even constraints upon bond lengths can, in certain cases, cause significant perturbations [@Watanabe:95], and that fixing bond angles may significantly affect the conformational statistics via the so-called metric tensor [@Fixman:74; @Fixman:78; @vanGunsteren:80]. Effects of each type of constraint upon the measured properties should, therefore, be thoroughly studied. These reservations, however, are not that important for many applications in structural refinement of biopolymers [@Rice:94], in conformational searches and in simulations of protein folding.
This study was initially motivated and partially based upon unpublished results obtained in collaboration with V. E. Dorofeyev [@Dor-phd:]. I wish to thank Dr. R. Lavery for useful discussions, without his support this work could have not be finished. I also would like to thank Dr. G. Rodriguez of the Jet Propulsion Laboratory for making the set of his reprints available for me in Russia.
Implicit Leapfrog Integrator {#implicit-leapfrog-integrator .unnumbered}
============================
To simplify notation let us consider the case of a single variable. Let $\theta$ and $p$ denote the generalized coordinate and the corresponding conjugate momentum. Eq. (19) may be rewritten as $$\dot{p}=g+f$$ where $g=dU/d\theta$ and $f$ denotes the inertial term. Calculation of $g$ may be denoted as $\left(\theta\right)\rightarrow g$, calculation of $f$ as $\left(\theta,\dot{\theta}\right)\rightarrow f$, etc. The on-step values are denoted as $\theta_n,\theta_{n+1},...$, etc. and the half-step values as $\theta_{n-\frac 12},\theta_{n+\frac 12},...$, etc. Then the sequence of steps in the leapfrog integrator is expressed as $$\left(\theta_n\right)\rightarrow g_n$$ $$p_{n+\frac 12}=p_{n-\frac 12}+\left( g_n+f_{n-\frac 12}\right) h$$ $$\theta_{n+\frac 12}=\theta_{n-\frac 12}+h\dot{\theta}_{n-\frac 12}$$ $$\left(\theta_{n+\frac 12},p_{n+\frac 12}\right)\rightarrow\dot{\theta}
_{n+\frac 12}$$ $$\left\lbrace
\begin{array}{rclr}
\theta_{n+\frac 12}&=&\theta_{n-\frac 12}+\left(\dot{\theta}
_{n-\frac 12}+\dot{\theta}_{n+\frac 12}\right)\frac h2&\ \ (a)\\
\left(\theta_{n+\frac 12},\dot{\theta}_{n+\frac 12}\right)&\rightarrow&
f_{n+\frac 12}&\ \ (b)\\
p_{n+\frac 12}&=&p_{n-\frac 12}+g_nh+\left(f_{n-\frac 12}
+f_{n+\frac 12}\right)\frac h2&\ \ (c)\\
\left(\theta _{n+\frac 12},p_{n+\frac 12}\right)&\rightarrow&
\dot{\theta}_{n+\frac 12}&\ \ (d)\\
\end{array}
\right.$$ $$\theta_{n+1}=\theta_n+\dot\theta_{n+\frac 12}h$$ where $h$ denotes the step size.
The steps enclosed in the brace are repeated iteratively until convergence of Eq. (6A). The iterative cycle involves two stages with non-trivial calculations given by Eqs (7A) and (9A). The half-step coordinates are corrected in Eq. (6A) therefore the inversion of the mass matrix is implicitly involved in Eq.(9A), which makes the algorithm relatively costly and leaves some room for possible improvements.
Some estimates of the CPU time made in the course of the numerical tests are given in Table 1A. The measurements were made on a Silicon Graphics Crimson R4000-50 work station. DY was complied by a standard C-compiler with the lowest level of optimization. In all calculations a relative tolerance of 10$
^{-6}$ was used as the criterion of convergence of Eq.(6A). The cost of the matrix inversion calculations scales roughly linearly with the number of variables therefore it is much less expensive in the standard geometry calculations. The average number of iterations depends upon the molecular model and also upon the step size. With the optimal time steps it was minimal in the standard geometry simulations (3.9) and maximal in simulations with fixed bond lengths to hydrogens (9.5). Thus the standard geometry approximation is the cheapest both because of the reduced number of variables and faster convergence. Evaluation of forces takes similar time in AMBER and in DY.
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A B C D
------- ------ ------ ------ ------
AMBER 1.10 1.12 1.20 -
DY 1.60 2.33 1.97 1.12
: Timing Comparisons. CPU time in seconds per one step of dynamics.
|
---
author:
- 'Oluwasanmi Koyejo, Cheng Lee, and Joydeep Ghosh, [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'rank\_GP.bib'
title: 'The trace norm constrained matrix-variate Gaussian process for multitask bipartite ranking'
---
\[0pt\]\[0pt\]
Introduction {#sec:introduction}
============
is the task of learning an ordering for a set of items. In bipartite ranking, these items are drawn from two sets, known as the positive set and the negative set. Bipartite ranking involves learning an ordering that ranks the positive items ahead of the negative items [@cortes04; @agarwal05; @kotlowski11; @gao12]. This paper proposes a generative model for bipartite ranking and an extension of bipartite ranking to the multitask domain. Our approach combines a latent multitask regression function with task-wise ordered observation variables. We employ a non-parametric matrix-variate Gaussian process prior for the multitask regression. Further, we propose a novel trace constrained variational inference approach that imposes useful low rank structure on the multitask regression.
Multitask learning (MTL) exploits inter-task relationships to improve the prediction quality over single task learning[@evingou04; @evingou05]. An important class of methods in this domain are based on the matrix-variate Gaussian process (MV-GP) and closely related models for vector valued reproducing kernel Hilbert space (RKHS) function estimation [@alvarez11]. The MV-GP is an extension of the matrix-variate Gaussian distribution [@stegle11] to (possibly) infinite dimensional feature spaces. Alternatively, the MV-GP may be understood as an extension of the scalar valued Gaussian process [@rasmussen05] to vector valued responses. The MV-GP is a useful model for learning multiple correlated tasks, as it jointly models the correlations across examples, and across tasks. The MV-GP has been applied to link analysis, transfer learning [@yu08], collaborative prediction [@kai09] and multitask learning [@bonilla2008] among other applications.
Our motivating application is the prioritization of disease genes. Genes are segments of DNA that determine specific characteristics; over 20,000 genes have been identified in humans, which interact to regulate various functions in the body. Researchers have identified thousands of diseases, including various cancers and respiratory diseases such as asthma [@gene11], caused by mutations in these genes. The standard approach for discovering disease-causing genes are genetic association studies [@mccarthy08]. However, these studies are often tedious and expensive to conduct. Hence, computational methods that can reduce the search space by predicting a prioritized list of candidate genes for a given disease are of significant scientific interest.
The disease-gene prioritization task has received a significant amount of study in recent years [@vanunu10; @li10; @mordelet11; @natarajan11]. The task is challenging because all the observed responses correspond to known associations and the states of the unobserved associations are unknown, i.e., there are no reliable negative examples. Such problems are also known *single class* or *positive-unlabeled* (PU) learning tasks [@elkan08]. A common approach for this task is to learn a model that that maximizes the classification accuracy between the positive class and the unlabeled class [@rendle09]. In the collaborative filtering literature, such single class tasks have also been addressed using the low rank matrix factorization approach [@pan08].
Recent work suggests that a model trained to rank the positive class ahead of the unknowns can be effective for ranking the unknown positive items ahead of unknown negative items [@elkan08]. Further, the scientific use case for gene prioritization depends on (and is evaluated by) the accuracy of the ranked list produced [@vanunu10; @mordelet11]. For these reasons, disease-gene prioritization is well posed as a bipartite ranking task. A low rank model induces significant correlation between the predictions of different tasks. This assumption matches observations made by domain experts that the gene ranking profiles of diseases are exhibit strong correlations [@sun11; @natarajan11]. This assumption is further validated by the empirical performance of the low rank model.
Low rank structure is a typical constraint in several real world multitask and matrix learning problems. However, despite its use for multitask learning, the standard MV-GP does not model low rank structure. Further, memory requirements for the MV-GP scale quadratically with data size, and naïve computation scales cubically with data size. These computational properties limit the applicability of the MV-GP to large scale problems. The hierarchical factor Gaussian process (factor GP) has been proposed as an alternative for problems with low rank structure [@yu07; @zhu09]. Here, latent row and column factors are drawn from a Gaussian process prior. The result is a model with mean of user-selected rank. We argue that the factor GP is an unsatisfactory model for two reasons: (i) the resulting posterior mean function is the solution of a non-convex optimization problem, and (ii) the posterior covariance is often intractable. We will show that the proposed trace constrained MV-GP provides the same low rank structure benefits without the drawbacks of the factor GP model. The proposed variational inference is jointly convex in the mean and the covariance, and the posterior covariance is given in closed form.
As a computational model, the optimization problem for the mean function of the trace constrained MV-GP is equivalent to kernel multitask learning with the sum of squared errors cost function [@koyejo11] combined with a novel regularizer. We will show that this regularizer can be expressed as a weighted sum of the Hilbert and the trace norms. We call the resulting regularization the *spectral elastic net*, highlighting its relationship to elastic net regularization for variable selection in finite dimensional linear models [@zou05]. To the best of our knowledge, ours is the first application of the spectral elastic net regularizer to matrix estimation and kernel multitask learning.
This paper proposes a novel generative model for multitask bipartite ranking and a novel constrained variational inference approach for the matrix variate Gaussian process applied to the disease-gene prioritization task. The main contributions of this paper are as follows:
- We propose a novel variational inference approach for matrix-variate Gaussian process regression using a trace norm constraint (). This constraint typically results in a regression matrix of low rank.
- We propose a novel generative model for bipartite ranking (). To our knowledge, ours is the first such generative model proposed in the literature.
- We show that variational inference for the latent regression model combined with maximum likelihood parameter estimation for the bipartite ranking is jointly convex ().
- We evaluate the proposed model empirically and show that it outperforms the state of the art domain specific model for the disease-gene prioritization task ().
[**The Kronecker product and the vec operator:** ]{} We will make significant use of the Kronecker product and the $\vec{\cdot}$ operator. Given a matrix $\bA
\in \bbR^{P \times Q}$, $\vec{\bA} \in \bbR^{PQ}$ is the vector obtained by concatenating columns of $\bA$. Given matrices $\bA \in \bbR^{P \times Q}$ and $\bB \in \bbR^{P' \times Q'}$, the Kronecker product of $\bA$ and $\bB$ is denoted as $\bA \kron \bB\in \bbR^{PP'\times QQ'}$. A useful property of the Kronecker product is the identity: $\vec{\bA \bX \bB} = (\trans{\bB} \kron \bA)
\vec{\bX}$, where $\bX \in \bbR^{Q \times P'}$.
The matrix-variate Gaussian process {#sec:mvgp}
===================================
The matrix-variate Gaussian process (MV-GP) is a collection of random variables defined by their joint distribution for finite index sets. Let $\sfM \!\ni\! m$ be the set representing the rows (tasks) and $\sfN \!\ni\! n$ be the set representing the columns (examples), with sizes $|\sfM|=M$ and $|\sfN|=N$. Let $X \sim \gp{\phi}{\cK}$, where $\gp{\phi}{\cK}$ denotes the MV-GP with mean function $\phi$ and covariance function $\cK$. As with the scalar GP, the MV-GP is completely specified by its mean function and its covariance function. These are defined as: $$\begin{aligned}
&\phi(m,n) = \bbE[X(m,n)] \\
&\cK((m,n), (m',n')) =\\
&\quad \quad \quad \quad \quad \bbE[(X(m,n)-\phi(m,n))(X(m',n')-\phi(m',n'))],\end{aligned}$$ where $\bbE[\cdot]$ is the expected value. For a finite index set $\sfM
\times \sfN$, define the matrix $\bX \in \bbR^{M\times N}$ such that $x\smn =
X(m,n)$, then $\vec{\bX}$ is a distributed as a multivariate Gaussian with mean $\vec{\bPhi}$ and covariance matrix $\bK$, i.e., $\vec{\bX}\sim
\g{\vec{\bPhi}}{\bK}$, where $\phi\smn = \phi(m,n)$, $\bPhi \in \bbR^{M\times
N}$, and $k_{(m,n), (m',n')}=\cK((m,n)(m', n'))$, $\bK \in \bbR^{MN \times MN}$.
The covariance function of the prior MV-GP is assumed to have Kronecker product structure [@kai09; @alvarez11]. The Kronecker product prior covariance captures the assumption that the prior covariance between matrix entries can be decomposed as the product of the row and column covariances. The Kronecker prior covariance assumption is a useful restriction as: (i) it improves computational tractability, enabling the model to scale to larger problems than may be possible with a full joint prior covariance, (ii) the regularity imposed by the separability assumption improves the reliability of parameter estimates even with significant data sparsity, e.g., when the observed data consists of a single matrix , and (iii) row-wise and column-wise prior covariance functions are often the only prior information available. A closely related concept is kernel MTL with *separable kernels*. This is a special case of vector valued regularized function estimation where the joint kernel decomposes into the product of the row kernel and the column kernel [@evingou05; @alvarez11; @koyejo11]. Learning in these models is analogous to inference for the MV-GP, with the prior row (resp. column) covariance matrix used as row (resp. column) kernels.
Define the row covariance (kernel) function $\ckm: \sfM \times \sfM \mapsto
\bbR$ and the column covariance function $\ckn: \sfN \times \sfN \mapsto \bbR$. The joint covariance function of the MV-GP with Kronecker covariance decomposes into product form as $\cK\!\left((m,n),(m',n')\right) =
\ckm(m,m')\ckn(n,n')$, or equivalently, $\cK = \cknm$. Hence, for the random variable $X \sim \gp{\phi}{\ckn \kron \ckm}$ and a finite index set $\sfM \times
\sfN$, $\vec{\bX} \sim \g{\vec{\bPhi}}{\kn \kron \km}$, where $\km \in \bbR^{M
\times M}$ is the row covariance matrix and and $\kn \in \bbR^{N \times N}$ is the column covariance matrix.
This definition also extends to finite subsets that are not complete matrices. Given any finite subset $\sfT\!=\!\{t\!=\!(m, n)\, |\, m\in \sfM, n \in \sfN
\}$, where $T=|\sfT|\le M\times N$, the vector $\bx = \left[ x_{t_1} \ldots
x_{t_{T}} \right]$ is distributed as $\bx \sim \g{ \bPhi_{\sfT} }{\bK}$. The vector $\bPhi_{\sfT} =[\phi(1) \ldots \phi(T)] \in \bbR^T$ are arranged from the entries of the mean matrix corresponding to the set $t \in \sfT$, and $\bK$ is the covariance matrix evaluated only on pairs $t,t' \in \sfT \times \sfT$.
Our goal is to estimate an unknown response matrix $\bR \in \bbR^{M \times N}$ with $M$ rows and $N$ columns. We assume observed data consisting of a subset of the matrix entries $\vr = \left[ r_{t_1} \ldots r_{t_{T}} \right]$ collected into a vector. Note that $\sfT \subset \sfM \times \sfN$; hence, the data represents a partially observed matrix. Our generative assumption proceeds as follows (see ):
1. Draw the function $Z$ from a zero mean MV-GP $Z \sim
\gp{ {0} }{\cknm}$.
2. Given $z\smn = Z(\mn)$, draw each observed response independently: $r\smn \! \sim \g{z\smn}{\sigmad}$.
Hence, $\bZ \in \bbR^{M \times N}$ with entries $z\smn = Z(\mn)$ may be interpreted as the latent noise-free matrix. The inference task is to estimate the posterior distribution $Z|\data$, where $\data=\{\vr, \sfT\}$. The posterior distribution is again a Gaussian process, i.e., $Z|\data \sim \gp{\phi}{\Sigma}$, with mean and covariance functions: $$\begin{aligned}
&\phi(m, n) = \kt(m,n)\inv{[\bK + \sigmad \bI_T ]}\vr \label{eq:post_mean}\\
&\Sigma\left((m, n), (m',n')\right) = \label{eq:post_var}\\
& \quad k((m,n),(m',n')) - \kt(m,n)\inv{[\bK + \sigmad \bI_T ]}
\trans{\kt(m,n)}.\notag\end{aligned}$$ where $\vr \in \bbR^T$ corresponds to the vector of responses for all training data indexes $(m,n)\in \sfT$. The covariance function $\kt(m,n)$ corresponds to the sampled covariance matrix between the index $(m,n)$ and all training data indexes $(m',n')\in \sfT$, $\bK$ is the covariance matrix between all pairs $(m,n),(m', n') \in \sfT \times \sfT$, and $\bI_T$ is the $T \times T$ identity matrix. The closed form follows directly from the definition of a MV-GP as a scalar GP [@rasmussen05] with appropriately vectorized variables. The model complexity scales with the number of observed samples $T$. Storing the kernel matrix requires $\cO(T^2)$ memory, and the naïve inference implementation requires $\cO(T^3)$ computation.
Although the matrix-variate Gaussian process approach results in closed form inference, it does not model in low rank matrix structure. The factor GP is a hierarchical Gaussian process model that attempts to address this deficiency in the MV-GP [@zhu09; @yu07]. With a fixed model rank $F$, the generative model for the factor GP is as follows (see ):
1. For each $f\in \{1\ldots F\}$, draw row functions: $U^f \sim \gp{ {0}
}{\ckm}$. Let $\vu_m \in \bbR^F$ with entries $u^f_m = U^f(m)$.
2. For each $f\in \{1\ldots F\}$, draw column functions: $V^f \sim \gp{
{0} }{\ckn}$. Let $\vv_n \in \bbR^F$ with $v^f_n
= V^f(n)$.
3. Draw each matrix entry independently: $r\smn \! \sim
\g{\innere{\vu_m}{\vv_n}}{\sigmad} \; \forall \,(m,n)\in \sfT$.
where $\vu_m$ is the $m^{\text{th}}$ row of $\bU=[\vu^1 \ldots \vu^F]\in \bbR^{M\times
F}$, and $\vv_n$ is the $n^{\text{th}}$ row of $\bV=[\vv^1 \ldots \vv^F]\in
\bbR^{N\times F}$. The maximum-a-posteriori (MAP) estimates of $\bU$ and $\bV$ can be computed as the solution of the following optimization problem: $$\begin{gathered}
\label{eq:factor_gp}
\bU^*,\bV^* = \underset{\bU, \bV}{\arg\min} \fracl{\sigmad}\sum_{(\mn)\in T}
(r\smn-\innere{\vu_m}{\vv_n})^2 \\
+ \tr(\trans{\bU}\inv{\km}\bU) + \tr(\trans{\bV}\inv{\kn}\bV)\end{gathered}$$ where $\tr(\bX)$ is the trace of the matrix $\bX$. However, the joint posterior distribution of $\{\bU, \bV\}$ and the distribution of $\bZ=\bU\trans{\bV}$ are not Gaussian, and the required expectations and posterior distributions are quite challenging to characterize. A Laplace approximation by proposed by [@yu07] and [@zhu09] utilized sampling techniques. Statistically, the factor GP may be interpreted as the sum of rank-one factor matrices. Hence, as the rank $F \to \infty$, the law of large numbers can be used to show that the distribution of $Z$ converges to $\gp{0}{\ckn\kron\ckm}$ [@yu07].
Spectral norms of compact operators {#sec:tracenorm}
-----------------------------------
The mean function of the MV-GP is an element of the Hilbert space defined by the kernels (covariances). We provide a brief overview of some relevant background required for defining this representation and for defining relevant spectral norms of compact operators. We will focus on the MV-GP with Kronecker prior covariance. Our exposition is closely related to the approach outlined in [@abernethy09]. Further details may be found in [@berlinet07].
Let $\hm$ denote the Hilbert space of functions induced by the row kernel $\ckm$. Similarly, let $\hn$ denote the Hilbert space of functions induced by the column kernel $\ckn$. Let $\vx \in \hm$ and $\vy \in \hn$ define (possibly infinite dimensional) feature vectors. The mean function the MV-GP is defined by a linear map $W: \hm \mapsto \hn$, i.e., the bilinear form on $\hk=\hm \times \hn$ given by: $$\phi(m,n) = \inner{\vx_m}{W \vy_n}_{\hm}$$ Let $\cB$ denote the set of compact bilinear operators mapping $\hm \mapsto
\hn$. A compact linear operator $W \in \cB$ admits a spectral decomposition [@abernethy09] with singular values given by $\{\xi_i(W)\}$. [**The trace norm**]{} is given by the *ell*-1 norm on the spectrum of $W$: $$\label{eq:trace_norm}
\normht{\phi}{\hk} = \sum_{i=1}^D \xi_i(W)$$ When the dimensions are finite, is the trace norm of the matrix $\vW \in \bbR^{D\sm \times D\sn}$. This norm has been widely applied to several machine learning tasks including multitask learning [@pong2010; @dudik12] and recommender systems [@srebro05b]. In addition to the trace norm, a common regularizer is the induced [**Hilbert norm**]{} given by the *ell*-2 norm on the spectrum of $W$: $$\label{eq:hilbert_norm}
\normhs{\phi}{\hk} = \sum_{i=1}^D \xi_i^2(W)$$ is equivalent to the matrix Frobenius norm for finite dimensional $\bW \in \bbR^{D\sm \times D\sn}$ computed as: $$\normf{\bW}^2 = \sum_{i=1}^{\min(D\sm, D\sn)} \xi_i^2(\bW)$$
Let $L(\phi, \vr, \sfT)$ represent the loss function for a finite set of training data points $\sfT \in \sfM \times \sfN$ and $Q(\phi)$ be a spectral regularizer. We define the regularized risk functional: $$L(\phi, \vr, \sfT) + \lambda Q(\phi)$$ where $\lambda \ge 0$ is the regularization constant. A representer theorem exists, i.e., the function $\phi$ that optimizes the regularized risk can be represented as a finite weighted sum of the kernel functions evaluated on training data[@abernethy09]. Employing this representer theorem, the optimizing function can be computed as: $$\begin{aligned}
\label{eq:representer}
\phi(m,n)
&=\sum_{m' \in \sfM} \sum_{n' \in \sfN} \alpha_{m',n'} \,\ckm(m, m')\ckn(n,
n')\notag\\
&=\km(m)\bAlpha\trans{\km(n)}\end{aligned}$$ where $\bAlpha \in \bbR^{M \times N}$ is a parameter matrix, $\km(m)$ is the kernel matrix evaluated between $m$ and $m' \in \sfM$, i.e., the $m^{\text{th}}$ row of $\km$, and $\kn(n)$ is the kernel matrix evaluated between $n$ and all $n' \in \sfN$.
[**Computing the norms:**]{} The Hilbert norm can be computed as: $$\begin{aligned}
\label{eq;hilber_norm}
\normhs{\phi}{\hk}
&=\trans{\vec{\bAlpha}}(\kn \kron \km)\vec{\bAlpha} \notag\\
&=\tr(\trans{\bAlpha}\km \bAlpha\kn).\end{aligned}$$ The trace norm can be computed using a basis transformation approach[@abernethy09] or by using the low rank “variational” approximation [@abernethy09; @dudik12].
[**Basis transformation:**]{} With the index set fixed, define bases $\gm\in
\bbR^{M \times D\sm}$ and $\gn\in \bbR^{N \times D\sn}$ such that $\km =
\gm\trans{\gm}$ and $\kn = \gn\trans{\gn}$. One such basis is the square root of the kernel matrix $\gm=\km^\half$ and $\gn=\kn^\half$. When the feature space is finite dimensional, the feature matrices $\xm \in \bbR^{M \times D\sm}$ and $\xn
\in \bbR^{N \times D\sn}$ are also an appropriate basis. The mean function can be re-parametrized as $\phi(m,n) = \gm(m)\bB\trans{\gn(n)}$, where $\bB \in
\bbR^{D\sm \times D\sn}$. Now, the trace norm is given directly by the trace norm of the parameter matrix, i.e., $\normht{\phi}{\hk}=\normt{\bB}$.
[**Low rank “variational” approximation:**]{} The trace norm can also be computed using the low rank approximation. This is sometimes known as the variational approximation of the trace norm [@dudik12]. $$\label{eq:low_rank}
\normht{\phi}{\hk} = \underset{\phi =
\inner{u}{v} }{\arg\min} \; \half\sum_{f=1}^F \left(\normhs{u^f}{\hm} +
\normhs{v^f}{\hn} \right)$$ where $\inner{u}{v}=\sum_{f=1}^F u^f v^f$. This approach is exact when $F$ is larger than the true rank of $\phi$. Note that this is the same regularization that is required for MAP inference with the factor GP model . Hence, when $F$ is sufficiently large, the regularizer in the factor GP model is the trace norm. Unfortunately, it is difficult to select an appropriate rank a-priori, and no such claims exist when $F$ is insufficiently large. With finite dimensions, the variational approximation of the trace norm is given by: $$\normt{\bW} = \underset{\bW =
\bU\trans{\bV} }{\arg\min} \; \half \left(\normf{\bU}^2 +
\normf{\bV}^2 \right)$$ where $\bU \in \bbR^{D\sm \times F}$ and $\bV \in \bbR^{D\sn \times F}.$ This sum of factor norms has proven effective for the regularization of matrix factorization models and other latent factor problems [@ruslan08].
Trace norm constrained inference for the MV-GP {#sec:tracegp}
==============================================
A generative model for low rank matrices has proven to be a challenging problem. We are unaware of any (non-hierarchical) distributions in the literature that generate sample matrices of low rank. Hierarchical models have been proposed, but such models introduce issues such as non-convexity and non-identifiability of parameter estimates. Instead of seeking a generative model for low rank matrices, we propose a variational inference approach. We constrain the inference of the MV-GP such that expected value of the approximate posterior distribution has a constrained trace norm (and is generally of low rank). In contrast to standard variational inference, this constraint is enforced in order to extract structure.
The goal of inference is to estimate of the posterior distribution $p(Z|\data)$. We propose approximate inference using the log likelihood lower bound [@bishop06]: $$\label{eq:var_bnd}
\ln p(\by|\data) \ge \bbE_{\bZ}[\ln p(\by, \bZ)] - \bbE_{\bZ}[\ln p(\bZ)]$$ Our approach is to restrict the search to the space of Gaussian processes $q(Z)=\gp{\psi}{S}$ subject to a trace norm constraint $\normht{\psi}{\cK} \le
C$ as defined in . With no loss of generality, we assume a set of rows $\sfM$ and columns $\sfN$ of interest so $\sfT \in \sfM
\times \sfN$. Let $\bZ \in \bbR^{M \times N}$ be the matrix of hidden variables. Given finite indexes, $q$ is a Gaussian distribution $q(\bz) = \g{\bpsi}{\bS}$ where $\bz = \vec{\bZ} \in \bbR^{M \times N}$, $\bphi = \vec{\bPsi} \in \bbR^{M
\times N}$, and $\bS\in \bbR^{MN \times MN}$. Let $\bP \in \bbR^{T \times MN}$ be a permutation matrix such that $\bS_{T} =
\bP\bS\trans{\bP}$ is the covariance matrix of the subset of observed entries $t
\in \sfT$, and $\bK_{T} = \bP\bK\trans{\bP}$ is the prior covariance of the corresponding subset of entries. Evaluating expectations, the lower bound results in the following inference cost function (omitting terms independent of $\bpsi$ and $\bS$): $$\begin{gathered}
\underset{\bpsi, \bS}{\max} \;
-\fracl{2\sigmad} \sum_{\mn \in
\sfT}(r\smn-\psi\smn)^2 -\fracl{2\sigmad} \tr( \bS_{T}) \notag \\
-\half \trans{\bpsi}\inv{\bK}\bpsi - \half
\tr(\inv{\bK}\bS) + \ln|\bS| \notag\\
\quad \quad \text{s.t.}\; \normht{\psi}{\cK} \le C
\label{eq:all_cost}\end{gathered}$$ where $|\bX|$ is the determinant of matrix $\bX$.
First, we compute gradients with respect to $\bS$. After setting the gradients to zero, we compute: $$\begin{aligned}
\label{eq:qsigma}
\bS^* &= \inv{\left({ \inv{\bK} + \fracl{\sigmad} \trans{\bP}\bP
}\right)} \notag \\
&= \bK - \bK\trans{\bP}\inv{\left({\bK_T +
\fracl{\sigmad} \bI_T }\right)} \bP\bK,\end{aligned}$$ The second equality is a consequence of the matrix inversion lemma. Interestingly, is identical the posterior covariance of the unconstrained MV-GP .
Next, we collect the terms involving the mean. This results in the optimization problem: $$\begin{gathered}
\bpsi^*= \underset{\bpsi }{\arg\min} \;\fracl{2\sigmad} \sum_{m,n \in
\sfT}(r\smn-\psi\smn)^2 +\half \trans{\bpsi}\inv{\bK}\bpsi \notag \\
\quad \quad \text{s.t.}\; \normht{\psi}{\hk} \le C \label{eq:qmean}\end{gathered}$$ This is a convex regularized least squares problem with a convex constraint set. Hence, is convex, and $\bpsi^*$ is unique. Using the Kronecker identity, we can re-write the cost function in parameter matrix form. We can also replace the trace constraint with the equivalent regularizer weighed by $\mu$. Multiplying through by $\sigmad$ leads to the equivalent cost: $$\begin{aligned}
\label{eq:qmean_reg}
\bPsi^*
&=\underset{\Psi }{\arg \min} \; \half \sum_{m,n \in
\sfT}(r\smn-\psi \smn)^2 \notag\\
&\quad+\frac{\sigmad}{2} \tr(\trans{\bPsi}\inv{\km}\bPsi\inv{\kn})
+\mu\sigmad \normht{\psi}{\hk},\end{aligned}$$ Applying the representation , we recover the parametric form of the mean function $\psi \in \cknm$ as $\bPsi = \km \bAlpha
\kn$ where $\bAlpha \in \bbR^{M \times N}$. We may also solve for $\bAlpha$ directly: $$\begin{aligned}
\label{eq:qmean_kernel}
\bAlpha^*
&=\underset{ \bAlpha }{\arg \min} \; \half \sum_{\mn \in
\sfT} \left(r\smn-(\km\bAlpha\kn)\smn \right)^2 \notag\\
&\quad+\frac{\sigmad}{2} \tr(\trans{\bAlpha}\km\bAlpha \kn)
+\mu\sigmad \normht{\psi(\bAlpha)}{\hk}.\end{aligned}$$ where $\psi(\bAlpha)$ is the mean function corresponding to the parameter $\bAlpha$ (see ). The representation of the mean function in functional form is useful for avoiding repeated optimization when testing a trained model with different evaluation sets.
The approximate posterior distribution is itself a finite index set representation of an underlying Gaussian process.
\[thrm:gp\] The posterior distribution $q=\g{\bpsi}{\bS}$ is the finite index set representation of the Gaussian process $\gp{\psi}{S}$ where the mean function $\psi$ is given by and the covariance function $S$ is given by . $g=\gp{\psi}{S}$ is the unique posterior distribution that maximizes the lower bound of the log likelihood subject to the trace constraint $\normht{\psi}{\cK} \le C$.
[**Sketch of proof:**]{} Uniqueness of the solution follows from , which is jointly convex in $\{\bpsi, \bS\}$. To show that the posterior distribution is a Gaussian process, we simply need to show that for a fixed training set $\data$, the posterior distribution of the superset $(\sfM \times \sfN)\cup (m',n')$ has the same mean function and covariance function. These follow directly from the solution of and from (see ). In addition to showing uniqueness, shows how the trained model can be extended to evaluate the posterior distribution of data points not in training.
In the case where a basis for $\km$ and $\kn$ can be found, may be solved using the matrix trace norm approach directly (see ): $$\begin{aligned}
\label{eq:qmean_matrix}
\bB^*=\underset{ \bB }{\arg \min} \;
&\half \sum_{\mn \in
\sfT}\left(r\smn-(\gm\bB\trans{\gn})\smn \right)^2 \notag\\
&\quad+\frac{\sigmad}{2} \normfs{\bB}+ \mu\sigmad \normt{\bB}.\end{aligned}$$ where $\gm\in \bbR^{M \times D\sm}$ is the basis for $\km$ and $\gn\in \bbR^{N
\times D\sn}$ is the basis for $\kn$. $\bB \in \bbR^{D\sm \times D\sn}$ is the estimated parameter matrix. The mean function is then given by $\psi\smn = (\gm\bB\trans{\gn})\smn$.
[**Spectral elastic net regularization:**]{} The regularization that results from the constrained inference has an interesting interpretation as the spectral elastic net norm. As discussed in , the mean function may be represented as $\psi(m,n) = \inner{\vx_m}{W \vy_n}_{\hm}$ for $\vx \in \hm$ and $\vy \in \hn$. The spectral elastic net is given as a weighed sum of the *ell*-2 norm and the *ell*-1 norms on the spectrum $\{\xi_i(W)\}$: $$\label{eq:enet_norm}
Q_{\alpha, \beta}(\psi) = \alpha \sum_{i=1}^D \xi_i^2(W) +
\beta \sum_{i=1}^D \xi_i(W)$$ where $\alpha$ and $\beta \ge 0$ are weighting constants. The naming is intentionally suggestive of the analogy to the elastic net regularizer [@zou05]. The elastic net regularizer is a weighted sum of the *ell*-2 norm and the *ell*-1 norms of the parameter vector in a linear model. The elastic net is a tradeoff between smoothness, encouraged by the *ell*-2 norm, and sparsity, encouraged by the *ell*-1 norm. The elastic net is particularly useful when learning with correlated features. The spectral elastic net has similar properties. The Hilbert norm encourages smoothness over the spectrum, while the trace norm encourages spectral sparsity, i.e., low rank. To the best of our knowledge, this combination of norms is novel, both in the matrix estimation literature and in the kernelized MTL literature. When the dimensions are finite, is given by a weighted sum of the trace norm and the Frobenius norm of the parameter matrix.
We propose a parametrization of the mean function inference inspired by the elastic net [@zou05]. Let $\lambda = \sigmad(1+\mu)$ and $\alpha=\frac{\mu\sigmad}{\sigmad(1+\mu)}$ where $\lambda\ge0$ and $\alpha \in
[0, 1]$. The loss function can be parametrized as: $$\begin{aligned}
\label{eq:qmean_reparam}
\bB^*=\underset{ \bB }{\arg \min} \;
&\half \sum_{\mn \in \sfT}\left(r\smn-(\gm\bB\trans{\gn})\smn \right)^2
\notag\\
&\quad+\frac{\lambda(1-\alpha)}{2} \normfs{\bB}+ \lambda\alpha \normt{\bB}.\end{aligned}$$ The same parametrization can also be applied to the equivalent representations given in and . This spectral elastic net parametrization clarifies the tradeoff between the trace norm and the Hilbert norm. The trace norm is recovered when $\alpha=1$, and the Hilbert norm is recovered for $\alpha=0$. The spectral elastic net approach is also useful for speeding up the computation with warm-start i.e. for a fixed $\alpha$, we may employ warm-start for decreasing values of $\lambda$. Computation of the spectral elastic net norm follows directly from the Hilbert and trace norms. From the variational approximation of the trace norm , it is clear that MAP inference for the factor GP is equivalent to inference for the mean of the trace constrained MV-GP in the special case where $\alpha = 1$ (assuming that the non-convex optimization achieves the global maximum).
[**Non-zero mean prior:**]{} To simplify the explanation, we have assumed so far that the prior Gaussian process has a zero mean. The non-zero mean case is a straightforward extension [@rasmussen05]. We include a short discussion for completeness. Let $b\smn$ represent the mean parameter of the Gaussian process prior, i.e., $Z \sim \gp{b}{\cknm}$. The posterior covariance estimate remains the same, and the posterior mean computation requires the same optimization, but with the observation $r\smn$ replaced by $\tilde{r}\smn = r\smn - b\smn$. The resulting posterior mean must then be shifted by the bias, i.e., $\bbE[Z\smn|\data] = \psi\smn + b\smn$. If desired, this parameter may be easily estimated. Suppose we choose to model a row-wise bias. Let $\sfT_m = \{(m, n) | (m', n) \in T, \,m'=m\}$, then solving the straightforward optimization, we find that the the row bias estimate is given by: $$b_m = \fracl{|\sfT_m|}\sum_{n | \mn \in T_m} r\smn - \psi\smn.$$
Bipartite ranking {#sec:binary}
=================
Bipartite ranking is the task of learning an ordering for items drawn from two sets, known as the positive set and the negative set, such that the items in the positive set are ranked ahead of the items in the negative set [@cortes04; @agarwal05; @kotlowski11; @gao12]. Many models for bipartite ranking attempt to optimize the pair-wise mis-classification cost, i.e., the model is penalized for each pair of data points where the positive labeled item is ranked lower than the negative labeled item. Although this approach has proven effective, the required computation is quadratic in the number of items. This quadratic computation cost limits the applicability of pair-wise bipartite ranking to large scale problems.
More recently, researchers have shown that it may be sufficient to optimize a classification loss, such as the exponential loss or the logistic loss, directly to solve the bipartite ranking problem [@kotlowski11; @gao12]. This is also known as the point-wise approach in the ranking literature. In contrast to the point-wise and pair-wise approach, we propose a *list-wise* bipartite ranking model. The list-wise approach learns a ranking model for the entire set of items and has gained prominence in the learning to rank literature [@pradeep11; @acharyya12] as it comes with strong theoretical guarantees and has been shown to have superior empirical performance.
![Monotone re-targeting searches for an order preserving transformation of the target scores that may be easier for the regressor to fit.[]{data-label="fig:mr"}](mr_partial_order){width="0.8\columnwidth"}
Our approach is inspired by monotone retargeting (MR) [@acharyya12], a recent method for adapting regression to ranking tasks. Although many ranking models are trained to predict the relevance scores, there is no need to fit scores exactly. Any scores that induce the correct ordering will suffice. MR jointly optimizes over monotonic transformations of the target scores and an underlying regression function (see ). We will show that maximum likelihood parameter estimation in the proposed model is equivalent to learning the target scores in MR. Though we show this equivalence for the special case of bipartite ranking with square loss, the relation holds for more general Bregman divergences and ranking tasks. This extension is beyond the scope of this paper. In addition to improving performance, MR has favorable statistical and optimization theoretic properties, particularly when combined with a Bregman divergence such as squared loss. To the best of our knowledge, ours is the first generative model for list-wise bipartite ranking.
Background {#sec:background}
----------
Let $\bbB = \{ +1, -1\}$, and let $\mB{d}$ be the set of binary isotonic vectors (binary vectors in sorted order), i.e., any $\vv \in \mB{d}$ satisfies $\vv \in
\bbB^d$ and $v_i \ge v_j \; \forall j > i$. Similarly, let $\mR{d}$ be the set of real valued isotonic vectors, i.e., any $\vv \in \mR{d}$ satisfies $\vv \in
\bbR^{d}$ and $v_i \ge v_j \; \forall j > i$, then $\vv$ satisfies *partial* order. We state that $\vv$ satisfies *total* order or *strict isotonicity* when the ordering is a strict inequality, i.e., $v_i > v_j \; \forall j > i$. We denote a vector in sorted order as $\sv =
\sort{\vv}$.
Compatibility is a useful concept for capturing the match between the sorted order of two vectors.
\[def:compatilibity\] $\vu$ is compatible with the sorted order of $\sv$ (denoted as $\compat{\vu}{\sv}$) if for every pair of indexes $(i,j)$, $\sv_{i} \ge
\sv_{j}$ implies $\vu_{i} \ge \vu_{j}$.
Compatibility is an asymmetric relationship, i.e., $\compat{\vu}{\sv}
\nRightarrow \compat{\sv}{\vu}$. It follows that sorted vectors always satisfy compatibility, i.e., if $\su$ and $\sv$ are two sorted vectors, then by definition \[def:compatilibity\], $\compat{\su}{\sv}$. Compatibility is straightforward to check when the target vector is binary. Let $\vu \in \bbR^d$ and $\sv \in \mB{d}$, and let $k$ be the be the number of $+1$’s in the sorted vector $\sv$, i.e., the threshold for transition between $+1$ and $-1$. Then $\compat{\vu}{\sv}$ implies that: $$\label{eq:compat_binary}
\exists \,b\in \bbR \quad
\suchthat \quad \underset{1\le i \le k}{\min} u_i
\ge b \ge \underset{k< j \le d}{\max}u_j.$$
There are several ways to permute a sorted binary vector $\sy \in \mB{d}$ while keeping all its values the same. These are permutations that separately re-order the $+1$s at the top of $\sy_m$ and the $-1$s at the bottom. Given $\sy =
\sort{\vy}$, we represent the set of permutations that do not change the value the sorted $\sy$ as $\Gamma = \{ \gamma(\cdot) \; \vert \;\gamma(\sy) = \sy
\}$. It follows that the set $\Gamma$ contains all permutations that satisfy $\compat{\gamma(\sv)}{\sy}$. In other words, all $\vv$ that satisfy $\compat{\vv}{\sy}$ can be represented as $\vv = \gamma(\sv)$ for some $\gamma \in \Gamma$.
We propose a representation for compatible vectors that reduces to permutations of isotonic vectors.
\[prep:represent\] Let $\sv \in \mB{d}$. Any $\vu \in \bbR^d$ that satisfies $\compat{\vu}{\sv}$ can be represented by $\vu = \gamma(\su)$ where $\gamma\in \Gamma$, the set $ \Gamma= \{ \gamma(\cdot) \; \vert \;\gamma(\sv) =
\sv \}$ and $\su \in \mR{d}$.
[**Sketch of proof:**]{} First, we note that by definition of compatibility for binary vectors , any permutation $\gamma\in \Gamma$ satisfies $\compat{\gamma(\su)}{\sv}$. Next, we note that the sorted order is a member of the permutation set. This representation is unique when $\su$ satisfies strict ordering.
The set $\mR{d}$ is a convex cone. To see this, note that the convex composition $\vx = \alpha \vu + (1-\alpha) \vv, \, \alpha \in [0, 1] $ of two isotonic vectors $\vu \in \mR{d}$ and $\vv \in \mR{d}$ preserves isotonicity. Further, any scaling $\alpha \vx$ where $\alpha >0$ preserves the ordering. Let $\Do{d}$ be the set of probability distributions, i.e., $\forall \, \vx \in
\Do{d}$, $x_i \ge 0$ and $\sum_{i=1}^d x_i = 1$. The set of probability distributions in sorted order is given by $\mDo{d} = \mDob{d}$ so for each $\vx
\in \mDo{d}$, $\vx \in \Do{d}$ and $x_i \ge x_j\, \forall i > j$.
\[lem:order\_simplex\] The set $\mDo{d}$ of all discrete probability distributions of dimension $d$ that are in descending order is the image $\bC\vx$ where $\vx \in \Do{d}$ and $\bC$ is an upper triangular matrix generated from the vector $\vv = \{1, \frac{1}{2} \cdots \frac{1}{T}\}$ such that $\bC(i,:)= \{0\}^{i-1} \times \vv(i:)$
Generative model
----------------
Let $y_{m, n} \in \bbB$ be the label for item $n$ in $m^{\text{th}}$ task and let $\sfT_m = \{n\, | \, (m, n) \in \sfT\}$ be the set of items in $m^{\text{th}}$ task so $|\sfT_m|
= T_m$. We define the negative set as $\dzero = \{(m, n) \in \sfT \,| \, y\smn
=-1 \}$ and the positive set as $\dones = \{(m, n) \in \sfT \,| \, y\smn =+1
\}$. For the $m^{\text{th}}$ task, the negative set is defined as $\dzero_m = \{n | (m,
n) \in \dzero \}$ and the positive set as $\dones_m = \{n | (m, n) \in \dones
\}$ so that $\sfT_m = \dones_m \cup \dzero_m$. The vector of labels for the $m^{\text{th}}$ task are given by $\vy_m \in \bbB^{T_m}$. We propose the following generative model for $\vy_m$: $$\label{eq:generative}
p(\vy_m | \vr_m) \propto\prod_{l \in \dones_m}
\prod_{l' \in \dzero_m} \indicator{r_{m, l} \ge r_{m, l'}}.$$ where $\indicator{\cdot}$ is the indicator function defined as: $$\indicator{b}
=\begin{cases}
1& \text{if $b$ evaluates to true },\\
0& \text{otherwise}.
\end{cases}$$ For clarity, we have suppressed the dependence of $p(\vy_m | \vr_m)$ on the sets $\{ \dones_m, \dzero_m\}$.
It is instructive to compare the form of the generative model to the area under the ROC curve (AUC) given by the fraction of correctly ordered pairs: $$\label{eq:auc}
\fracl{\abs{\dones_m}\abs{\dzero_m}}
\sum_{l \in \dones_m}
\sum_{l' \in \dzero_m}
\left(\indicator{r_{m, l} > r_{m, l'}}
+ \half \indicator{r_{m, l} = r_{m, l'}}\right).$$ We note that $p(\vy_m | \vr_m)$ is nonzero if and only if $\vr_m$ satisfies the ordering defined by $\{ \dones_m, \dzero_m\}$. It follows that any vector $\vr_m \;
\suchthat \; p(\vy_m | \vr_m)$ is nonzero also maximizes the AUC.
We can now combine the bipartite ranking model with the latent regression model. The full generative model proceeds as follows (see ):
1. Draw the latent variable $Z$ from a zero mean MV-GP as $Z
\sim \gp{ {0} }{\cknm}$.
2. Given $z\smn = Z(\mn)$, draw each score vector independently as $r\smn \! \sim \g{z\smn}{\sigmad}$.
3. For each task $m \in \sfM$, draw the observed response vector $p(\vy_m |
\vr_m)$ as given by .
Inference and parameter estimation {#sec:inference_and_learning}
----------------------------------
We utilize variational inference to train the underlying multitask regression model and maximum likelihood to estimate the parameters of the bipartite ranking model. This is equivalent to the variational approximation $q(\vr, Z) = \indicator{\vr = \vr^*} q(Z)$ where $\vr = \{\vr_m\}$. The variational lower bound of the log likelihood is given by: $$\ln p(\vy|\data)
\ge \ln p(\vy|\vr) + \bbE_{\bZ}[\ln p(\vr, \bZ)] - \bbE_{\bZ}[\ln p(\bZ)]$$ where $\vy = \{\vy_m\}$. As outlined in , we restrict our search to the space for $q(Z)$ of Gaussian processes $q(Z)=\gp{\psi}{S}$ subject to a trace norm constraint $\normht{\psi}{\cK} \le C$. Evaluating expectations (and ignoring constant terms independent of $\{\vr, \psi, S\}$) results in the following optimization problem: $$\begin{gathered}
\underset{\vr, \bpsi, \bS}{\min} \;
-\sum_{m \in \sfM} \sum_{l \in \dones_m} \sum_{l' \in \dzero_m} \ln
\left( \indicator{r_{m, l} \ge r_{m, l'}} \right) \notag \\
+\fracl{2\sigmad} \sum_{\mn \in
\sfT}(r\smn-\psi\smn)^2 +\fracl{2\sigmad} \tr( \bS_{T}) \notag \\
+\half \trans{\bpsi}\inv{\bK}\bpsi + \half
\tr(\inv{\bK}\bS) - \ln|\bS| \notag\\
\quad \quad \text{s.t.}\; \normht{\psi}{\cK} \le C
\label{eq:all_cost}\end{gathered}$$
Inference and parameter estimation follow an alternating optimization scheme. We alternately optimize each of the parameters $\{\vr, \psi, S\}$ till a local optima is reached. Following , it is straightforward to show that the optimal $S$ is given in closed form and is independent of $\{\vr, \psi\}$. Hence, model training requires alternating between optimizing $\vr^* | \psi$ and optimizing $\psi^* | \vr$. We will show that is convex, and the alternating optimization approach achieves the global optimum. The optimization for $\psi^* | \vr$ follows directly from the discussion in . Hence, we will focus our efforts on the optimization of $\vr^* | \psi$.
Collecting the terms of that are dependent on $\vr$ results in the following loss function for $\vr | \psi$: $$\begin{gathered}
\underset{\vr}{\min} \;
\overbrace{-\sum_{m \in \sfM}\sum_{l \in \dones_m}
\sum_{l' \in \dzero_m} \ln \left( \indicator{r_{m, l} \ge r_{m, l'}} \right)
}^\text{Order violation penalty}\\
+ \underbrace{\fracl{2\sigmad} \sum_{\mn \in \sfT}
(r\smn-\psi\smn)^2}_\text{Square loss}\end{gathered}$$ The first term in the loss penalizes violations of order. In fact, the first term evaluates to infinity if any of the binary order constraints are violated. Hence, to maximize the log likelihood, the variables $\vr$ must satisfy the order constraints $\{ \compat{\vr_m}{\sy_m} \; \forall m \in
\sfM\}$. This interpretation suggests a constrained optimization approach: $$\label{eq:min_compat}
\underset{\{\vr_m| \compat{\vr_m}{\sy_m} \}}{\min} \;
\half \sum_{l \in \sfT_m}(r_{m, l}-\psi_{m,l})^2 \quad \forall m \in \sfM$$ Note that this loss decomposes task-wise. Hence, the proposed approach results in a list-wise ranking model. We also note that the independence between tasks means that the optimization is embarrassingly parallel. The constrained score vectors $\{\vr_m| \compat{\vr_m}{\sy_m}\}$ can be optimized efficiently using the inner representation outlined in . One issue that arises is that the cost function is not invariant to scale. Hence, the loss can be reduced just by scaling its arguments down. To avoid this degeneracy, we must constrain the score vectors away from $\mathbf{0}$. We achieve this by constraining the score vectors to the ordered simplex $\mDo{T_m}$, as it is a convex set and satisfies the requirement $\mathbf{0} \notin \mDo{T_m}$. Applying Lemma \[lem:order\_simplex\], the score is given by $\vr_m =
\gamma(\sr_m) = \gamma(\bC\vx_m)$ for $\vx_m \in \Do{T_m}$.
Let $\bpsi_m \in \bbR^{T_m}$ be the score vector ordered to satisfy $\bpsi_m =
[ \{\psi(m,l) \, \forall \,l \in \dones_m\} |\{\psi(m,l') \, \forall \,l' \in
\dzero_m\}] $. The ordering of the score vector is not unique. The loss function can now be written as: $$\label{eq:min_represent}
\underset{\vx \in \Do{T_m}}{\min} \;
\underset{\pi \in \Gamma_m}{\min} \;
\half \big\| \pi(\bC\vx) - \bpsi_m \big\|_2^2
\quad \forall\, m \in \sfM$$ This is exactly equivalent to: $$\label{eq:min_representb}
\underset{\vx \in \Do{T_m}}{\min} \;
\underset{\gamma \in \Gamma_m}{\min} \;
\half \big\| \bC\vx - \gamma(\bpsi_m) \big\|_2^2
\quad \forall\, m \in \sfM$$ The equivalence can be shown simply by setting $\gamma(\cdot) =
\inv{\pi}(\cdot)$. We present both forms as it provides some flexibility when implementing the algorithm. We optimize by alternating optimization. We first optimize the vector $\vx_m^*$ and then optimize the permutation order $\gamma_m^*$. The overall optimization combining the variational inference and maximum likelihood parameter estimation is presented in . The probability vector $\vx_m$ can be optimized efficiently using the exponentiated gradient (EG) algorithm [@kivinen95] or other simplex-constrained least squares solvers and can be embarrassingly parallelized over the tasks $\sfT_m$. Optimization of $\gamma_m$ requires optimizing over all permutations of the vector. This may be naïvely solved by expensive enumeration or by solving a combinatorial assignment problem.
\[lem:sort\] If $x_1 \geq x_2$ and $y_1 \geq y_2$, then $\normbs{ \prta{x} - \prta{y}} \leq \normbs{ \prta{x} - \prtb{y} }$ and $\normbs{ \prta{y} - \prta{x}} \leq \normbs{ \prtb{y} - \prta{x} }$.
Lemma \[lem:sort\] implies that selecting the pair-wise sorted order for equivalent items minimizes the loss. Lemma \[lem:sort\] can then be extended to $\vy \in \bbR^d$ using induction over $d$. Hence, selecting $\gamma_m$ as the sorted ordering in each block of equivalent values $\{
\dones_m, \dzero_m\}$ minimizes the loss.
**initialize** $\psi$, $\{\vx_m\}$, $\{\gamma_m\}$ Update $\psi^*| \vr$ by solving . Update $\gamma_m^* | \bpsi_m$ by block sorting (). Update $\vx_m^* | \gamma_m, \bpsi_m$ by solving . $\psi$, $\{\vx_m\}$, $\{\gamma_m\}$ Compute $S$ using
The set of compatible vectors as defined by is a convex cone, i.e., the convex composition $\vz = \alpha \vu + (1-\alpha) \vv, \, \alpha
\in [0, 1] $ of vectors in the set remains in the set, and any scaling $\alpha
\vz$ where $\alpha >0$ preserves compatibility. To show convexity of parameter estimation, it remains to show that the combination of optimizing $\vx_m$ and optimizing $\gamma_m$ minimizes . This is shown using the following lemma.
\[lem:r\_convex\] Let $\vr$ be partitioned into two sets such that $\vr_1 = \{r_k,\; \forall k\in \dones\}$ and $\vr_2 =
\{ r_k,\; \forall k \in \dzero\}$, and let: $$\prta{\vr}^* = \underset{\substack{\vr'_i \in \Pi(\vr_i)\\
\vr'_1 \geq \vr'_2}}{\arg \min} \normbs{\prtnb{\vr}{1}{2}{'} - \prta{\vz} },$$ where $\Pi(\vr_i)$ is the set of all permutations of the vector $\vr_i$ and $\vr'_1 \geq \vr'_2$ represents element wise inequality, then $\vr_i^*$ is isotonic with $\vz_i \; \forall i=1,2$.
We can now consider the global properties of the variational inference and maximum likelihood parameter estimation.
\[lem:all\_convex\] The variational inference and parameter estimation given by is jointly convex in $\{ \psi, S, \vr \}$. Alternating optimization () recovers the global optimum.
[**Sketch of proof:**]{} Recall that squared loss is jointly convex in both of its arguments. In addition, the components $\{\psi, S\}$ and $ \vr$ are in separate convex sets. Hence to show global convexity, it is sufficient to show that is convex separately in $\{\psi, S\}$ (by ) and $\vr$ (by ). It follows from the joint convexity of that the alternating optimization of recovers the global optimum.
The proposed model is trained to estimate bipartite ranking scores for each task and the underlying multitask latent regression distribution. Item rankings are predicted by sorting the expected noise-free scores of the trained model $\bbE[z\smn| \data] = \psi(m, n)$.
Experiments {#sec:experiments}
===========
This section details the experiments comparing the performance of the proposed model applied to the disease-gene prioritization task. We evaluated the modeling performance on association data curated from the OMIM database [@mckusick07] by the authors of [@mordelet11] and data we curated ourselves. We partitioned each dataset into five-fold cross validation sets. The model was trained on $4$ of the $5$ sets and tested on the held out set. The results presented are the averaged $5$-fold cross validation performance. Great care was taken to train all the models on the same datasets. Hence the results represent performance differences due to either the low rank modeling, the list-wise bipartite ranking model, or both.
[**Baseline (ProDiGe [@mordelet11]):**]{} We compared our proposed model to ProDiGe which, to the best of our knowledge, is the state of the art in the disease-gene prioritization literature. ProDiGe estimates the prioritization function using multitask support vector machines trained with gene kernel and disease kernel information. Parameter selection for ProDiGe was performed as suggested by the authors [@mordelet11].
[**OMIM dataset:**]{} The OMIM dataset [@mckusick07] is a curated database of known human disease-gene associations ($4178$ associations in the provided dataset). We derived the gene-gene interaction graph using data from HumanNet [@lee11]. We selected all genes with one or more connections in the network and all diseases with one or more genetic associations. This resulted in a disease-gene matrix with $M = 3,210$ diseases, $N=13,614$ genes and $T=3,636$ known associations (data sparsity $.0083\%$). In addition, the gene-gene graph contained $433,224$ known gene-gene links. We note the extreme sparsity of this matrix, and the resulting difficulty of the ranking task. Such sparse datasets are typical in the disease-gene domain. The OMIM dataset did not contain a disease graph; hence, we were unable to test the generalization of the methods to new diseases.
[**Curated dataset:**]{} We curated a large disease-gene association dataset. The set of genes were defined using the NCBI ENTREZ Gene database [@maglott11], and the set of diseases were defined using the “Disease” branch of the NIH Medical Subject Heading (MeSH) ontology [@mesh]. We extracted co-citations of these genes and diseases in the PubMed/Medline database [@pubmed] to identify positive instances of disease-gene associations. We derived our gene-gene interaction graph using data from HumanNet [@lee11] and our disease-disease similarity graph from the MeSH ontology. This resulted in a set of $250,190$ observed interactions, $21,243$ genes and $4,496$ diseases. We selected all genes with one or more connections in the gene-gene graph and all diseases with with one or more connections in the disease-disease graph. This resulted in a dataset with $M = 4,495$ diseases, $N=13,614$ genes and $T=224,091$ known associations (data sparsity $0.36\%$). The resulting disease network contained $13,922$ links, and the gene network contained $433,224$ links.
We were unable to run ProDiGe on the full dataset due to insufficient memory for storing the kernel matrix. Instead, we trained ProDiGe and the MV-GP models on a randomly selected 5% subsample of the associations. We also provide results for the MV-GP models trained on the full dataset. We performed two kinds of experiments for the curated dataset. The first experiment ([**known diseases**]{}) tests the ranking ability of the model for associations selected randomly over the matrix. The second experiment ([**new diseases**]{}) tests the generalization ability of the model for new diseases not observed in the training set. For the known disease experiments, the cross validation associations were randomly selected over the matrix. For the new disease experiments, the cross validation was performed row-wise, i.e., we selected training set diseases and test set diseases.
[**Model Setup:**]{} The proposed model was trained using the alternating optimization approach (). The trace constrained mean function was estimated using the cost function . The model was trained using our implementation of the algorithm outlined in [@laue12]. Like other large scale trace constrained matrix optimizers, [@laue12] maintains a low rank representation. The rank is estimated automatically by the optimizer. We found that employing a row bias improved the model performance, so we learned row biases while training. Note that the row offsets do not change the ranking and hence are not required for testing.
We selected the hyperparameter $\lambda = s*\lambda_{\max}$ with $30$ values of $s$ logarithmically spaced between $10^{-3}$ and $1.0$. Let $F(\bB)$ be the loss function. Then $\lambda_{\max}$ is the maximum singular value of $\D{F(\bB)}{\bB}\big|_{\bB = \mathbf{0} }$. The optimization returns the zero matrix for any $\lambda > \lambda_{\max}$. [@dudik12]. We used warm start to speed up the computation for decreasing values of $s$. We selected $\alpha \in
\{1,\, 0.8,\, 0.6,\, 0.4,\, 0\}$.
We implemented the full rank Gaussian process model ($\alpha=0$) by keeping the kernels as separate row and column kernels. This allowed us to scale the model to the larger datasets at the expense of more computations. We observed that the full rank model required a significant amount of computation time. This observation provides further motivation for the low rank approach. The full rank Gaussian process was trained directly using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.
[**Sampling unknown negative items:**]{} Recall that the observed data only consists of known associations. Following ProDiGe [@mordelet11] we sampled “negative” observations randomly over the disease-gene association matrix. We sampled $10$ different negatively labeled item sets. All models are trained with the positive set combined with one of the negative labeled sets. The model scores are computed by averaging the scores over the 10 trained models. All algorithms were trained using the same samples.
[**Covariance/Kernels:**]{} The covariances for the MV-GP prior and the kernels for ProDiGe were computed from gene graph $\cG\sm$ and the disease graphs $\cG\sn$. We performed preliminary experiments with a large class of graph kernels [@smola03] and selected the exponential kernel. We briefly outline kernel generation for the gene kernel. Let $\bA\sm$ be the adjacency matrix for the gene-gene graph. We computed the normalized Laplacian matrix as $\bL\sm = \bI - \bD^{-\half}\bA\sm\bD^{-\half}$, where $\bI$ is the identity matrix and $\bD$ is a diagonal matrix with entries $\bD_{i,i}=
(\bA\sm\mathbf{1})_i$. The exponential kernel is given by $\bK'\sm =
\exp{(-\bL\sm)}$. Following the suggestion in [@mordelet11] and preliminary experiments, we observed an improvement in performance with the identity matrix added to the exponential kernel. Hence the final kernel is given by $\bK\sm =
\exp{(-\bL\sm)} + \bI$. The disease kernel generation was obtained using the same approach when a disease-disease graph was available. All algorithms were trained using the same kernel matrices.
[**Metrics:**]{} Experimental validation of disease-gene associations in a laboratory can be time consuming and costly, so only a small set of the top ranked predictions are of practical interest. Hence, we focus on metrics that capture the ranking behavior of the model at the top of the ranked list. In addition, all metrics are computed on the test set after removing all relevant genes that had been observed in the training set removed. All metrics are computed per disease and then averaged over all the diseases in the test set. Let $\sg_l$ denote the labels of item (gene) $l$ as sorted by the predicted scores of the trained regression model, and let $G_m = |\dones_m| = \sum_{l}
\indicator{\sg_l=1}$ be the total number of relevant genes for disease $m$ in the test data after removing relevant genes observed in the training data. The metrics computed are as follows:
1. Area under the ROC curve (AUC) . This measures the overall ranking performance of the model.
2. The precision at $k\in\{1, 2, \ldots , 100\}$ computes the fraction of relevant genes retrieved out off all retrieved genes at position $k$. $$\pres{k} = \frac{\sum_{l=1}^k \indicator{\sg_l=1} }{k}.$$
3. The recall at $k\in\{1, 2, \ldots , 100\}$ computes the fraction of relevant genes retrieved out off all relevant genes. $$\recall{k} = \frac{\sum_{l=1}^k \indicator{\sg_l=1} }
{\min(G_m, k) }.$$
4. Mean average precision at $k=100$ ($\map{100}$) computed as the mean of the average precision at $k=100$. The average precision is given as: $$\ap{k} = \frac{\sum_{l=1}^{k} \indicator{\sg_l=1} \pres{l} }
{\min(G_m, k) }$$
The $\map_{@100}$ metric was used for model selection over the cross validation runs. To reduce notation, $\map$ refers to $\map_{@100}$ in all results. Higher values reflect better performance for the $\auc,\, \pres{k},\,
\recall{k}$ and $\map$ metrics, and their maximum value is $1.0$.
Discussion
----------
We present performance results for ProDiGe, the standard MV-GP (*Hilbert*, $\alpha=0$), the trace norm regularized MV-GP (*Trace*, $\alpha=1$), and the best overall MV-GP model (*Best*).
Our first experiment was on the known disease prediction with the 5% subset of the curated data. This task is very challenging as the training data consisted of an average of less than $3$ known associations out the possible $13,614$ per disease. The difficulty of this task is reflected in the performance results shown in and . We found that the trace model had the same performance as the best MV-GP model, suggesting that the trace norm is an effective regularization in this case. We found that the trace regularization resulted in a significant improvement in performance across metrics compared to ProDiGe and the Hilbert models. We also experimented with predicting the gene ranking of new diseases not seen during training and found similar performance as shown in and . As this is new disease prediction, none of the known genes on are removed from the test diseases. Interestingly, we found that this seems to improve the model performance as compared to the in-matrix prediction.
Next, we experimented with prediction on the full curated dataset predicting known diseases. We were unable run this experiment with ProDiGe due to memory limitations. Hence, only results from the proposed model are shown. The results are as shown in and . As expected, we observed a significant improvement in performance by using the entire dataset. Similar results were observed for the new disease prediction as shown in and . In all models, we observed that the trace norm constrained approach out-performed the standard full rank MV-GP model. We especially note the performance improvement at the top of the list, as these are the most important to the domain.
Our final experiment was on the OMIM dataset. The results are as shown in and . These results are especially interesting as we found that the best overall model outperformed the trace model, and significantly outperformed all other model in terms of ranking at the top of the list. This suggests that the spectral elastic net regularizer may be most useful with significant data sparsity. ProDiGe out-performed the Hilbert model in terms of recall, but Hilbert model had the best overall ranking performance as measured by AUC. We are investigating this observation further, but preliminary investigation suggests that the metrics are more sensitive to small changes in order when the data is very sparse. The sparsity also explains the significant drop in $\pres{k}$ as k grows.
![Curated data (known diseases, 5% subsample) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap.[]{data-label="fig:our_mat_p5_prec_rec"}](our_mat_pt5_precision "fig:"){width="0.8\columnwidth"} ![Curated data (known diseases, 5% subsample) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap.[]{data-label="fig:our_mat_p5_prec_rec"}](our_mat_pt5_recall "fig:"){width="0.8\columnwidth"}
![Curated data (new diseases, 5% subsample) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap. []{data-label="fig:our_cold_p5_prec_rec"}](our_cold_pt5_precision "fig:"){width="0.8\columnwidth"} ![Curated data (new diseases, 5% subsample) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap. []{data-label="fig:our_cold_p5_prec_rec"}](our_cold_pt5_recall "fig:"){width="0.8\columnwidth"}
![Curated data (known diseases, full dataset) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap. []{data-label="fig:our_mat_prec_rec"}](our_mat_all_precision "fig:"){width="0.8\columnwidth"} ![Curated data (known diseases, full dataset) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap. []{data-label="fig:our_mat_prec_rec"}](our_mat_all_recall "fig:"){width="0.8\columnwidth"}
![Curated data (new diseases, full dataset) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap.[]{data-label="fig:our_cold_prec_rec"}](our_cold_all_precision "fig:"){width="0.8\columnwidth"} ![Curated data (new diseases, full dataset) experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$. Best and Trace curves overlap.[]{data-label="fig:our_cold_prec_rec"}](our_cold_all_recall "fig:"){width="0.8\columnwidth"}
![OMIM data experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$.[]{data-label="fig:omim_prec_rec"}](Marcotte_mat_all_precision "fig:"){width="0.8\columnwidth"} ![OMIM data experiment results: precision (top) and recall (bottom) curves $@k=1, 2, \ldots, 100$.[]{data-label="fig:omim_prec_rec"}](Marcotte_mat_all_recall "fig:"){width="0.8\columnwidth"}
Conclusion {#sec:conclude}
==========
This paper proposes a novel hierarchical model for multitask bipartite ranking that combines a trace constrained matrix-variate Gaussian process and a bipartite ranking model. We showed that the trace constraint led to a mean function with low rank and discussed the spectral elastic net as the MAP regularizer that arises from this model. We showed that constrained variational inference for the Gaussian process combined with maximum likelihood parameter estimation for the ranking model was jointly convex. We applied the proposed model to the prioritization of disease-genes and found that the proposed model significantly improved performance over strong baseline models.
We plan to explore the trace norm constrained MV-GP and the spectral elastic net further and analyze their theoretical properties. We also plan to explore parameter estimation using the resulting constrained posterior distribution. In addition, we plan to investigate the applications of the constrained MV-GP to other tasks including multitask regression and collaborative filtering.
Acknowledgments {#acknowledgments .unnumbered}
===============
Acknowledgment {#acknowledgment .unnumbered}
==============
Authors acknowledge support from NSF grant IIS 1016614. We thank Sreangsu Acharyya for helpful discussions on bipartite ranking. We also thank U. Martin Blom and Edward Marcotte for providing the OMIM data set.
[Oluwasanmi Koyejo]{} is a PhD student working on data mining and machine learning at The University of Texas at Austin and advised by Dr. Joydeep Ghosh. He received his B.S. Degree in Electrical Engineering with a minor in Statistics from the New Jersey Institute of Technology (NJIT) and completed a M.S. in Electrical Engineering at the University of Texas at Austin with a focus on machine learning applied to wireless communications.
[Cheng Lee]{} is a PhD student at the University of Texas at Austin advised by Dr. Joydeep Ghosh. His research work focuses on applications of data mining and machine learning algorithms in biology and medicine. He received his B.S. degrees in computer science and electrical engineering and his M.S. in electrical engineering from the University of Texas at Dallas. He was also a computational biologist in McDermott Center for Human Growth and Development at the University of Texas Southwestern Medical Center.
[Joydeep Ghosh]{} received the B.Tech. degree from the Indian Institute of Technology Kanpur in 1983 and the Ph.D. degree from the University of Southern California in 1988. He is currently the Schlumberger Centennial Chair Professor with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, where he has been with the faculty since 1988. He has published more than 250 refereed papers and 50 book chapters, coedited 20 books, and received 14 “best paper” awards.
[^1]:
|
---
author:
- 'B.Altieri , S.Berta , D.Lutz , J.-P.Kneib , L.Metcalfe , P.Andreani , H.Aussel , A.Bongiovanni , A.Cava , J.Cepa , L.Ciesla , A.Cimatti , E.Daddi , H.Dominguez , D.Elbaz , N.M.Förster Schreiber , R.Genzel , C.Gruppioni , B.Magnelli , G.Magdis , R.Maiolino , R.Nordon , A.M.Pérez García , A.Poglitsch , P.Popesso , F.Pozzi , J.Richard , L.Riguccini , G.Rodighiero , A.Saintonge , P.Santini , M.Sanchez-Portal, L.Shao , E.Sturm , L.J.Tacconi , I.Valtchanov, M.Wetzstein , and E.Wieprecht [^1]'
date: 'Received ; accepted '
title: '$Herschel$ deep far-infrared counts through Abell 2218 cluster-lens'
---
=1
Introduction
============
The discovery of the cosmic infrared background (CIB) [@puget96; @fixsen96; @lagache99] has opened new perspectives on galaxy formation and evolution. A large number of the sources contributing to this far-infrared (FIR) CIB have been resolved in the mid-infrared (mid-IR) at 15$\mu$m with the ISOCAM instrument on ISO [@genzel00; @elbaz02], and later at 24$\mu$m with MIPS on [*Spitzer*]{} [@papovich04; @dole04; @frayer06]. A striking result concerns the evolution of the infrared and submillimeter (sub-mm) galaxy population: the infrared source counts are higher than no-evolution or moderate-evolution models and provide strong constraints on the evolution of the bolometric energy output from distant galaxy populations.
However, at FIR and sub-mm wavelengths, a much lower fraction has been resolved so far, because of the small aperture of telescopes, the prohibitive confusion limits, and the low sensitivity of available instruments. Nevertheless, stacking results (Dole et al. 2006) show that most or all of the FIR background is due to known high-z IR galaxies.
{width="50.00000%"} {width="50.00000%"}
The [*Herschel Space Observatory*]{}$\textquoteright$s [@pilbratt10] compact point-spread-function (PSF) enables probes of the FIR emission of large samples of galaxies near their spectral energy distribution (SED) peak, over a wide z range. The sensitive [*Photometer Array Camera & Spectrometer*]{} (PACS) [@poglitsch10] achieves imaging surveys of unrivalled depth at 70, 100, and 160$\mu$m in the PEP (PACS Extragalactic Probe) Key Program (PI D. Lutz).
As pioneered by @smail97 in the sub-mm, and later in the mid-IR [@altieri99; @metcalfe03], we are conducting a survey toward massive clusters as part of PEP, to resolve the FIR beyond the field-survey limit and penetrate the $Herschel$ confusion limit with powerful gravitational lenses. This letter reports initial observations on the cluster Abell 2218.
Observations {#sec:data}
============
The rich and massive lensing cluster Abell 2218 was targeted in the PEP science demonstration phase. This field has been intensively studied, including deep archival Hubble Space Telescope(HST)/ACS and [*Spitzer*]{} (IRAC & MIPS) observations, which allow simple and direct identification of the sources detected with $Herschel$/PACS.
Our 13 hours of observations, centered on the cD galaxy at RA=16h35m51.84s, Dec=62d14m19.2s (J2000), used scan mapping (20$^{\prime\prime}$/s scan speed; 4$^{\prime}$ scan leg length; 20$^{\prime\prime}$ cross-scan steps) over an area of about 6$^{\prime}$x6$^{\prime}$, with coverage strongly peaked toward the center. Though affected by relatively high overheads (reduced later in the mission), they are the deepest observations so far at 100 and 160$\mu$m.
Data analysis {#sec:analysis}
=============
The data were reduced using HIPE [@ott10] v2.0. build 1328. The 10Hz data cubes were processed with the standard PACS pipeline, along with custom procedures: 1) to remove interference patterns, transients from calibration blocks and tracking anomalies, 2) to perform re-centering of positional offsets. Moreover, to remove detector drifts and $1/f$ noise, a sliding high-pass filter was run on the pixel timelines with an iterative masking of the brighter sources. For more information, refer to @berta10. Final maps are displayed in Figure \[fig:a2218maps\].
Source extraction was performed with the StarFinder PSF-fitting code [@diolaiti00], down to the 3$\sigma$ level. The total number of sources is reported in Table \[tab:tabsources\]. Based on random extractions from the residual images, the averaged 1$\sigma$ noise levels in the maps are 0.92mJy at 100$\mu$m and 1.61mJy at 160$\mu$m.
[llc]{} & 100$\mu$m & 160$\mu$m\
PSF FHWM (arcsec) & 7.5 & 11.0\
$1\sigma$ average sensitivity (mJy) & 0.92 & 1.61\
$1\sigma$ central sensitivity (mJy) & 0.6 & 1.2\
number of detected sources ($>3\sigma$) & 98 & 94\
number of lensed sources ($>3\sigma$) & 70 & 78\
lensed sources with spectroscopic redshift & 9& 13\
lensed sources with photometric redshift & 20 &17\
We inspected all sources in the maps by eye. The source catalog was cross-correlated with various redshift surveys for Abell 2218 (@leborgne92 [@ziegler01; @ebbels98; @metcalfe03]; Kneib, priv. comm.) and with photometric redshifts of the Abell 2218 MIPS 24$\mu$m sources computed using HST/ACS images, near-IR images and the 4 [*Spitzer*]{}/IRAC bands [@ciesla09]. All PACS sources were detected in the [*Spitzer*]{}/MIPS observation. This cross-correlation allowed us to classify sources as foreground, cluster, or background galaxies. More than 90% of background sources with confirmed spectroscopic or photometric redshift have $F_{160}/F_{100} \gtrsim 1$, while 11 out of the 13 confirmed cluster galaxies detected both at 100 and 160$\mu$m have $F_{160}/F_{100} < 1$. This allows classification of sources with $F_{160}/F_{100} \gtrsim 1$, but without redshift, as lensed-background galaxies. This is as also supported by their extremely faint magnitudes in the optical and/or their disturbed morphologies.
Less than 30% (20%) of the sources at 100$\mu$m (160$\mu$m) are classified as cluster galaxies. This situation is very similar to the mid-IR, where e.g. @altieri99 [@metcalfe03; @egami06; @hopwood10], showed that the large majority of far-IR sources are background, hence lensed, sources. The cluster core is virtually transparent in the FIR and acts as a natural telescope to provide a magnified view of the background sky. This increases the sensitivity and at the same time reduces the effects of source confusion; in particular, the cD galaxy is undetected in both bands.
Four of the 7 sub-mm sources in the ultra-deep map of Abell 2218 [@knudsen06] are detected in the 160$\mu$m map, including the triple z=2.516 sub-mm source SMMJ16359+6612 [@kneib04a], the sources SMMJ163555.2+661150 at $z=1.034$, and SMMJ163541.2+661144 at $z=3.1824$ [@knudsen09]. However, the highest redshift sub-mm source (z=4.048) SMMJ163555.5+661300 and the two faintest sub-mm sources have not been detected, which is also true for the z=5.56 source of @ellis01 and the $z\sim7$ triple source of @kneib04b.
Lensing inversion and source counts {#sec:counts}
===================================
![The area of the source plane at 100$\mu$m, as a function of the 3$\sigma$ sensitivity, for redshifts z=0.7, 1, 1.5, and 2.5, compared to the corresponding area in the image plane (i.e. apparent sky - the solid line).[]{data-label="fig:areasens"}](14634fg2.pdf){width="47.00000%"}
Lensing acts in two ways on the background sky: [*i)*]{} it amplifies source brightness, typically by a factor of 2, but by as much as 10 near critical lines; [*ii)*]{} it magnifies the area probed - and both the flux amplification and the space magnification are stronger toward the cluster core and increase with source-plane redshift. We exploited the detailed mass model of Abell 2218 [@kneib96; @kneib04b], which considers 8 multiply imaged systems, of which 7 have spectroscopically confirmed redshifts (among these the several high-z multiply-lensed sources mentioned at the end of Section 3).
We used the publicly available software, LensTool, described in @jullo07 to compute the source amplification factors and the surface magnification due to lensing. Where redshifts were not available for sources judged by morphology or intrinsic faintness to be in the background, a value of $z=0.7$ was assigned, corresponding to the median redshift of field sources having spectroscopic or photometric redshift. This has little impact on the lensing amplification factor and area magnification, as illustrated in Figure \[fig:areasens\]. This figure shows the area mapped to a given sensitivity in the image plane (i.e. on the apparent sky), compared to the corresponding areas in source planes at redshifts 0.7, 1, 1.5, and 2.5. For redshifts above 0.7, the lensing surface magnification depends only weakly on redshift. The lensing amplification factor for a given source on the other hand may be more sensitive to the redshift. However, setting all (morphologically classified) background sources without known redshift to z=1 (instead of z=0.7) did not change the shape of the counts significantly, because it shifted only a few sources among adjacent flux bins.
Figure \[fig:areasens\] expresses the fact that some small, highly-lensed regions of the source plane are mapped onto larger areas in the image plane (the apparent sky), with the result that, over a few arcmin$^{2}$, flux densities in the range 1 to 3mJy become accessible by virtue of the lensing effect.
Ten sources (12 apparent sources, including all images of SMM J16359+6612) have lens-corrected fluxes below 3mJy at 100$\mu$m and 13 (15 apparent) below 5.7mJy at 160$\mu$m, with these quoted limits the 3$\sigma$ sensitivities achieved on the GOODS-N field [@berta10]. Such sources would most likely not be detected even in upcoming deeper blank-field surveys, like GOODS-S (deeper by a factor of 2). The triple source SMMJ16359+6612, for instance, is amplified by a factor of 45, in total. It was counted only once, as number counts refer to the source plane.
By correcting for lensing amplification, surface magnification effects, contamination by cluster galaxies, and non-uniform sensitivity of our maps, we can derive number counts at 100$\mu$m and 160$\mu$m. Because of the non-uniform sensitivity of the maps on the sky and the lensing effect, different areas on the sky are surveyed to different depths in the source plane. The object density per flux bin was computed using gain-dependent surface areas. Incompleteness affects the counts progressively below apparent fluxes of 6mJy at 100$\mu$m and 9mJy at 160$\mu$m. We restricted these counts to 4$\sigma$ detections where the completeness of measurement is typically 80%. This avoids potentially complicated folding of completeness correction with lensing correction, not justified by the relatively small numbers of sources and relatively large statistical error bars associated with these sample counts on a single lensing cluster. A comprehensive completeness analysis will be required when combining the observations of a sample of massive cluster lenses to extract the galaxy number counts below the 1mJy level.
{width="45.00000%"} {width="45.00000%"}
Our counts are barely affected by confusion at 100$\mu$m, with a source density of 20 beams/source in the image plane (for the @lagache03 definition of the beam). At 160$\mu$m the density is as high as 10 beams/source, with even fewer beams per source in the central area of high-lensing. Hence, the Abell 2218 catalog is affected by source confusion at 160$\mu$m (16.7 beams/source from @dole03) and the high density of detected sources prevents the extraction of fainter objects.
The source counts, corrected for cluster contamination and lensing effects, in both the 100$\mu$m and 160$\mu$m bands, are presented in Figure \[fig:counts100\], normalized to the Euclidean slope ($dN/dS \propto S^{-2.5}$). Error bars consider Poisson statistics only, as flux uncertainties are minimal by comparison.
Discussion {#sec:discussion}
==========
The Abell 2218 $Herschel$/PACS maps are the deepest FIR maps to date. Covering an area of $6\times 6$ arcmin$^{2}$ over the cluster core, we have surveyed the region where strongly lensed background sources are present. At fluxes higher than 5mJy, the lens-corrected normalized number counts are consistent, within error bars, with the GOODS-N counts [@berta10] showing evidence of a peak between 5-10 and 30mJy. Thanks to gravitational lensing, we could extend the counts down to $\sim$1mJy at 100$\mu$m and $\sim$2mJy at 160$\mu$m, confirming the negative sub-Euclidean slope below the turnover.
The Abell 2218 differential counts (not normalized to the Euclidean slope) at 100$\mu$m show a faint-end slope similar to GOODS-N, but they are higher at fainter fluxes. At 160$\mu$m the differential counts show a steeper slope than GOODS-N ($-\,1.82$ instead of $-1.67$ assuming the functional form of the counts: $dN/dS \propto S^\alpha$). This is also reflected in the Euclidean-normalized counts. The high error bars seen at the faint end come from the poor statistics. The counts could be slightly underestimated at 160$\mu$m at the lowest fluxes because the source density is approaching confusion with no correction made for this.
The counts are reproduced well by the models of @valiante09 or @rowanrobinson09, both at 100 and 160$\mu$m, but other backwards evolutionary models similarly reproduce the downward turn below 10mJy. The summed contribution of resolved galaxies provides a lower limit to the IR background and can be compared to the estimation of the CIB. Here we have adopted the latest measure of its surface brightness from COBE/DIRBE maps [@dole06]: $14.4\pm6.3$ nWm$^{-2}$sr$^{-1}$ at 100$\mu$m and $12.75\pm6.4$ nWm$^{-2}$sr$^{-1}$ at 160$\mu$m (which is an interpolated value).
[|l c c c c|]{} & flux-range & CIB & d(CIB) & fraction\
& (mJy) & (nW m$^{-2}$ sr$^{-1}$) & & (%)\
100$\mu$m A2218 & 0.94-35 & 7.9 & 3.4 & 55 $\pm$ 24\
100$\mu$m PEP & 0.94-142 & 8.8 & 3.5 & 62 $\pm$ 25\
160$\mu$m A2218 & 1.47-36 & 9.9 & 3.9 & 77 $\pm$ 31\
160$\mu$m PEP & 1.47-179 & 11.3 & 4.1 & 88 $\pm$ 32\
The contribution to the CIB by Abell 2218 background galaxies above the 4$\sigma$ detection threshold was computed by simply integrating the observed number counts (Table\[tab:tabCIB\]). Errors were computed by integrating the envelope of the counts with their uncertainties. More than half of the DIRBE CIB [@dole06] has been directly resolved, which is also consistent with the surface brightness found by @bethermin10 in a stacking analysis of [*Spitzer*]{}/MIPS sources.
Combining the deep counts in Abell 2218 with the results obtained by Berta et al (2010) in wider/shallower PEP fields (e.g. COSMOS), we were able to extend the integration flux range to 142mJy at 100$\mu$m and 179mJy at 160$\mu$m. Consequently, the resolved CIB fractions increase to $62\pm25$% and $88\pm32$% in the two bands. One must keep in mind that not only is the CIB surface brightness from PEP affected by large uncertainties (cosmic variance), but the reference values by @dole06 are also defined only within a factor of $\sim$2.
Another 9 massive lensing clusters will be targeted as part of PEP and another 40 more in the open-time $Herschel$ Lensing Survey KP [@egami10]. We expect that, in the coming years, combined results from many lensing clusters will greatly improve the statistics of highly amplified sources and constrain source densities around $\sim$1mJy or below. By penetrating below the unlensed $Herschel$ confusion limit and probing the high-redshift galaxy populations beyond the sensitivity limit of blank-field surveys, the fraction of the resolved CIB will increase.
PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).
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[^1]: $Herschel$ is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
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---
abstract: 'We study the behaviour of a Symmetric Exclusion Process (SEP) in presence of stochastic resetting where the configuration of the system is reset to a step-like profile with a fixed rate $r.$ We show that the presence of resetting affects both the stationary and dynamical properties of SEP strongly. We compute the exact time-dependent density profile and show that the stationary state is characterized by a non-trivial inhomogeneous profile in contrast to the flat one for $r=0.$ We also show that for $r>0$ the average diffusive current grows linearly with time $t,$ in stark contrast to the $\sqrt{t}$ growth for $r=0.$ In addition to the underlying diffusive current, we identify the resetting current in the system which emerges due to the sudden relocation of the particles to the step-like configuration and is strongly correlated to the diffusive current. We show that the average resetting current is negative, but its magnitude also grows linearly with time $t.$ We also compute the probability distributions of the diffusive current, resetting current and the total current (sum of the diffusive and the resetting currents) using the renewal approach. We demonstrate that while the typical fluctuations of both the diffusive and reset currents around the mean are typically Gaussian, the distribution of the total current shows a strong non-Gaussian behaviour.'
author:
- Urna Basu
- Anupam Kundu
- Arnab Pal
title: Symmetric Exclusion Process under Stochastic Resetting
---
Introduction
============
Stochastic resetting, which refers to intermittent interruption and restart of a dynamical process, has been a subject of immense interest in recent years. It has found applications in a wide range of areas starting from search problems [@search1; @search4; @Arnab2017; @Arnab2019], population dynamics [@population1; @population2], enzymatic catalysis [@catalysis; @bio4] to computer science[@search2; @search3], stock markets [@stock] and biological processes [@bio1; @bio2; @bio3]. Stochastic resetting of a single Brownian particle is the paradigmatic example where the position of the particle is reset to a fixed point in space with a certain rate [@Brownian]. This simple act has drastic consequences on the statistical properties of the particle — it results in a nontrivial stationary state, anomalous relaxation behaviour, as well as finite mean first passage time.
Several variations and extensions of this simple model have been explored in recent years[@Brownian2; @Brownian3; @absorption; @highd; @Mendez2016; @Puigdellosas; @deepak; @Arnab2019]. Specific examples include: resetting in presence of an external potential [@Arnab2015; @potential], in a confinement [@circle; @interval] or to an extended region [@CCRW], and resetting to already excursed positions [@Boyer2014; @Sanjib2015]. Stochastic resetting has also been studied in more general nonequilibrium contexts – in reaction processes [@bio4; @catalysis], Lévy flights [@Levy], coagulation-diffusion process [@coagulation], telegraphic process [@telegraphic], for run-and-tumble particles [@RTP] and to model nonequilibrium baths [@Maes2017]. Studies were not only limited to a constant rate resetting, other protocols have also been investigated in great details. These include deterministic resetting [@deterministic], space [@Roldan2017] or time dependent [@time-dep1; @time-dep2] resetting rate, resetting followed by a refractory period[@refractory; @refractory2], non-Markovian resetting [@ShamikPRE; @nonmarkov2; @nonmarkov3] and resetting sensitive to internal dynamics [@Evans2017].
![Typical snapshots of time evolution of a system of size $L=400$ for two different values of the resetting rate $r=0.01$ (left) and $r=0.1$(right). The dark blue points indicate presence of particles while the light Grey ones indicate empty sites. For a small value of $r$ (left) the typical duration between two consecutive resetting events are longer, and the particles spread deeper into the empty half of the lattice whereas for larger $r$ (right) the resetting occurs more frequently and the density profile remains close to the step-like initial state. []{data-label="fig:snapshot"}](s.pdf){width="7.5"}
An important question that naturally arises is how the presence of resetting dynamics affects the systems with many interacting degrees of freedom. This issue has not been explored much so far except for a few handful of models. These studies include dynamics of KPZ-like fluctuating interfaces[@kpz-reset; @Shamik2016], one-dimensional quantum spin chain[@KS2018], and a pair of interacting Brownian particles [@Evans2017; @Silva2018]. In all these cases, resetting leads to nonequilibrium stationary states, characterized by non-Gaussian fluctuations. However, the effect of resetting on the behaviour of current, which plays an important role in characterizing the nonequilibrium stationary state, has not been studied yet. This question is of paramount importance, because presence of stochastic resetting introduces an additional time-scale and is expected to modify the behaviour of current significantly. The exclusion processes [@Liggett], which are simple well known models of interacting particles, provide a natural playground for exploring these questions.
In this article we study the effect of stochastic resetting on Symmetric Exclusion Process (SEP) [@Spitzer; @Liggett] and explore how the presence of resetting changes the dynamical and stationary properties of SEP. The stochastic resetting is implemented by interrupting the time-evolution at some rate $r,$ and restarting the process from a specific configuration. It turns out that the incorporation of the resetting mechanism introduces an extra current $\Jres$ in addition to the usual diffusive particle current $J_\text{d}.$ We show that, for $r>0,$ the average diffusive current increases linearly with time $t,$ in contrast to the $\sqrt{t}$ behaviour in the absence of the resetting [@Derrida1]. Additionally, the average resetting current also shows a linear temporal growth in magnitude, although it remains negative. We also compute the distribution of the diffusive current $J_\text{d}$, resetting current $\Jres,$ as well as the total current $J_r=J_\text{d}+\Jres$. We observe that, while the diffusive and resetting current show Gaussian behaviour, the fluctuations of the total current are characterized by a strongly non-Gaussian distribution.
The article is organized as follows: In the next section we define our system and summarize our main results. Sec \[sec:density\] is devoted to the computation of the time evolution of the density profile under resetting. In Sec. \[sec:cur\] we investigate the behaviour of the particle current – Sec. \[sec:J\_dif\] and \[sec:J\_reset\] focus on the diffusive and resetting currents respectively, whereas the behaviour of the total current $J_r$ is explored in Sec. \[sec:J\_tot\]. We conclude with some open questions in Sec. \[sec:conc\].
Model and Results {#sec:model}
=================
The symmetric exclusion processes (SEP) is a paradigmatic model for interacting particle systems [@Spitzer; @Liggett] which have been used to describe a wide range of physical phenomena including particle transport in narrow channels, motion of molecular motors, ion transport through porous medium etc. This process describes unbiased motion of particles on a lattice which interact via mutual local exclusion. In this section we define the dynamics of SEP with stochastic resetting and present a brief summary of our main results.
Let us consider a periodic lattice of size $L$ where each lattice site can contain at most one particle. The state of a site, say $x$, is characterized by a variable $s_x$ which takes values $1$ and $0$ depending on whether the site $x$ is occupied or not, respectively. The configuration of the system is characterized by ${\cal C}=\{s_x; x=0,1,2,\dots L-1 \}.$ We consider the case of half-filling, [*i.e.*]{}, the total number of particles $\sum_x s_x = \frac L2.$ The system evolves according to the following two dynamical moves:
- [**Hopping**]{}: A particle randomly hops to one of its nearest neighbouring sites with unit rate, provided the target site is empty.
- [**Resetting**]{}: In addition, the system is ‘reset’ to some specific configuration ${\cal C_0}$ with rate $r.$ In the following we consider ${\cal C_0}$ to be a step like state where all the particles are in the left-half of the lattice: :={
s\_x=1 & 0 x L2-1,\
s\_x=0 &
. \[eq:C0\]
Both the hopping and resetting dynamics conserve the total number of particles, so that the half-filling condition is respected at all times, and the global particle density remains fixed at $1/2.$ Between two resetting events the time evolution of the system is governed by the hopping dynamics only. The time-scale associated with the resetting mechanism is given by $r^{-1},$ which also gives a measure of the typical duration between two consecutive resetting events. Figure \[fig:snapshot\] shows typical examples of the time evolution for two different values of the resetting rate $r.$
In the absence of resetting, the master equation governing the time-evolution of the probability ${\cal P}_0({\cal C},t)$ for the system to be in the configuration ${\cal C}$ at time $t$ is given by \_0([C]{},t) = [L]{}\_0 [P]{}\_0([C]{},t). Here ${\cal L}_0$ is the Markov matrix in absence of the resetting, [*i.e.*]{}, ${\cal L}_0 {\cal P}_0({\cal C},t) = \sum_{\cal C'}[W_\cal{C' \to C}{\cal P}_0({\cal C',t}) - W_\cal{C \to C'} {\cal P}_0({\cal C},t)]$ where $W_\cal{C' \to C}$ denotes the rate for the jump $\cal {C \to C'}$ due to hopping dynamics [*only*]{}. Note that, $W_\cal{C' \to C}=1$ only if the two configurations $\cal C$ and $\cal {C'}$ are connected by a single hop of a particle to a neighbouring site.
Let ${\cal P}({\cal C},t)$ denote the probability of finding the system in the configuration ${\cal C}$ at time $t$ in presence of resetting. In this case, the master equation reads, ([C]{},t) &=& [L]{}\_0 [P]{}([C]{},t) &&+ r \_[\_0]{} [P]{}([C’]{},t) \_[\_0]{} - r [P]{}([C]{},t) (1-\_[\_0]{}) &=& ([L]{}\_0-r) [P]{}([C]{},t) + r \_[C,C\_0]{}, \[eq:Master\_r\] where $\delta_{\cal{C,C}_0}$ is the Kronecker delta symbol, which takes the value unity when $\cal C$ is same as $\cal C_0,$ and is zero otherwise. It is straightforward to write a formal solution of Eq. , ([C]{},t) &=& e\^[(L\_0-r)t]{} [P]{}(C,0) + r \_0\^t s e\^[(L\_0-r)s]{} \_[C,C\_0]{} &=& e\^[-rt]{} [P]{}\_0(C,t) + r \_0\^t s e\^[-rs]{} [P]{}\_0(C,s). \[eq:renewal\_PC\] Here ${\cal P}_0(\cal C,t)=e^{\cal L_0 t}{\cal P}(\cal C,0)$ is the probability of finding the system in configuration $\cal{C}$ at time $t$ in the [*absence of resetting*]{} given that the system was initially at $\cal{C}_0,$ [*i.e.*]{}, $\cal P(\cal C,0)={\cal P}_0(\cal C,0)=\delta_{\cal C, \cal C_0}$. Equation is nothing but the renewal equation for the configuration probability, which has been obtained earlier and used to study resetting phenomena in various other contexts [@Brownian; @kpz-reset]. Note that Eq. holds true irrespective of the specific choice of $\cal C_0$ given in Eq. .
In the absence of resetting the ordinary SEP on a ring relaxes to an equilibrium state with flat density profile and zero current. The approach to the equilibrium state, starting from the step-like initial configuration ${\cal C}_0,$ is characterized by a diffusive current flowing through the system. It has been shown that, for an infinitely large system, the time-integrated current measuring the net particle flux through the central bond up to time $t,$ grows as $\sqrt{t}$ for large $t$ [@Derrida1; @Derrida2]. Presence of resetting is expected to affect these characteristics of SEP which we investigate in detail in this paper. A brief summary of our results is presented below.
- First, we compute an exact expression for the evolution of the average density profile $\rho(x,t) = \la s_x(t) \ra$ for any arbitrary value of the resetting rate $r,$ which is given in Eq. . We observe that the evolution is non-trivially modified due to the presence of resetting which leads to an inhomogeneous stationary density profile \[see Fig. \[fig:rho\](b)\] in contrast to the flat one for $r=0$.
- This inhomogeneous density profile provides some characterization of the non-equilibrium state of the system. It is, however, also important to look at how the particle currents in the system are affected by the introduction of resetting. In addition to the usual diffusive current $J_{\textrm{d}}(t)$ created due to the local hopping of the particles, there is also a contribution $\Jres(t)$ to the total current due to the global movements of the particles during the resetting events.
We show that the behaviour of the diffusive current changes drastically in presence of resetting. In particular, we compute the average diffusive current $\la J_\text{d}(t) \ra$ exactly, which, in the long time limit, shows a linear growth with time $t,$ $$\la J_\text{d}(t) \ra \simeq t\sqrt{\frac r{r+4}}.$$ This behaviour is in stark contrast to the $\sqrt{t}$ growth which is seen in absence of resetting [@Derrida1]. Similar change in the dynamical behaviour is also observed for the variance of the diffusive current, which also grows as $\sim t$ in presence of resetting, as opposed to $\sqrt{t}.$ We explore the behaviour of the resetting current $\Jres$ too and show that, its average and variance also grow linearly with time. We also investigate the probability distribution of $\Jd(t)$ and demonstrate that, in the long-time regime, the typical fluctuations of $J_\text{d}(t)$ around its mean is characterized by a Gaussian distribution. Similar Gaussian fluctuations are also expected for the resetting current $\Jres.$
- Finally, we study the behaviour of the total current $J_r = \Jd + \Jres$ and calculate the average $\la J_r(t)\ra$ and the second moment $\la J_r^2(t)\ra$ as functions of time $t.$ In the long-time limit the moments reach stationary values. In particular, we show that the average stationary current is given by J\_r = 1. We also compute the stationary probability distribution of the total current $P_r^\text{st}(J_r),$ for small values of $r,$ using a renewal approach. Interestingly, it turns out that, this distribution is non-Gaussian, and has very asymmetric behaviour at the two tails.
Density profile {#sec:density}
===============
Presence of repeated resetting to the inhomogeneous configuration $\cal C_0$ destroys the translational invariance in the system and a non-trivial density profile can be expected, even in the stationary state. The average density $\rho(x,t) = \la s_x(t) \ra$ is given by the probability that the site $x$ is occupied any time $t.$ The time-evolution equation for the density profile can be derived by multiplying Eq. by $s_x$ and summing over all configurations $\cal C,$ (x,t) &=& (x+1,t)+ (x-1,t)- 2(x,t) && -r(x,t) + r (x) \[eq:drho\] Here $\phi(x)$ is the density profile corresponding to the resetting configuration ${\cal C}_0$ which, as mentioned before, is also taken as the initial profile. The exact time-dependent density profile $\rho(x,t)$ can be obtained by solving Eq. . To this end we introduce the discrete Fourier transform (n,t) = \_[x=0]{}\^[L-1]{} e\^[i ]{} (x,t), n=0,1,2 …L-1. \[eq:rho\_dft\] Substituting Eq. in Eq. , we get, (n,t) = - (\_n+r) (n,t) + r (n) \[eq:dtilde\_rho\] with $\lambda_n= 2\left(1- \cos \frac{2 \pi n}{L}\right)$ and $\tilde \phi(n)$ is the Fourier transform of the resetting (and initial) profile $\phi(x).$ Equation can immediately be solved, (n,t) = + e\^[-(r+\_n)t]{} The density profile is then obtained by inverting the Fourier transform, (x,t) &=& rL \_[n=0]{}\^[L-1]{} e\^[-i L]{} &&+ 1L \_[n=0]{}\^[L-1]{} e\^[-(r+\_n)t]{} e\^[-i L]{}\[eq:rhor\_xt\]
![Density profile: (a) Time evolution of the density profile $\rho(x,t),$ starting from a step-like initial condition, for resetting rate $r=0.05,$ and for different values of time $t.$ The symbols correspond to the data obtained from numerical simulations and the solid lines correspond to the analytical result \[see Eq. \]. (b) The stationary density profile $\rho(x)$ for different reset rates $r.$ The symbols correspond to the data obtained from numerical simulations and the solid lines correspond to the analytical result \[see Eq. \]. The lattice size $L=40$ for both (a) and (b).[]{data-label="fig:rho"}](profile.pdf){width="8.8"}
In the stationary state, the second term decays exponentially and the stationary density profile is given by, (x) &=& rL \_[n=0]{}\^[L-1]{} e\^[-i L]{} &=& 12 + rL \_[n=1]{}\^[L-1]{} e\^[-i L]{} \[eq:rhor\_stat\] where we have used the fact that $\tilde \phi(0) = \sum_x \phi(x) = L/2.$ Clearly, in absence of resetting, [*i.e.*]{}for $r=0,$ we get the flat profile which corresponds to the equilibrium scenario. For non-zero $r,$ however, the stationary profile is non-trivial and corresponds to a non-equilibrium stationary state, carrying non-zero current. We will explore that in the next section.
It is worth mentioning that $\rho(x,t)$ also satisfies a renewal equation (following directly from Eq. ) in terms of the density profile $\rho_0(x,t)$ in absence of resetting, (x,t) = e\^[-rt]{} \_0(x,t) + r \_0\^t e\^[- r ]{} \_0(x,). \[eq:rhoreset\] For the sake of completeness we have added a brief review of the density profile and its evolution for ordinary SEP in the Appendix \[app:sep\]. Using the explicit form of $\rho_0(x,t)$ given in Eq. it is straightforward to check that Eq. leads to Eq. .
We have not used any specific form of $\phi(x)$ so far; in fact, the results above are valid for resetting to any generic profile. For the specific choice of the step-like configuration given in Eq. we have $\phi(x)=1- \Theta(x+ 1 -\frac L2)$ and (n) = = {
1+i L & n 0 & n
. In that case, the density profile takes the form, (x,t) &=& (x) + 1L \_[n=1,3]{}\^[L-1]{} e\^[- i ]{} e\^[-(r+\_n)t]{}&& \[eq:rhor\_t\_C0\] where (x) = 12 + rL \_[n=1,3]{}\^[L-1]{} e\^[- i ]{} \[eq:rhor\_stat\_C0\] is the stationary profile.
Figure \[fig:rho\](a) shows the time-evolution of the density profile $\rho(x,t)$ for a specific resetting rate $r$ and Fig. \[fig:rho\](b) shows stationary profiles $\rho(x)$ for different values of $r.$ In both cases, the analytical results (solid lines) are compared with the data obtained from numerical simulations (symbols). An excellent match confirms our analytical prediction.
Particle current {#sec:cur}
================
The behaviour of current plays an important role in characterizing the interacting particle systems like exclusion processes. For ordinary SEP, there is no particle current flowing through the system in the stationary (equilibrium) state. However, starting from a step-like initial configuration, the relaxation to equilibrium is characterized by the presence of a non-vanishing particle current. In particular, the behaviour of the time-integrated current, [*i.e.*]{}, the net particle flux through the central bond up to time $t,$ has been studied extensively in the past and it was shown that at long time limit, the average flux grows $\sim \sqrt{t}$ [@Derrida1; @Derrida2].
In presence of resetting, there are two different kinds of particle motions, consequently the total current can be expressed as, J\_r(t) = J\_(t) + (t) \[eq:Jtot\_def\] Here $\Jd$ is net diffusive flux, [*i.e.*]{}, the net number of particles which crossed the central bond due to the nearest neighbour hopping. $\Jres$ denotes the contribution due to the sudden reset to the step-like configuration $\cal C_0$. Note that, after a resetting, the system is brought back to $\cal C_0$, [*i.e.*]{}, there are no particles to the right of the central bond, implying that the total current $J_r$ is also reset to zero after each resetting event. Figure \[fig:traj\] shows the time evolution of $J_\text{d}$ and $J_r$ for a typical trajectory of the system. The sudden jumps in $J_r$ indicate the resetting events.
![Time evolution of the diffusive current $\Jd(t)$ and the total current $J_r(t)$ along a typical trajectory of the system. On an average, the diffusive current increases with time $t.$ The total current vanishes after each resetting event –indicated by the vertical lines on the light green curve – and reaches a stationary state in the long time limit. []{data-label="fig:traj"}](./traj.pdf){width="8"}
In the absence of resetting, the only source of current is the diffusive hopping motion. In the following we explore the behaviours of all these three different currents, in presence of resetting.
{width="11.4"} {width="5.6"}
Diffusive current {#sec:J_dif}
-----------------
The diffusive current $\Jd(t)$ measures the total number of particles which crossed the central bond $(\frac L2 -1, \frac L2)$ during the time interval $[0,t],$ and can be expressed as a time integral, J\_(t) &=& \_0\^t s j(s) \[eq:Jdif\_intdefn\] Here, $j(t)$ denotes the instantaneous diffusive current, [*i.e.*]{}, the number of particles crossing the central bond during the time-interval $t$ and $t+{\textrm{d}}t.$ The average instantaneous current is given by, j(t) &= & s\_[L2-1]{}(1-s\_[L2]{})- (1- s\_[L2-1]{})s\_[L2]{}&= &(L2 -1,t) - (L2,t ) Using the explicit expression for the density from eq. one gets, j(t) = 2L \_[n=1,3]{}\^[L-1]{}. In the limit of thermodynamically large system size, [*i.e.*]{}$L \to \infty$, the sum in the above expression can be converted to an integral over continuous variable $q= 2 \pi n/L$ and we get, j(t) = \_0\^[2 ]{} where $\lambda_q = 2(1- \cos q).$ In the long-time regime, the second term decays exponentially and $\la j(t)\ra$ reaches a stationary value, \_[t]{} j(t) = \_0\^[2 ]{} r[r+ 2(1- q)]{} = . \[eq:jr\_stat\] The average net flux $\la J_\text{d}(t)\ra$ up to time $t$ can be found by integrating the instantaneous current, J\_(t)&=& t & +& \_0\^[2 ]{} (1- e\^[-(r+\_q)t]{}). \[eq:avJdif\_q\] Clearly, in the long-time regime, the second term goes to a constant and the first term dominates the behaviour of the average current which grows linearly with time, J\_(t) t. This equation is one of our main results, which shows that the behaviour of the diffusive current changes drastically by the presence of resetting; instead of the standard $\sqrt{t}$ growth in a diffusive system, resetting yields a much faster, linear, temporal growth of the diffusive current. The average current $\la \Jd(t)\ra$ at any time $t,$ [*i.e.*]{}, before reaching the $\sim t$ behaviour, can be obtained from Eq. by evaluating the $q$-integral numerically. In fact, one can also derive an alternative expression which lends itself more easily to numerical evaluation. Let us recall that the density profile $\rho(x,t)$ satisfies a renewal equation for any $x$. Then, clearly, $\la j(t)\ra$ must also satisfy the same renewal equation, j(t)&=& e\^[-r t]{} j\_0(t) + r \_0\^t e\^[-r ]{} j\_0() \[eq:jrt\_defn\] where $j_0(t)$ denotes instantaneous current through the central bond in the absence of resetting. The average instantaneous current is given by $\la j_0(t) \ra = e^{-2t} I_0(2t)$ where $I_0$ is the Modified Bessel function of the first kind (see Appendix \[sec:J0\] for the details). The average diffusive net current is obtained by integrating the above equation w.r.t. time \[see Eq. \], (t) &=& \_0\^t e\^[-r ]{} (1+rt - r) j\_0() . \[eq:avJdif\_t\] It is straightforward to show that Eq. is equivalent to Eq. . Average current $\la \Jd(t) \ra$ computed from Eq. , for different values of $r,$ are plotted together with the same obtained from simulation in Figure \[fig:J\_r\](a).
An explicit form for $\la J_\text{d}(t)\ra$ can be derived for small $r \ll 1.$ Using a variable transformation $w=r \tau,$ and using the exact form for $\la j_0(\tau) \ra$ we get, (t) &=& 1r \_0\^[rt]{} w (1 + rt - w) e\^[-w]{} e\^[-r]{} I\_0(r). For small $r,$ the argument of $I_0$ is large and one can use the asymptotic form for the Modified Bessel function given in Eq. , (t) & & 1r \_0\^[rt]{} w (1 + rt - w) e\^[-w]{} 1[2 ]{} &=&1[2 ]{} . \[eq:Jd\_av\] In the short time-regime this function grows as $\sqrt{t}$ which is reminiscent of the free SEP and crosses over to the linear behaviour for $t \gg r^{-1}.$ Figure \[fig:J\_r\](b) shows plot of $\sqrt{r}\la \Jd(t) \ra$ as a function of $r t$ for different small values of $r$ which shows a perfect collapse and matches with the scaling function given by the above equation.
To characterize the fluctuation of the diffusive current we next calculate the second moment of $J_\text{d}.$ The above renewal equation method cannot be applied directly to compute higher order moments. To this end, we now adopt a different approach. Let us assume that there are $n$ resetting events during the time interval $[0,t];$ moreover, let $t_i$ denote the interval between $(i-1)^{th}$ and $i^{th}$ events, so that $\sum_{i=1}^{n+1} t_i = t.$ Note that, $t_{n+1}$ denotes the time-interval between the last reset and the final time $t.$ Let us also recall that between two consecutive resetting events the system evolves following ordinary SEP dynamics. The diffusive current during the interval $[0,t]$ can, then, be expressed as, J\_ = \_[i=1]{}\^[n+1]{} J\_0(t\_i), \[J\_d-sum\] where, $J_0(t_i)$ are independent of each other. For notational convenience, we denote $J_i \equiv J_0(t_i).$ The probability density that the diffusive current will have a value $\Jd$ in time $t$ is then given by P(J\_,t) &=& \_[n=0]{}\^ \_[0]{}\^t \_[i=1]{}\^[n+1]{}t\_i [P]{}\_n({t\_i };t) && \_[i=1]{}\^[n+1]{}J\_i P\_0(J\_i,t\_i) (J\_-\_i J\_i), [P]{}\_n({t\_i };t) &=& r\^n e\^[-r\_[i=1]{}\^[n+1]{} t\_i]{} (t-\_i t\_i) \[eq:P\_Jd\] denotes the probability of having $n$ resetting events with duration $t_i$ within the interval $[0,t].$ The distribution of the individual $J_i$-s are denoted by $P_0(J_i,t_i)$ which is exactly the distribution of the diffusive current in SEP, in the absence of resetting.
To handle the constraints presented by the $\delta$-functions, it is convenient to calculate the Laplace transform w.r.t. time $t$ of the moment generating function $\la e^{\lambda J_\text{d}} \ra,$ Q(s,) &=& [L]{}\_[t s]{} \[e\^[J\_]{} \]= \_0\^ t e\^[-s t]{}e\^[J\_]{} &=& \_0\^ t e\^[-s t]{} J\_ e\^[J\_]{} P(J\_,t). Using Eq. , and performing the integrals over $\Jd$ and $t,$ we get, Q(s,)&=& \_[n=0]{}\^r\^n \_0\^\_[i]{}\^[n+1]{} t\_i\
\[0.2em\] &&\_[i]{}\^[n+1]{} J\_i P\_0(J\_i,t\_i) &=& \_[n=0]{}\^r\^n h(s,)\^[n+1]{} \[eq:Q\_sum\] where, we have denoted, h(s,) = \_0\^ e\^[-(r+s)]{} J\_0 e\^[J\_0]{} P\_0(J\_0,)\[eq:hsl\] Performing the sum in Eq. , we get, Q(s,) = \[eq:Qsl\] which gives a simple relation between the moment generating functions of the current in the presence and absence of resetting. To calculate $h(s,\lambda)$ we need the current distribution $P_0(J_0,\tau)$ for the ordinary SEP, which is not known in general for arbitrary values of $\tau$. However, for small values of $r$ and $s,$ the $\tau$-integral in Eq. is dominated by large values of $\tau,$ and in that case one can use the result of Ref. [@Derrida1] where the authors have derived an expression for the moment generating function of $J_0(\tau)$ in the large time limit. Adapting their result to our specific case (see Appendix \[sec:J0\_dist\] for the details), we have, J\_0 e\^[J\_0]{} P\_0(J\_0,) =e\^[J\_0]{} e\^[F()]{}, \[eq:J0\_gen\] with, F() = -1 \_[3/2]{}(1-e\^).\[eq:F\_lam\] Here $\text{Li}_\alpha(z)$ denotes the Poly-Logarithm function (see Ref. [@polylog], Eq. 25.12.10). Substituting Eq. in Eq. and performing the integral over $\tau,$ we get, for small $r$ and $s,$ h(s,) &=& && \[eq:hsl\_F\] One can easily extract the Laplace transforms of the moments using Eq. along with . First, we have, \_[t s]{} &=& Q(s,)|\_[=0]{} &=& The average current can be obtained by inverting the Laplace transform, \_(t) (t) &=& [L]{}\^[-1]{}\_[s t]{} The inversion can be performed exactly using Mathematica, and yields, \_(t)&=& 1[2 ]{} \[eq:Jd\_av2\] Note that the above equation is the same as Eq. , which was obtained using a different method.
![Distribution of the diffusive current $P(J_{\textrm{d}},t)$ for $r=0.01$: (a) Plot of $P(J_{\textrm{d}},t)$ [*vs*]{} $\Jd$ for different values of $t,$ the left most curve corresponding to the smallest value of $t.$ The solid lines correspond to the predicted Gaussian form . (b) The same data are plotted as a function of $(J_{\textrm{d}}- \mu_{\textrm{d}})/\sigma_d.$ The solid black line corresponds to a standard normal distribution ${\cal N}(0,1).$ []{data-label="fig:dist_r0.01"}](dist_r_01.pdf){width="8.8"}
The Laplace transform of the second moment is obtained from the second derivative of $Q(s,\lambda),$ \_[t s]{} \^2(t) &=& Q(s,)|\_[=0]{}&=& 1[s\^2]{}+ + r[2 s\^3]{} where $b=(1-1/\sqrt{2}).$ Fortunately, the inverse Laplace transform can be performed exactly in this case also, and it yields, for small $r,$ \^2(t) &=& 1[4 ]{}\[eq:Jdsq\_av\] Note that, the above expression is expected to be valid for large $t,$ as we have assumed $s$ to be small. The variance of the diffusive current $\sigma^2_\text{d}(t) =\la \Jd^2(t) \ra -\la \Jd(t) \ra^2$ can be obtained using Eqs. and . In particular, in the long time limit, the variance increases linearly with time $t$, and is given by, \_\^2(t) t \[eq.sigma\_J\_d\] Figure \[fig:J\_r\](c) shows a plot of $\sigma^2_\text{d}(t)$ vs $t$ for different values of $r,$ obtained from numerical simulations; all the curves show linear growth in the long time regime. The curves corresponding to small values of $r \ll 1$ are compared with the analytical result (solid lines) which shows a perfect match for $t > 10.$\
[**Distribution of $\Jd$**]{}: It is interesting to investigate the probability distribution of the diffusive current $J_d(t)$. From Eq. we observe that $J_d(t)$ is the sum of the hopping currents $J_0(t_i)$ between successive resetting events. Since the time-evolution of the system is Markovian and after each resetting event the system is brought back to the initial configuration, the variables $J_0(t_i)$ are independent and distributed identically [^1]. As mentioned earlier, the distribution of $J_0(t_i)$ is known [@Derrida1] and has finite moments. Over a large time interval $t$, the number $n$ of resetting events, which is also a random quantity, is typically large and on an average grows linearly with time $t;$ in fact, $\la n\ra = rt.$ For $t \gg r^{-1},$ $\Jd(t)$ is a sum of large number of independent random variables. Hence, by central limit theorem, one can expect that for large $t$, the typical distribution of $J_d$ would be a Gaussian: P(,t) (-) \[eq:PJd\_gauss\] where the mean $\mu_{\textrm{d}}(t)$ and the variance $\sigma_d^2(t)$ are given in Eqs. and respectively. This prediction is verified in Fig. \[fig:dist\_r0.01\](a) where the Gaussian form of $P(\Jd,t)$ is compared to the data obtained from numerical simulations for a set of (large) values of $t$ and fixed $r.$ Clearly, the analytical curves are indistinguishable from the simulation data, which confirms our prediction. Fig. \[fig:dist\_r0.01\](b) shows the same data plotted against the scaled variable $(\Jd-\mu_d(t))/\sigma_d(t)$ and compared with the standard normal distribution (solid black line).
Resetting current {#sec:J_reset}
-----------------
Presence of the resetting dynamics gives rise to a resetting current $\Jres$ \[see Eq. \] which measures the flow of particles due to the sudden change in the configuration of the system. In this Section we investigate the properties of this resetting current. Let us remember that the number of particles crossing the central bond (from right to left) at the resetting event is exactly same as the hopping current (from left to right) during the period after the previous resetting event. Net resetting current during a time interval $[0,t],$ then, can be expressed as, = - \_[i=1]{}\^n[J\_0(t\_i)]{} \[eq:Jreset\_def\] where, as before, $n$ denotes the number of resetting events in time $t$ and $t_i$ denotes the interval between $(i-1)^{th}$ and $i^{th}$ resetting events. Note that, the upper limit of the sum is $n$ in the above Eq. as there is no contribution to the resetting current after the last resetting event.
![Behaviour of the resetting current $\Jres :$ (a) Plot of $-\la \Jres(t)\ra$ as a function of time for different values of $r$ obtained from numerical simulations. The lowermost curve corresponds to the smallest value of $r.$ The black solid lines correspond to the analytical prediction Eq. for small values of $r.$ (b) Variance of the resetting current $\la \sigma^2_\text{reset}(t) \ra$ as a function of time $t$ for different values of $r.$ The curves corresponding to small values of $r$ (three lower curves) are compared with the analytical result (black solid lines). A lattice of size $L=1000$ is used for all the numerical simulations.[]{data-label="fig:Jreset"}](cur_reset.pdf){width="8.8"}
To calculate the moments of $\Jres$ we follow the same method as in Sec. \[sec:J\_dif\] and calculate the Laplace transform of the moment generating function of $\Jres,$ K(s,) = \_0\^ t e\^[-s t]{} e\^ P(,t). \[eq:K-def\] Here $\mathscr P(\Jres,t)$ denotes the probability that the resetting current has a value $\Jres$ at time $t,$ and is given by, P(,t) &=& \_[n=0]{}\^ \_[0]{}\^t \_[i=1]{}\^[n+1]{}t\_i [P]{}\_n({t\_i };t) & & \_[i=1]{}\^[n]{}J\_i P\_0(J\_i,t\_i) (+\_[i=1]{}\^n J\_i). \[eq:P\_Jreset\] with ${\cal P}_n(\{t_i \};t)$ given in Eq. . As before, we have used $J_i \equiv J_0(t_i).$ Using Eq. in Eq. and performing the integrals over $t$ and $\Jres,$ we get, K(s,) &=& 1[(r+s)]{} \_[n=0]{}\^r\^n h(s,-)\^n&=& 1[(r+s)\[1- rh(s,-)\] ]{}\[eq:Ksl2\] where $h(s,\lambda)$ is given by Eq. . As mentioned already, it can computed exactly for small values of $r, s$ and is given by Eq. .
Next we calculate the moments of the resetting current using Eqs. along with Eq. . First, we have the Laplace transform of the average resetting current, L\_[ts]{} \[(t)\] &=& K(s,)|\_[=0]{}&=& -. The inverse transform can be performed exactly to obtain, (t)= - 1[2]{}. \[eq:Jreset\_av\] Note that the above expression is expected to be valid for small values of $r \ll 1$ and large $t \gg 1.$ Equation is very similar to Eq. , which gives the average diffusive current $\la \Jd(t)\ra$, except, of course, the fact that the average reseting current is negative. In fact, at very long-times $t \gg r^{-1},$ we see a linear growth in magnitude, (t)= -(t)- 2, At short-times, however, a different behaviour is seen. From Eq. , for $t \ll r^{-1},$ we have, (t)= -+ O(t\^[5/2]{}). Clearly, at short-times, the resetting current grows much faster than the diffusive current. Figure \[fig:Jreset\](a) shows a plot of $\la \Jres(t)\ra$ as a function of $t$ for different values of $r$ which illustrates these features.
It is also interesting to look at the fluctuations of $\Jres.$ From Eq. we can find the Laplace transform of the second moment, L\_[ts]{} \[\^2(t)\]&=& K(s,)|\_[=0]{}&=& + where, as before, we have used $b= 1- \frac 1{\sqrt 2}.$ Once again, the Laplace transform can be inverted exactly and yields, for $r \ll 1$ and $t \gg 1,$ \^2(t)&=& 1[4 r]{}.\[eq:Jreset\_sqav\] The variance of the resetting current $\sigma_\text{reset}^2(t)=\la \Jres^2(t)\ra -\la \Jres(t)\ra^2$ can be computed from Eqs. and and it turns out that the variance also increases linearly at the long time limit $t \gg r^{-1}$. In fact, it is straightforward to show that, in this limit, $\sigma_\text{reset}^2(t)=\sigma_{\textrm{d}}^2(t)$ \[see Eq. \]. Figure \[fig:Jreset\](b) shows $\sigma_\text{reset}^2(t)$ for different values of $r$ obtained from numerical simulations together with the analytical prediction for small $r.$
We conclude the discussion about the resetting current with a brief comment about the probability distribution $\mathscr P(\Jres,t).$ Since $\Jres,$ similar to $\Jd$, is also a sum of a set of independent variables $J_0(t_i)$, we can use the central limit theorem to predict the behaviour of the corresponding distribution. In fact, for $rt \gg 1,$ one can expect that $\mathscr P(\Jres,t)$ is similar to $P(\Jd,t)$ and has a Gaussian behaviour around the mean value, P(,t) 1.
Total current {#sec:J_tot}
-------------
In this section we investigate the behaviour of the total current $J_r,$ as defined in Eq. . $J_r(t)$ measures the net number of particles which have crossed the central bond towards right (by hopping, or due to resetting) up to time $t.$ As already mentioned, $J_r$ is set to zero after every resetting event; the contribution to the total current comes only from the diffusion of the particles after the last resetting event. Consequently, one can write a renewal equation for $P_r(J_r,t),$ the probability that, at time $t,$ the total current will have a value $J_r,$ P\_r(J\_r,t) = e\^[-r t]{} P\_0(J\_r,t) + r \_0\^t s e\^[-rs]{} P\_0(J\_r,s). \[eq:PJtot\_renewal\] Here $P_0(J_r,s)$ denotes the probability that, starting from $\cal C_0,$ in absence of resetting, $J_r$ number of particles cross the central bond until time $s.$ We will use the above equation to explore $P_r(J_r,t),$ but, first it is useful to investigate the mean and the variance of the total current.
It is easy to see that all moments of $J_r$ should also satisfy a renewal equation similar to Eq. . In particular, the average total current must satisfy, J\_r(t) = e\^[-rt]{} J\_0(t)+ r\_0\^t e\^[-r]{} J\_0() \[eq:Jtot\] where $\la J_0(t) \ra$ is the average current in absence of resetting, and is given by Eq. . Unfortunately, the above integral in Eq. cannot be computed analytically. However, it is possible to numerically evaluate the integral and get $\la J_r (t)\ra$ for any time $t.$ This is shown in Fig. \[fig:Jtot\_r\] for different values of $r$ and compared with numerical simulations (symbols) which matches perfectly at all times.
{width="11.4"} {width="5.8"}
For small values of $r,$ a more explicit expression for the average total current $\la J_r(t) \ra$ can be derived. In that case, it is convenient to rewrite Eq. as J\_r(t) = e\^[-rt]{} J\_0(t) + \_0\^[rt]{} u e\^[-u]{} J\_0 (u r ) The integral is dominated by the contribution from small $u \sim \cal{O}(1);$ consequently, $u/r$ is large for small $r,$ and we can use the asymptotic expression $\la J_0(u/r) \ra \simeq \sqrt{u/r \pi}.$ Substituting that in the above equation, and performing the integral, we get, for large $t,$ J\_r (t) = 1[2]{} (). \[eq:Jtot\_scaling\] Equation provides an explicit expression for the average total current for small $r,$ and in the large time regime. Note that, $\la J_r(t)\ra$ given by the above equation is same as $\la \Jd(t) \ra + \la \Jres(t)\ra,$ as clearly seen from Eqs. and . This is expected as the total current is a sum of the diffusive current and the resetting current \[see Eq. \].
We have also measured the total current $J_r$ from numerical simulations. Figure \[fig:Jtot\_r\](b) shows a plot of $\sqrt{r}\la J_r (t) \ra$ as a function of $rt$ for different (small) values of $r,$ as obtained from numerical simulation; the solid line corresponds to $\text{erf}(\sqrt{rt}).$ The perfect collapse of all the curves verifies our analytical prediction.
From Eq. it can be seen that for $t \ll r^{-1}$ the average total current grows as $\sqrt{t},$ which is a signature of the ordinary SEP. On the other hand, in the large time limit $\la J_r \ra$ reaches a stationary value $1/2 \sqrt{r}.$
In fact, the stationary value of the average total current $\la J_r \ra$ can be calculated exactly from Eq. for any value of $r.$ As we have already seen, at large times $t,$ $\la J_0(t) \ra \sim \sqrt{t},$ hence, the first term in Eq. decays exponentially and the large-time behaviour of the average total current is dominated by the second integral in the above equation. Recalling Eq. and using the series expansion of the Modified Bessel functions $I_0$ and $I_1,$ (see Ref. [@polylog], Eq. 10.25.2) we have, && \_0\^t e\^[-r]{} J\_0() &=& \_[m=0]{}\^1[m!(m+1)!]{}1[(r+2)\^[2m+3]{}]{}, where $\Gamma_n$ and $\Gamma_n(x)$ are the Gamma function and the incomplete Gamma function, respectively. $\Gamma_n(x)$ decays to zero for large $x$ for all values of $n,$ and hence, in the long time limit we have the contributions only from the $t$-independent terms, \_[t ]{}J\_r && r\_[m=0]{}\^ &=& 1. \[eq:Jtot\_larget\] Clearly, in the long-time limit the average total current reaches a stationary value $\mu_r = 1/{\sqrt{r(r+4)}}$ which decreases as the resetting rate $r$ is increased. For small $r,$ $\mu_r \approx \frac 1{2\sqrt{r}}$ which is same as what we obtained by taking $t \to \infty$ limit in Eq. . On the other hand, for large $r \gg 1,$ $\mu_r$ approaches $1/r.$
Physically, the limiting behaviours of the stationary value of the average total current can be understood from the following argument. Since the value of $J_r$ is reset to zero after each resetting event, the final contribution to $J_r$ comes only from the diffusion of particles after last resetting event. Moreover, the typical duration since the last resetting event is $\tau_r \sim 1/r.$ For small $r,$ this typical duration is long, and the average diffusive current (without resetting) during this period is $\sim \sqrt{\tau_r} = 1/\sqrt{r}.$ On the other hand, for large $r,$ $\tau_r$ is small, the diffusive current is $\sim \tau_r = 1/r.$
Next we calculate the second moment of the total current $\la J_r(t)^2 \ra.$ As mentioned already, the second moment also satisfies a renewal equation of the form, J\_r(t)\^2 = e\^[-rt]{} J\^2\_0(t)+ r \_0\^t e\^[-r ]{} J\^2\_0(). \[eq:Jr\_sq\] The above equation is valid at all times and for all values of $r.$ Unfortunately, however, the behaviour of $\la J^2_0(\tau)\ra$ is known only for long time $\tau$ (see Eq. ) so we are not able to calculate an exact analytical expression for $\la J_r(t)^2 \ra$ for any arbitrary time $t.$ Nevertheless, one can use Eq. along with Eqs. and to calculate $\la J_r(t)^2 \ra$ for small values of $r,$ where the integral in Eq. is dominated by the contribution from large $\tau \gg r^{-1}.$ This exercise leads to a simple analytical formula for the second moment for small $r$ (and large time $t$), J\^2\_r(t)1[r]{}(1 - e\^[-rt]{} )+ 1[2]{} (1- 1[2]{}) () \[eq:Jrsq\_t\]
Figure \[fig:Jtot\_r\](c) shows the plot of $\la J_r^2(t) \ra$ as a function of $t$ for different values of $r.$ The curves for small $r$ are compared with the analytical result Eq. which show an excellent match. Similar to the average total current, the second moment $\la J_r^2(t) \ra$ also eventually reaches a stationary value which, for small values of $r,$ can be obtained by taking $\lim t \to \infty$ in Eq. , J\_r\^2 = 1 [r]{} + 1[2]{} (1- 1[2]{}). \[eq:Jtotsq\_larget\] One can immediately calculate the stationary value of the variance $\sigma_r^2 = \la J_r^2 \ra - \la J_r \ra^2;$ as $r \to 0,$ $\sigma_r^2 \simeq (4- \pi)/ 4 \pi r.$
On the other hand, for short time $t,$ we have, J\_r\^2 (t)t + (1- 1[2]{})\[eq:Jsq\_shortt\] At very short-times, one expects a $\sqrt{t}$ behaviour which crosses over to a linear behavior as $t$ is increased. This is also seen in Figure \[fig:Jtot\_r\](c), where the approach to the stationary value appears predominantly linear.\
[**Correlation between $\Jd$ and $\Jres$:**]{} The computation of the second moment of the total current $J_r$ provides a way to estimate the correlation between the diffusive and resetting components of the current. From the definition of the total current Eq. , we get, J\_r\^2(t) = \^2(t) + \^2(t) + 2 (t)(t) . The connected correlation $C(t)= \la \Jd(t) \Jres(t) \ra - \la \Jd(t)\ra \la \Jres(t) \ra$ is then given by, C(t)= 12 where $\sigma_r^2,\sigma_{\textrm{d}}^2$ and $\sigma_\text{reset}^2$ are the variances of the total, diffusive and resetting currents, respectively. Using Eqs. , , and along with Eqs. , and , we get, for small values of $r,$ C(t) &=&1[4 r]{}. \[eq:Ct\] Clearly, the diffusive and resetting currents are strongly correlated. To understand the nature of this correlation we look at the limiting behaviour of $C(t).$ At long times $t \gg r^{-1},$ we get a linear temporal growth from Eq. , C(t) - \_\^2(t) -t . On the other hand, for short-times $t \ll r^{-1}$ (but $t \gg 1$) we get, C(t) = - t\^[3/2]{} + O(t\^2). In fact, the correlation remains negative at all times. Figure \[fig:J\_cor\] shows a plot of $-C(t)$ [*vs*]{} $t$ for different values of $r$ obtained from numerical simulations (symbols) along with the analytical prediction (solid lines) for small values of $r.$
![Correlation between diffusive and resetting currents: Plot of $-C(t)$ as a function of time $t$ for different values of $r;$ the lowest curve corresponds to the smallest value of $r.$ The data from the numrical simulations (symbols) are compared to the analytical prediction from Eq. (solid lines) for small values of $r.$ Lattice size $L=1000$ is used for the simulations.[]{data-label="fig:J_cor"}](corrJ.pdf){width="6.5"}
The presence of a non-trivial correlation between the diffusive and resetting currents suggests that even though the fluctuations of both these components of current are Gaussian in nature, the distribution of the total current need not be so. In the following we investigate this issue and show that, indeed the fluctuations of $J_r$ are characterized by a strongly non-Gaussian distribution.\
[**Probability distribution of $J_r$:** ]{} In this Section we explore the behaviour of the probability distribution of the total current $P_r(J_r,t)$ using the renewal equation . In the absence of resetting, the fluctuations of the total (diffusive) current are characterized by a Gaussian distribution in the long-time limit (see Appendix \[sec:J0\_dist\] for more details). Using the Gaussian form of $P_0(J_r,t)$ one can calculate the total current distribution $P_r(J_r,t)$ for small values of $r$ (for small $r$ the integral is dominated by the large $t$ contribution). It is particularly interesting to look at the stationary distribution,
P\_r\^(J\_r) = r \_0\^ \[eq:P\_Jtot\] where $\mu_\tau = \sqrt{\tau/\pi}$ and $\sigma_\tau^2= \sqrt{\tau/\pi}(1-1/\sqrt{2})$ are the mean and the variance of the current in absence of resetting, respectively. Clearly, for any finite value of $J_r,$ the Gaussian part of the integrand, [*i.e.*]{}, $\exp{[-(J_r-\mu_\tau)^2/2\sigma_\tau^2]},$ vanishes both at $\tau \to 0$ and $\tau \to \infty$ limits, ensuring that the integral is convergent. One can then use the series expansion of $e^{-r\tau}$ in Eq. to express $P_r^\text{st}(J_r)$ as an infinite sum of integrals, P\_r\^(J\_r) = r \_[n=0]{}\^ \_0\^.Because of the asymptotic properties of $\exp{[-(J_r-\mu_\tau)^2/2\sigma_\tau^2]}$ mentioned above, each of these integrals converge. It turns out that, these integrals can be evaluated exactly for all values of $n$ and yields an explicit expression for the stationary distribution in the form of an infinite series, P\_r\^(J\_r) &=& && \_[n=0]{}\^ (J\_r)\^[2n+32]{}K\_[2n+32]{}(b).\[eq:P\_Jrst\] We here have used $b=(1-1/\sqrt{2})$ for brevity, and $K_\nu(z)$ is the Modified Bessel function of the second kind [@polylog] (see Eq. 10.31.1 therein). Convergence of the original integral in Eq. ensures that the series is also convergent for any finite $J_r.$ Hence, the stationary distribution $P_r^\text{st}(J_r)$ can be computed to arbitrary accuracy using Eq. . This is demonstrated in Fig. \[fig:P\_Jtot\](a) where the theoretical computation is plotted together with the simulation results.
![Stationary probability distribution of the total current $P_r^\text{st}(J_r)$: (a) Plot of $P_r^\text{st}(J_r)$ for different (small) values of $r;$ symbols represent the data obtained from numerical simulations and solid lines correspond to the analytical result obtained using Eq. . (b) Plot of the same data as in (a) as a function of $Y=(J_r - \mu_r)/\sigma_r$. For simulations we have used $L=1000.$[]{data-label="fig:P_Jtot"}](./PJ.pdf){width="8.8"}
The stationary distribution has some interesting features which are visible from Fig. \[fig:P\_Jtot\](a). First, it is apparent that $P_r^\text{st}(J_r)$ is vanishingly small for negative values of $J_r.$ This can be understood in the following way. Let us recall that, at any time, the total current is nothing but the net number of particles hopping across the central bond since the last resetting event, [*i.e.*]{}, after being brought to the configuration ${\cal C}_0$ where the left half of the lattice is filled-up. To produce a negative current, the number of particles crossing the central bond from left to right should be lower than that from right to left [*i.e.*]{}, there should be a net flux of the particles to the left. Since, the particles are allowed to hop only to empty neighbouring sites, starting from the configuration ${\cal C}_0,$ this is an extremely unlikely event and has a vanishingly small probability.
Secondly, it also appears that $P_r^\text{st}(J_r)$ is strongly non-Gaussian which is manifest in the asymmetric behaviour of the two tails, as seen in Fig. \[fig:P\_Jtot\](a). To characterize this asymmetry and the non-Gaussian nature quantitatively we look at the decay of $P_r^\text{st}(J_r)$ at the two tails, namely, near $J_r=0$ and large $J_r.$ Near $J_r=0,$ for small values of $r,$ the behaviour is dominated by the $n=0$ term in Eq. . One can then use the asymptotic behaviour of $K_{\frac 32}(z)$ near $z=0$ to get P\_r\^(J\_r) 2 r (J\_r + 1 - 1 ) + O(r\^2). \[eq:Pjtot\_smallj\] Clearly, for small values of $r,$ the probability distribution of the total current $J_r$ decays linearly near $J_r=0.$ To determine how $P_r^\text{st}(J_r)$ decays for large $J_r$ we use the asymptotic behaviour of $K_\nu(z);$ for large values of the argument $z,$ we have (see Ref. [@polylog], Eq. 10.40.2), \_[z ]{} K\_(z) e\^[-z]{}. \[eq:K\_largez\] Using that in Eq. and performing the sum over $n,$ we get, P\_r\^(J\_r) 2 r J\_r e\^[-r J\_r\^2]{} + (r\^2). \[eq:Pjtot\_largej\] Note that the above expression holds true to the leading order in $r,$ higher order corrections can be systematically calculated by including higher order terms in .
We conclude the discussion about $P_r^\text{st}(J_r)$ with one final remark. From our numerical data, we observe a surprising collapse of the current distribution when plotted as a function of the scaled variable $Y=(J_r-\mu_r)/\sigma_r$ where $\mu_r$ and $\sigma_r$ are, respectively, the mean and the variance of $J_r.$ The collapse is shown in Fig. \[fig:P\_Jtot\](b) where the scaled distribution $\tilde P(Y)$ appears to be independent of $r$ as the curves corresponding to different values of $r$ from Fig. \[fig:P\_Jtot\](a) fall on top of each other. To understand this collapse, let us look at $\tilde P(Y)$ predicted from Eqs. and . Recalling that for small values of $r,$ $\mu_r \simeq \frac1{2\sqrt{r}}$ and $\sigma_r \simeq \sqrt{\frac{(4-\pi)}{4 \pi r}},$ we get from Eq. , P(Y) && 12 (+ (4-)Y) e\^[-14 (+ Y)\^2]{}&& + (r\^[3/2]{}).\[eq:PY\_large\] Clearly, to the leading order in $r,$ $\tilde P(Y)$ calculated from Eq. (corresponding to large values of $J_r$) is independent of $r,$ and is consistent with the scaling collapse observed in Fig. \[fig:P\_Jtot\](b). On the other hand, it can be easily seen, that Eq. does not lend itself to a similar form; $\tilde P(Y)$ derived from Eq. depends explicitly on $r,$ P(Y) && 12 (1+ Y + (2- )) && + (r\^[3/2]{}).\[eq:PY\_small\] Hence, while for large positive $J_r$ ($\gtrsim \mu_r+\sigma_r$), the distribution $\tilde P(Y)$ becomes independent of $r$, it is not the case in the $J_r \to 0$ limit. Indeed, as seen from Eq. , $\tilde P(Y)$ explicitly depends on $r.$ However, notice that the $r$-dependence in Eq. comes in the form of an additional term proportional to $\sqrt{r},$ which is vanishingly small for $r \ll 1.$ This makes the expected mismatch in the collapse at the left tail in Fig. \[fig:P\_Jtot\](b) practically invisible where an apparent collapse is also observed.
Conclusion {#sec:conc}
==========
In this article, we explore the effect of stochastic resetting on interacting many particle systems. To this end, we study the dynamical properties of a canonical set-up, namely, the symmetric exclusion process in the presence of stochastic resetting. The resetting is implemented by interrupting the dynamical evolution of the exclusion process with some rate $r$ and restarting it from a step-like configuration where all the particles are clustered together in the left-half of the system.
We find that the presence of resetting strongly affects the behaviour of the system. The key findings are as follows. First, in a finite size system, the density profile evolves to an inhomogeneous stationary profile in contrast to the flat profile in the absence of resetting. We have exactly calculated the full time-dependent density profile for arbitrary resetting rate $r.$ Secondly, we find that, in a thermodynamically large system the resetting mechanism drastically changes the $\sqrt{t}$ growth of the diffusive current to linear in $t$. We have explicitly computed the mean and variance of the diffusive current, the latter is also shown to have a linear growth in the long-time regime. Apart from the diffusive current, we also identify the another component of the current which arises due the resetting move and show that this resetting current is negative, with a linear temporal growth in magnitude. The moments of the total current, [*i.e.*]{}, the sum of the diffusive and resetting current, are also calculated using the renewal approach.
We also have investigated the probability distribution of the diffusive current $\Jd,$ resetting current $\Jres,$ as well the total current $J_r.$ We have found that that while the typical fluctuations of $\Jd$ and $\Jres$ are Gaussian in nature, the distribution of $J_r$ is strictly non-Gaussian. The non-Gaussian nature is manifest in the asymmetric asymptotic behaviour of the distribution at the two tails, which we also demonstrate.
Our study opens up a new direction in the area of stochastic resetting and gives rise to a wide range of further questions. For example, it would be interesting to study the effect of stochastic resetting in other interacting particle systems, e.g., the asymmetric exclusion process, driven and equilibrium lattice gas models etc. Furthermore, it would also be interesting to study behavior of these interacting particle systems under various other resetting mechanisms like resetting at power-law times or time-dependent resetting etc.
Apart from these theoretical questions, the framework of stochastic resetting in exclusion processes can also be relevant in the context of certain biophysical systems. For example, stochastic motion of backtracked RNA polymerases can be modelled as an interacting many particle random walk on the DNA template, with RNA cleavage playing the role of resetting dynamics [@bio3; @Lisica].\
Similarly, motion of two-headed molecular motors such as kinesin and Myosin V moving on a polymeric track can be modelled as an energy driven hopping process in the presence of backward jumps (or resetting)[@Astumian]. We believe that the formalism introduced in the present work will be useful in understanding such systems.
The authors acknowledge useful discussions with Christian Maes and Sanjib Sabhapandit. U.B. acknowledges support from Science and Engineering Research Board (SERB), India under Ramanujan Fellowship (Grant No. SB/S2/RJN-077/2018). A. K. acknowledges support from DST grant under project No. ECR/2017/000634. A.P. gratefully acknowledges support from the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel-Aviv University.
Density profile of SEP in absence of resetting {#app:sep}
==============================================
In this section we present a brief account of the dynamical evolution of the density profile and current for ordinary SEP, starting from the step-like configuration $\cal C_0.$ In absence of resetting, the time evolution of the system is governed by the free Markov matrix ${\cal L}_0$ which yields, for the density profile, \_0(x,t) &=& \_0(x+1,t)+ \_0(x-1,t)- 2\_0(x,t). The corresponding Fourier components $\tilde \rho_0(n,t)$ evolve following, \_0(n,t) = - \_n \_0(n,t) where, as before, $\lambda_n= 2\left(1- \cos \frac{2 \pi n}{L}\right),$ with $n=0,1,2,\dots L-1.$ The above equation is immediately solved to obtain, \_0(n,t) = e\^[-\_n t]{} (n). \[eq:rho0\_n\] where $\tilde \phi(n)$ corresponds to the initial profile $\phi(x).$ Note that $\lambda_0=0$ and hence $\tilde \rho_0(0,t)= \tilde \phi(0) = \frac L2$ does not evolve with time.
The spatial density profile is obtained by taking inverse Fourier transform of Eq. . In particular, for the step-like initial profile $\phi(x)=1- \Theta(x+1-\frac L2)$ we have, \_0(x,t) = 12 + 1L \_[n=1,3]{}\^[L-1]{} e\^[-i ]{} (1+i ) e\^[-\_n t]{} \[eq:rho0xt\]
Behaviour of current in absence of resetting {#sec:J0}
============================================
In the absence of resetting the only source of current in SEP is the hopping dynamics of the particles. The average instantaneous current across the initial step, [*i.e.*]{}, across the central bond $\left(\frac L2 -1, \frac L2 \right)$ is given by, j\_0(t) &=& \_0(L2-1,t) - \_0(L2,t) &=& 2L \_[n=1,3]{}\^[L-1]{} e\^[-\_n t]{} \[eq:j0t\] where we have used Eq. to calculate the average densities at the sites $x=\frac L2-1$ and $x=\frac L2.$ Clearly, in the long-time limit $t \to \infty,$ the instantaneous current vanishes as the density profile becomes flat.
We are interested in the time-integrated current $J_0(t) = \int_0^t {\textrm{d}}s~ j_0(s)$ which measures the net number of particles crossing the central bond towards right. The average time-integrated current is obtained by integrating Eq. , J\_0(t) = 2L \_[n=1,3]{}\^[L-1]{} (1-e\^[-\_n t]{}). For any finite $L,$ the average time-integrated current $J_0(t)$ saturates to an $L$-dependent constant value in the long-time limit.
To understand the behaviour of a thermodynamically large system, one has to take the limit $L \to \infty$ first. In this case, the sum in Eq. can be can be converted to an integral by denoting $q= 2 \pi n/L,$ and we have the mean instantaneous current, j\_0(t) &=& \_0\^[2 ]{} e\^[-2(1-q)t]{}\
\[0.25em\] &=& e\^[-2 t]{} I\_0(2 t). Here $I_0(x)$ is the Modified Bessel function of the first kind [@polylog] (see Eq. 10.25.2 therein). In this limit, the average time-integrated current becomes, J\_0(t) = e\^[-2t]{} t \[I\_0(2t)+I\_1(2t)\].\[eq:J0\_t\] For large values of the argument $x,$ both $I_0(x)$ and $I_1(x)$ have the same asymptotic behaviour (see [@polylog], Eq. 10.40.1 ), \_[x]{}I\_[0,1]{}(2x) \~ \[eq:I01\_large\] which yields, in the long-time regime, J\_0(t) . \[eq:J0\_larget\] This result has been obtained in Ref. [@Derrida1], albeit using a different method. In fact, it has also been shown [@Derrida1] that, in the long-time regime, all the higher moments of $J_0$ show a similar behaviour. In particular, the variance is given by, J\_0\^2(t) - J\_0(t) \^2 (1- 1). \[eq:J0\_var\] The above equation is used in Eq. in the main text to calculate $\la J_r^2\ra.$
Probability Distribution of $J_0$ {#sec:J0_dist}
---------------------------------
For ordinary SEP, the probability distribution of the time-integrated current $J_0$ was explored in Ref. [@Derrida1]. There the authors considered a scenario where, initially, each site to the left (respectively, right) of the origin ($x \le 0$ and $x>0$ respectively) is occupied with probability $\rho_a$ (respectively $\rho_b$). It was shown that, for large $t,$ the moment generating function of the total particle flux $J_0(t)$ through the origin is given by, e\^[J\_0(t)]{} \~e\^[ F()]{} where $\omega = \rho_a(e^\lambda -1) + \rho_b(e^{-\lambda}-1) + \rho_a \rho_b(e^\lambda -1)$ and F() &=& 1 \_[n=1]{}\^ && -1 \_[3/2]{}(-).
In our case, we have $\rho_a=1$ and $\rho_b=0$ which simplifies $\omega$ and in turn $F(\omega);$ we get $\omega = e^{\lambda}-1$ and F() = -1 \_[3/2]{}(1-e\^) which is quoted in Eq. in the main text.
It has been shown in Ref. [@Derrida1] that the corresponding probability distribution $P_0(J_0,t),$ in the long-time limit, is of the form, P\_0(J\_0,t) \~e\^[G(J\_0/)]{}. \[eq:PJ0\_LD\] The large deviation function $G(q=J_0/\sqrt{t})$ is related to $F(\lambda)$ through a Legendre transform, G(q) &=& \_\[F() - q\] &=& F(\^\*) - \^\* q, \[eq:Gq\] where $\lambda^*$ corresponds to the minimum of the function $F(\lambda)- \lambda q$ and is obtained by solving $\frac{{\textrm{d}}F(\lambda)}{{\textrm{d}}\lambda} = q.$ It is easy to see that for small values of $q,$ $\lambda^*$ is also small. Hence, it is convenient to use the series expansion of $F(\lambda)$ near $\lambda=0$ F()= + (1- 1) + O(\^3), to find $\lambda^*$ for small values of $q.$ Restricting ourselves to the quadratic order in $\lambda,$ we get $\lambda^*=\frac{(q \sqrt{\pi}- 1)\sqrt{2}}{(\sqrt{2}-1)}.$ Substitution of this $\lambda^*$ in Eq. yields, G(q) = Using the above $G(q)$ in Eq. results in a Gaussian form for the current distribution, P\_0(J\_0,t) = 1 \[eq:P\_J0\] where the prefactor is just a normalization constant. Here, $\mu_0(t)=\sqrt{\frac t{\pi}}$ is nothing but the average hopping current $\la J_0(t)\ra$ and $\sigma_0^2(t)=\sqrt{\frac{t}{\pi}}\left(1- \frac 1{\sqrt{2}}\right)$ is the variance \[see Eq. \]. Note that, this Gaussian distribution is expected only in the long-time limit, as Eq. holds true in this limit only.
![(a) Plot of $P_0(J_0,t)$ [*vs*]{} $J_0$ for different (large) values of $t.$ The symbols indicate the data obtained from numerical simulation of a system of size $L=1000,$ whereas the solid black lines correspond to the Gaussian distribution (see Eq. ). (b) The same data plotted as function of $(J_0-\mu_0(t))/\sigma_0(t);$ the solid line indicates the standard normal distribution.[]{data-label="fig:J0_dist"}](J0_dist.pdf){width="8.8"}
Figure \[fig:J0\_dist\] (a) shows a comparison of $P_0(J_0,t)$ obtained from numerical simulations (symbols) with the that predicted from Eq. (solid lines) for different (large) values of $t.$ Fig. \[fig:J0\_dist\] (b) shows the same data but plotted against the scaled variable $y=\frac{J_0 - \mu_0(t)}{\sigma_0(t)};$ the solid line corresponds to the standard normal distribution $\frac 1{\sqrt{2 \pi}}e^{-y^2/2}.$ The numerical data shows a very good match with the predicted Gaussian curve for typical values of $J_0,$ there are deviations only at the regime $|y| \gg 1$ which are visible only at a logarithmic scale. The large deviation function calculated in Ref [@Derrida1] describes the distribution for these atypical values. However, as shown in Sec. \[sec:J\_tot\], for our purposes it suffices to consider the typical fluctuations and we use the Gaussian distribution to calculate the distribution of the diffusive current $\Jd$ in presence of resetting.
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[^1]: Of course, the time durations $\{ t_i\}$ are different.
|
---
bibliography:
- 'references.bib'
---
Astro2020 Science White Paper
Stellar Feedback in the ISM Revealed by Wide-Field Spectral-Imaging
**Thematic Areas:** $\boxtimes$ Resolved Stellar Populations and their Environments
**Principal Author:**
Name: Institution: Instituto de Física Fundamental, CSIC, Madrid, Spain. Email: javier.r.goicoechea$@$csic.es **Co-authors:** Observatoire de Paris & CNRS, France.\
Observatoire de Paris-Meudon, France.\
**Abstract:** The radiative and mechanical interaction of stars with their environment drives the evolution of the ISM and of galaxies as a whole. The far-IR emission ($\lambda \simeq$30 to 350$\mu$m) from atoms and molecules dominates the cooling of the warm gas in the neutral ISM, the material that ultimately forms stars. lines are thus the most sensitive probes of stellar *feedback* processes, and allow us to quantify the deposition and cycling of energy in the ISM. While *ALMA* (in the (sub)mm) and *JWST* (in the IR) provide astonishing sub-arcsecond resolution images of point sources and their immediate environment, they cannot access the main interstellar gas coolants, nor are they designed to image entire star-forming regions (SFRs). far-IR photometric images of the interstellar dust thermal emission revealed the ubiquitous filamentary structure of SFRs, their mass content, and the location of thousands of prestellar cores and protostars. These images, however, provide a *static* view of the ISM: not only they dont constrain the cloud dynamics, moreover they cannot reveal the chemical composition and energy transfer within the cloud, thus giving little insight into the regulation process of star formation by stellar feedback. In this *WP* we emphasize the need of a space telescope with wide-field spectral-imaging capabilities in the critical far-IR domain.
 \[fig:cover\]
**1. Far-IR Tracers of Stellar Feedback: Warm Molecular Gas**
Massive stars dominate the injection of radiative and mechanical energy into the ISM through ionizing radiation, stellar winds, supernova explosions, and merger encounters [@Beuther07; @Zinnecker07; @Krumholz14; @Motte18; @Pabst19]. This energy input shakes the environment, heats the gas, disrupts star-forming regions (SFRs), and creates the cloud and intercloud phases of the ISM. Massive stars are born inside giant molecular cloud (GMC) cores. Protostars of different masses develop inside these star-forming cores . Their outflows shock the ambient cloud, heating and compressing the immediate environment around them. This is an example of small spatial scale protostellar feedback. Young stellar objects (YSOs) can be detected in photometric images by the bright far-IR/submm luminosity of their heavily obscured dusty cocoons.
On the large-spatial scales of an entire GMC, however, most of the gas and dust emission does not arise from individual YSOs but from the extended and more massive cloud component – the poorly known star-forming *environment*. Once a new massive star or cluster is formed, the energy and momentum injected by UV photoionization, radiation pressure, and stellar winds, ionize and erode the parental molecular cloud, creating [H[ii]{}]{} regions and blowing expanding bubbles [see simulations in e.g., @Krumholz14; @Rahner17]. feedback can thus determine the gas physical conditions and chemical composition at large scales, drive the evolution of the parental cloud itself, and regulate the formation of new stars. However, due to the lack of wide-field spectroscopic observations, there are fundamental aspects that remain to be understood. UV radiation in particular, is a very important player in the interaction of stars and ISM.
regions (PDRs) develop at the interfaces between the hot ionized gas and the cold molecular gas, and more generally, at any slab of neutral gas ( atoms not ionized) illuminated by stellar far-UV (FUV) photons with [@Hollenbach97]. The famous far-IR \[[C[ii]{}]{}\]158$\upmu$m fine-structure line is typically the brightest emission from PDR gas, and it dominates the cooling of the neutral ISM at the scales of an entire galaxy. It thus provides key information on the energy deposited in the ISM [see *COBE*’s low resolution maps of the Milky Way, @Bennett94]. With an ionization potential of 11.3eV, however, C$^+$ can also be abundant in both the warm neutral or ionized gas . Hence, it is not always trivial to delimitate the origin of the \[[C[ii]{}]{}\]158$\upmu$m emission and fully exploit its diagnostic power. Because of their specific chemistry, *certain molecular species that we discuss below, emitting in the far-IR, are very good *filters* of the different feedback processes*.
***Finding observational tracers of the feedback processes acting on the molecular gas, the fuel to form new stars, is crucial not only locally, but also in the framework of star formation across cosmic time***.\
**Only in the far-IR and only from space.** While the low-energy molecular lines that can be detected from (sub)mm-wave ground-based radio telescopes such as ALMA typically trace cold and quiescent interstelar gas, the extended ***warm molecular gas*** () affected by feedback processes (i.e., heated by stellar UV fields, enhanced doses, cosmic-ray particles, or affected by shocks and turbulence dissipation) naturally emits at higher frequencies, in the far-IR. The far-IR domain hosts a plethora of bright atomic fine-structure lines (from the neutral and ionized phases of the ISM) and high-energy (rotationally-excited) lines from CO, H$_2$O, CH$^+$, HD, and other hydrides that cannot be detected from the ground. These lines, often the most luminous lines emitted by the ISM of galaxies as a whole, prove to be unique diagnostics of the different types of energy and momentum input deposited into the ISM. That is, of the different feedback mechanisms. The launch of ESA’s *Infrared Space Observatory* (*ISO*) in 1995 opened the complete far-IR domain to spectroscopic observations, and demonstrated that the gas properties and energetics of YSOs can be constrained by studying their rotationally excited CO and H$_2$O line emission . These lines are major coolants of the $hot$ ($T_{\rm k}$$>$500 K) molecular gas and the unambiguous signature of shocked gas . lines, observed by *ISO*, NASA’s *Spitzer* and soon again with *JWST*, are also excellent tracers of the hot molecular gas, but their emission is often heavily affected by dust extinction toward embedded protostars. *ISO* had a primary mirror of $\sim$60cm size and carried primitive far-IR detectors. This limited the angular resolution and sensitivity of these pioneering detections. The launch of ESA’s *Herschel* in 2009, with a much larger $\sim$3.5m telescope and equipped with more sophisticated far-IR instrumentation (direct detection detectors and heterodyne receivers) allowed a more robust characterization of (a rather short list of) low-and high-mass YSOs in different stages of evolution . However, robust statistics and spectroscopic observations of much larger protostellar samples and their environment are largely missing. More precisely, ***[Herschel]{} did not have instrumentation to carry out spectroscopic maps of entire star-forming regions***.
Figure \[fig:OMC1-maps\] shows $\sim$85 arcmin$^2$ images of key far-IR line obtained with the HIFI receiver toward the Orion molecular cloud core , the closest high-mass SFR (at $\sim$414pc). These are one of the few spectroscopic-maps obtained by *Herschel* [@Goi19]. These images show the emission from CH$^+$ (strongly FUV-irradiated gas), H$_2$O (hot shocked gas from protostellar outflows), CO $J$$=$10–9 and CO $J$$=$2–1 (observed from the ground). The four emission lines show remarkably different spatial distributions, ***emphazising the distinctive diagnostic power of different molecules and far-IR lines***. In the following, we briefly review the information that can be obtained *only* from far-IR line observations, and we apply it to our case: the radiative feedback from massive stars.\
**H$_2$O and OH** only reach high abundances in hot/warm shocked gas. H$_2$O and OH rotational lines cover a broad range of energies and excitation conditions. Thus, they are excellent diagnostics of YSOs, and of shocks due to outflows in general: protostellar, supernovae or extragalactic [@Sturm11]. Owing to the water vapor on Earth’s atmosphere, these lines can not be observed from the ground or with *SOFIA*.\
**CO rotational ladder** can be used as a “thermometer" of the hot/ warm molecular gas. Far-IR lines have been detected by *ISO* and *Herschel* at different spatial scales: from outflows driven by nearby low-mass protostars to luminous active galactic nuclei (AGN) galaxies. In these energetic sources affected by shocks, the CO spectral line energy distribution (SLED) peaks at $J>15$ ($\lambda <$173.6 $\upmu$m) and shows detectable emission at $J>30$ [$\lambda <$87.2 $\upmu$m, @Hailey12; @Karska13]. The CO line emission measured by *Herschel* toward less extreme SFRs, however, typically peaks at $J\lesssim10$ and shows large-scale emission as seen in nearby SFRs like Orion (see Fig. \[fig:OMC1-maps\]). When observed at high spectral resolution ($<$1 kms$^{-1}$), lines show narrow line-profiles, demonstrating that they arise from FUV-illuminated extended warm gas and not from fast shocks [e.g., @Goi19].\
**Hydrides such as CH$^+$** have their rotational lines in the far-IR. These molecules are not only important because their formation represents the first steps of interstellar chemistry . *Herschel* showed that their abundances are also unique tracers of the H$_2$ column density (HF and CH), the ionization rate by cosmic-ray particles or (OH$^+$, H$_2$O$^+$ or ArH$^+$), and the stellar FUV field (CH$^+$ or SH$^+$). In addition, the hydride emission from dense star-forming clouds ($n_{\rm H}$$>$1000 cm$^{-3}$) traces ***gas components and physical conditions that cannot be studied from the ground***. One outstanding example is CH$^+$. The main gas-phase pathway producing detectable quantities of CH$^+$ is reaction , where $v$ refers to the specific H$_2$ vibrational level [@Sternberg_1995; @Agundez10]. For H$_2$ molecules in =0 state, this reaction is very endothermic, , much higher than the typical gas temperatures of molecular clouds. However, the reaction becomes exothermic and fast for $v$$\geq$1. Observations and theory show that UV photons from nearby massive stars can radiatively pump H$_2$ to vibrationally excited states $v$$\geq$1 over large spatial-scales. In consequence, CH$^+$ can be abundant at large scales too, and its emission can be used to characterize the stellar UV radiation field . In particular, CH$^+$ probes a narrow (low A$_V$) but very extended component of UV-irradiated molecular cloud surfaces (where not all carbon is locked in CO). CH$^+$ is clearly a unique tracer of harsh interstellar conditions. Not surprisingly, CH$^+$ ($J$=1–0) emission has been detected by ALMA toward high-redshift ULIRG galaxies . Hence, CH$^+$ can also be used to constrain the energetics of the primitive ISM.\
**Radiative Feedback and PDRs**: Most of the mass contained in GMCs resides at low visual extinction depths (A$_V$). Therefore, most of the gas is permeated by stellar FUV photons. A strong FUV radiation field from nearby massive stars induces a plethora of poorly understood dynamical effects and chemical changes in the cloud. The high thermal presures inferred from *Herschel* observations toward a few [H[ii]{}]{}/PDR interfaces, , are consistent with the expected dynamical response of molecular clouds to strong radiation: the cloud edges are heated and compressed, and photoevaporate if the high pressures are not balanced by those of the environment [@Bertoldi96; @Bron18]. Cloud photoevaporation models predict that the thermal pressure at the irradiated cloud edges scales with the strength of the flux impinging the cloud (see Fig. 2). CH$^+$, a very reactive and short-lived molecule, proves to be as a unique tracer of these narrow layers, clearly revealing feedback processes. In particular, @Goi19 found a spatial correlation between the intensity of the CH$^+$ emission and $G_0$, the flux of FUV photons (Fig. 2). The observed correlation is supported by PDR models in the parameter space where many observed PDRs lie [@Joblin18].
**2. Need and feasibility of far-IR spectral-imaging capabilities**\
The ISM is a central component of galaxies; it provides the fuel to form new stars and it keeps a relic of previous star generations in the form of metal enrichment. Dust extinction hides the star and planet formation processes from UV, visible-light, and often near-IR observations. Stellar UV radiation, winds and supernova explosions heat and disrupt the interstellar environment, sometimes enhancing, sometimes quenching the star formation processes. The main gas coolants in these environments emit in the far-IR [e.g., @Schneider18]. The peak of the dust emission also lies in the far-IR ($\lambda \simeq$100$\upmu$m for $T_{\rm d}$$\simeq$30 K). Therefore, ***the far-IR is the natural domain to study the ISM lifecycle and stellar feedback. Since these proceses take place at large-spatial scales, spectral-imaging capabilities are desperately needed***. *space telescopes are the obvious choice for mapping and accessing the complete wavelength domain at high sensitivity and stability*. The spectacular large-scale photometric-images of the dust emission taken by far-IR cameras provided a static “snapshot” of the star-formation processes. However, these images say less about the stellar feedback processes in the ISM, and how to quantify them. It is only by carrying out large spatial scale far-IR observations of the critical gas cooling lines, and of other key astrochemical probes such as CH$^+$, that we will be able to quantify these proceses and their energetics. In addition, by using velocity-resolution spectrometers similar to *Herschel*/HIFI or *SOFIA*/GREAT receivers, we will also be able constrain the gas kinematics and turbulence properties .\
**The future.** Neither *ISO* nor *Herschel* could carry out wide-field spectral-imaging observations. The new generation multi-beam instruments on board *SOFIA* or balloon experiments, combined with efficient (OTF) mapping techniques, will allow faster and larger maps. However, these stratospheric telescopes do not cover the entire far-IR domain (e.g., they can’t observe H$_2$O or CH$^+$), they are severely limited by the available flying time (e.g., they can’t map the Galactic plane in reasonable times), and they are restricted to the detection of a few bright lines. To overcome these issues, several far-IR space-telescope concepts have been conceived to probe the ISM lifecycle and stellar feedback, some adapted to the bright and extended emission of the Milky Way and nearby galaxies, others pushing beyond to characterize the ISM of distant and primitive galaxies [see e.g., the EU far-IR Space Roadmap, @Rigopoulou17]:\
(1.) A $\sim$1m-class space telescope equipped with multi-beam heterodyne receivers (thus providing spectral resolution) and employing OTF techniques to scan very large areas of the Milky Way’s galactic plane and provide velocity-resolved images of the interstellar gas in the brightest gas cooling lines simultaneously [see e.g., @Rigopoulou16 for the *FIRSPEX* concept].\
(2.) A cooled $\sim$2.5m-class space telescope equipped with a very sensitive grating spectrometer, accessing the entire far-IR band at medium spectral resolution, but providing very little spectral multiplexing capabilities [see e.g., @Roelfsema18 for the *SPICA* concept].\
(3.) A cooled $\sim$6m-class space telescope equipped with ultra-sensitive grating spectrometers accessing the entire far-IR band for wide-field spectral-imaging at low-spectral resolution ($R$ of a few hundred) or at higher $R$ for single beam observations [see e.g., @Battersby18 for the *OST* concept].\
Concept (1.) is specifically designed for mapping the galactic ISM, and is only limited by sensitivity and available bandwidth to simultaneously map $N$ lines. Concept (3.), if equipped with multi-beam heterodyne receivers, will fullfil the science cases presented in this WP. Concept (2.) or (3. without receivers) will be able to detect all relevant far-IR gas cooling lines, retrieving their line intensities (thus accessing the gas energetics) but not the gas kinematics or line-profile information. Even at low to medium $R$, they will detect the far-IR atomic and molecular lines from the warm interstellar gas and from hundreds of protostars, and propoplanetary disks. This has been demonstrated by previous observations toward Orion with the grating spectrometers *ISO*/LWS ($R \simeq 200$) and *Herschel*/PACS ($R \simeq 2000$). Future instruments of this kind, however, will have to develop spectral-imaging techniques as efficient as possible (e.g., using steering mirrors).\
**Science case:** With the exception of a few far-IR spectral-maps toward nearby and bright high-mass SFRs, almost all square-degree areas of the Milky Way covered by *Herschel* far-IR dust photometric images do not have spectroscopic counterpart in the main gas cooling lines (\[[C[ii]{}]{}\]158$\upmu$m, \[[O[i]{}]{}\]63$\upmu$m) and in critical molecular line tracers of stellar feedback (high-$J$ CO lines, H$_2$O, CH$^+$...). In order to fill this gap, we propose a program to carry out spectral-imaging of the most relevant high-mass SFRs of the Galaxy (typically located a kpc distances) and also to map young superclusters in nearby galaxies . As an example, here we focus on the CO $J$=12–11 ($E_u/k =$431 K) line at $\lambda$=217$\upmu$m. This is the first CO rotational line that could be detected by a instrument such as *SPICA*/SAFARI, but we note that the highest-energy lines at would be detectable too. In addition to the many CO and H$_2$O lines available in the , other key target lines are CH$^+$ $J$=1-0, 2-1, 3-2 and 4-3 (at $\lambda$=359, 179.6, 119.8, and 90.0$\upmu$m respectively) and HD $J$=1-0 and 2-1 (at $\lambda$=112.0 and 56.2$\upmu$m respectively).\
In this example, we require to map a $\sim$75pc$^2$ area of several template GMCs. This would imply scanning angular sizes of 50$'$$\times$50$'$ for massive SFRs like W51 (at a distance of 5 kpc), but sizes of 5$'$$\times$5$'$ toward GMCs of the LMC. *Herschel* observations show that the *extended* warm molecular gas emission in Orion (at only 0.5 kpc) produces surface brightness of $\sim$10$^{-8}$ Wm$^{-2}$sr$^{-1}$ in the mid-$J$ CO lines (see Fig. 1). Assuming extended emission filling the beam of the different types of telescope concepts ($\sim$1, $\sim$2.5, and $\sim$6m telescope sizes), we compute the line flux sensitivity needed to detect this emission level toward SFRs of increasing distance (of course some regions will have a brighter or a fainter line emission level depending on the particular excitation conditions and dominant feedback proceses). Expectations are shown in . Just for reference, we note that the line sensitivity of *Herschel*/PACS was several times (5$\sigma$/1hr). We see that in order to detect the warm molecular gas emission beyond the Galactic Center with a $\sim$2.5m telescope, we require line sensitivities of several , and an order of magnitude better to image the molecular ISM of nearby galaxies. While the smaller telescope concept will be more flexible to map degree-size areas of the Milky Way at parsec-scale spatial resolution, a $\sim$6m telescope improves by a factor of $\sim$2 the angular resolution provided by *SOFIA* or *SPICA*. This will allow us reaching, for the first time in far-IR spectral line maps, a $\sim$0.1pc ($\sim$20,000 AU) spatial resolution for all SFRs at distances below 2.5kpc. This scale has been suggested to be the universal width of the filaments that build up the structure of star-forming clouds [e.g., @Arzoumanian11]. In addition, such kind of telescope will allow us to spatially resolve the emission from hundreds of protostars and their outflows.
|
---
abstract: 'Apostol’s Möbius functions $\mu_k(n)$ of order $k$ are generalized to depend on a second integer parameter $m\geq k$. Asymptotic formulas are obtained for the partial sums of these generalized functions.'
author:
- |
Antal Bege\
“Babeş-Bolyai” University\
Faculty of Mathematics\
Str. Kogâlniceanu, Nr. 1\
3400 Cluj-Napoca, Romania
date: 'September 12, 2000'
title: '**A generalization of Apostol’s Möbius functions of order $k$**'
---
Introduction
============
Möbius functions of order $k$, introduced by T. M. Apostol [@apostol1], are defined by the formulas $$\mu_k(n)=
\left\{
\begin{array}{cl}
1& \mbox{ if }n=1,\\
0& \mbox{ if }p^{k+1}\mid n\mbox{ for some prime } p,\\
(-1)^r& \mbox{ if } n=p_1^k\cdots p_r^k\prod\limits_{i>r}p_i^{\alpha_i},
\quad \mbox{ with } 0\leq \alpha_i<k,\\
1& \mbox{ otherwise. }
\end{array}
\right.$$ In [@apostol1] Apostol obtained the asymptotic formula $$\label{11}
\sum_{n\leq x}\mu_k(n)=A_kx+O(x^{\frac{1}{k}}\log x),$$ where $$A_k=\prod_p\left(1-\frac{2}{p^k}+\frac{1}{p^{k+1}}\right).$$ Later, Suryanarayana [@suryanarayana1] showed that, on the assumption of the Riemann hypothesis, the error term in (\[11\]) can be improved to $$\label{12}
O\left(x^{\frac{4k}{4k^2+1}}\omega (x)\right),$$ Where $$\omega (x)=\mbox{exp}\{A\log x (\log \log x)^{-1}\}$$ for some positive constant $k$.\
\
This paper generalizes Möbius functions of order $k$ and establishes asymptotic formulas for their partial sums.
Preliminary lemmas
==================
The generalization in question is denoted by $\mu_{k,m}(n)$, where $1<k\leq m$.\
If $m=k$, $\mu_{k,k}(n)$ is defined to be $\mu_k(n)$, and if $m>k$ the function is defined as follows: $$\label{21}
\mu_{k,m}(n)=
\left\{
\begin{array}{cl}
1& \mbox{ if } n=1,\\
1& \mbox{ if } p^{k}\nmid n\mbox{ for each prime } p,\\
(-1)^r& \mbox{ if } n=p_1^m\cdots p_r^m\prod\limits_{i>r}p_i^{\alpha_i},
\quad \mbox{ with } 0\leq \alpha_i<k,\\
0& \mbox{ otherwise. }
\end{array}
\right.$$ This generalization, like Apostol’s $\mu_k(n)$, is a multiplicative function of $n$, so it is determined by its values at the prime powers. We have $$\mu_k(p^{\alpha })=\left\{
\begin{array}{rcl}
1& \mbox{ if }& 0\leq \alpha <k,\\
-1& \mbox{ if }& \alpha =k,\\
0& \mbox{ if }& \alpha >k,
\end{array}
\right.$$ whereas $$\label{22}
\mu_{k,m}(p^{\alpha })=\left\{
\begin{array}{rcl}
1& \mbox{ if }& 0\leq \alpha <k,\\
0& \mbox{ if }& k\leq \alpha <m,\\
-1& \mbox{ if }& \alpha =m,\\
0& \mbox{ if }& \alpha >m,
\end{array}
\right.$$
[**Lemma 2.1**]{} [*For $k\leq m$ we have $$\label{23}
\mu_{k,m}(n)=\sum_{
\begin{array}{l}
\delta d^m=n\\
(d,\delta )=1
\end{array}}
\mu (d)\; q_k(\delta )$$ where $\mu (n)$ is the Möbius function and $q_k(n)$ is the caracteristic function of the k-free integers:*]{} $$q_k(n)=\left\{
\begin{array}{lcl}
0& \mbox{ if }& p^k|n \mbox{ for some prime }p\\
1& \mbox{ if }& p^{\alpha }|n \mbox{ implies }\alpha <k
\end{array}
\right. .$$
[**Proof**]{}\
Because $\mu (n)$ and $q_k(n)$ are multiplicative functions of $n$, the sum in the lemma is also multiplicative, so to complete the proof we simply note that when $n=p^{\alpha }$ the sum has the values indicated in (\[22\]).\
The next two lemmas, proved in [@suryanarayana2], involve the following functions: $$\theta(n)=\mbox{ the number of square-free divisors of }n,$$ $$\psi_k(n)=n\prod_{p|n}\left(1+\frac{1}{p}+\cdots +\frac{1}{p^{k-1}}\right),$$ where $k$ is an integer $\geq 2$, $$\delta _k(x)=\mbox{ exp }\{-A\; k^{-\frac{8}{5}}\log^{\frac{3}{5}}x\;
(\log \log x)^{-\frac{1}{5}}\},$$ where $A>0$ is an absolute constant, $$\omega_k(x)=\mbox{ exp }\{B_k\; \log x\; (\log \log x)^{-1}\},$$ where $B_k$ is a positive constant.\
[**Lemma 2.2**]{} [*For $x\geq 3$ we have $$\label{24}
Q_k(x,n)=\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
q_k(r)=
\frac{xn}{\zeta (k) \psi_k(n)}+
0\left(\theta (n)x^{\frac{1}{k}}\delta_k(x)\right)$$ uniformly in $x$, $n$ and $k$.*]{}\
[**Lemma 2.3**]{} [*If the Riemann hypothesis is true, then for $x\geq 3$ we have $$\label{25}
Q_k(x,n)=\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
q_k(r)=
\frac{xn}{\zeta (k) \psi_k(n)}+
0\left(\theta (n)x^{\frac{2}{2k+1}}\omega_k(x)\right)$$ uniformly in $x$, $n$ and $k$.*]{}\
Our derivation of an asymptotic formula for the summatory function of $\mu_{k,m}(n)$ will also make use of the following lemma.\
[**Lemma 2.4**]{} [*For $k\geq 2$ we have $$\label{26}
\sum_{d|n}\frac{\mu(d)\psi_{k-1}(d)}{d\psi_k(d)}=
\frac{n}{\psi_k(n)}.$$*]{}
[**Proof**]{}\
Both sides of (\[26\]) are multiplicative functions of $n$ so it suffices to verify the equation when $n$ is a prime power. If $n=p^{\alpha }$ we have $$\sum_{d|p^{\alpha }}\frac{\mu(d)\psi_{k-1}(d)}{d\psi_k(d)}=
1-\frac{1+\frac{1}{p}+\cdots +\frac{1}{p^{k-2}}}
{p\left(1+\frac{1}{p}+\cdots +\frac{1}{p^{k-1}}\right)}=$$ $$=\frac{p^{\alpha }}
{p^{\alpha }\left(1+\frac{1}{p}+\cdots +\frac{1}{p^{k-1}}\right)}=
\frac{n}{\psi_k(n)}.$$
The next lemma is proved in [@suryanarayana3].\
\
[**Lemma 2.5**]{} [ *For $x\geq 3$, $n\geq 1$, and every $\epsilon >0$ we have*]{} $$\label{27}
L_n(x)\equiv \sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{r}=
O\left(\sigma ^*_{-1+\epsilon}(n)\delta (x)\right)$$ [*uniformly in $x$ and $n$, where $\sigma ^*_{\alpha }(n)$ is the sum of the $\alpha $th powers of the square-free divisors of $n$, and $$\delta (x)=\mbox{ exp }\{-A\; \log^{\frac{3}{5}}x\;
(\log \log x)^{-\frac{1}{5}}\},$$ for some absolute constant $A>0$.*]{}\
\
We note that if $\alpha <0$ we have $\sigma^*_{\alpha }(n)\leq \sigma^*_0(n)=
\theta (n)$. Also, $x^{\epsilon }\delta (x)$ is an increasing function of $x$ for every $\epsilon >0$ and $x>x_0(\epsilon )$. Using the method described in [@suryanarayana3], it can be shown that if the Riemann hypothesis is true the factor $\delta (x)$ in the error term in (\[27\]) can be replaced by $\omega (x)x^{-\frac{1}{2}}$, where $$\omega (x)=\mbox{ exp }\{A\; \log x\; (\log \log x)^{-1}\},$$ for some absolute constant $A>0$.\
\
[**Lemma 2.6**]{} [*For $x\geq 3$ and every $\epsilon >0$ we have $$\label{28}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{\psi_k(r)}=
0\left(\sigma^*_{-1+\epsilon}(n)\delta(x)\right).$$ uniformly in $x$, $n$ and $k$*]{}.\
\
[**Proof**]{} We write $$\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{\psi_k(r)}=
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{r}\frac{r}{\psi_k(r)},$$ then use (\[26\]) to obtain $$\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{\psi_k(r)}=
\sum_{
\begin{array}{c}
d\delta\leq x\\
(d,\delta )=1\\
(d\delta ,n)=1
\end{array}}
\frac{\mu^2(d)\mu (\delta )\psi_{k-1}(d)}{d^2\delta \psi_k(d)}=$$ $$=\sum_{
\begin{array}{c}
d\leq x\\
(d,n)=1
\end{array}}
\frac{\mu^2(d)\psi_{k-1}(d)}{d^2\psi_k(d)}\;
\sum_{
\begin{array}{c}
\delta\leq \frac{x}{d}\\
(\delta ,nd)=1
\end{array}}
\frac{\mu(\delta )}{\delta}=$$ $$=\sum_{
\begin{array}{c}
d\leq x\\
(d,n)=1
\end{array}}
\frac{\mu^2(d)\psi_{k-1}(d)}{d^2\psi_k(d)}\;
L_{dn}\left(\frac{x}{d}\right).$$ Using (\[27\]) and the inequality $\psi_{k-1}(d)\leq \psi_k(d)$ we find that the last sum is $$O(\sigma^*_{-1+\epsilon}(n)\;
\sum_{
\begin{array}{c}
d\leq x\\
(d,n)=1
\end{array}}
\frac{\mu^2(d)\delta\left(\frac{x}{d}\right)
\sigma^*_{-1+\epsilon }(d)}{d^2}).$$ Because $x^{\epsilon '}\delta (x)$ increases for every $\epsilon '>0$, we have $\left(\frac{x}{d}\right)^{\epsilon '}\delta\left(\frac{x}{d}\right)
\leq
x^{\epsilon '}\delta (x)$ so $\delta \left(\frac{x}{d}\right)\leq d^{\epsilon '}\delta (x)$ and the foregoing $0$-term is $$O (\sigma^*_{-1+\epsilon}(n)\delta(x)\;
\sum_{
\begin{array}{c}
d\leq x\\
(d,n)=1
\end{array}}
\frac{\mu^2(d)\sigma^*_{-1+\epsilon }(d)}{d^{2-\epsilon'}}).$$ But $\sigma^*_{-1+\epsilon}(d)\leq \tau(d)=O(d^{\epsilon '})$ for every $\epsilon '>0$. If we choose $\epsilon '<\frac{1}{2}$ the last sum is $O(1)$ and we obtein (\[28\]).\
\
Applying (\[28\]) together with the remark following Lemma 2.5 we obtain\
\
[**Lemma 2.7**]{} [ *If the Riemann hypothesis is true, then for $x\geq 3$, $n\geq 1$ and every $\epsilon >0$ we have $$\label{29}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{\psi_k(r)}=
O\left(\sigma^*_{-1+\epsilon}(n)\omega(x)x^{-\frac{1}{2}}\right).$$ uniformly in $x$, $n$ and $k$*]{}.\
\
Applying partial summation in (\[28\]) we obtain\
\
[**Lemma 2.8**]{} [*For $x\geq 3$, $k\geq 2$, and every $\epsilon >0$ we have $$\label{210}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\frac{\mu(r)}{\psi_k(r)r^{k-1}}=
O\left(\sigma^*_{-1+\epsilon}(n)\delta(x)x^{1-k}\right).$$ uniformly in $x$, $n$ and $k$*]{}.\
\
[**Note**]{}: If the Riemann hypothesis is true, the error term in Lemma 2.8 holds with $\delta (x)x^{1-k}$ replaced by $\omega (x) x^{\frac{1}{2}-k}$.
Main results
============
[**Theorem 3.1**]{} [*For $x\geq 3$ and $m>k\geq 2$ we have $$\label{31}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\mu_{k,m}(r)=
\frac{xn^2\; \alpha_{k,m}}{\zeta(k)\psi_k(n)\alpha_{k,m}(n)}+
0\left(\theta(n)x^\frac{1}{k}\delta(x)\right).$$ uniformly in $x$, $n$ and $k$, where $$\alpha_{k,m}=\prod_p\left(1-\frac{1}{p^{m-k+1}+p^{m-k+2}+\cdots +p^m}\right)$$ and $$\alpha_{k,m}(n)=n\prod_{p|n}\left(1-
\frac{1}{p^{m-k+1}+p^{m-k+2}+\cdots +p^m}\right).$$*]{}\
\
[**Proof**]{} By (\[23\]) and (\[24\]) we have $$\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\mu_{k,m}(n)=
\sum_{
\begin{array}{l}
\delta d^m\leq x\\
(d,\delta )=1\\
(d\delta ,n)=1
\end{array}}
\mu (d)\; q_k(\delta )=$$ $$=
\sum_{
\begin{array}{l}
d\leq x^{\frac{1}{m}}\\
(d,n)=1
\end{array}}
\mu (d)
\sum_{
\begin{array}{l}
\delta \leq \frac{x}{d^m}\\
(\delta ,dn)=1
\end{array}}
q_k(\delta )=$$ $$=
\sum_{
\begin{array}{l}
d\leq x^{\frac{1}{m}}\\
(d,n)=1
\end{array}}
\mu (d)
Q_k\left( \frac{x}{d^m}, dn\right)=$$ $$=\sum_{
\begin{array}{l}
d\leq x^{\frac{1}{m}}\\
(d,n)=1
\end{array}}
\mu (d)\left\{
\frac{\left(\frac{x}{d^m}\right)dn}{\zeta (k) \psi_k(dn)}+
0\left(\theta (dn)\frac{x^{\frac{1}{k}}}{d^{\frac{m}{k}}}
\delta \left(\frac{x}{d^m}\right)
\right)
\right\}=$$ $$=
\frac{xn}{\zeta(k)\psi_k(n)}
\sum_{
\begin{array}{c}
d=1\\
(d,n)=1
\end{array}}
^{\infty }
\frac{\mu(d)}{d^{m-1}\psi_k(d)}-
\frac{xn}{\zeta (k)}{\psi_k(n)}
\sum_{
\begin{array}{c}
d>x^{\frac{1}{m}}\\
(d,n)=1
\end{array}}
\frac{\mu (d)}{d^{m-1}\psi_k(d)}+$$ $$+
O(
\theta (n)x^{\frac{1}{k}-\epsilon}
\sum_{
\begin{array}{c}
d\leq x^{\frac{1}{m}}\\
(d,n)=1
\end{array}}
\frac{\delta\left(\frac{x}{d^k}\right)\left(\frac{x}{d^k}\right)\mu^2(d)
\theta(d)}
{d^{\frac{m}{k}-\epsilon 'k}}
).$$ Using the Euler product representation for absolutely convergent series of multiplicative terms [@apostol2] we have $$\sum_{
\begin{array}{c}
d=1\\
(d,n)=1
\end{array}}
^{\infty }
\frac{\mu(d)}{d^{m-1}\psi_k(d)}=
\prod_{p\mid n}\left(1-
\frac{1}{p^{m-k+1}+\cdots +p^m}\right)=
\frac{\alpha_{k,m}}{\alpha_{k,m}(n)}.$$ Now use (\[210\]) and the fact that $\delta(x)x^{\epsilon '}$ is increasing for all $\epsilon '>0$, then choose $\epsilon >0$ so that $\frac{m}{k}-\epsilon 'k>1+\epsilon $ and we obtain (\[31\]).\
When $n=1$, Theorem 3.1 gives the following corollary for $x\geq 3$ and $m>k\geq 2$: $$\label{32}
\sum_{r\leq x}
\mu_{k,m}(r)=
\frac{x}{\zeta(k)}\alpha_{k,m}+
0\left(x^{\frac{1}{k}}\delta(x)\right)$$ uniformly in $x$ and $k$.\
Applying the method used to prove Theorem 1, and making use of (\[25\]) and Lemma 2.9 we get\
\
[**Theorem 3.2**]{}\
[*If the Riemann hypothesis is true, then for $x\geq 3$ and $m>k\geq 2$ we have $$\label{33}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\mu_{k,m}(r)=
\frac{xn^2\; \alpha_{k,m}}{\zeta(k)\psi_k(n)\alpha_{k,m}(n)}+
0\left(\theta(n)x^{\frac{2}{2k+1}}\omega(x)\right).$$ uniformly in $x$, $n$ and $k$.*]{}\
\
In particular, if n=1 we have $$\label{34}
\sum_{r\leq x}
\mu_{k,m}(r)=
\frac{x}{\zeta(k)}\alpha_{k,m}+
0\left(x^{\frac{2}{2k+1}}\omega(x)\right)$$ uniformly in $x$ and $k$.
Conjectures
===========
Suryanarayana raised the question of improving the error term in Apostol’s asymptotic formula (\[11\]), and notes that no improvement seems possible by this method. Our method gives no improvement in the error term but it does suggest the following conjectures:\
For $x\geq 3$, $n\geq 1$ and $k\geq 2$ we have $$\label{41}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\mu_{k}(r)=
\frac{xn^2\; \alpha_{k,k}}{\zeta(k)\psi_k(n)\alpha_{k,k}(n)}+
0\left(\theta(n)x^{\frac{1}{k}}\delta(x)\right).$$ uniformly in $x$, $n$ and $k$.\
In particular, when $n=1$ the conjecture is $$\label{42}
\sum_{r\leq x}
\mu_{k}(r)=
\frac{x}{\zeta(k)}\alpha_{k,k}+
0\left(x^{\frac{1}{k}}\delta(x)\right)$$ uniformly in $x$ and $k$.\
If the Riemann hypothesis is true, the conjectured formulas are $$\label{43}
\sum_{
\begin{array}{c}
r\leq x\\
(r,n)=1
\end{array}}
\mu_{k}(r)=
\frac{xn^2\; \alpha_{k,k}}{\zeta(k)\psi_k(n)\alpha_{k,k}(n)}+
0\left(\theta(n)x^{\frac{2}{2k+1}}\omega(x)\right).$$ uniformly in $x$, $n$ and $k$, for $x\geq 3$, $n\geq 1$ and $k\geq 2$.\
In particular, when $n=1$ the conjecture is $$\label{44}
\sum_{r\leq x}
\mu_{k}(r)=
\frac{x}{\zeta(k)}\alpha_{k,k}+
0\left(x^{\frac{2}{2k+1}}\omega(x)\right)$$ uniformly in $x$ and $k$.\
\
It should be noted that $\alpha_{k,k}=\zeta (k)A_k$, where $A_k$ is Apostol’s constant in (\[11\]), so the leading term in (\[42\]) and (\[44\]) is the same as that in (\[11\]).
ACKNOWLEDGEMENT
The author wishes to thank Professor Tom M. Apostol for his help in preparing this paper for publication.
[99]{} T. M. Apostol, Möbius functions of order $k$, [*Pacific Journal of Math.*]{}, [**32**]{} (1970), 21-17. T. M. Apostol, [*Introduction to Analytic Number Theory*]{}, Undergraduete texts in Mathematics, Springer Verlag, New-York, 1976. D. Suryanarayana, On a theorem of Apostol concerning Mobius functions of order $k$, [*Pacific Journal of Math.*]{}, [**68**]{} (1977), 277-281. D. Suryanarayana, Some more remarks on uniform O-estimates for k-free integers, [*Indian J. Pure Appl. Math.*]{}, [**12**]{} (11) (1981), 1420-1424. D. Suryanarayana and P. Subrahmanyam, The maximal k-free divisor of m which is prime to n, [*Acta Math. Acad. Sci. Hung.*]{}, [**33**]{} (1979), 239-260.
|
---
abstract: 'We determine the complexity of several constraint satisfaction problems using the heuristic algorithm, WalkSAT. At large sizes $N$, the complexity increases exponentially with $N$ in all cases. Perhaps surprisingly, out of all the models studied, the hardest for WalkSAT is the one for which there is a polynomial time algorithm.'
author:
- Marco Guidetti
- 'A. P. Young'
bibliography:
- 'refs.bib'
title: ' Complexity of several constraint satisfaction problems using the heuristic, classical, algorithm, WalkSAT'
---
Introduction {#sec:intro}
============
There is considerable interest, in different fields of science, in finding efficient methods to solve optimization problems. For a few such problems there are clever algorithms which enable one to find the solution, for all cases, in a time which only grows with a power of the size $N$ of the problem. Problems with such a polynomial time algorithm are said to be in the complexity class P. In many cases of interest, however, no polynomial time algorithm is known, though a solution, if given, can be *verified* in polynomial time. Decision problems for which which “yes" instances can be verified in polynomial time are said to be [@Garey:97] in complexity class NP. There is no proof that P $\ne$ NP, though it is generally assumed that they are different, i.e. there exist problems which cannot be solved in polynomial time algorithm, at least in the worst case. There is a subset of NP problems which have the property that any problem in NP can be mapped into them in polynomial time. These are called NP-complete [@Garey:97]. Consequently if a polynomial time algorithm could be found for *one* NP-complete problem, all problems in NP could be solved in polynomial time and so we would have P = NP (which, as stated above, is felt to be unlikely).
Several heuristic algorithms to solve optimization problems have been proposed. (“Heuristic" means that the answer provided is not guaranteed to be exact. This is in contrast to “complete” algorithms which guarantee to find the solution, if one exists.) One well known example is simulated annealing [@kirkpatrick:83] (SA) in which an artificial temperature is introduced and gradually set to zero. Another popular algorithm is WalkSAT [@WalkSAT] which is similar in spirit to simulated annealing in that both make moves which reduce the “energy”, but also sometimes make moves which increase it to avoid being trapped in the nearest local minimum. It has also been proposed to solve optimization problems on a quantum computer using the quantum adiabatic algorithm (QAA) [@farhi:01], which is based on quantum annealing [@kadawoki:98]. To assess whether a quantum computer could solve optimization problems more efficiently than a classical computer, it is valuable to compare the efficiency of the QAA to solve a range of optimization problems with that of classical heuristic algorithms, in particular to see whether those problems which are harder classically are also harder quantum mechanically.
As part of this project we report here results of the efficiency of a classical algorithm to study several optimization problems of the “constraint satisfaction" type. We chose WalkSAT rather than SA because the implementation is simpler since there is just one parameter (the strength of the “noise") whereas in SA one has to decide on the whole annealing schedule of temperature against time. It is not obvious how to choose the best annealing schedule, and different choices could lead to significant differences in the number of sweeps needed to find the ground state, which would be a disadvantage for us. Also, there is a publicly available code for WalkSAT which can be downloaded and easily compiled on different machines.
We study three NP-complete models and two versions of one model that is in the P class (known as XORSAT). For WalkSAT, all models we study require a computer time which increases exponentially with $N$. Some are harder than others, in that the coefficient of $N$ in the exponent is larger. Curiously the hardest of all is XORSAT even though there exists a polynomial time algorithm for this problem. Hence, the difficulty of a problem using a heuristic algorithm does not, in general, depend on whether it is in the P or NP complexity class. Interestingly, it is also known that XORSAT is very hard using the QAA [@jorg:09]. The plan of this paper is as follows. Section \[sec:models\] describes the four models that will be studied. Results are presented in Sec. \[sec:results\] and our conclusions summarized in Sec. \[sec:conclusions\].
Models {#sec:models}
======
We shall consider problems of the “constraint satisfaction” type, in which there are $N$ bits (or equivalently Ising spins) and $M$ “clauses” where each clause is a logical condition on a small number of randomly chosen bits. A configuration of the bits is a “satisfying assignment” if it satisfies all the clauses. Frequently, in statistical mechanics approaches to these problems, one converts each clause to an energy function, which depends on the bits in the clause, such that the energy is zero if the clause is satisfied and is positive if it is not. However, we will not need to do this here since WalkSAT uses the logical structure of the clauses rather than an energy function.
Clearly, it is easy to satisfy all clauses if the ratio $\alpha \equiv M/N$ is small enough. In fact one expects an exponentially large number of satisfying assignments in this region. Conversely, if $M/N$ is very large, with high probability there will be a conflict between different clauses. Hence there is a “satisfiability transition” at some value $\alpha_s$ where the number of satisfying assignments goes to zero. It is believed that it is particularly hard to solve satisfiability problems close to the transition [@kirkpatrick:94], and so we will work in this region. Furthermore, when studying the efficiency of the QAA numerically [@farhi:01; @young:08; @young:10], it is convenient to consider instances with a unique satisfying assignment (USA). Since we intend eventually to compare the results presented here with results for the QAA, here we will also only consider instances with a USA (which of courses, forces the system to be close the transition).
We now discuss the different models that will be investigated in this paper.
Exact Cover (Unlocked 1-in-3 SAT) {#sec:unlocked}
---------------------------------
Several numerical studies of the QAA [@farhi:01; @young:08; @young:10] consider the “Exact Cover” problem, in which each clause consists of three bits chosen randomly, and the clause is satisfied if one bit is one and the others are zero. In order to increase the probability of a USA, Young et al. [@young:08; @young:10] removed isolated bits, and clauses which are only connected to the others by one bit. Here we use exactly the same instances as in Refs. [@young:08; @young:10]. For each value of $N$, the number of clauses $M$ is chosen to maximize the probability of a USA. The values of $N$ and $M$ are shown in Table \[tab:EC\]. It is found that the probability of a USA decreases with $N$, apparently exponentially.
N 16 32 64 128 192 256
---------- -------- -------- -------- -------- -------- -------- --
M 12 23 44 86 126 166
$\alpha$ 0.7500 0.7188 0.6875 0.6719 0.6563 0.6484
: The number of clauses $M$ and number of bits $N$ used in the study of the Exact Cover problem, from Ref. [@young:08]. The ratio $M/N$ is expected to approach the value at the satisfiability phase transition $\alpha_s \simeq 0.625$ [@knysh:04; @raymond:07] for $N
\to \infty$.
\[tab:EC\]
This Exact Cover problem is sometimes called 1-in-3 SAT, for obvious reasons. In subsections \[sec:locked\_1-in-3\] and \[sec:locked\_2-in-4\], we will discuss models with a special property called “locked”, defined by Zdeborová and Mézard [@zdeborova:08a; @zdeborova:08b]. To distinguish the present model from the locked models, we will refer to it as “unlocked 1-in-3 SAT" from now on.
Locked 1-in-3 SAT {#sec:locked_1-in-3}
-----------------
Recently Zdeborová and Mézard [@zdeborova:08a; @zdeborova:08b] have proposed that it is useful to study a set of models, which they call “locked”, which have the following two properties:
1. Every variable is in at least two clauses.
2. Whether or not a clause is satisfied only depends on the sum of the bits in it (occupation problem), and two successive values of the sum are not allowed. Thus 1-or-3-in-4 is allowed, but 1-or-2-in-4 is not.
It follows that one can not get from one satisfying assignment to another by flipping a single bit. In fact, Zdeborová and Mézard argue that typically order $\ln N$ bits needs to be flipped. They also argue that, locked instances are analytically “simple” (or at least simpler than previously studied models such as random K-SAT) but are computationally hard. They are therefore eminently suitable as benchmarks.
If the sites are chosen at random to form the clauses, the distribution of the degree of the sites (i.e. the number of clauses involving a site) would be Poissonian. However, locked instances have a minimum degree of two, so instead we use a truncated Poissonian distribution [@zdeborova:08b] which is Poissonian except that the probabilities for zero and one are set to zero. We fix the ratio $M/N$ to be the critical value for the satisfiability transition. According to Table I of Ref. [@zdeborova:08b], this is equal to $\alpha_s = 0.789$. Since $M$ has to be an integer we take $M$ to be the nearest integer to $\alpha_s N$.
Having generated these instances we run them through a (complete) Davis-Putnam-Logemann-Loveland (DPLL) [@davis:60; @davis:62] code to select those with a unique satisfying assignment (USA).
The probability of a USA only decreases slowly with $N$ and may tend to a non-zero value as $N \to \infty$, see Fig. \[P\_USA\_1-in-3\]. This is in contrast to the unlocked instances in Sec. \[sec:unlocked\] for which the probability decreases exponentially with $N$. The locked problem therefore has the advantage that instances with a USA should be a good representation of *randomly chosen* instances.
![(Color online) Probability of a unique satisfying assignment (USA) as a function of $1/N$ for locked instances of 1-in-3 SAT at the satisfiability threshold. The line is a guide to the eye. []{data-label="P_USA_1-in-3"}](P_USA_1-in-3.eps){width="\figurewidth"}
Locked 2-in-4 SAT {#sec:locked_2-in-4}
-----------------
We also consider locked 2-in-4 instances, in which a clause has four bits, and a clause is satisfied if two are zero and two are one. Unlike the other models discussed in this paper, this one has a symmetry under flipping all the bits.
We fix the ratio $M/N$ to be the critical value for the satisfiability transition. According to Table I of Ref. [@zdeborova:08b], this is equal to $\alpha_s = 0.707$. Since $M$ has to be an integer we take $M$ to be the nearest integer to $\alpha_s N$. As with the locked 1-in-3 instances in Sec. \[sec:locked\_1-in-3\] the probability of a USA seems to tend to a non-zero value for $N \to \infty$, see Fig. \[P\_USA\_2-in-4\].
![(Color online) Probability of a unique satisfying assignment (USA) as a function of $1/N$ for locked instances of 2-in-4 SAT at the satisfiability threshold. The line is a guide to the eye. []{data-label="P_USA_2-in-4"}](P_USA_2-in-4.eps){width="\figurewidth"}
XORSAT {#sec:XORSAT}
------
The exclusive-or of a set of bits is their sum (mod 2). In the K-XORSAT problem, K bits are chosen to form a clause and the clause is satisfied if their sum (mod 2) is a specified value (either 0 or 1). Again, the problem to be solved is whether there is an assignment of the $N$ bits which satisfies all $M$ clauses. In fact, since the problem just involves linear algebra (mod 2) the satisfiability problem can be solved in polynomial time using, for example, Gaussian elimination. However, if there is *not* a satisfying assignment, there is no known polynomial time algorithm to determine the minimal number of unsatisfied clauses, a problem known as MAX-XORSAT.
For XORSAT instances with a USA, it is not difficult to show that one can gauge transform any instance into one in which the sum of the bits of every clause is equal to 0 (mod 2). The USA is then all bits equal to 0, (a “ferromagnetic" ground state in statistical physics language). Although this ground state is “trivial", we shall see that it is very hard to find using a heuristic algorithm.
Here we will take the case of $K = 3$, and study two variants of the model.
### 3-regular 3-XORSAT
Firstly, we will follow Jörg et al. [@jorg:09] in taking the “3-regular” case where every bit is in exactly three clauses, a model which turns out to be *precisely* at the satisfiability threshold. We denote this model as 3-regular 3-XORSAT. As usual, we consider instances with a USA. Fortunately, these are *a non-zero fraction*, about 0.285 [@jorg:09], of the total, so the USA instances should be a good representation of randomly chosen ones. Note that $M = N$ for this model.
### 3-XORSAT {#xorsat}
Secondly, following the suggestion of Lenka Zdeborová, we will consider instances in which the distribution of the number of clauses attached to a bit is truncated Poissonian with *mean* degree three. For this model, which we denote as 3-XORSAT, we find numerically that the fraction of instances with a USA is also non-zero, about 0.25. As usual, we restrict our attention to these instances.
Results {#sec:results}
=======
We study the models in Sec. \[sec:models\] using the WalkSAT [@WalkSAT] algorithm. WalkSAT picks at random a clause which is currently unsatisfied and flips a variable in that clause. With some probability the variable is chosen to be the one which causes the fewest previously satisfied clauses to become unsatisfied, and otherwise it is chosen at random. The probability that the variable is chosen at random is called the “noise parameter". We know that there is a (unique) satisfying assignment for each instance (since we selected instances to have this condition using a DPLL algorithm). We determine how many elementary WalkSAT “flips” are needed to find it, for the different models for a range of sizes.
The overall logical condition which must be satisfied is the logical AND of each clause. Most algorithms for solving satisfiability problems, including WalkSAT, require that the problem is expressed in conjunctive normal form (cnf), in which each clause is written entirely in terms of logical OR’s. Note that OR just excludes one possible state of the bits. The cnf representation of the problems studied here is not unique. For example, in 1-in-3 SAT, a clause requires $x_1 + x_2 + x_3 =
1$, i.e. one of the bits is 1 and the others are 0 (so there are 3 allowed configurations). This has to be written as a logical AND of *several* cnf clauses (each of which is comprised of OR’s). A natural choice is to use five clauses as $$\begin{aligned}
&(x_1 \lor x_2 \lor x_3) \land (\lnot x_1 \lor \lnot x_2 \lor \lnot x_3) \land \\
& (x_1 \lor \lnot x_2 \lor \lnot x_3) \land (\lnot x_1 \lor x_2 \lor \lnot x_3) \land \\
& (\lnot x_1 \lor \lnot x_2 \lor x_3) \, ,\end{aligned}$$ in which each clause disallows one of the $8-3\ (= 5)$ forbidden configurations. However, we can actually combine two of the clauses together using a clause with only two variables, for example as $$\begin{aligned}
&(x_1 \lor x_2 \lor x_3) \land (\lnot x_1 \lor \lnot x_2 ) \land \\
& (x_1 \lor \lnot x_2 \lor \lnot x_3) \land (\lnot x_1 \lor x_2 \lor \lnot x_3)
\, .\end{aligned}$$ We always chose the cnf representation that used the smallest number of clauses.
First, we show results using the default value of the noise parameter (0.5). Figure \[all\_no\_opt\_log-lin\] plots the median number of “flips” to find a solution as a function of $N$ for the different models. Note the logarithmic vertical scale. The data fits a straight line at large $N$ indicating a number of flips (which is proportional to the CPU time) increasing exponentially, as expected.
![(Color online) A log-lin plot of the median number of flips needed to solve the various problems as a function of $N$ using WalkSAT with the default noise parameter. The straight lines are fits to Eq. for the larger sizes. []{data-label="all_no_opt_log-lin"}](all_no_opt_log-lin.eps){width="\figurewidth"}
model $\mu$
--------------------- -------
unlocked 1-in-3 SAT 0.027
locked 1-in-3 SAT 0.052
locked 2-in-4 SAT 0.086
3-XORSAT 0.116
3-regular 3-XORSAT 0.124
: The rate of exponential increase of the number of flips, according to Eq. for the different models using the default value of the noise.
\[tab:mu\]
![(Color online) A log-lin plot of the median number of flips needed to solve the various problems as a function of $N$ using WalkSAT with optimized noise parameters. The straight lines are fits to Eq. for the larger sizes. The results are quite similar to those for the default noise shown in Fig. \[all\_no\_opt\_log-lin\] []{data-label="all_opt_log-lin"}](all_opt_log-lin.eps){width="\figurewidth"}
We see that the easiest model is unlocked 1-in-3 SAT (Exact Cover), while the hardest is 3-XORSAT. Both versions of 3-XORSAT are harder than any of the other problems that we looked at, with the 3-regular version being somewhat harder than the version with a Poissonian degree distribution.
Writing the median number of flips as $$\label{Nflip}
N_\text{flips} = A e^{\mu N}$$ we present the values of $\mu$ in Table \[tab:mu\]. It is curious that the one problem in the set (XORSAT) which is in the polynomial time complexity class, P, is the hardest for a heuristic algorithm.
We have also investigated the extent to which WalkSAT can be improved by optimizing with respect to the noise parameter. For each size, and each instance, we try different values of the noise and home in on the value which minimizes the number of flips. Results for the median number of flips are shown in Fig. \[all\_opt\_log-lin\]. Naturally, the numbers are a bit smaller than for the unoptimized case in Fig. \[all\_no\_opt\_log-lin\], but we still see an exponential growth at large sizes, and the ordering of the difficulty of the different models is the same (e.g. unlocked 1-in-3 SAT is the easiest and and the two versions of 3-XORSAT are the hardest). The fits in Fig. \[all\_opt\_log-lin\] correspond to the coefficients of $N$ in the exponential growth shown in Table \[tab:mu\_opt\], see Eq. . Unsurprisingly, these are somewhat less than the values for the unoptimized case shown in Table \[tab:mu\].
model $\mu$
--------------------- -------
unlocked 1-in-3 SAT 0.021
locked 1-in-3 SAT 0.034
locked 2-in-4 SAT 0.053
3-XORSAT 0.085
3-regular 3-XORSAT 0.107
: The rate of exponential increase of the number of flips, according to Eq. for the different models using optimized values of the noise. These values for $\mu$ are somewhat smaller than those for unoptimized noise, in Table \[tab:mu\], but the overall trend between the different models is the same.
\[tab:mu\_opt\]
Conclusions {#sec:conclusions}
===========
We have studied the complexity of the WalkSAT algorithm for four constraint satisfaction problems, three of them in the NP complexity class and one in the P complexity class. All show exponential complexity for large sizes. Curiously, the hardest problem for WalkSAT is the one in P. It will be interesting, in future work, to compare the relative hardness of these problems for the classical WalkSAT algorithm that we found here, with their relative hardness when using the quantum adiabatic algorithm.
One of us (APY) thanks Florent Krzakala, Lenka Zdeborová, Francesco Zamponi, Eddie Farhi and David Gosset for helpful discussions. We also thank Zdeborová and Farhi for their comments on an earlier version of this paper. This work is supported in part by the National Security Agency (NSA) under Army Research Office (ARO) contract number W911NF-09-1-0391, and in part by the National Science Foundation under Grant No. DMR-0906366. MG has been partially supported by the Doctoral Program of the IUSS (University of Ferrara) and by INFN.
|
---
abstract: 'While first-order optimization methods such as stochastic gradient descent (SGD) are popular in machine learning (ML), they come with well-known deficiencies, including relatively-slow convergence, sensitivity to the settings of hyper-parameters such as learning rate, stagnation at high training errors, and difficulty in escaping flat regions and saddle points. These issues are particularly acute in highly non-convex settings such as those arising in neural networks. Motivated by this, there has been recent interest in second-order methods that aim to alleviate these shortcomings by capturing curvature information. In this paper, we report detailed empirical evaluations of a class of Newton-type methods, namely sub-sampled variants of trust region (TR) and adaptive regularization with cubics (ARC) algorithms, for non-convex ML problems. In doing so, we demonstrate that these methods not only can be computationally competitive with hand-tuned SGD with momentum, obtaining comparable or better generalization performance, but also they are highly robust to hyper-parameter settings. Further, in contrast to SGD with momentum, we show that the manner in which these Newton-type methods employ curvature information allows them to seamlessly escape flat regions and saddle points.'
author:
- 'Peng Xu [^1]'
- 'Farbod Roosta-Khorasani [^2]'
- 'Michael W. Mahoney [^3]'
bibliography:
- 'references.bib'
title: 'Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study'
---
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to acknowledge ARO, DARPA, Cray, and NSF for providing partial support of this work. FR gratefully acknowledges the support of the Australian Research Council through a Discovery Early Career Researcher Award (DE180100923). We also sincerely thank Profs. Dominique Orban, Nicholas I.M. Gould, and Coralia Cartis for kindly helping us with the code for adaptive cubic regularization as well as setting up GALAHAD package. We also greatly appreciate Dr. Felix Lenders’ help with the installation of the trlib package. We would like to thank Dr. Amir Gholaminejad for valuable comments on our empirical evaluations and suggestions for improving them.
[^1]: Institute for Computational and Mathematical Engineering, Stanford University, Email: pengxu@stanford.edu
[^2]: School of Mathematics and Physics, University of Queensland, Brisbane, Australia, and International Computer Science Institute, Berkeley, USA, Email: fred.roosta@uq.edu.au
[^3]: International Computer Science Institute and Department of Statistics, University of California at Berkeley, Email: mmahoney@stat.berkeley.edu
|
---
abstract: 'A positive integer $A$ is called a [*congruent number*]{} if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a [*congruent number*]{} if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in{\mathds{Q}}^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.'
---
[**A Theorem of Fermat**]{}\
[**on**]{}\
[**Congruent Number Curves**]{}
[Lorenz Halbeisen]{}\
[Department of Mathematics, ETH Zentrum, Rämistrasse101, 8092 Zürich, Switzerland\
lorenz.halbeisen@math.ethz.ch]{}\
[Norbert Hungerbühler]{}\
[Department of Mathematics, ETH Zentrum, Rämistrasse101, 8092 Zürich, Switzerland\
norbert.hungerbuehler@math.ethz.ch]{}
Introduction
============
A positive integer $A$ is called a [**congruent number**]{} if $A$ is the area of a right-angled triangle with three rational sides. So, $A$ is congruent if and only if there exists a rational Pythagorean tripel $(a,b,c)$ ([[*i.e.*]{}]{}, $a,b,c\in{\mathds{Q}}$, $a^2+b^2=c^2$, and $ab\neq 0$), such that $\frac{ab}2=A$. The sequence of integer congruent numbers starts with $$5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37,\ldots$$ For example, $A=7$ is a congruent number, witnessed by the rational Pythagorean triple $$\Bigl(\frac{24}{5}\,,
\frac{35}{12}\,,\frac{337}{60}\Bigr).$$ It is well-known that $A$ is a congruent number if and only if the cubic curve $$C_A:\ y^2=x^3-A^2 x$$ has a rational point $(x_0,y_0)$ with $y_0\neq 0$. The cubic curve $C_A$ is called a [**congruent number curve**]{}. This correspondence between rational points on congruent number curves and rational Pythagorean triples can be made explicit as follows: Let $$C({\mathds{Q}}):= \{(x,y,A)\in {\mathds{Q}}\times{\mathds{Q}}^*\times {\mathds{Z}}^*:y^2=x^3-A^2x\},$$ where ${\mathds{Q}}^*:={\mathds{Q}}\setminus\{0\}, {\mathds{Z}}^*:={\mathds{Z}}\setminus\{0\}$, and $$P({\mathds{Q}}):=\{(a,b,c,A)\in {\mathds{Q}}^3\times{\mathds{Z}}^*:a^2+b^2=c^2\ \textsl{and\/}\ ab=2A\}.$$ Then, it is easy to check that $$\label{psi}
\begin{aligned}
\psi\ :\ \quad P({\mathds{Q}})&\ \to\ C({\mathds{Q}})\\
(a,b,c,A)&\ \mapsto \ \Bigl(\frac{A(b+c)}{a}\,,\,\frac{2A^2(b+c)}{a^2}\,,\,A\Bigr)
\end{aligned}$$ is bijective and $$\label{psi-1}
\begin{aligned}
\psi^{-1}\ :\qquad C({\mathds{Q}})&\ \to\ P({\mathds{Q}})\\
(x,y,A)&\ \mapsto\ \Bigl(\frac{2x A}{y}\,,\;\frac{x^2-A^2}{y}\,,\;\frac{x^2+A^2}{y}\,,\,A\Bigr).
\end{aligned}$$
For positive integers $A$, a triple $(a,b,c)$ of non-zero rational numbers is called a [[**rational Pythagorean $\boldsymbol{A}$-triple**]{}]{} if $a^2+b^2=c^2$ and $A=\big{|}\frac{ab}{2}\big{|}$. Notice that if $(a,b,c)$ is a [[rational Pythagorean $A$-triple]{}]{}, then $A$ is a congruent number and $|a|,|b|,|c|$ are the lengths of the sides of a right-angled triangle with area $A$. Notice also that we allow $a,b,c$ to be negative.
It is convenient to consider the curve $C_A$ in the projective plane ${\mathds{R}}P^2$, where the curve is given by $$C_A :\ y^2z = x^3-A^2xz^2.$$ On the points of $C_A$, one can define a commutative, binary, associative operation “$+$”, where ${\mathscr{O}}$, the neutral element of the operation, is the projective point $(0,1,0)$ at infinity. More formally, if $P$ and $Q$ are two points on $C_A$, then let $P{\#}Q$ be the third intersection point of the line through $P$ and $Q$ with the curve $C_A$. If $P=Q$, the line through $P$ and $Q$ is replaced by the tangent in $P$. Then $P+Q$ is defined by stipulating $$P+Q\;:=\;{\mathscr{O}}{\#}(P{\#}Q),$$ where for a point $R$ on $C_A$, ${\mathscr{O}}{\#}R$ is the point reflected across the $x$-axis. The following figure shows the congruent number curve $C_A$ for $A=5$, together with two points $P$ and $Q$ and their sum $P+Q$.
(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211) (0,0)(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211) (-9.0,-9.0)(9.0,10.0)[1.0\*y\^2+25.0\*x\^1-1.0\*x\^3]{} (-4.388182328551535,-7.588280903625012)(-4.388182328551535,9.360344080789211)
(5.824738905621608,7.211160924647724) (5.3,7.4)[$Q$]{} (-1.3905453803081107,5.663466525103901) (-1.3,6)[$P$]{} (-4.388182328551535,5.020466785549749) (-6.,5.2)[$P{\#}Q$]{} (-4.388182328551535,-5.020466785549749) (-6.3,-5.4)[$P+Q$]{}
More formally, for two points $P=(x_0,y_0)$ and $Q=(x_1,y_1)$ on a congruent number curve $C_A$, the point $P+Q=(x_2,y_2)$ is given by the following formulas:
- If $x_0\neq x_1$, then $$x_2=\lambda^2-x_0-x_1,\qquad y_2=\lambda(x_0-x_2)-y_0,$$ where $$\lambda:=\frac{y_1-y_0}{x_1-x_0}.$$
- If $P=Q$, [[*i.e.*]{}]{}, $x_0=x_1$ and $y_0=y_1$, then $$\label{eq:2P}
x_2=\lambda^2-2x_0,\qquad y_2=3x_0\lambda-\lambda^3-y_0,$$ where $$\label{eq:lambda}
\lambda:=\frac{3x_0^2-A^2}{2y_0}.$$ Below we shall write $2*P$ instead of $P+P$.
- If $x_0=x_1$ and $y_0=-y_1$, then $P+Q:={\mathscr{O}}$. In particular, $(0,0)+(0,0)=(A,0)+(A,0)=(-A,0)+(-A,0)={\mathscr{O}}$.
- Finally, we define ${\mathscr{O}}+P:=P$ and $P+{\mathscr{O}}:=P$ for any point $P$, in particular, ${\mathscr{O}}+{\mathscr{O}}={\mathscr{O}}$.
With the operation “$+$”, $(C_A,+)$ is an abelian group with neutral element ${\mathscr{O}}$. Let $C_A({\mathds{Q}})$ be the set of rational points on $C_A$ together with ${\mathscr{O}}$. It is easy to see that $\bigl(C_A({\mathds{Q}}),+\bigr)$. is a subgroup of $(C_A,+)$. Moreover, it is well known that the group $\bigl(C_A({\mathds{Q}}),+\bigr)$ is finitely generated. One can readily check that the three points $(0,0)$ and $(\pm A,0)$ are the only points on $C_A$ of order $2$, and one easily finds other points of finite order on $C_A$. But do we find also rational points of finite order on $C_A$? This question is answered by the following
\[thm:main\] If $A$ is a congruent number and $(x_0,y_0)$ is a rational point on $C_A$ with $y_0\neq 0$, then the order of $(x_0,y_0)$ is infinite. In particular, if there exists one [[rational Pythagorean $A$-triple]{}]{}, then there exist infinitely many such triples.
The usual proofs of [[Theorem]{}]{}\[thm:main\] are quite involved. For example, Koblitz [@Koblitz Ch.I, §9, Prop.17] gives a proof using Dirichlet’s theorem on primes in an arithmetic progression, and in Chahal [@Chahal Thm.3], a proof is given using the Lutz-Nagell theorem, which states that rational points of finite order are integral. However, both results, Dirichlet’s theorem and the Lutz-Nagell theorem, are quite deep results, and the aim of this article is to provide a simple proof of [[Theorem]{}]{}\[thm:main\] which relies on an elementary theorem of Fermat.
A Theorem of Fermat
===================
In [@Fermat], Fermat gives an algorithm to construct different right-angled triangles with three rational sides having the same area (see also Hungerbühler [@Noebi]). Moreover, Fermat claims that his algorithm yields infinitely many distinct such right-angled triangles. However, he did not provide a proof for this claim. In this section, we first present Fermat’s algorithm and then we show that this algorithm delivers infinitely many pairwise distinct rational right-angled triangles of the same area.
\[algo\] Assume that $A$ is a congruent number, and that $(a_0,b_0,c_0)$ is a [[rational Pythagorean $A$-triple]{}]{}, [[*i.e.*]{}]{}, $A=\big{|}\frac{a_0 b_0}2\big{|}$. Then $$\label{eq:fermat}
a_1:=\frac{4c_0^2a_0b_0}{2c_0(a_0^2-b_0^2)},\quad
b_1:=\frac{c_0^4-4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},\quad
c_1:=\frac{c_0^4+4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},$$ is also a [[rational Pythagorean $A$-triple]{}]{}. Moreover, $a_0b_0=a_1b_1$, [[*i.e.*]{}]{}, if $(a_0,b_0,c_0,A)\in P({\mathds{Q}})$, then $(a_1,b_1,c_1,A)\in P({\mathds{Q}})$.
Let $m:=c_0^2$, let $n:=2a_0b_0$, and let $$X:=2 m n,\quad Y:=m^2-n^2,\quad Z:=m^2+n^2,$$ in other words, $$X=4c_0^2a_0b_0,\quad
Y=c_0^4-4a_0^2b_0^2,\quad
Z=c_0^4+4a_0^2b_0^2.$$ Then obviously, $X^2+Y^2=Z^2$, and since $a_0,b_0,c_0\in{\mathds{Q}}$, $\bigl(|X|,|Y|,|Z|\bigr)$ is a rational Pythagorean triple, where the area of the corresponding right-angled triangle is $${\tilde A}=\bigg{|}\frac{X Y}{2}\bigg{|}=
\big{|}2 a_0 b_0 c_0^2 (c_0^4-4 a_0^2 b_0^2)\big{|}.$$ Since $a_0^2+b_0^2=c_0^2$, we get $c_0^4=(a_0^2+b_0^2)^2=a_0^4+2a_0^2b_0^2+b_0^4$ and therefore $$c_0^4-4 a_0^2 b_0^2\;=\;a_0^4-2a_0^2b_0^2+b_0^4\;=\;(a_0^2-b_0^2)^2>0.$$ So, for $$a_1=\frac{X}{2c_0(a_0^2-b_0^2)},\quad
b_1=\frac{Y}{2c_0(a_0^2-b_0^2)},\quad
c_1=\frac{Z}{2c_0(a_0^2-b_0^2)},$$ we have $a_1^2+b_1^2=c_1^2$ and $$\frac{a_1b_1}{2}\;=\;
\frac{XY}{2\cdot 4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{2 a_0 b_0 c_0^2 (c_0^4-4 a_0^2 b_0^2)}{4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{2 a_0 b_0 c_0^2 (a_0^2-b_0^2)^2}{4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{a_0 b_0}{2}.$$
\[thm:FermatClaim\] Assume that $A$ is a congruent number, that $(a_0,b_0,c_0)$ is a [[rational Pythagorean $A$-triple]{}]{}, and for positive integers $n$, let $(a_n,b_n,c_n)$ be the [[rational Pythagorean $A$-triple]{}]{} we obtain by [[Fermat’s Algorithm]{}]{} from $(a_{n-1},b_{n-1},c_{n-1})$. Then for any distinct non-negative integers $n,n'$, we have $|c_n|\neq |c_{n'}|$.
Let $n$ be an arbitrary but fixed non-negative integer. Since $A=\big{|}\frac{a_n b_n}2\big{|}$, we have $2A=|a_nb_n|$, and consequently $$\label{*}
a_n^2b_n^2=4A^2.$$ Furthermore, since $a_n^2+b_n^2=c_n^2$, we have $$(a_n^2+b_n^2)^2=a_n^4+2a_n^2b_n^2+b_n^4=a_n^4+8A^2+b_n^4=c_n^4,$$ and consequently we get $$c_n^4-16A^2=a_n^4-8A^2+b_n^4=a_n^4-2a_n^2b_n^2+b_n^4=(a_n^2-b_n^2)^2>0.$$ Therefore, $$\sqrt{(a_n^2-b_n^2)^2}=|a_n^2-b_n^2|=\sqrt{c_n^4-16A^2},$$ and with (\[eq:fermat\]) and (\[\*\]) we finally have $$|c_{n+1}|=\frac{c_n^4+16A^2}{2c_n\sqrt{c_n^4-16A^{2\mathstrut}}}\,.$$ Now, assume that $c_n=\frac uv$ where $u$ and $v$ are in lowest terms. We consider the following two cases:
[*$u$ is odd*]{}: First, we write $v=2^k\cdot{\tilde v}$, where $k\ge 0$ and ${\tilde v}$ is odd. In particular, $c_n=\frac{u}{2^{k\mathstrut}\cdot{\tilde v}}$. Since $c_{n+1}$ is rational, $\sqrt{c_n^4-16A^2}\in{\mathds{Q}}$. So, $$\sqrt{c_n^4-16A^2}=\sqrt{\frac{u^4-16A^2v^4}{v^4}}=\frac{{\tilde u}}{v^2}$$ for a positive odd integer ${\tilde u}$. Then $$|c_{n+1}|=\frac{\frac{u^4+16A^2v^4}{v^4}}{\frac{2u{\tilde u}}{v^3}}=
\frac{\bar{u}}{2u{\tilde u}v}=
\frac{\bar{u}}{2u{\tilde u}2^{k\mathstrut}{\tilde v}}=\frac{\bar{u}}{2^{k+1}u{\tilde u}{\tilde v}}
=\frac{u'}{2^{k+1\mathstrut}\cdot v'}$$ where $\bar{u},u',v'$ are odd integers and $\gcd(u',v')=1$. This shows that $$c_n=\frac{u}{2^{k\mathstrut}\cdot{\tilde v}}\quad\Rightarrow\quad
|c_{n+1}|=\frac{u'}{2^{k+1\mathstrut}\cdot v'}$$ where $u,{\tilde v},u',v'$ are odd.
[*$u$ is even*]{}: First, we write $u=2^k\cdot{\tilde u}$, where $k\ge 1$ and ${\tilde u}$ is odd. In particular, $c_n=\frac{2^k\cdot{\tilde u}}{v}$, where $v$ is odd. Similarly, $A=2^l\cdot{\tilde A}$, where $l\ge 0$ and ${\tilde A}$ is odd. Then $$c_n^4\pm 16A^2=\frac{2^{4k}\cdot{\tilde u}^4\pm 2^{4+2l}{\tilde A}^2 v^4}{v^4},$$ where both numbers are of the form $$\frac{2^{2m}\bar{u}}{v^4}\,,$$ where $\bar{u}$ is odd and $4\le 2m\le 4k$, [[*i.e.*]{}]{}, $2\le m\le 2k$. Therefore, $$|c_{n+1}|=\frac{2^{2m}u_0\cdot v^3}{2\cdot 2^k{\tilde u}\cdot
v^4\cdot2^{m\mathstrut}u_1}=\frac{2^{m-k-1\mathstrut}\cdot u'}{v'},$$ where $u_0,u_1,u',v'$ are odd. Since $m<2k+1$, we have $m-k-1<k$, and therefore we obtain $$c_n=\frac{2^k\cdot{\tilde u}}{v}\quad\Rightarrow\quad
|c_{n+1}|=\frac{2^{k'}\cdot u'}{v'}$$ where ${\tilde u},v,u',v'$ are odd and $0\le k'<k$.
Both cases together show that whenever $c_n=2^k\cdot\frac uv$, where $k\in{\mathds{Z}}$ and $u,v$ are odd, then $|c_{n+1}|=2^{k'}\cdot\frac{u'}{v'}$, where $u',v'$ are odd and $k'<k$. So, for any distinct non-negative integers $n$ and $n'$, $|c_n|\neq |c_{n+1}|$.
The proof of [[Theorem]{}]{}\[thm:FermatClaim\] gives us the following reformulation of [[Fermat’s Algorithm]{}]{}:
\[cor:FA\] Assume that $A$ is a congruent number, and that $(a_0,b_0,c_0)$ is a [[rational Pythagorean $A$-triple]{}]{}, [[*i.e.*]{}]{}, $A=\big{|}\frac{a_0 b_0}2\big{|}$. Then $$a_1=\frac{4Ac_0}{\sqrt{c_0^4-16A^2}},\quad
b_1=\frac{\sqrt{c_0^4-16A^2}}{2c_0},\quad
c_1=\frac{c_0^4+16A^2}{2c_0\sqrt{c_0^4-16A^2}},$$ is also a [[rational Pythagorean $A$-triple]{}]{}.
Notice that $c_0^4-4a_0^2b_0^2=c_0^4-16A^2$ and recall that $|a_0^2-b_0^2|=\sqrt{c_0^4-16A^2}$.
Doubling points with Fermat’s Algorithm
=======================================
Before we prove [[Theorem]{}]{}\[thm:main\] ([[*i.e.*]{}]{}, that congruent number curves do not contain rational points of finite order), we first prove that [[Fermat’s Algorithm]{}]{}\[algo\] is essentially doubling points on congruent number curves.
\[lem:doubling\] Let $A$ be a congruent number, let $(a_0,b_0,c_0)$ be a [[rational Pythagorean $A$-triple]{}]{}, and let $(a_1,b_1,c_1)$ be the [[rational Pythagorean $A$-triple]{}]{} obtained by [[Fermat’s Algorithm]{}]{} from $(a_0,b_0,c_0)$. Furthermore, let $(x_0,y_0)$ and $(x_1,y_1)$ be the rational points on the curve $C_A$ which correspond to $(a_0,b_0,c_0)$ and $(a_1,b_1,c_1)$, respectively. Then we have $$2*(x_0,y_0)=(x_1,-y_1).$$
Let $(a_0,b_0,c_0)$ be a [[rational Pythagorean $A$-triple]{}]{}. Then, according to (\[eq:fermat\]), the [[rational Pythagorean $A$-triple]{}]{} $(a_1,b_1,c_1)$ which we obtain by [[Fermat’s Algorithm]{}]{} is given by $$a_1:=\frac{4c_0^2a_0b_0}{2c_0(a_0^2-b_0^2)},\quad
b_1:=\frac{c_0^4-4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},\quad
c_1:=\frac{c_0^4+4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)}.$$ Now, by (\[psi\]), the coordinates of the rational point $(x_1,y_1)$ on $C_A$ which corresponds to the [[rational Pythagorean $A$-triple]{}]{} $(a_1,b_1,c_1)$ are given by $$\begin{aligned}
x_1&=\frac{a_0b_0\cdot(b_1+c_1)}{2\cdot a_1}=\frac{a_0b_0\cdot 2c_0^4}{2\cdot 4c_0^2a_0b_0}=
\frac{c_0^2}{4}\,,\\
y_1&=\frac{2(\frac{a_0b_0}2)^2(b_1+c_1)}{a_1^2}=\frac18 (a_0^2 - b_0^2) c_0.\end{aligned}$$ Let still $(a_0,b_0,c_0)$ be a [[rational Pythagorean $A$-triple]{}]{}. Then, again by (\[psi\]), the corresponding rational point $(x_0,y_0)$ on $C_A$ is given by $$x_0=\frac{b_0(b_0+c_0)}{2}\,,\qquad
y_0=\frac{b_0^2(b_0+c_0)}{2}\,.$$ Now, as we have seen in (\[eq:2P\]) and (\[eq:lambda\]), the coordinates of the point $(x'_1,y'_1):=2*(x_0,y_0)$ are given by $x'_1=\lambda^2-2x_0$, $y'_1=3x_0\lambda-\lambda^3-y_0$, where $$\begin{gathered}
\lambda=\frac{3x_0^2-(\frac{a_0b_0}2)^2}{2y_0}=
\frac{\frac{3 b_0^2(b_0+c_0)^2-a_0^2b_0^2}{4}}{b_0^2(b_0+c_0)}=
\frac{3 (b_0+c_0)^2-a_0^2}{4(b_0+c_0)}=
\frac{3 (b_0+c_0)^2+(b_0^2-c_0^2)}{4(b_0+c_0)}=\\[3ex]
\frac{(3b_0^2+6b_0c_0+3c_0^2)+(b_0^2-c_0^2)}{4(b_0+c_0)}=
\frac{4b_0^2+6b_0c_0+2c_0^2}{4(b_0+c_0)}=
\frac{2b_0^2+3b_0c_0+c_0^2}{2(b_0+c_0)}=\\[3ex]
\frac{(2b_0+c_0)(b_0+c_0)}{2(b_0+c_0)}=
\frac{(2b_0+c_0)}{2}\,.\end{gathered}$$ Hence, $$x_1'=\lambda^2-2x_0=\frac{(2b_0+c_0)^2}{4}-b_0(b_0+c_0)=
\frac{(4b_0^2+4b_0c_0+c_0^2)-(4b_0^2+4b_0c_0)}{4}=\frac{c_0^2}{4}\,$$ and $$y'_1=3x_0\lambda-\lambda^3-y_0=\frac18 (2 b_0^2 c_0 - c_0^3)=\frac18(b_0^2-a_0^2)c_0,$$ [[*i.e.*]{}]{}, $x_1=x'_1$ and $y_1=-y'_1$, as claimed.
With [[Lemma]{}]{}\[lem:doubling\], we are now able to prove [[Theorem]{}]{}\[thm:main\], which states that for a congruent number $A$, the curve $C_A:y^2=x^3-A^2x$ does not have rational points of finite order other than $(0,0)$ and $(\pm A,0)$.
Assume that $A$ is a congruent number, let $(x_0,y_0)$ be a rational point on $C_A$ which $y_0\neq 0$, and let $(a_0,b_0,c_0)$ be the [[rational Pythagorean $A$-triple]{}]{} which corresponds to $(x_0,y_0)$ by (\[psi-1\]). Furthermore, for positive integers $n$, let $(a_n,b_n,c_n)$ be the [[rational Pythagorean $A$-triple]{}]{} we obtain by [[Fermat’s Algorithm]{}]{} from $(a_{n-1},b_{n-1},c_{n-1})$, and let $(x_n,y_n)$ be the rational point on $C_A$ which corresponds to the [[rational Pythagorean $A$-triple]{}]{} $(a_n,b_n,c_n)$ by (\[psi\]).
By the proof of [[Lemma]{}]{}\[lem:doubling\] we know that the $x$-coordinate of $2*(x_n,y_n)$ is equal to $\frac{c_n^2}{4}$, and by [[Theorem]{}]{}\[thm:FermatClaim\] we have that for any distinct non-negative integers $n,n'$, $|c_n|\neq |c_{n'}|$. Hence, for all distinct non-negative integers $n,n'$ we have $$(x_n,y_n)\neq (x_{n'},y_{n'}),$$ which shows that the order of $(x_0,y_0)$ is infinite.
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---
abstract: 'In topological Weyl semimetals, the low energy excitations are comprised of linearly dispersing Weyl fermions, which act as monopoles of Berry curvature in momentum space and result in topologically protected Fermi arcs on the surfaces. We propose that these Fermi arcs in Weyl semimetals lead to an anisotropic magnetothermal conductivity, strongly dependent on externally applied magnetic field and resulting from entropy transport driven by circulating electronic currents. The circulating currents result in no net charge transport, but they do result in a net entropy transport. This translates into a magnetothermal conductivity that should be a unique experimental signature for the existence of the arcs. We analytically calculate the Fermi arc-mediated magnetothermal conductivity in the low-field semiclassical limit as well as in the high-field ultra-quantum limit, where only the chiral Landau levels are involved. By numerically including the effects of higher Landau levels, we show how the two limits are linked at intermediate magnetic fields. This work provides the first proposed signature of Fermi arc-mediated thermal transport and sets the stage for utilizing and manipulating the topological Fermi arcs in experimental thermal applications.'
author:
- 'Timothy M. McCormick'
- 'Sarah J. Watzman'
- 'Joseph P. Heremans'
- Nandini Trivedi
bibliography:
- 'arcThermo\_ref.bib'
title: Fermi arc mediated entropy transport in topological semimetals
---
Introduction
============
The discovery of topological band insulators has led to a new paradigm in condensed matter physics[@hasanKane; @qiZhang]. The recent theoretical prediction[@burkBal; @wanTurnVish; @PhysRevLett.107.186806; @Volovik2014514] and subsequesnt experimental discovery[@Xu613; @PhysRevX.5.031013; @Lv2015] of topological Weyl semimetals has expanded the list of topologically nontrivial quadratic Hamiltonians. These Weyl semimetals possess nodal Fermions composed of non-degenerate linear band crossings. In order to satisfy the condition of nondegeneracy, Weyl semimetals must break either inversion-symmetry or time-reversal symmetry. These so-called Weyl nodes come in pairs of opposite chirality[@Nielsen1981219] and, unlike the Dirac fermions in graphene[@graphene], they are robust against the formation of a gap due to their three-dimensional nature.
The low energy linear dispersing modes of a Weyl semimetal carry monopole charges of Berry curvature. There have been extensive predictions of novel electronic transport resulting from these Weyl nodes[@vishChargexport; @sonSpivakWeyl; @PhysRevB.88.125105; @PhysRevLett.111.246603] as well as promising experimental signatures of some of these transport properties, such as negative longitudinal magnetoresistance [@PhysRevX.5.031023; @Zhang2016; @Arnold2016]. This negative magnetoresistance in the presence of closely parallel electric and magnetic fields is a result of the local non-conservation of charge in the Brillouin zone as a consequence of the chiral anomaly in Weyl semimetals.
In addition to the plethora of novel transport phenomena exhibited by Weyl semimetals, these monopole charges of Berry curvature are responsible for topological Fermi arcs in Weyl semimetals, perhaps their most fascinating feature. Fermi arcs form open contours of surface states which terminate on the projections of Weyl nodes, or in the case of doped Weyl semimetals, terminate on the projections of Fermi pockets enclosing Weyl nodes. Like the bulk Weyl nodes, the states comprising the Fermi arcs also disperse linearly. These gapless boundary modes provide the key signature for Weyl semimetals in spectroscopy experiments.
Fermi arcs in Weyl semimetals are known to lead to exotic quantum oscillations involving mixed real and momentum space orbits[@pkv; @Zhang2016a; @Moll2016] as well as resonant transparency[@PhysRevX.5.041046]. However, the effects of Fermi arcs on thermal transport so far remains a completely unexplored frontier. Preliminary theoretical studies of thermal transport in Weyl semimetals have so far only considered contributions from the bulk Weyl fermions[@fieteThermoelec; @PhysRevB.93.035116; @doubleWeyl].
In this Letter, we explore the role of Fermi arcs in entropy transport for the first time. We demonstrate that Fermi arcs in topological semimetals lead to a thermal conductivity which depends strongly on an externally applied magnetic field. This external magnetic field is shown to enhance the heat current in topological semimetals by creating “conveyor belt” motion of charge leading to a coherent transmission of heat. We propose that this Fermi arc-mediated entropic transport will lead to a dramatically anisotropic magnetothermal conductivity. In the low-field semiclassical limit, it is shown that a Lorentz force on the electrons comprising the Fermi arcs necessarily leads to a bulk flow of electrons, carrying a net energy current in the presence of a temperature gradient. When the field strength reaches the ultraquantum limit, we find that the electrons from the arcs can only hybridize with the chiral Landau levels in the bulk, which allow for a dissipationless flow of energy. In the intermediate quantum regime, where several quantized Landau levels become involved in Fermi arc-mediated magnetothermal transport, we calculate the specific heat and entropy density and the magnetothermal conductivity for various temperatures and magnetic fields. We conclude by discussing the experimental implications of our work. This work sets the stage for utilizing and manipulating the topological Fermi arcs in experimental magnetothermal applications.
Fermi arc-mediated magnetothermal transport
===========================================
![ Energy dispersion of a pair of Weyl nodes given by Eqn. (\[contham\]). The Weyl nodes are separated by a distance of $2 \Lambda$ in the Brillouin zone. Areas of the band structure colored in darker blue have a larger magnitude of Berry curvature. []{data-label="weylPlot"}](weylPlot.pdf){width="50.00000%"}
We consider a linearized model of Weyl fermions in the continuum limit. For a given Weyl node of chirality $\chi = \pm 1$, the Hamiltonian that describes states near the node is given by $$\label{contham}
\hat{H}_\chi = \chi \hbar v_F
\Big(
k_x \hat{\sigma}_x
+k_y \hat{\sigma}_y
+k_z \hat{\sigma}_z
\Big),$$ where the Fermi velocity of the Weyl electrons is given by $v_F$ and where the Pauli matrices $\hat{\sigma}_j$ span a spin or orbital degree of freedom. These Weyl nodes come in pairs of opposite chirality and we consider $N_p$ of such pairs of nodes. We will take each pair of Weyl nodes to be separated in the Brillouin zone by a characteristic distance set by some momentum cutoff $\Lambda$. We take all pairs of Weyl nodes to lie at the same energy $E = 0$, but our calculations generalize to cases where sets of Weyl nodes lie at different energies.
Semiclassical limit
-------------------
We consider a Weyl semimetal in a slab geometry such that the Fermi arcs reside in the surface Brillouin zone labeled by $k_x$ and $k_y$ in the presence of a perpendicular magnetic field $\mathbf{B} = B\mathbf{e}_z$. Then, assuming the Fermi arc states all have a uniform magnitude of velocity $v_F$, a semiclassical wavepacket of electrons on a Fermi arc will follow the trajectory $$\label{arckdot}
\dfrac{d\mathbf{k}}{dt} = \dfrac{e}{\hbar c}{\left(}\mathbf{v}_{\mathbf{k}}\times \mathbf{B} {\right)}= \dfrac{e}{\hbar c} v_F B \mathbf{e}_t,$$ where $\mathbf{e}_t$ is the unit tangent vector to the Fermi arc. This configuration has been predicted[@pkv] to give rise to unusual Fermi arc-mediated quantum oscillations in Weyl semimetals, but similar effects remain unexplored in thermal transport.
We consider the collective motion of the electrons comprising the Fermi arcs and note that the change in the number $N$ of electrons on a given surface with one Fermi arc is $$\label{dndtfa}
\dfrac{dN}{dt} = A \dfrac{dn}{d{\mathcal{E}}}\dfrac{d{\mathcal{E}}}{dk}\dfrac{dk}{dt},$$ where $A$ is the area of the surface on which the Fermi arcs reside, $\frac{dn}{d{\mathcal{E}}}$ is the density of states of the Fermi arcs $g_{A}({\mathcal{E}})$, the magnitude of the Fermi velocity is of course $\frac{d{\mathcal{E}}}{dk} =\hbar v_F$ and, from Eqn. (\[arckdot\]), we know that $\frac{dk}{dt} = \frac{e}{\hbar c} v_F B$. The density of states on the Fermi arcs is given by $$\label{gofefa}
g_{A}({\mathcal{E}}) = \dfrac{k_0}{\hbar v_F},$$ where $k_0$ is the length of the Fermi arc. Here, we have made the reasonable assumption that the magnitude of $v_F$ is constant along the arcs. Hence we have that the total rate of electrons moving along each arc is given by $$\label{dndtfa2}
\dfrac{dN}{dt} = A \dfrac{e}{\hbar c}k_0 v_F B.$$
{width="99.00000%"}
In Fig. \[slabCartoon\]a, we show a mixed real-space and momentum-space cartoon of a Weyl semimetal in a slab geometry. We see that the bulk Weyl nodes are separated in the $x$-direction, shown in purple and labeled with their chirality $\chi = \pm 1$. The magnetic field induced flow of electrons along the arcs described by Eqn. (\[dndtfa2\]) will lead to a transfer of electrons from right to left along the blue Fermi arc on the top surface in Fig. \[slabCartoon\] and similarly from left to right along the red Fermi arc along the bottom surface. In the absence of electric fields or temperature gradients, the only way the system can maintain this steady-state circulation of electrons is from bulk transport through the Fermi pockets upon which the Fermi arcs terminate. The projections of these Fermi pockets in the surface Brillouin zone are shown in purple in Fig. \[slabCartoon\]a. Thus, by the continuity equation, through each bulk pocket surrounding a node with chirality $\chi$, we must have a real-space current density $\mathbf{J}_B^{\chi}$ given by $$\label{sscurdens}
\mathbf{J_B^\chi} = \chi \dfrac{ e}{A} \dfrac{dN}{dt}\mathbf{e}_z,$$ where $\frac{dN}{dt}$ is given by Eqn. (\[dndtfa2\]). These circulating currents are the low-field analog of those explored by the authors of Ref.[@pkv].
Since the Weyl nodes come in pairs of opposite chirality, that these steady state circulating currents do not cause any net current flow along the $z$-direction in the absence of external potentials. However, in the presence of an applied temperature gradient, entropy contributions from these circulating currents cause additive contributions to the heat current. To see this, we consider an applied temperature gradient $\nabla T = \frac{dT}{dz}\mathbf{e}_z$ for the slab depicted in Fig. \[slabCartoon\], which we take to have a thickness $L$ in the $z$-direction. Now, electrons on the top surface $z = \frac{L}{2}$ have a thermal energy density $u\left(\frac{L}{2}\right)$ and those on the bottom at $z =- \frac{L}{2}$ have thermal energy density $u\left(-\frac{L}{2}\right)$. For a single pair of nodes, in the presence of a magnetic field $\mathbf{B} = B\mathbf{e}_z$, we have shown above that a bulk number current of electrons given by Eqn. (\[dndtfa2\]) circulates through the bulk, driven by the Lorentz force on the top and bottom Fermi arcs.
The flow of electrons from the Fermi arcs is topological in nature and so we only assume that scattering occurs at the surfaces of the sample. The heat current density for $N_p$ pairs of Weyl nodes at some point $z$ in the bulk is given by $$\label{fajqkzz}
J_z^{Q} = N_p L \dfrac{dN}{dt}
\left(
u\left(z+\dfrac{L}{2}\right) -
u\left(z-\dfrac{L}{2}\right)
\right).$$ We expand the thermal energies $u(z\pm\frac{L}{2})$ to obtain $$\label{fajqkzz2}
J_z^{Q} = \dfrac{N_p L^2}{2}\dfrac{dN}{dt}\dfrac{du}{dT}\dfrac{dT}{dz},$$ where $\frac{du}{dT}$ is the specific heat of the bulk Weyl pockets that these thermally excited electrons from the surfaces must traverse. We can calculate the energy density to find $$\label{utot}
u = N_p \int d{\mathcal{E}}\dfrac{{\mathcal{E}}\ g({\mathcal{E}})}
{1+e^{\beta({\mathcal{E}}-\mu(T))}} = \dfrac{N_p}{\pi^2(\hbar v_F)^3} \int d{\mathcal{E}}\dfrac{{\mathcal{E}}^3}{1+e^{\beta({\mathcal{E}}-\mu(T))}}.$$ We first consider the case of $T \gtrsim T_W$ where $\mu(T) \rightarrow 0$. In this case, we have that $$\label{hightutot}
u \approx \dfrac{N_p (k_B T)^4}{\pi^2(\hbar v_F)^3} \int dx \dfrac{x^3}{1+e^{x}},$$ where $\int dx \dfrac{x}{1+e^{x}} \equiv c_0 $ is a constant of order 1. Therefore we obtain that $$\label{hightdudt}
\dfrac{du}{dT} = c_0 \dfrac{N_p k_B^4 T^3}{\pi^2(\hbar v_F)^3},$$ and so for $T \gtrsim T_W$, we obtain that $$\label{hightjq}
J_z^{Q} = \dfrac{2c_0 N_p}{\pi^2}\dfrac{e k_B^4}{\hbar^4 c} \dfrac{L^2 A k_0 T^3 B}{v_F^2}\dfrac{dT}{dz}.$$ From Eqn. (\[hightjq\]), we obtain the contribution of the Fermi arcs to the thermal conductivity and find it to be $$\label{hightkzzz}
\kappa^{SC}_{zzz} = \dfrac{2c_0 N_p}{\pi^2}\dfrac{e k_B^4}{\hbar^4 c} \dfrac{L^2 A k_0 }{v_F^2} T^3 B,$$ for temperatures $T \gtrsim T_W$.
In Fig. (\[scLimitKappa\]), we show the thermal conductivity in the semiclassical limit as a function of temperature. We see the cubic dependence at low temperature saturates as the temperature is raised. This saturation occurs since the bulk degrees of freedom taking part in the transport of heat increases with temperature. Eventually, all available bulk states become involved and the magnetothermal conductivity saturates. We constrast our results here with classical Drude theory for a metal where $\kappa \sim T$ and is independent of magnetic field.
![Fermi arc-mediated magnetothermal conductivity in the semiclassical limit as a function of temperature. We see that at low temperatures $\kappa_{zzz}^{\textrm{SC}} \sim T^3$. At higher temperatures, saturation of $\kappa_{zzz}^{\textrm{SC}}$ occurs as more and more bulk degrees of freedom take part in the transport of heat. At high enough temperatures, all available bulk states become involved and the magnetothermal conductivity saturates.[]{data-label="scLimitKappa"}](scLimitKappa.pdf){width="50.00000%"}
Ultra-quantum limit
-------------------
Next we analyze the limit of large magnetic fields where the energies of the Weyl nodes split into discrete Landau levels. Due to the linear dispersion of the Weyl nodes, the Landau levels are not evenly spaced and because of their three dimensional nature, they disperse in the direction of the field in the following way $${\mathcal{E}}_{n}(k_z) = \pm \hbar v_{F} \sqrt{2|n|l_B^{-2} + k_z^2},
\label{ellnn0}$$ for any integer $n \neq 0$ and $${\mathcal{E}}_{0}(k_z) = \chi \hbar v_{F} k_z,
\label{elln0}$$ for $n = 0$. Here we have aligned the magnetic field again along the $z$-direction such that $\mathbf{B} = B \mathbf{e}_z$. The magnetic length is given by $l_B = \sqrt{\frac{\hbar c}{eB}}$ and $v_F$ again denotes the magnitude of the Fermi velocity. We show a schematic of this dispersion in Fig. (\[chiralLLs\]) where the chiral Landau levels are shown in red. The slope of the $n = 0$ Landau level depends on the chirality of the Weyl node.
![Schematic of the Landau levels. Chiral $n = 0$ Landau levels are shown in red. The sign of the slope of the $n=0$ Landau levels is positive (dashed line) for $\chi = 1$ nodes and negative (solid line) for $\chi = -1$. Non-chiral Landau levels ($n \neq 0$) are shown in black. []{data-label="chiralLLs"}](chiralLLs.pdf){width="40.00000%"}
As in the semiclassical case above, on a given Fermi arc, an electron will follow the semiclassical motion of Eqn. (\[arckdot\]). From Ref. [@pkv], we know that, in the high field limit with Landau levels in the bulk, electrons will engage in mixed momentum and real space orbits which traverse the bulk, parallel to the direction of the magnetic field. As an electron moves along the arc, we know that it will terminate on the projection of a Weyl node which, in the presence of a large magnetic field, will be comprised of the Landau levels in Eqns. (\[ellnn0\]-\[elln0\]). For $k_B T \ll \frac{\hbar v_F}{l_B}$, the only state available is the zeroth Landau level given by Eqn. (\[elln0\]). Because of the chiral nature of these states, electrons from the arc will traverse the bulk without dissipation and emerge on the other surface. After it reaches the other side, it will move along another Fermi arc until it reaches the opposite chirality bulk Weyl node. It will then traverse the bulk in the opposite direction and complete the loop.
We again consider a thermal gradient in the same direction as the magnetic field such that $\nabla T = \frac{\partial T}{\partial z}\mathbf{e}_z$. The conveyor belt motion of the electrons will again enhance the magnetothermal conductivity. In this case however, unlike the semiclassical limit explored above, the $n = 0$ Landau levels will provide single quantum channels for heat transport. Taking the thickness of the slab to again be $L$, we find that the heat current along the $z$-direction is given by $$\label{heatCurrqlim}
J^{q}_{z} =N_p L \dfrac{dN}{dt}
\left(
u_{0}\left(z+\dfrac{L}{2} \right) - u_{0}\left(z - \dfrac{L}{2}\right)
\right)
=\dfrac{1}{2}N_p L^2 \dfrac{dN}{dt} \dfrac{du_{0}}{dT} \dfrac{dT}{dz},$$ where $u_0$ is the energy density of the $n = 0$ Landau levels and $\frac{du_{0}}{dT}$ is the specific heat of a pair opposite chirality $n = 0$ Landau levels. Again, we have assumed that continuity of charge leads to a particle current $\frac{dN}{dt}$ defined by Eqn. (\[dndtfa2\]) through the bulk. However, energy current is only carried by the chiral $n = 0$ Landau level.
We can calculate the internal energy $$\label{intnrgqlim}
u_0 = \int d{\mathcal{E}}g_{0} ({\mathcal{E}}) f({\mathcal{E}}){\mathcal{E}},$$ where the density of states of the zero energy Landau levels is given by $$\label{ll0dos}
g_{0}({\mathcal{E}}) = CB \int_{-\Lambda}^{\Lambda} \dfrac{dk_z}{2\pi}
\bigg(
\delta({\mathcal{E}}- |{\mathcal{E}}_0| ) + \delta({\mathcal{E}}+ |{\mathcal{E}}_0| )
\bigg),$$ with ${\mathcal{E}}_0$ give by Eqn. (\[elln0\]). Here $C = \frac{N_p}{\Phi_0}$ is a normalization factor where $N_p$ is the number of pairs of Weyl nodes and $\Phi_0 = \frac{hc}{e}$ is the quantum of magnetic flux. $\Lambda$ is a momentum regularization which is set by the separation of the Weyl nodes in momentum-space. Then we have that $$\label{intnrgqlim2}
u_0 = CB\hbar v_F\int_{-\Lambda}^{\Lambda} \dfrac{dk_z}{2\pi}
\Bigg(
\dfrac{k_z}{1+e^{\beta(\hbar v_F \Lambda-\mu)}}-\dfrac{k_z}{1+e^{\beta(-\hbar v_F \Lambda-\mu)}}
\Bigg),$$ which can be evaluated to obtain $$\label{dudtfull}
\dfrac{du_0}{dT} =\dfrac{N_p B}{\pi\Phi_0}\Bigg(4\dfrac{\hbar v_F \Lambda^2}{T}
\dfrac{1}{1+e^{-\frac{\hbar v_F \Lambda}{k_B T}}} - 8k_B \Lambda\textrm{ln}\Big( 1+e^{\frac{\hbar v_F \Lambda}{k_B T}} \Big)+
\dfrac{2k_B^2 T}{\hbar v_F}
\bigg(
\dfrac{\pi^2}{3}+4
\textrm{Li}_{2}\Big(-e^{\frac{\hbar v_F \Lambda}{k_B T}} \Big)
\bigg)
\Bigg),$$ where $\textrm{Li}_s(z)$ is the polylogarithm function of order $s$, defined as $$\label{polyln}
\textrm{Li}_s(z) = \sum_{k=1}^{\infty}\dfrac{z^k}{k^s},$$ for complex $z$ such that $|z| < 1$.
In the low temperature limit $\dfrac{k_B T}{\hbar v_F \Lambda} \ll 1$, we find that the heat capacity goes like $$\label{dudtlowt}
\dfrac{du_0}{dT} \approx \dfrac{2\pi}{3} \dfrac{N_p k_B^2}{\hbar v_F \Phi_0}T B$$ and so we find that at low temperatures, the thermal conductivity in the ultra-quantum limit goes like $$\label{kappaqlowt}
\kappa^{UQ}_{zzz} \approx \dfrac{\pi}{3} \dfrac{N_p^2 L^2 A k_0 k_B^2}{\hbar \Phi_0^2}B^2 T.$$ Unlike the semiclassical case, we see that the thermal conductivity is quadratic in field, and now has a linear temperature dependence. This linear dependence on temperature holds for temperatures $k_B T \ll \frac{\hbar v_F}{l_B}$. Close to that temperature, the higher Landau levels become thermally populated, allowing for them to engage in Fermi arc-mediated magnetothermal transport as well.
The result in Eqn. (\[kappaqlowt\]) holds up to fields high enough that the inverse magnetic length approaches the scale of the momentum cutoff $l_B^{-1} \sim \Lambda$. For magnetic fields above $B_{c} = \frac{\hbar c \Lambda^2 }{e}$, the coupling between the nodes cannot be neglected and a magnetic breakdown of the mixed real and momentum space orbits occurs. In experimental realizations of Weyl semimetals, a typical value for the node separation is approximately $\Lambda \approx 0.1 \AA^{-1}$. This allows us to estimate a typical value of $B_{c} \approx 10^3\ \textrm{T}$, well outside of the range of any laboratory fields.
Heat capacity and entropy transport of higher Landau levels
-----------------------------------------------------------
In order to investigate the crossover from the low-field semiclassical limit to the ultraquantum limit only involving chiral Landau levels, we include quantum effects of nonchiral Landau levels. In this intermediate field regime, the density of states used to calculate specific heat calculated in Eqn. (\[intnrgqlim\]) must include the higher Landau levels. This means that Eqn. (\[ll0dos\]) for a pair of Weyl nodes in a magnetic field becomes $$\label{allLLdos}
g_{LL}({\mathcal{E}}) = CB \int_{-\Lambda}^{\Lambda} \dfrac{dk_z}{2\pi}
\bigg(
\delta({\mathcal{E}}- |{\mathcal{E}}_0| ) + \delta({\mathcal{E}}+ |{\mathcal{E}}_0|) +2 \sum_{n}
\big(
\delta({\mathcal{E}}- |{\mathcal{E}}_n| )+
\delta({\mathcal{E}}+ |{\mathcal{E}}_n| )
\big)
\bigg),$$ where ${\mathcal{E}}_n$ is given by Eqn. (\[ellnn0\]). Then the additional internal energy can be calculated by $$\label{llallnewnrg}
\widetilde{u} = 2 CB \sum_{n} \int_{-\Lambda}^{\Lambda} \dfrac{dk_z}{2\pi}
\Bigg(
\dfrac{\hbar v_{F} \sqrt{|n|l_B^{-2} + k_z^2}}{1+e^{\beta(\hbar v_{F} \sqrt{|n|l_B^{-2} + k_z^2}-\mu)}}
-
\dfrac{\hbar v_{F} \sqrt{|n|l_B^{-2} + k_z^2}}{1+e^{\beta(-\hbar v_{F} \sqrt{|n|l_B^{-2} + k_z^2}-\mu)}}
\Bigg),$$ where the total energy is now given by the sum of Eqns. (\[intnrgqlim2\]) and (\[llallnewnrg\]) to obtain $u_\textrm{tot} = u + \widetilde{u}$. Unlike Eqn. (\[intnrgqlim2\]), the expression above for $\widetilde{u}$ cannot be evaluated analytically. Instead, we evaluate Eqn. (\[llallnewnrg\]) numerically by introducing a regularization $N_\textrm{max}$ which cuts off the sum over Landau levels. In the calculations below, we take the system to be at charge neutrality where $\mu = 0$.
{width="99.00000%"}
We can then calculate the heat capacity by numerically evaluating derivatives with respect to temperature. In Fig. \[interMedQcv\]a, the total specific heat of all Landau levels $c_v = \frac{du_{\textrm{tot}}}{dT}$ is calculated as function of temperature for several fixed magnetic fields. For low temperatures, $c_v$ increases linearly with temperature. Around $\frac{k_B T}{\hbar v_F \Lambda} \sim 0.01$, the temperature dependence becomes cubic, before again becoming linear at high temperature. In Fig. \[interMedQcv\]b, $c_v$ is similarly calculated as a function of field for various fixed temperatures. At both high and low temperatures, the field dependence of $c_v$ is linear but with different slopes. This is precisely what we would expect from the high field ultraquantum limit in Eqn. (\[dudtlowt\]), which is linear in field.
After the heat capacity has been obtained, we can then calculate the arc-mediated magnetothermal conductivity. Unlike in the case of the ultraquantum limit above, higher Landau levels do not provide the one-way channels of heat transport that the chiral $n = 0$ states do. We now find the total heat current to be the sum of the heat current from the $n = 0$ Landau level as well as all higher Landau levels. The total heat current for all Landau levels is given by $$\label{allLLheatcurr}
J^{q}_{z} = \dfrac{1}{2}N_pL^2 \dfrac{dN}{dt}\dfrac{du_{\textrm{tot}}}{dT}\dfrac{dT}{dz},$$ where $c_v = \frac{du_{\textrm{tot}}}{dT}$ is the specific heat of all Landau levels as defined above. Then the magnetothermal conductivity is given by $$\label{allLLkappazzz}
\kappa_{zzz} = \dfrac{1}{2}N_pL^2 \dfrac{dN}{dt}\dfrac{du_{\textrm{tot}}}{dT},$$ which we evaluate numerically as above and show in Fig. \[interMedQ\_kappa\]. In the temperature dependence of $\kappa_{zzz}$ shown in Fig. \[interMedQ\_kappa\]a, we see that the magnetothermal conductivity has a linear dependence on temperature for low and high temperatures. In Fig. \[interMedQ\_kappa\]b, we see that the field dependence is linear for the temperature ranges shown here, where multiple Landau levels are involved in thermal transport. This matches the calculations which we obtained semiclassically above.
{width="99.00000%"}
Discussion and Summary
======================
We have seen in the presence of a magnetic field and temperature gradient, each applied perpendicular to the surface Brillouin zone of a Weyl semimetal, the Lorentz force on the Fermi arcs will lead to a conveyor belt motion of charge and a net flow of heat. This heat flow leads to a highly anisotropic magnetothermal conductivity which has distinct behavior in the semiclassical and quantum regimes. For relatively high temperatures and small fields, the fermi-arc mediated magnetothermal conductivity $\kappa_{zzz}$ is found to be linear in magnetic field and cubic in temperature. On the other hand, in the ultra-quantum limit where the magnetic field is strong and temperatures are low such that only the chiral $n= 0$ Landau level is involved, we find that the Fermi arc-mediated magnetothermal conductivity is instead linear in temperature and quadratic in field. The difference in temperature dependences can be understood by noting that in the semiclassical regime, there are many more degrees of freedom in the bulk available to thermal transport. Once the system is the ultraquantum limit, the sole degrees of freedom available are the chiral $n =0$ Landau levels, and $\kappa_{zzz}$ is much less sensitive to changes in temperature. In the semiclassical limit, the linear magnetic field dependence comes only from the Lorentz force on the arcs. In fields sufficiently strong enough, the quantization of the Landau levels results in a density of states which also depends on the magnetic field. In the ultra-quantum limit, the lone chiral $n=0$ Landau levels result in a specific heat which is quadratic in field. As more Landau levels become populated at higher temperatures, $\kappa_{zzz}$ again becomes linear and tends toward the semiclassical limit.
In this Letter, we have focused on the so-called type I Weyl semimetals, but recently a second class, known as type II Weyl semimetals, have been theoretically predicted [@Soluyanov2015; @Sun2015; @PhysRevLett.117.056805; @PhysRevB.93.201101] and observed in several candidate materials[@Huang2016; @Wu2016; @Bruno2016; @Wang2016; @Change1600295]. Type II Weyl semimetals have non-vanishing density of states at the Weyl points and can be understood as the limiting case of an indirect gap semiconductor where the gap closes at the Weyl nodes. The tilted nodes in type II Weyl semimetals lead to several distinct characteristics, perhaps most notably the lack of chiral Landau levels when the magnetic field is applied outside of the tilt cone[@Soluyanov2015; @PhysRevLett.117.086402]. Due to the extended nature of the Fermi pockets that meet at the Weyl nodes, lattice models[@mkt] are needed to accurately describe their properties rather than the continuum models used for the type I case above. We leave an investigation of Fermi arc mediated magnetothermal conductivity in type II Weyl semimetals for future work.
Acknowledgments
===============
T. M. M. and J. P. H. were supported by the Center for Emergent Materials, an NSF MRSEC, under grant DMR-1420451. S. J. W. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0822215. N. T. acknowledges funding from NSF-DMR-1309461.
|
---
abstract: 'Let ${\mathbb{F}_{\Theta}}=U/K_\Theta$ be a generalized flag manifold, where $K_\Theta$ is the centralizer of a torus in $U$. We study $U$-invariant almost Hermitian structures on ${\mathbb{F}_{\Theta}}$. The classification of these structures are naturally related with the system $R_t$ of t-roots associated to ${\mathbb{F}_{\Theta}}$. We introduced the notion of connectedness by triples zero sum in a general set of linear functional and proved that t-roots are connected by triples zero sum. Using this property, the invariant $\mathcal{G}_1$ structures on ${\mathbb{F}_{\Theta}}$ are completely classified. We also study the Kähler form and classified the invariant quasi Kähler structures on ${\mathbb{F}_{\Theta}}$, in terms of t-roots.'
author:
- 'Luciana A. Alves [^1] and Neiton Pereira da Silva [^2]'
title: '$\mathcal{G}_1$ structures on flag manifolds'
---
Mathematics Subject Classifications: 53C55; 53D15; 22F30 .
Keywords: Flag manifolds; t-roots; connectedness by triples zero sum; almost Hermitian manifold; $\mathcal{G}_1$ structures.
Introduction
============
An almost Hermitian manifold is a differentiable manifold of even dimension $M$ endowed with a almost complex structure $J$ and a Riemannian metric $g(\cdot, \cdot )$ such that $g( JX, JY ) =g( X, Y )$, for all $X, Y \in \mathfrak{X}(M)$. Let $\Omega (X,Y)=g(JX,Y)$ be the Kähler form and $\nabla$ the Riemannian connection. The pair $(g,J)$ is a K" ahler structure on $M$ if $J$ is integrable and the associated Kähler form is closed (i.e. $d\Omega=0$) or equivalently $\nabla J=0$. According to [@GH], various authors have studied certain types of almost Hermitian manifolds with the aim of generalizing geometry K" ahler. A pre-Kählerian structure has linear type if $$\begin{aligned}
\label{pre k}
(\nabla_X J)Y+A_1 J(\nabla_{JX}J)Y +A_2 J(\nabla_X J)Y + A_3(\nabla_{JX}J)Y + A_4 (\nabla_Y J)X\nonumber \\+ A_5 J (\nabla_{JY}J)X + A_6 J(\nabla_Y J)X + A_7 (\nabla_{JY}J)X=0\end{aligned}$$ with $A_i\in\mathbb{R}$, (see [@Vidal-Hervella]). For instance, the structure $(g,J)$ is quasi-Kähler if $(\nabla_XJ)Y-J(\nabla_{JX}J)Y=0$ for all $X,Y \in \mathfrak{X}(M)$; nearly Kähler if $(\nabla_X J)Y+J(\nabla_Y J)X=0$; almost Kähler if $d\Omega=0$ and semi-Kählerian if $\delta\Omega(X)=0$, where $\delta\Omega(X)$ denotes co-derivative of the Kähler form, (cf. [@G]). It is usual to denotes these structures by QK, NK, AK and SK, respectively.
Hervella and Vidal, found in [@Vidal-Hervella] a new pre-Käherian structures, which they called by *$\mathcal{G}_1$ structures*. The pair $(g,J)$ is a $\mathcal{G}_1$ structure if it satisfies $g(N(X,Y),X)=0$, where $N(X,Y)$ denotes the Nijenhuis tensor of the structure $J$. They also proved that any polynomial of the form \[pre k\], with $A_i\in \mathbb{R}$, gives a known pre-Kählerian structure or the $\mathcal{G}_1$ structures.
The $\mathcal{G}_1$ structures have an interested property: Let $\varphi\colon (M, g, J)\longrightarrow(M^0, g^0, J^0)$ be a conformal diffeomorphism, where $J^0X^0=(JX)^0$. Then SK, NK and AK are not conserved by $\varphi$, (see [@G]). On the other hand, $\mathcal{G}_1$ structures are conformally invariant, (cf. [@F-H]).
Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak{g}$ and $P_\Theta$ a parabolic subgroup of $G$, a generalized flag manifold (or simply flag manifold) is the homogeneous space ${\mathbb{F}_{\Theta}}=G/P_\Theta$. If $U$ is a compact real form of $G$, then ${\mathbb{F}_{\Theta}}=U/K_\Theta$, where $K_\Theta$ is the centralizer of a torus in $U$. We denote by $\mathfrak{u}$ and $\mathfrak{k}_\Theta$ the Lie algebra of $U$ and $K_\Theta$ respectively. We study $U$-invariant almost Hermitian structures on flag manifolds ${\mathbb{F}_{\Theta}}$. Such structures consists of a pair $(g,J)$ where $g$ is a $U$-invariant Riemannian metric and $J$ is a $U$-invariant almost complex structure on ${\mathbb{F}_{\Theta}}$, which is compatible with the metric $g$. It is well known that for each invariant complex structure $J$ on ${\mathbb{F}_{\Theta}}$, there exist a unique, up to homotheties, invariant metric $g$ such that the pair is $(g, J)$ is a Kähler structure and $g$ is an Einstein metric (see [@BH]; [@Besse] 8.95). In [@GH], all almost Hermitian structures were classified into sixteen classes. In the case of flag manifolds some of these classes coincide, for example, in [@SM; @N] San Martin-Negreiros proved that for most of full flag manifolds the sixteen classes of Gray-Hervella reduce to three classes: Kähler (K), Quasi Kähler - QK (or (1,2)-symplectic) and $\mathcal{G}_1$ structures (or $W_1\oplus W_3\oplus W_4$). Besides they proved that the only exception is the full flag of type $A_2$, where there exist an invariant almost Hermitian structure nearly Kähler which is not Kähler. Thus, in the case of full flag manifolds, they got a positive answer for the Wolf-Gray conjecture, which is: *Let $U/K$ be a homogeneous space of a compact Lie group $U$ which is not Hermitian symmetric and such that the isotropy $K$ has maximal rank. Then there are invariant almost Hermitian structures on $U/K$ which are nearly Kähler but not Kähler if and only if the isotropy subalgebra is the fixed point set of an autormorphism of order three*, (cf. [@Wolf-Gray]). This conjecture was also proved for generalized flag manifolds in [@SM; @R].
Concerning the three classes of invariant almost Hermitian structures on flag manifolds, the Kähler structures on flag manifolds were extensively studied by several authors (see for example [@Alek; @e; @Perol] and [@Besse] p.224), class (1,2)-symplectic was completed classified and well understood in [@SM; @N]. In this paper we obtain classify completely and obtain a well understood of $\mathcal{G}_1$ structures on flag manifolds ${\mathbb{F}_{\Theta}}$.
An important invariant of a flag manifold ${\mathbb{F}_{\Theta}}$ is the system of t-roots $R_t$, which is a set of linear functional defined as restriction of the root system $R$ of $\mathfrak{g}$ to real form $\mathfrak{t}$ of the center $Z({\mathfrak}{k}_{\Theta}^\mathbb{C})$ of the complexification of the subalgebra ${\mathfrak}{k}_{\Theta}$. The system of t-roots $R_t$ provides a unique decomposition of tangent space $T_o{\mathbb{F}_{\Theta}}={{\mathfrak}{m}}_1\oplus\cdots\oplus{{\mathfrak}{m}}_s$ into irreducible and inequivalent $ad({\mathfrak}{k}_{\Theta})$-submodules ([@Sie]). Thus all $G$-invariant tensor on ${\mathbb{F}_{\Theta}}$ can be described by means of t-roots. For example, any invariant almost complex structure $J$ on a flag manifold ${\mathbb{F}_{\Theta}}$ is determined by a set of signs $\{\varepsilon_\delta = \pm 1, \delta\in R_t\}$, satisfying $-\varepsilon_\delta=\varepsilon_{-\delta}$, (see Proposition \[iacs\]) and any invariant metric $g$ on ${\mathbb{F}_{\Theta}}$ has the form $g(\cdot, \cdot)=- \lambda_1 B(\cdot,\cdot)|_{{{\mathfrak}{m}}_1\times{{\mathfrak}{m}}_1}-\cdots-\lambda_s B(\cdot,\cdot)|_{{{\mathfrak}{m}}_s\times{{\mathfrak}{m}}_s}$, where $B(\cdot,\cdot)$ denotes the Killing form of ${\mathfrak}{g}$. Thus the definition condition of any class from the sixteen classes of Gray - Hervella reduces to a algebraic expression depending on the parameters $\lambda_\delta$ and $\varepsilon_\delta$.
We define the term connected by triple zero sum for a general set of linear functional, which for the system of t-roots $R_t$ can be state as following: *we say that two t-roots $\eta, \delta \in R_t$, $\eta\neq \pm \delta$, are connected by triples zero sum (tzs) if there exists a chain of triples $T_{i,j,k}^p=\{\xi_i,\xi_j,\xi_k\}\subset R_t$ with $\xi_i+\xi_j+\xi_k=0$ such that $\pm\eta\in T_{i,j,k}^1$, $\pm\delta\in T_{i,j,k}^n$ and $T_{i,j,k}^p\cap T_{i^{\prime},j^{\prime},k^{\prime}}^{p+1}\neq \emptyset$, $1\leq p\leq n$.* The key of our study on $\mathcal{G}_1$ -structures on ${\mathbb{F}_{\Theta}}$ is the following property we prove: For any flag manifold the associated system $R_t$ of t-roots is connected by tzs, i.e., any two (non symmetric) t-roots in $R_t$ can be connected by tzs (see Theorem \[connected tzs t-roots\]). Using this result we prove, in section 10, that the metric induced by the Killing form on ${\mathbb{F}_{\Theta}}$ (the normal metric) is the unique invariant metric on ${\mathbb{F}_{\Theta}}$ which is a $\mathcal{G_1}$ structure with respect to any invariant almost complex structure (see Theorem \[metric G1\]). We think that the notion of connectedness by tzs can be very useful for the study of others invariants geometric aspect on flag manifolds.
In section 2 we give the description of flag manifold as complex and real homogeneous spaces using Lie theory. In section 3 we introduce the notion of connection by triples zero sum for roots of the system root $R$ of ${\mathfrak}{g}$ and proved that in any irreducible root system, any two roots are connected by tzs. This section is a preparation to investigate the connection by tzs in t-roots.
In section 6 we study the invariant almost complex structures (abbreviated by iacs) on ${\mathbb{F}_{\Theta}}$. We get a description of iacs on flag manifolds in terms of t-roots, which extends the similar result in [@SM; @N] and [@SM; @R]. As a consequence of this we obtain that every iacs $J$ on a isotropy irreducible flag manifold is integrable. We also give an example of a iacs on a flag manifold with two isotropy summand which is not integrable.
In section 9 we present a study of the Kähler form $\Omega$ on ${\mathbb{F}_{\Theta}}$ associated to the structure $(g, J)$, which extends the results obtained for full flag manifolds obtained in [@SM; @N]. In particular, we obtain that classes almost Kähler and Kähler coincide on ${\mathbb{F}_{\Theta}}$.
In the last section we show how two structures $(g, J)$ and $(g^0, J^0)$ can be equivalent by means of the action of the Weyl group of $G$ on these structures. Equivalence here means that $(g, J)$ and $(g^0, J^0)$ are associated by a bi-homorphic map ${\mathbb{F}_{\Theta}}\longrightarrow{\mathbb{F}_{\Theta}}$. Thus equivalent structures share the same class of invariant almost Hermitian structures.
Flag manifolds {#sec:1}
==============
In this section we set up our notation and present the standard theory of partial (or generalized) flag manifolds associated with semisimple Lie algebras (see for example [@SM; @N] for similar description).
Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $G$ a Lie group with Lie algebra $\mathfrak{g}$. Consider a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. We denote by $R$ the system of roots of $(\mathfrak{g},\mathfrak{h})$. A root $\alpha\in R$ is a linear functional on $\mathfrak{g}$. It determines uniquely an element $H_{\alpha}\in\mathfrak{h}$ by the Riesz representation $\alpha(X)=B(
X,H_\alpha)$, $X\in\mathfrak{g}$, with respect to the Killing form $B(\cdot,\cdot)$ of ${\mathfrak}{g}$. The Lie algebra $\mathfrak{g}$ has the following decomposition $$\mathfrak{g}=\mathfrak{h}\oplus\sum_{\alpha\in R}\mathfrak{g}_{\alpha}$$ where $\mathfrak{g}_{\alpha}$ is the one-dimensional root space corresponding to $\alpha$. Besides the eigenvectors $E_{\alpha}\in\mathfrak{g}_{\alpha}$ satisfy the following equation
$$\left[ E_{\alpha},E_{-\alpha}\right] =B\left(
E_{\alpha},E_{-\alpha}\right) H_{\alpha}. \label{igualdade}$$
We fix a system $\Sigma$ of simple roots of $R$ and denote by $R^+$ and $R^{-}$ the corresponding set of positive and negative roots, respectively. Let $\Theta\subset\Sigma$ be a subset, define $$\begin{aligned}
R_{\Theta}:=\langle\Theta\rangle\cap R \quad \text{and}\quad
{R_{\Theta}}^\pm:=\langle\Theta\rangle\cap R^{\pm}.\nonumber\end{aligned}$$ We denote by $R_M:=R\setminus{R_{\Theta}}$ the complementary set of roots. In general, $R_M$ is not a root system.
\[ex1\] In the Lie algebras $A_3$, $\Sigma=\{\alpha_1,\alpha_2, \alpha_3\}$. If $\Theta=\{\alpha_2,\alpha_3\}$ then $R_M=\{\pm\alpha_1,\pm(\alpha_1+\alpha_2), \pm(\alpha_1+\alpha_2+\alpha_3)\}$ and it is not a root system.
Recall that $R$ is irreducible if and only if $R$ (or, equivalently, $\Sigma$) cannot be partitioned into two proper, orthogonal subsets (see [@H]). Equivalently, $R$ is irreducible iff the Dynkin diagram of $\Sigma$ is connected. From the classification of the connected Dynkin diagrams, if ${\mathfrak}{g}$ is a simple Lie algebra then its root system $R$ is irreducible.
\[ponte\] Let $R$ be a irreducible root system with a simple root system $\Sigma$. Let $\Theta\subset \Sigma$ be a subset such that $\Sigma-\Theta=\Delta_1\cup\Delta_2$ is decomposed into two proper, orthogonal subsets, $\Delta_1$ and $ \Delta_2 $. Then there exists a root $\beta \in R_M$ of the form $$\beta=\alpha^1+\phi+\alpha^2$$ where $\alpha^i\in \Delta_i$, $i=1,2$, and $\phi\in R_{\Theta}$.
Since the Dynkin diagram of $\Sigma$ is connected, there exists $\alpha^i\in \Delta_i$, ($i=1, 2$) which $(\alpha^1,\phi_1)\neq 0$ and $(\alpha^2,\phi_k)\neq 0$ for some $\phi_1,\phi_k\in \Theta$, where $(\alpha^i,\phi_j)=B(H_{\alpha^i},H_{\phi_j})$. Note that $(\alpha^1,\phi_1)< 0$ and $(\alpha^2,\phi_k)< 0$ then $\alpha^1+\phi_1,\alpha^2+\phi_k\in R$, because $\alpha^1,\alpha^2,\phi_1$ and $\phi_k$ are simple roots (see [@H], Lemma 9.4).
From the connectedness of the Dynkin diagram of $\Theta$ we can see that there exist $\phi_2,\phi_{k-1}\in \Theta$ which satisfy $$(\phi_1,\phi_2)<0\quad\text{and} \quad (\alpha^1,\phi_2)=0 \quad \text{then}\quad (\alpha^1+\phi_1,\phi_2)<0 \quad \text{and}\quad \alpha^1+\phi_1+\phi_2\in R$$ $$(\phi_{k-1},\phi_k)<0\quad\text{and} \quad (\phi_{k-1},\alpha^2)=0 \quad \text{then}\quad (\phi_{k-1}, \phi_k+\alpha^2)<0 \quad \text{and}\quad \phi_{k-1}+ \phi_k+\alpha^2\in R.$$ Continuing with this process we obtain $\phi_3,\dots,\phi_{k-2}\in \Theta$ such that $$(\alpha^1+\phi_1+\phi_2)+(\phi_3+\dots+\phi_{k-2})+(\phi_{k-1}+\phi_k+\alpha^2)\in R.$$ Note that $\phi=\phi_1+\dots+\phi_k\in R_{\Theta}$ and $\alpha^1+\phi+\alpha^2\in R\setminus R_{\Theta}=R_M$.
Next we discuss the flag manifolds as homogeneous spaces. Note that $${{\mathfrak}{p}}_{\Theta}:={\mathfrak}{h}\oplus\sum_{\alpha\in R^+}{\mathfrak}{g}_{\alpha}\oplus\sum_{\alpha\in R_{\Theta}^-}{\mathfrak}{g}_{\alpha}$$ is a parabolic subalgebra, since it contains the Borel subalgebra ${\mathfrak}{b}^+={\mathfrak}{h}\oplus\sum\limits_{\alpha\in R^+}{\mathfrak}{g}_{\alpha}$.
The partial flag manifold determined by the choice $\Theta\subset R$ is the homogeneous space ${\mathbb{F}_{\Theta}}=G/P_{\Theta}$, where $P_{\Theta}$ is the normalizer of ${\mathfrak}{p}_{\Theta}$ in $G$. In the special case $\Theta=\emptyset$, we obtain the *full* (or maximal) flag manifold $\mathbb{F}=G/B$ associated with $R$, where $B$ is the normalizer of the Borel subalgebra ${\mathfrak}{b}^+={\mathfrak}{h}\oplus\sum\limits_{\alpha\in R^+}{\mathfrak}{g}_{\alpha}$ in $G$.
Now we will see the construction of any flag manifold as the quotient $U/K_{\Theta}$ of a semisimple compact Lie group $U\subset G$ modulo the centralizer $K_{\Theta}$ of a torus in $U$. We fix once and for all a Weyl base of ${\mathfrak}{g}$ which amounts to giving $X_\alpha\in{\mathfrak}{g_{\alpha}}$, $H_{\alpha}\in{\mathfrak}{h}$ with $\alpha\in R$, with the standard properties: $$\label{base weyl}
\begin{tabular}
[c]{lll}
$B( X_{\alpha},X_{\beta}) =\left\{
\begin{array}
[c]{cc}1, & \alpha+\beta=0,\\
0, & \text{otherwise};
\end{array}
\right. $ &\hspace{-.5cm} & $\left[ X_{\alpha},X_{\beta}\right] =\left\{
\begin{array}
[c]{cc}H_{\alpha}\in\mathfrak{h}, & \alpha+\beta=0,\\
n_{\alpha,\beta}X_{\alpha+\beta}, & \alpha+\beta\in R,\\
0, & \text{otherwise.}\end{array}
\right. $
\end{tabular}$$ The real constants $n_{\alpha,\beta}$ are non-zero if and only if $\alpha+\beta\in R$. Besides that it satisfies $$\left\{
\begin{array}
[c]{cc} n_{\alpha,\beta}=-n_{-\alpha,-\beta}=-n_{\beta,\alpha}& \\
\hspace*{-1cm} n_{\alpha,\beta}=n_{\beta,\gamma}=n_{\gamma,\alpha},&\mbox{if}\quad\alpha+\beta+\gamma=0.
\end{array}
\right.$$
We consider the following two-dimensional real spaces ${\mathfrak}{u}_{\alpha}=\text{span}_\mathbb{R}\{A_{\alpha},iS_{\alpha}\}$, where $\hspace{0.3cm}A_{\alpha}=X_{\alpha}-X_{-\alpha}$ and $\hspace{0.3cm}S_{\alpha}=X_{\alpha}+X_{-\alpha}$, with $\alpha\in R^{+}$. Then the real Lie algebra ${\mathfrak}{u}=i{\mathfrak}{h}_\mathbb{R}\oplus\sum{\mathfrak}{u}_{\alpha}$, with $\alpha\in R^{+},$ is a compact real form of ${\mathfrak}{g}$, where ${\mathfrak}{h}_\mathbb{R}$ denotes the real space vector spanned by $\{H_\alpha; \alpha\in R\}$.
Let $U=\exp{\mathfrak}{u}$ be the compact real form of $G$ corresponding to ${\mathfrak}{u}$. By the restriction of the action of $G$ on ${\mathbb{F}_{\Theta}}$, we can see that $U$ acts transitively on ${\mathbb{F}_{\Theta}}$, then ${\mathbb{F}_{\Theta}}=U/K_{\Theta}$, where $K_{\Theta}=P_{\Theta}\cap U$. The Lie algebra ${\mathfrak}{k}_{\Theta}$ of $K_\Theta$ is the set of fixed points of the conjugation $\tau\colon X_{\alpha}\mapsto-X_{-\alpha}$ of ${\mathfrak}{g}$ restricted to ${{\mathfrak}{p}}_{\Theta}$ $${\mathfrak}{k}_{\Theta}={\mathfrak}{u}\cap{{\mathfrak}{p}}_{\Theta}=i{\mathfrak}{h}_\mathbb{R}\oplus\sum_{\alpha\in{R_{\Theta}}^+}{\mathfrak}{u}_{\alpha}.$$
The tangent space of ${\mathbb{F}_{\Theta}}=U/K_{\Theta}$ at the origin $o=eK_{\Theta}$ can be identified with the orthogonal complement (with respect to the Killing form) of ${\mathfrak}{k}_{\Theta}$ in ${\mathfrak}{u}$ $$T_o{\mathbb{F}_{\Theta}}={{\mathfrak}{m}}=\sum\limits_{\alpha\in R_{M}^{+}}{\mathfrak}{u}_{\alpha}$$ with $R_{M}^{+}=R_{M}\cap R^+$. Thus we have ${\mathfrak}{u}={\mathfrak}{k}_{\Theta}\oplus{\mathfrak}{m}$.
It is known that there is a one-to-one correspondence between flag manifolds ${\mathbb{F}_{\Theta}}$ of a compact semisimple Lie group (up to isomorphism) and painted Dynkin diagrams (up to equivalence) (cf. [@Alek; @e; @Perol] or [@Arv; @art2]). Next we present a briefly description of this correspondence.
Two flag manifolds $\mathbb{F}=G/K$ and $\mathbb{F}^{\prime}=G/K^{\prime}$ are *equivalent* if there exist an automorphism $\Phi\in Aut\left( G\right)$ such that $\Phi\left( K\right) =K^{\prime}$. This automorphism $\Phi$ induce a diffeomorphism $\ \widetilde{\Phi}\colon \mathbb{F}\longrightarrow
\mathbb{F}^{\prime}$ defined by $\ \widetilde{\Phi}\left(
gK\right) =\Phi\left( g\right) K^{\prime}$.
Let ${\mathbb{F}_{\Theta}}$ be a flag manifold determined by $\Theta \subset \Sigma$. Let $\Gamma=\Gamma(\Sigma)$ be the Dynkin diagram of the root system $R$. In $\Gamma$ we obtain the painted Dynkin diagram of ${\mathbb{F}_{\Theta}}$ by painting the nodes $\Sigma - \Theta$ in black. Thus the simple roots $\Theta$ correspond to the subdiagram of white nodes.
The flag manifold ${\mathbb{F}_{\Theta}}=\frac{SU(5)}{S(U(3)\times
U(2))}$ corresponds to the painted diagram
[2]{}
(-1,0) node\[anchor=east\] ;
\(1) at (0,0) ; (2) at (1,0) ; (3) at (2,0) ; (4) at (3,0) ; (1) edge\[sedge\] (2) (2) edge\[sedge\] (3) (3) edge\[sedge\] (4) ;
Note that maximal flag manifolds correspond to Dynkin diagrams with all roots painted in black.
Conversely, let $\Gamma$ be the Dynkin diagram of a semisimple Lie algebra $\mathfrak{g}$. Suppose that some nodes in $\Gamma$ are painted in black. Then $${\mathfrak}{k}_{\Theta}= \mathfrak{u}(1)\oplus\cdots\oplus\mathfrak{u}(1)\oplus{\mathfrak}{k}_{\Theta}^{\prime}$$ where the semisimple part of ${\mathfrak}{k}_{\Theta}$, denoted by ${\mathfrak}{k}_{\Theta}^{\prime}$, is yielded by the set of white nodes together with the connected line between them, and each black nodes in $\Gamma$ yields a $\mathfrak{u}(1)$ component.
The painted Dynkin diagram of the Lie algebra $\mathfrak{e_8}$ below
[2]{}
(-1,0) node\[anchor=east\] ;
\(1) at (0,0) ; (2) at (1,0) ; (3) at (2,0) ; (4) at (3,0) ; (5) at (4,0) ; (6) at (4,1) ; (7) at (5,0) ; (8) at (6,0) ; (1) edge\[sedge\] (2) (2) edge\[sedge\] (3) (3) edge\[sedge\] (4) (4) edge\[sedge\] (5) (5) edge\[sedge\] (6) (5) edge\[sedge\] (7) (7) edge\[sedge\] (8) ;
corresponds to the flag manifold $\dfrac{E_8}{U(1)^2\times SU(3)\times SO(8)}$.
Connectedness by triples zero sum
=================================
In this section we define connectedness by triples zero sum for a general set of linear functional. In the case of root system we prove that this property is equivalenty to the connectedness of the Dynkin diagram. We will show that the study of this notion on certain sets of linear functional is the key for a complete classification of $\mathcal{G}_1$ structures on ${\mathbb{F}_{\Theta}}$.
Let $\Gamma$ be a non empty set of linear functional. Two linear functinal $\alpha, \beta$ in $\Gamma$ such that $\alpha\neq \pm \beta$ are connected by triples zero sum (in $\Gamma$), if there exists a chain of triples $T_{i,j,k}^p=\{\gamma_i,\gamma_j,\gamma_k\}\subset \Gamma$ with $\gamma_i+\gamma_j+\gamma_k=0$ such that $\pm\alpha\in T_{i,j,k}^1$, $\pm\beta\in T_{i,j,k}^n$ and $T_{i,j,k}^p\cap T_{i^{\prime},j^{\prime},k^{\prime}}^{p+1}\neq \emptyset$ with $1\leq p \leq n-1$. We will say that $\Gamma$ is connected by triples zero sum (or connected by tzs) if any pair $\alpha, \beta \in \Gamma$ are connected by triples zero sum.
If we adopt that each linear functional $\alpha \in \Gamma$ is connected by tzs with itself, then connectedness by tzs becomes a equivalence relation in set of linear functional $\Gamma$.
Consider the Lie algebras of type $A_5$. Its Dynkin diagram is given by
[2]{}
(-1,0) node\[anchor=east\] [$A_5$]{};
\(1) at (0,0) ; (2) at (1,0) ; (3) at (2,0) ; (4) at (3,0) ; (5) at (4,0) ;
\(1) edge\[sedge\] (2) (2) edge\[sedge\] (3) (3) edge\[sedge\] (4) (4) edge\[sedge\] (5);
If $\Gamma$ is the set of roots generated by $\alpha_1, \alpha_3$ and $\alpha_4$ then $\Gamma$ is not connected by tzs.
\[tzs for root system\] All irreducible root system is connected by tzs.
Let $R$ be a irreducible root system with basis $\Sigma=\{\alpha_1,\dots,\alpha_n\}$ and consider the correspondent set $R^+$ of positive roots. Since $R$ is irreducible, $(\alpha_1,\alpha_j)\neq 0$ for some $2\leq j\leq n$. For simplicity, let $(\alpha_1,\alpha_2)\neq 0$. The roots $\alpha_1,\alpha_2$ are simple then $(\alpha_1,\alpha_2)< 0$ and $\alpha_1+\alpha_2\in R$ ( see [@H], Lemma 9.4). Then $\alpha_1$ and $\alpha_2$ are connected by the triple zero sum $\{\alpha_1,\alpha_2,-(\alpha_1+\alpha_2)\}$. Applying the same idea to $\alpha_2$, we conclude that $\alpha_2$ and $\alpha_3$ are connected by the tzs $\{\alpha_2,\alpha_3,-(\alpha_2+\alpha_3)\}$. (If $(\alpha_2,\alpha_j)= 0$ for $3\leq j\leq n$ then $(\alpha_1,\alpha_j)\neq 0$ for some $3\leq j\leq n$.) Thus $\alpha_1$ and $\alpha_3$ are connected by the triple zero sum. Continuing with this process we obtain that any pair $\alpha_i, \alpha_j$ are connected by the tzs.
Now let $\beta\in R$ not simple. It is easy to see that $\beta\pm\alpha_i\in R$ for some simple root $\alpha_i$. Then $\beta$ and $\alpha_i$ are connected by tzs. Thus every root in $R$ is connected by triple zero sum to some simple root and any pair of simple roots are connected by tzs, so any pair of roots $\beta, \gamma \in R$ are connected by tzs.
If the Dynkin diagram of $R$ is not connected then $R$ cannot be connected by tzs. Since a root system $R$ is irreducible if and only if its Dynkin diagram is connected, we obtain:
The Dynkin diagram of $R$ is connected iff $R$ is connected by tzs. In particular, the Dynkin diagram of semisimple Lie algebras are connected by tzs.
t-roots
=======
In order, to describe invariant tensors and to set the notation, we present the fundamental theory on the isotropy representation for flag manifolds (see, for example, [@Alek; @e; @Perol] or [@Sie]).
It is known that ${\mathbb{F}_{\Theta}}$ is a reductive homogeneous space, this means that the adjoint representation of ${\mathfrak}{k}_{\Theta}$ (and $K_{\Theta}$) leaves ${\mathfrak}{m}$ invariant, i.e. ${\mbox{ad}}({\mathfrak}{k}_{\Theta}){\mathfrak}{m}\subset{\mathfrak}{m}$. Thus we can decompose ${\mathfrak}{m}$ into a sum of irreducible ${\mbox{ad}}({\mathfrak}{k}_{\Theta})$ submodules ${\mathfrak}{m}_i$ of the module ${\mathfrak}{m}$: $${{\mathfrak}{m}}= {{\mathfrak}{m}}_1\oplus\cdots\oplus{{\mathfrak}{m}}_s.$$
Now we will see how to obtain each irreducible ${\mbox{ad}}({\mathfrak}{k}_{\Theta})$ submodules ${{\mathfrak}{m}}_i$. By complexifying the Lie algebra of $K_{\Theta}$ we obtain $${\mathfrak}{k}_{\Theta}^{\mathbb{C}}={\mathfrak}{h}\oplus\sum_{\alpha\in{R_{\Theta}}}{\mathfrak}{g}_{\alpha}.$$ The adjoint representation ${\mbox{ad}}({\mathfrak}{k}_{\Theta}^{\mathbb{C}})$ of ${\mathfrak}{k}_{\Theta}^{\mathbb{C}}$ leaves the complex tangent space ${{\mathfrak}{m}}^{\mathbb{C}}$ invariant. Let $${\mathfrak}{t}:=Z({\mathfrak}{k}_{\Theta}^\mathbb{C})\cap i{\mathfrak}{h}_\mathbb{R}$$ the real form of the center $Z({\mathfrak}{k}_{\Theta}^\mathbb{C})$. It is easy to see that $\mathfrak{t}$ is a subalgebra of $i{\mathfrak}{h}_\mathbb{R}$ orthogonal (with respect to the Killing form on $i{\mathfrak}{h}_\mathbb{R}$) to $H_{\alpha}$, for all $\alpha$ in ${R_{\Theta}}$, i.e., $${\mathfrak}{t}=\{H\in i{\mathfrak}{h}_{{\mathbb R}}:\alpha(H)=0,\,\text{for all} \,\alpha \in {R_{\Theta}}\}.$$
Let $i{\mathfrak}{h}_{\mathbb R}^{\ast}$ and ${\mathfrak}{t}^{\ast}$ be the dual vector space of $i{\mathfrak}{h}_{\mathbb R}$ and ${\mathfrak}{t}$, respectively, and consider the map $k\colon i{\mathfrak}{h}_{\mathbb R}^{\ast}\longrightarrow{\mathfrak}{t}^{\ast}$ given by $k(\alpha)=\alpha|_{{\mathfrak}{t}}$. The linear functional of $R_t:=k(R_{M})$ are called *t-roots*. Denote by $R_{{\mathfrak}{t}}^+=k(R_M^+)$ the set of positive t-roots. Note that the map $k$ is not a 1-1 correspondence in general.
A basis of real space ${\mathfrak}{t}^{\ast}$ can be obtained as following: we fix a basis $\Sigma=\{\alpha_1, \dots \alpha_k,\beta_1, \dots, \beta_r \}$ of $R$ associated to ${\mathbb{F}_{\Theta}}$ where $\Theta=\{\alpha_1, \dots \alpha_k\}$ is a basis of $R_\Theta$ and $\Sigma_M=\Sigma\setminus\Theta=\{\beta_1, \dots, \beta_r \}$ . Let $\bar{\beta_1},\dots, \bar{\beta_r}$ be the fundamental weights corresponding to the simple roots of $\Sigma_M$, defined by $$\left< \bar{\beta_i},\beta_j\right>\colon =\dfrac{2(\bar{\beta_i},\beta_j)}{(\beta_j, \beta_j)}=\delta_{ij},\quad \left<\bar{\beta_i}, \alpha_j\right>=0,$$ where $(\alpha, \beta)=B(H_\alpha, H_\beta)$. Then $\{\bar{\beta_1},\dots, \bar{\beta_r}\}$ is a basis of ${\mathfrak}{t}^{\ast}$. Indeed, let $\lambda$ in ${\mathfrak}{t}^{\ast}$ be a t-root, since the Killing form $(\cdot,\cdot)$ on $i{\mathfrak}{h}_\mathbb{R}$ is non degenerated and negative definite, each t-root $\lambda$ determines $H_\lambda\in{\mathfrak}{t}\subset i{\mathfrak}{h}_\mathbb{R}$, such that $\beta(H_\lambda)=(\lambda,\beta)$, for each root $\beta$ in $R$, by the Riez representation. Then $0=\alpha_i(H_{\lambda})=( \lambda,\alpha_i)$, for $i=1,\dots,k$. Now let $m_j=\left< \lambda,\beta_j\right>$ then $0=\left< \lambda-\sum m_j\bar{\beta_j},\gamma\right>$, for each simple root $\gamma \in \Sigma$, which implies that $(\lambda-\sum m_j\bar{\beta_j},\gamma)=0$ as well, or that $\lambda=\sum m_j\bar{\beta_j}$.
According to [@Sie], there exists a 1-1 correspondence between positive t-roots and irreducible submodules of the adjoint representation of ${\mathfrak}{k}_{\Theta}$. This correspondence is given by $$\xi\longleftrightarrow{\mathfrak}{m}_{\xi}=\sum_{k(\alpha)=\xi}{\mathfrak}{u}_{\alpha}$$ with $\xi\in R_{{\mathfrak}{t}}^+$. Besides these submodules are inequivalents. Hence the tangent space can be decomposed as follows $$\label{decomp}
{{\mathfrak}{m}}={{\mathfrak}{m}}_{\xi_1}\oplus\cdots\oplus{{\mathfrak}{m}}_{\xi_s}$$ where $R_t^+=\lbrace \xi_1, \ldots, \xi_s\rbrace$.
Following the idea of section 3, it is natural to ask if $R_t$ is connected by tzs. It is easy to see that if $R_M$ is connected by tzs then $R_t$ is connected by tzs. But the reciprocal is not true, as we can see in example \[ex1\], where ${R_{{\mathfrak}{t}}}=\{k(\alpha_1)\}$ and $R_M=\{\pm\alpha_1,\pm(\alpha_1+\alpha_2), \pm(\alpha_1+\alpha_2+\alpha_3)\}$. The next result shows that $R_t$ inherits the connectivity property by tzs from the root system $R$.
\[connected tzs t-roots\] The set of t-roots $R_t$ is connected by tzs.
Consider the Dynkin diagram of $\Sigma$. Let $N$ be the amount of connected components of the Dynkin diagram of $\Sigma-\Theta$. We will prove by mathematical induction on $N$. If $N=1$ then the diagram of $\Sigma-\Theta$ is connected, following the idea of Lemma \[tzs for root system\] proof, we can see that if $\alpha,\beta\in \Sigma-\Theta$ then $\alpha$ and $\beta$ are connected by tzs. Besides $\gamma\in R_M$ is not a simple root then $\gamma$ is connected by tzs to some $\alpha\in\Sigma-\Theta$. Thus any pair of roots in $R_M$ is connected by tzs. Then $R_t$ is connected by tzs.
Now suppose that $N=k+1$ then $\Sigma-\Theta=\Delta_1\cup\cdots\cup\Delta_k\cup\Delta_{k+1}$ where each connected component of the Dynkin diagram of $\Sigma-\Theta$ corresponds to the Dynkin diagram $\Delta_i$, for some $i=1, \dots,k+1$. From Lemma \[ponte\], there exist a complementary root $\beta\in R_M$ such that $\beta=\alpha^1+\phi+\alpha^2$, with $\alpha^1\in\Delta_k$, $\alpha^2\in\Delta_{k+1}$ and $\phi\in R_{\Theta}$. Then the correspondent t-root is $k(\beta)=k(\alpha^1)+k(\alpha^2)$ and $\left\langle \bar{\Delta}_k \right\rangle $ and $\left\langle \bar{\Delta}_{k+1} \right\rangle $ are connected by tzs, where $\left\langle \bar{\Delta}_i \right\rangle $ denotes the t-roots generated by $k(\Delta_i)$. By induction hypothesis, $$\{\left\langle \bar{\Delta}_{1} \right\rangle\oplus\cdots\oplus\left\langle \bar{\Delta}_{k} \right\rangle\}\cap{R_{{\mathfrak}{t}}}$$ is connected by tzs. So $${R_{{\mathfrak}{t}}}=\{\left\langle \bar{\Delta}_{1} \right\rangle\oplus\cdots\oplus\left\langle \bar{\Delta}_{k} \right\rangle\oplus\left\langle \bar{\Delta}_{k+1}\right\rangle \}\cap{R_{{\mathfrak}{t}}}$$ is connected by tzs.
Invariant Almost Hermitian Structures on ${\mathbb{F}_{\Theta}}$
================================================================
An *almost complex structure* on ${\mathbb{F}_{\Theta}}=U/K_{\Theta}$ is a tensor field of type $(1,1)$ that corresponds each $x\in {\mathbb{F}_{\Theta}}$ to a linear endomorphism $J_{x}\colon T_{x}{\mathbb{F}_{\Theta}}\rightarrow T_{x}{\mathbb{F}_{\Theta}}$ which satisfies $J_{x}^{2}=-Id$. The almost complex structure on ${\mathbb{F}_{\Theta}}$ is invariant (or $U$-invariant) if $$d \mathit{u}_{x}\circ J_{x}=J_{\mathit{u}x}\circ d\mathit{u}_{x}$$ for all $\mathit{u}\in U$. An invariant almost complex structure (*iacs* from now) is determined by a linear endomorphism $J\colon\mathfrak{m\longrightarrow}\mathfrak{m}$, which satisfies $J^{2}=-Id$ and commutes with the adjoint action of $K_\Theta$ on $\mathfrak{u}$, that is, $$Ad(k) J=J Ad(k), \quad \text{for all},\quad k\in K_\Theta,$$ or, equivalently $$ad(L)J=Jad(L), \quad \text{ for all}, \quad L\in \mathfrak{k}_\Theta$$ (cf. [@Kob]).
As it is common in the literature, we will use the same letter $J$ to denote its extension to the complexi-\
fication $\mathfrak{m}^{\mathbb{C}}$. Since $J^{2}=-Id$, its eigenvalues are $i$ and $-i$ and the correspondents eigenspaces are denoted by $$T^{(1,0)}_{o}{\mathbb{F}_{\Theta}}=\left\{X\in T_{o}\mathbb{F}^{\mathbb{C}}:
JX=iX\right\}
\quad \text{and} \quad
T^{(0,1)}_{o}{\mathbb{F}_{\Theta}}=\left\{X\in T_{o}\mathbb{F}^{\mathbb{C}}:
JX=-iX\right\}.$$
Thus $$\mathfrak{m}^{\mathbb{C}}=T_{o}{\mathbb{F}_{\Theta}}^{\mathbb{C}}=T^{(1,0)}_{o}{\mathbb{F}_{\Theta}}\oplus
T^{(0,1)}_{o}{\mathbb{F}_{\Theta}}.$$ The eigenvectors with eigenvalue $+i$ (resp. $-i$) are called *of type (1,0)* (resp. *of type (0,1)* ).
The next result is known for maximal flag manifolds (see [@SM; @N]). A similar result for generalized flag manifolds can be found in [@SM; @R]. Our work here is to classify iacs in terms of t-roots.
\[iacs\] Let ${\mathbb{F}_{\Theta}}$ be a generalized flag manifold and $R_t$ the correspondent set of t-roots. Then any iacs $J$ on ${\mathbb{F}_{\Theta}}$ is determined by a set of signals $\{\varepsilon_{\delta},\delta\in R_t\}$, where $\varepsilon_{\delta}=\pm 1$, satisfying $\varepsilon_{\delta}=-\varepsilon_{-\delta}$ for all $\delta\in R_t$. In particular, there exist $2^{|R_t^{+}|}$ iacs on ${\mathbb{F}_{\Theta}}$ and each iacs $J$ is determined by exactly $|R_t^{+}|$ signals.
Let $J$ be an iacs on ${\mathbb{F}_{\Theta}}$ and consider its complexification on $\mathfrak{m}^{\mathbb{C}}$. By the invariance of $J$, if $H\in\mathfrak{h}^\mathbb{C}$ it follows $ad(H)JX_{\alpha}=\alpha(H)JX_{\alpha}$, then $J(\mathfrak{g}_{\alpha}^{\mathbb{C}})=\mathfrak{g}_{\alpha}^{\mathbb{C}}$ with $\alpha\in R$. Since the eigenvalues of $J$ are $\pm i$ and the eigenvectors in $\mathfrak{m}^{\mathbb{C}}$ are $X_{\alpha}$, $\alpha\in R$, we can write $JX_{\alpha}=i\varepsilon_{\alpha}X_{\alpha}$, with $\varepsilon_{\alpha}=\pm 1$.
Computing $J$ on the basis $\{A_\alpha, iS_\alpha\}$ of the real vector space $\mathfrak{m}$ and using that $X_{-\alpha}=\frac{1}{2}\left(-i(iS_{\alpha})-A_{\alpha}\right)$ we obtain
$$i\varepsilon_{-\alpha}\frac{1}{2}\left(-i(iS_{\alpha})-A_{\alpha}\right)=i\varepsilon_{-\alpha}X_{-\alpha}
=
\frac{1}{2}\left(-iJ(iS_{\alpha})-JA_{\alpha}\right).$$ Then $$J(iS_{\alpha})=\varepsilon_{-\alpha}A_{\alpha} \quad\text{and}\quad JA_{\alpha}=-\varepsilon_{-\alpha}(iS_{\alpha}),$$ thus, using $A_{\alpha}=-A_{-\alpha}$ and $iS_{\alpha}=iS_{-\alpha}$ it follows $$-\varepsilon_{-\alpha}(iS_{\alpha})=JA_{\alpha}=-JA_{-\alpha}=\varepsilon_{\alpha}(iS_{-\alpha})=\varepsilon_{\alpha}(iS_{\alpha})$$ so $-\varepsilon_{-\alpha}=\varepsilon_{\alpha}$ for each $\alpha$ in $R$.
Now consider the decomposition of the complex tangent space $$\mathfrak{m}^{\mathbb{C}}=\mathfrak{m}^{\mathbb{C}}_{1}\oplus\mathfrak{m}^{\mathbb{C}}_{2}\oplus\cdots\oplus\mathfrak{m}^{\mathbb{C}}
_{2s}$$ in irreducible and inequivalent $\mathfrak{k}_\Theta^{\mathbb{C}}$-submodules determined by the t-roots $\delta_{1},-\delta_{1},\dots,-\delta_{s},\delta_{s}$, where $$\mathfrak{m}^{\mathbb{C}}_{j}=\mathfrak{m}^{\mathbb{C}}_{\delta_{j}}=\sum_{\alpha\in R,
k(\alpha)=\delta_{j}}\mathbb{C}X_{\alpha}.$$ Since $J$ comutes with $ad(L)$, for all $L\in\mathfrak{k}_\Theta^{\mathbb{C}}$, then we see that $J(\mathfrak{m}^{\mathbb{C}}_{\delta_{j}})=\mathfrak{m}^{\mathbb{C}}_{\delta_{j}}$, for all $\delta_{j}\in R_t$. Then by Schur Lemma $$J =
i\varepsilon_{1}Id|_{\mathfrak{m}^{\mathbb{C}}_{1}}\oplus\cdots\oplus
i\varepsilon_{s}Id|_{\mathfrak{m}^{\mathbb{C}}_{s}}$$ this means that, if $\alpha,\beta\in R$ are such that $k(\alpha)=k(\beta)=\delta\in R_t$ then $\varepsilon_{\alpha}=\varepsilon_{\beta}=\varepsilon_{\delta}$.
Finally, given a t-root $\delta$ we obtain $$\varepsilon_{\delta}=\varepsilon_{\alpha}=-\varepsilon_{-\alpha}=-\varepsilon_{-\delta}$$ where $\alpha$ is any root in $R$ such that $k(\alpha)=\delta$.
An iacs $J$ on ${\mathbb{F}_{\Theta}}$ is called an *invariant complex structure* (or *integrable complex struture*) if its Nijenhuis tensor, defined by $$-\frac{1}{2}N(X,Y)=-\left[JX,JY\right]+\left[X,Y\right]+J\left[X,JY\right]+J\left[JX,Y\right], \quad X,Y\in\mathfrak{m}.$$ is zero, (cf. [@Kob]). In the next result we obtain the necessary and suficient condition of integrability of $J$ in terms of t-roots.
\[J complex\] An iacs $J=\{\varepsilon_{\delta},\delta\in R_t\}$ is a complex structure if and only if $\varepsilon_{\delta}\varepsilon_{\eta}+1-\varepsilon_{\delta+\eta}(\varepsilon_{\delta}+\varepsilon_{\eta})=0$, whenever $\delta+\eta\in R_t$.
An easy computation of Nijenhuis tensor on the Weyl base shows that $$\label{nij}
-\dfrac{1}{2}N(X_{\alpha},X_{\beta}) =\left\{
\begin{array}
[c]{cc}0, &\quad \text{if}\quad \alpha+\beta\notin R,\\
n_{\alpha,\beta}\{\varepsilon_{\alpha}\varepsilon_{\beta}+1-\varepsilon_{\alpha+\beta}(\varepsilon_{\alpha}+\varepsilon_{\beta})\}X_{\alpha+\beta}, &\quad \text{if}\quad \alpha+\beta \in R.
\end{array}
\right.$$ Now if $k(\alpha)=\delta\in R_t$ and $k(\beta)=\eta\in R_t$ then $\varepsilon_{\alpha}=\varepsilon_{\delta}$, $\varepsilon_{\beta}=\varepsilon_{\eta}$ and $\varepsilon_{\alpha+\beta}=\varepsilon_{\delta+\eta}$, if $\alpha+\beta\in R_M$. Note that if $\alpha, \beta \in R_M$ are such that $\alpha+\beta\in R_\Theta$ then $N(X_{\alpha},X_{\beta})=0$.
\[J iso irr\] Every iacs $J$ on a isotropy irreducible flag manifold ${\mathbb{F}_{\Theta}}$, i.e. $|R_t^+|=1$, is an invariant complex structure.
The next example shows that if $|R_t^+|\geq 2$ it is easy to construct a iacs $J$ not complex.
Let ${\mathbb{F}_{\Theta}}$ be a flag manifold with two isotropy summands, i.e. the real tangent space decomposes into two irreducible submodules $\mathfrak{m}=\mathfrak{m_1}\oplus\mathfrak{m_2}$. In this case $R_t=\{\pm\delta, \pm 2\delta\}$, where $\delta\in t^*\setminus 0$, (cf. [@art; @1]). Using the Proposition \[J complex\] it is easy to see that the iacs given by $J: \varepsilon_{2\delta}=1,\varepsilon_{\delta}=-1$ is not complex.
In the other hand, there is natural manner to obtain invariant complex structures on ${\mathbb{F}_{\Theta}}$, as below. Consider the space $\mathfrak{t}$ with the collection of hyperplanes $\Gamma=\left\{H\in\mathfrak{t}:\delta(H)=0, \delta\in R_t\right\}$. The complementary set $ \mathfrak{t}\setminus\Gamma$ of the union of these hyperplanes is open and dense in $\mathfrak{t}$. Its connected components in $\mathfrak{t}$ are called a chamber in $\mathfrak{t}$ or a t-chamber. Each t-chamber is a cone in $\mathfrak{t}$ which is given by the inequalities $V=\{\delta_1>0, \dots,\delta_m>0\}$, where $\delta_i$ are t-roots corresponding to its faces.
A subset $\Sigma_t=\left\{\delta_{1},\ldots,\delta_{n}\right\}$ of t-roots is called a *basis of system $R_t$* if all vectors of $R_t$ have integer coordinates of same sign $\left(\geq 0\text{ or }\leq 0\right)$ with respect to $\Sigma_t$. Note that the basis given in Section 4 is a basis of system $R_t$.
From [@Alek; @e; @Perol] and [@BH] it is known that there exists natural one-to-one correspondence between:
1. Parabolic subalgebras $\mathfrak{p}_\Theta$ in $\mathfrak{g}$ with reductive part ${\mathfrak}{h}\oplus\sum_{\alpha\in{R_{\Theta}}}{\mathfrak}{g}_{\alpha}$;
2. Basis $\Sigma$ of the root system $R$ which contain a fixed basis $\Theta$ of the root system $R_{\Theta}$;
3. Basis of system $R_t$;
4. t-chambers;
5. A choice of positive roots $R^+_M$ in $R_M$ satisfying:
1. $R=\Theta\cup R^+_M\cup R^-_M$ (disjoint union), where $R^-_M=\{-\alpha; \alpha\in R_M^+\}$;
2. if $\alpha\in R_{\Theta}\cup R^+_M $, $\beta \in R^+_M$ with $\alpha+\beta\in R$, then $\alpha+\beta\in R^+_M$ (cf. [@Arv; @art2]).
6. Invariant complex structures on ${\mathbb{F}_{\Theta}}$, which are determined by $$\label{ics}
J X_{\pm\alpha}=\pm i X_{\pm\alpha}, \quad \text{with}\quad \alpha\in R^+_M.$$
7. Reflections $\omega$ of the Weyl group (of the root system $R$) which satisfy $\omega\Theta\subset\Sigma$.
Now we describe the invariant metrics on flag manifolds. A Riemannian invariant metric on ${\mathbb{F}_{\Theta}}$ is completely determined by a real inner product $g\left(\cdot,\cdot\right)$ on $\mathfrak{m}=T_{o}{\mathbb{F}_{\Theta}}$ which is invariant by the adjoint action of ${\mathfrak}{k}_{\Theta}$. Besides that any real inner product ${\mbox{ad}}({\mathfrak}{k}_{\Theta})$-invariant on ${\mathfrak}{m}$ has the form
$$\label{inner product}
g\left(\cdot,\cdot\right)=-\lambda_1 B\left(\cdot,\cdot\right)|_{\mathfrak{m}_{1}\times\mathfrak{m}_{1}}-\cdots
-\lambda_s B\left(\cdot,\cdot\right)|_{\mathfrak{m}_{s}\times\mathfrak{m}_{s}}$$
\
where ${\mathfrak{m}}_{i}={\mathfrak}{m}_{\xi_i}$ and $\lambda_i=\lambda_{\xi_i}>0$ with $\xi_{i}\in R^{+}_{\mathfrak{t}}$, for $i=1,\ldots,s$. So any invariant Riemannian metric on ${\mathbb{F}_{\Theta}}$ is determined by $|R_{{\mathfrak}{t}}^+|$ positive parameters. We will call an inner product defined by (\[inner product\]) as an invariant metric on ${\mathbb{F}_{\Theta}}$.
An invariant metric $g(\cdot, \cdot)$ is called [*normal*]{} if there exist a bi-invariant metric on $\mathfrak{g}$ such that the restriction to ${{\mathfrak}{m}}=\mathfrak{k}^{\perp}$ is $g(\cdot, \cdot)$. In other words, $g(\cdot, \cdot)$ is a normal metric is $$g(\cdot, \cdot)=-\lambda B\left(\cdot,\cdot\right)|_{\mathfrak{m}}\quad \text{or} \quad \lambda_1=\cdots=\lambda_s=\lambda.$$
Let ${\mathbb{F}_{\Theta}}$ be a flag manifold endowed an invariant metric $g$ and an almost complex structute $J$. Computing $g(JX,JY)$ for the Weyl basis chosen it is easy to see that $g$ is *almost Hermitian* with respect to $J$, i.e, $g(JX,JY)=g(X,Y)$. The pair $(g,J)$ is called an *invariant almost Hermitian structure* on ${\mathbb{F}_{\Theta}}$.
Kähler form
===========
Consider an invariant almost Hermitian structure $(g,J)$ on a flag manifold ${\mathbb{F}_{\Theta}}$. We denote by $\Omega=\Omega_{J,g}$ the correspondent Kähler form: $$\Omega\left(X,Y\right)=g\left(X,JY\right), \hspace{1cm}X,Y\in\mathfrak{m}.\label{forma de
kahler}$$
We also denote by $\Omega$, its natural extension to a $G$-invariant 2-form on $\mathfrak{m}^{\mathbb{C}}$. On the Weyl basis $\Omega$ is given by $$\Omega(X_{\alpha},X_{\beta})=g\left(
X_{\alpha},JX_{\beta}\right)=-i\lambda_{k(\alpha)}\varepsilon_{k(\beta)}B\left(X_{\alpha},X_{\beta}\right)=
\left\{
\begin{array}
[c]{c}-i
\varepsilon_{k(\alpha)}\lambda_{k(\alpha)}, \quad \text{
if
}\quad \beta=-\alpha, \\ \\
0, \quad \text{ otherwise},
\end{array}
\right.$$ where $k(\alpha)$, $k(\beta)$ are the correspondents t-roots to $\alpha$ and $\beta$ respectively. Then from decomposition $$\mathfrak{m}^{\mathbb{C}}=\mathfrak{m}^{\mathbb{C}}_{1}\oplus\mathfrak{m}^{\mathbb{C}}_{2}\oplus\cdots\oplus\mathfrak{m}^{\mathbb{C}}
_{2s}$$ in $\mathfrak{k}^{\mathbb{C}}$-irreducible and inequivalent submodules we obtain $$\Omega\left(\cdot,\cdot\right)=\sum_{\alpha\in R_{M}^+}-i\varepsilon_{\alpha}\lambda_{\alpha}
B\left(\cdot,\cdot\right)|_{\mathfrak{g}^{\mathbb{C}}_{\alpha}\times\mathfrak{g}^{\mathbb{C}}_{-\alpha}}
=\sum_{\delta\in R_{\mathfrak{t}}^+}-i\varepsilon_{\delta}\lambda_{\delta}\left(\sum\limits_{\substack{\alpha\in R_{M}^+\\k(\alpha)=\delta}}
B\left(\cdot,\cdot\right)|_{\mathfrak{g}^{\mathbb{C}}_{\alpha}\times\mathfrak{g}^{\mathbb{C}}_{-\alpha}}\right)
=\sum_{\delta\in R^{+}_{\mathfrak{t}}}
-i\varepsilon_{\delta}\lambda_{\delta}
B\left(\cdot,\cdot\right)|_{\mathfrak{m}^{\mathbb{C}}_{\delta}\times\mathfrak{m}^{\mathbb{C}}_{-\delta}}.$$
The exterior differential $d\Omega$ computed at $T_o{\mathbb{F}_{\Theta}}$ is $$3d\Omega(X,Y,Z)=-\Omega([X,Y],Z)+\Omega([X,Z],Y)-\Omega([Y,Z],X)$$ for $X,Y,Z\in\mathfrak{m}^{\mathbb{C}}$ (cf. [@Kob], 3.11). An easy computation on Weyl basis obtained in [@SM; @N] shows that if $\alpha,\beta,\gamma\in R_{M}$ then $$d\Omega(X_{\alpha},X_{\beta},X_{\gamma})=\left\{
\begin{array}
[c]{c}-3in_{\alpha,\beta}\left(\varepsilon_{\alpha}\lambda_{\alpha}
+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma}\right),\quad\text{
if
}\quad \alpha+\beta+\gamma=0, \\ \\
0,\quad \text{ otherwise.}
\end{array}
\right.$$
The exterior differential $d\Omega$ has a similar look in terms of t-roots. To prove it we need the following result.
([@Alek-Arv], Lemma 4)\[lema soma\] Let $\xi,\eta,\zeta$ be t-roots such that $\xi+\eta+\zeta=0$. Then there exist roots $\alpha,\beta,\gamma\in R_{M}$ with $k(\alpha)=\xi,k(\beta)=\eta,k(\gamma)=\zeta$, satisfying $\alpha+\beta+\gamma=0$.
\[12 simp\] Let $\delta,\zeta,\eta$ be t-roots then $d\Omega(\mathfrak{m}^{\mathbb{C}}_{\delta},\mathfrak{m}^{\mathbb{C}}_{\zeta},\mathfrak{m}^{\mathbb{C}}_{\eta})=\left\{0\right\}$, unless if $\delta,\zeta,\eta$ are tzs on $R_t$. In this case $$d\Omega(X,Y,Z)=-3ir\left(\varepsilon_{\delta}\lambda_{\delta}
+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}\right).$$ where $r$ is a non zero constant and $X,Y,Z$ are vectors in $\mathfrak{m}^{\mathbb{C}}_{\delta},\mathfrak{m}^{\mathbb{C}}_{\zeta}$ and $\mathfrak{m}^{\mathbb{C}}_{\eta}$, respectively.
If $\delta,\zeta,\eta$ are not tzs on $R_t$ it is easy to see that $d\Omega(\mathfrak{m}^{\mathbb{C}}_{\delta},\mathfrak{m}^{\mathbb{C}}_{\zeta},\mathfrak{m}^{\mathbb{C}}_{\eta})=\left\{0\right\}$.
On the other hand, if $\delta,\zeta,\eta$ are tzs on $R_t$ let $\alpha,\beta,\gamma$ be roots in $R_{M}$ with $k(\alpha)=\delta$, $k(\beta)=\zeta$ and $k(\gamma)=\eta$, satisfying $\alpha+\beta+\gamma=0$, according to the previous Lemma. Thus, from the description of invariant metrics and iacs, we obtain: $$d\Omega(X_{\alpha},X_{\beta},X_{\gamma})=-3in_{\alpha,\beta}\left(\varepsilon_{\alpha}\lambda_{\alpha}
+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma}\right)
=-3in_{\alpha,\beta}\left(\varepsilon_{\delta}\lambda_{\delta}
+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}\right).$$ Now note that if $\alpha',\beta',\gamma'\in R_{M}$ are such that $k(\alpha')=\delta$, $k(\beta')=\zeta$, $k(\gamma')=\eta$ and $\alpha'+\beta'+\gamma'=0$ then $$d\Omega(X_{\alpha'},X_{\beta'},X_{\gamma'})=
-3in_{\alpha',\beta'}\left(\varepsilon_{\delta}\lambda_{\delta}
+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}\right).$$ Now let $X$,$Y$ and $Z$ be non zero vector in $\mathfrak{m}^{\mathbb{C}}_{\delta},\mathfrak{m}^{\mathbb{C}}_{\zeta}$ and $\mathfrak{m}^{\mathbb{C}}_{\eta}$, respectively. Using the Weyl basis we can write $X=\sum_{\alpha}a_\alpha X_\alpha$, $Y=\sum_{\beta}b_\beta X_\beta$, $Z=\sum_{\gamma}c_\gamma X_\gamma$, where the sums is taken over all roots $\alpha$, $\beta$ and $\gamma$ in $R_M$ such that $k(\alpha)=\delta$, $k(\beta)=\zeta$ and $k(\gamma)=\eta$ satisfying $\alpha+\beta+\gamma=0$. Then, using the linearity of the exterior differential $$\begin{aligned}
d\Omega(X,Y,Z)&=&
-3i\sum_{\alpha, \beta, \gamma}a_\alpha b_\beta c_\gamma n_{\alpha,\beta}\left(\varepsilon_{\alpha}\lambda_{\alpha}
+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma}\right)\nonumber\\
&=& -3i\left(\varepsilon_{k(\alpha)}\lambda_{k(\alpha)}
+\varepsilon_{k(\beta)}\lambda_{k(\beta)}+\varepsilon_{k(\gamma)}\lambda_{k(\gamma)}\right)\sum_{\alpha, \beta, \gamma}a_\alpha b_\beta c_\gamma n_{\alpha,\beta}\nonumber\\
&=& -3ir\left(\varepsilon_{\delta}\lambda_{\delta}
+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}\right)\nonumber\\\end{aligned}$$ where $r=\sum_{\alpha, \beta, \gamma}a_\alpha b_\beta c_\gamma n_{\alpha,\beta}$. Note that $r$ is not zero because $\alpha+\beta$ is a root.
Let $J=\left\{\varepsilon_{\delta},\delta\in R_t\right\}$ be an iacs on ${\mathbb{F}_{\Theta}}$. Let $\{\delta,\zeta,\eta\}$ be a tzs of t-roots. The triple $\{\delta,\zeta,\eta\}$ is said to be
1. a $(0,3)$-triple of t-roots if $\varepsilon_{\delta}=\varepsilon_{\zeta}=\varepsilon_{\eta}$;
2. a $(1,2)$-triple of t-roots, otherwise.
The notion of $(0,3)$-triple and $(1,2)$-triple of roots in $R$ was introduced in [@SM; @N] for full flag manifolds and in [@SM; @R] for partial flag manifolds. It is easy see that a tzs $\alpha,\beta,\gamma$ in $R_M$ is a (0,3)-triple (resp. (1,2)-triple) iff $k(\alpha),k(\beta),k(\gamma)$ is a (0,3)-triple of t-roots (resp. (1,2)-triple of t-roots). In this sense, the above definition extends that one in [@SM; @N] or [@SM; @R].
\[prop ics\] Let ${\mathbb{F}_{\Theta}}$ endowed with the complex structure $J$ associated to a fixed choice of positive roots $R_M^+$ in $R_M$, then $J$ has no (0,3)-triple of t-roots. Reciprocally if an iacs $J$ has no (0,3)-triple of t-roots then $J$ is a complex structure on ${\mathbb{F}_{\Theta}}$.
Let $\{\delta, \zeta, \eta\}$ be a tzs of t-roots and $\{\alpha, \beta, \gamma\}$ a correspondent tzs in $R$, such that $k(\alpha)=\delta$, $k(\beta)=\zeta$ and $k(\gamma)=\eta$. Two of the roots $\{\alpha, \beta, \gamma\}$ are positive and the other is negative or two is negative and the other is positive, with respect to the choice of positive roots $R_M^+$ associated to $J$. Then, from (\[ics\]), the triple $\{\varepsilon_{\alpha},\varepsilon_{\beta},\varepsilon_{\gamma}\}$ has different signals. From Proposition \[iacs\], $\varepsilon_{\alpha}=\varepsilon_{\delta}$, $\varepsilon_{\beta}=\varepsilon_{\zeta}$, $\varepsilon_{\gamma}=\varepsilon_{\eta}$, then $\{\delta, \zeta, \eta\}$ is a (1,2)-triple of t-roots.
On the other hand, if $J$ has no (0,3)-triple of t-roots then from Proposition \[J complex\] it is clear that $J$ is a complex structure.
An almost Hermitian manifold $(M,g,J)$ is *(1,2)-sympletic (or quasi Kähler)* if $$d\Omega\left(X,Y,Z\right)=0$$ when one of the vectors $X,Y,Z$ is of type $(1,0)$ and the other two of type $(0,1)$.
The next result classify the quasi Kähler class on flag manifolds in terms of t-roots.
\[crit 12 simp\] An invariant almost Hermitian structure $\left(J=\left\{\varepsilon_{\delta}\right\},g=\left\{\lambda_{\delta}\right\}\right)$, $\delta\in R_t$, on ${\mathbb{F}_{\Theta}}$ is $(1,2)$-sympletic if and only if
$$\label{(1,2) criterio}
\varepsilon_{\delta}\lambda_{\delta}+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}=0$$
for every $(1,2)$-triple of t-roots $\{\delta,\zeta,\eta\}$.
Consider a $(1,2)$-triple of t-roots $\{\delta,\zeta,\eta\}$ and let $\alpha$, $\beta$ and $\gamma$ in $R_M$ such that $\alpha+\beta+\gamma=0$ and $k(\alpha)=\delta$, $k(\beta)=\zeta$ and $k(\gamma)=\eta$. Then $$\label{without 03 triples}
0=d\Omega(X_{\alpha},X_{\beta},X_{\gamma})=-3in_{\alpha,\beta}\left(\varepsilon_{\alpha}\lambda_{\alpha}
+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma}\right)
=-3in_{\alpha,\beta}\left(\varepsilon_{\delta}\lambda_{\delta}
+\varepsilon_{\zeta}\lambda_{\zeta}+\varepsilon_{\eta}\lambda_{\eta}\right).$$ Since $n_{\alpha,\beta}\neq 0$ if $\alpha+\beta+\gamma=0$, we get (\[(1,2) criterio\]).
An almost Hermitian manifold is said to be *almost Kähler* if $\Omega$ is sympletic, i.e., $d\Omega=0$. When $d\Omega=0$ and $J$ is integrable the manifold is said to be *Kähler*, (cf. [@Kob]). In the next result we used the idea performed in [@SM; @N] and [@SM; @R] for t-roots.
\[12 simp e kahler\] An invariant almost Hermitian structure on ${\mathbb{F}_{\Theta}}$ is almost Kähler if and only if it is Kähler.
Note that the a pair $(g,J)$ almost Kähler do not admits $(0,3)$-triples of t-roots. Because in this case, from (\[without 03 triples\]), we would have the following equation $$\lambda_{\delta}+\lambda_{\zeta}+\lambda_{\eta}=0$$ that is impossible since $\lambda_{\delta},\lambda_{\zeta},\lambda_{\eta}>0$. Thus the iacs $J$ admits only $(1,2)$-triples of t-roots and in this case, from Lemma \[prop ics\], $J$ is integrable and the pair $(g,J)$ is Kähler.
Of course if the pair $(g,J)$ is Kähler then it is almost Kähler.
$\mathcal{G}_1$ structures
==========================
An invariant almost Hermitian structure $(g,J)$ on ${\mathbb{F}_{\Theta}}$ is called a $\mathcal{G}_1$ structure (or $W_1\oplus W_3\oplus W_4$), according the sixteen classes in [@GH], if $$g(N(X,Y),X)=0,\quad X,Y\in \mathfrak{m}.$$
It is clear that if $J$ is a complex structure, i.e. $N(X,Y)=0$, then $(g,J)$ is a $\mathcal{G}_1$ structure for any invariant metric $g$ on ${\mathbb{F}_{\Theta}}$.
\[J for all\] Fixed a choice of positive roots $R_M^+$ in $R_M$, the correspondent complex structure $J$ on ${\mathbb{F}_{\Theta}}$ is a $\mathcal{G}_1$ structure with respect to all invariant metric $g$ on ${\mathbb{F}_{\Theta}}$.
\[G1 iso irr\] If ${\mathbb{F}_{\Theta}}$ is isotropy irreducible, then any invariant pair $(g,J)$ is a $\mathcal{G}_1$ structure.
Follows from Corollary \[J iso irr\] and the previous Theorem.
Now we study the annihilation of the tensor $g(N(X,Y),X)$ taking in account the vector roots from the Weyl basis. From (\[nij\]), if $\alpha, \beta, \gamma$ are roots in $R_M$ it easy to see that
$$\label{g1 s}
-\dfrac{1}{2}g(N(X_{\alpha},X_{\beta}),X_{\gamma})=\left\{
\begin{array}
[c]{c}\lambda_\gamma n_{\alpha,\beta}(\varepsilon_\alpha\varepsilon_\beta +\varepsilon_\alpha\varepsilon_\gamma+\varepsilon_\beta\varepsilon_\gamma+1),
\quad\text{
if
}\quad \alpha+\beta+\gamma=0, \\ \\
0, \hspace{0.5cm} \text{ otherwise},
\end{array}
\right.$$
in particular $g(N(X_\alpha, X_\beta), X_\alpha)=0$, for $\alpha, \beta \in R_M$.
Using similar arguments of [@SM; @N], in the next result we obtain a classification of $\mathcal{G}_1$ structures in terms of t-roots.
\[g1 criterio\] The pair $(g,J)$ is a $\mathcal{G}_1$ structures on ${\mathbb{F}_{\Theta}}$ if and only if $\lambda_{\delta}=\lambda_{\zeta}=\lambda_{\eta}$ if $\{\delta,\zeta,\eta\}$ is a (0,3)-triple of t-roots.
Let $\{\delta,\zeta,\eta\}$ is a tzs of t-roots and $\alpha, \beta, \gamma \in R_M$ a tzs of roots such that $k(\alpha)=\delta$, $k(\beta)=\zeta$ and $k(\gamma)=\eta$. Note that $$\begin{aligned}
-\dfrac{1}{2}\left\lbrace g(N(X_{\alpha},X_{\beta}),X_{\gamma})+g(N(X_{\gamma},X_{\beta}),X_{\alpha})\right\rbrace
&=& n_{\alpha,\beta} (\lambda_\gamma-\lambda_\alpha) (\varepsilon_\alpha\varepsilon_\beta +\varepsilon_\alpha\varepsilon_\gamma+\varepsilon_\beta\varepsilon_\gamma+1)\nonumber\\ \nonumber\\
&=& n_{\alpha,\beta} (\lambda_\eta-\lambda_\delta) (\varepsilon_\delta\varepsilon_\zeta +\varepsilon_\delta\varepsilon_\eta+\varepsilon_\zeta\varepsilon_\eta+1)\label{g1}\end{aligned}$$ where in first equality we used that $ n_{\alpha,\beta}=-n_{\gamma,\beta}$ if $\alpha+\beta+\gamma=0$. Now we note that $(\varepsilon_\delta\varepsilon_\zeta +\varepsilon_\delta\varepsilon_\eta+\varepsilon_\zeta\varepsilon_\eta+1)$ is not zero if and only if $\{\delta,\zeta,\eta\}$ is a (0,3)-triple of t-roots. Thus if $(g,J)$ is a $\mathcal{G}_1$ structure then $\lambda_\eta=\lambda_\delta$. Analogously one proves that $\lambda_\zeta=\lambda_\delta$. Reciprocally, for $X=\sum_\alpha a_\alpha X_\alpha$ with $\alpha\in R_M$, we obtain: $$g(N(X,X_\beta),X)=\sum_{\alpha\neq\gamma}a_\alpha a_\gamma\left\lbrace g(N(X_{\alpha},X_{\beta}),X_{\gamma})+g(N(X_{\gamma},X_{\beta}),X_{\alpha})\right\rbrace.$$ Then, from (\[g1\]), $g(N(X,X_\beta),X)=0$ if $\lambda_{\delta}=\lambda_{\zeta}=\lambda_{\eta}$ when $\{\delta,\zeta,\eta\}$ is a (0,3)-triple of t-roots.
As we saw, given a complex structure $J$ on ${\mathbb{F}_{\Theta}}$, for any invariant metric $g$, the pair $(g,J)$ is a $\mathcal{G}_1$ structure. Thus fixed any invariant metric $g$ on ${\mathbb{F}_{\Theta}}$, the $\mathcal{G}_1$ structures $(g,J)$ are in correspondence one-to-one with: parabolic subalgebras $\mathfrak{p}_\Theta$ in $\mathfrak{g}$; basis $\Sigma$ of the root system $R$ which contain a fixed basis $\Theta$ of the root system $R_{\Theta}$; basis of system $R_t$; t-chambers; a choice of positive roots $R^+_M$ in $R_M$ and reflections $\omega$ of the Weyl group (of the root system $R$) which satisfy $\omega\Theta\subset\Sigma$, according to Section 5.
Let $J=\{\varepsilon_\delta, \delta\in R_t\}$ be an iacs on ${\mathbb{F}_{\Theta}}$ and denote by $C(J)$ the subset of t-roots $\delta$ such that there exist $(0,3)$-triple of t-roots $\{\delta,\zeta,\eta\}$ containing $\delta$.
Given an iacs $J$ on ${\mathbb{F}_{\Theta}}$, a pair $(g,J)$ is a $\mathcal{G}_1$ struture if and only if $g=\{\lambda_\delta\}$ is constant on $C(J)$.
Follows immediately from Lemma \[g1 criterio\].
Analogously, let $g=\{\lambda_\delta, \delta\in R_t\}$ be an invariant metric on ${\mathbb{F}_{\Theta}}$ and denote by $C(g)$ the subset of t-roots $\delta$ such that there exist tzs $\{\delta,\zeta,\eta\}$ of t-roots containing $\delta$ and satisfying $\lambda_{\delta}=\lambda_{\zeta}=\lambda_{\eta}$.
A pair $(g,J)$ is a $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$ if and only if $C(J)\subset C(g)$.
Suppose that $(g,J)$ is a $\mathcal{G}_1$ structure and $C(g)=\emptyset$. Then $J$ has no $(0,3)$-triple of t-roots, from Lemma \[g1 criterio\]. Thus $C(J)=\emptyset$ or $J$ is a invariant complex structure.
Now let $(g,J)$ be a $\mathcal{G}_1$ structure such that $C(J)\neq \emptyset$ and let $\delta$ be a t-root in $C(J)$. Then there exist a $(0,3)$-triple $\{\delta,\zeta,\eta\}$ of t-roots containing $\delta$. From Lemma \[g1 criterio\], the metric $g$ satisfies $\lambda_{\delta}=\lambda_{\zeta}=\lambda_{\eta}$, so $\delta$ is a t-root in $C(g)$ and $C(J)\subset C(g)$.
Finally, if $C(J)\subset C(g)$ follows imediately from Lemma \[g1 criterio\] that $(g,J)$ is a $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$.
As we saw in Theorem \[J for all\], any invariant complex structure $J$ provides a $\mathcal{G}_1$ structure with respect to all invariant metric on ${\mathbb{F}_{\Theta}}$. Thus a natural question is: Is there an invariant metric $g_0$ on ${\mathbb{F}_{\Theta}}$ such that $(g_0,J)$ is a $\mathcal{G}_1$ structure for any iacs on ${\mathbb{F}_{\Theta}}$? The next result shows that, up to homotheties, there exists a unique invariant metric $g$ which is $\mathcal{G}_1$ structure with respect to every iacs $J$ on ${\mathbb{F}_{\Theta}}$.
\[metric G1\] The normal metric is the unique invariant metric which is a $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$ with respect to any iacs $J$.
If ${\mathbb{F}_{\Theta}}$ is isotropy irreducible the result follows from Corollary \[G1 iso irr\]. Let $|R_t^+|>1$, from Lemma \[g1 criterio\] it is easy to see that the normal metric is $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$ with respect to any iacs $J$. Reciprocally, suppose an invariant metric $g$ is $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$ with respect to any iacs $J$. Consider $\delta$ and $\eta$, $\eta\neq \pm \delta$, two t-roots and $\lambda_\delta$, $\lambda_\eta$ the parameters of the invariant metric $g$ associated to these two t-roots, respectively. By Theorem \[connected tzs t-roots\], there exists a chain $T^1,\dots, T^n $ of tzs of t-roots, connecting $\delta$ and $\eta$, such that $\delta\in T^1$ and $\eta\in T^n$. For each tzs $T_{i,j,k}^p=\{\xi_i,\xi_j,\xi_k\}\subset R_t$, we can take an iacs $J_p$ such that $\{\xi_i,\xi_j,\xi_k\}$ is a (0,3)-triple of roots, then $\lambda_{\xi_i}=\lambda_{\xi_j}=\lambda_{\xi_k}$, since $g$ is $\mathcal{G}_1$ structure on ${\mathbb{F}_{\Theta}}$ with respect to any iacs $J_p$. Continuing with this finite process we get that $\lambda_\delta=\lambda_\eta$.
Equivalent structures
=====================
Consider the Weyl group $\mathcal{W}$ of the root system $R$. Recall that if $\omega\in \mathcal{W}$, then $\omega^2=1$ and $\omega^{-1}=\omega$. Let $A_\Theta$ be the subgroup of $\mathcal{W}$ of the reflection which preserves $R_\Theta \subset R$. Let $\omega\in\mathcal{W}$ be a reflection, it is not difficult to see that $\omega R_M\subset R_M$ if and only if $\omega \in A_\Theta$. Thus a reflection $\omega$ of $A_\Theta$ permites the roots $\beta$ in $R_M$ and the related eigenspaces $\mathfrak{g}_\beta$. If $\beta\in R_M$, we denote by $k(\beta)$ the correspondent t-root. The group $A_\Theta$ acts on the set of iacs by $$\omega\cdot J=\omega\cdot \{\varepsilon_{k(\beta)}\}=\{\varepsilon_{k(\omega\beta)}\}, \quad \beta \in R_M,\quad \omega\in A_\Theta.$$ This action is well defined because if $\alpha, \beta \in R_M$ are such that $k(\alpha)=k(\beta)$ and $\omega \in A_\Theta$ then $k(\omega\alpha)=k(\omega\beta)$. Indeed, let $\Sigma=\{\alpha_1, \dots \alpha_k,\beta_1, \dots, \beta_r \}$ be a basis of $R$ associated to ${\mathbb{F}_{\Theta}}$ where $\Theta=\{\alpha_1, \dots \alpha_k\}$ is a basis of $R_\Theta$ and $\Sigma_M=\Sigma\setminus\Theta=\{\beta_1, \dots, \beta_r \}$. Then $\alpha=\sum p_i\alpha_i+\sum m_j \beta_j$, $\beta=\sum q_i\alpha_i+\sum m_j \beta_j$ and $k(\omega \alpha)=k(\omega \beta)$, since $\omega \alpha_i\in R_\Theta$ for all $\omega\in A_\Theta$. It is known that the iacs $J$ and $\omega\cdot J$ are associated by a bi-holomorphic map on ${\mathbb{F}_{\Theta}}$. In this sense, if $\omega\in A_\Theta$ we say that the iacs $J$ and $\omega\cdot J$ are equivalents. In particular, $J_c$ and $\omega\cdot J_c$ are equivalents structures, (see [@Alek; @e; @Perol]).
In a similiar way, the $A_\Theta$ act on the set of invariant metrics on ${\mathbb{F}_{\Theta}}$ by $$\omega\cdot g=\omega\cdot \{\lambda_{k(\beta)}\}=\{\lambda_{k(\omega\beta)}\},\quad \beta \in R_M,\quad \omega\in A_\Theta.$$ Following [@SM; @N], if $\omega\in A_{\Theta}$ we say that the invariant almost Hermitian structures $(g, J)$ and $(\omega\cdot g, \omega\cdot J)$ are equivalent. The structures $(g, J)$ and $(\omega\cdot g, \omega\cdot J)$ share the same class of invariant almost Hermitian structures.
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A.Arvanitoyeorgos, Geometry of Flag Manifolds, International Journal of Geometric Methods in Modern Physics, Vol.3, Nos. 5 e 6 957-974 (2006) . Besse, A.L., Einstein Manifolds. Springer-Verlag (1987).
A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces I. Amer. J. Math. 80, N.2, 458-538 (1958). Da Silva, N. P., Grama, L. and Negreiros, C. J. C., Variational results on flag manifolds: harmonic maps, geodesics and Einstein metrics. Journal of Fixed Point Theory and its Applications, vol. 10, 307-325 (2011).
Da Silva N. P., Alves L. A. , Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$. Bol. Soc. Paran. Mat. (2020) To appear.
Dashevich O. V., Characteristic Properties of Almost Hermitian Structures on Homogeneous Reductive Spaces. Mathematical Notes, Vol. 73, n. 5, (2003).
M. Fernández, L. M. Hervella, Un exemple de variété pseudo-kählérienne ($G_1$-variété). C. R. Acad. Sci. Paris Sér. A-B 283, no. 4, Aiii, A203-A205, (1976).
A. Gray, Some examples of almost Hermitian manifolds, Illinois J. Math., 10, pp. 353-366,(1969).
A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123, 35-58,(1980).
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Wiley (Intercience), New York, Vol. 1, 1962; Vol. 2, (1969).
San Martin L. A. B., Negreiros C. J. C., Invariant almost Hermitian structures on flag manifolds, Advances in Mathematics, 178, 277-310 (2003).
San Martin L. A. B., Silva R. de C. J., Invariant nearly-Kähler structures, Geometriae Dedicata, Vol. 121, 1, p.p. 143-154, (2006). J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, (1972).
Siebenthal, J., Sur certains modules dans une algèbre de Lie semisimple. Comment. Math. Helv. 44(1) 1-44 (1964).
J. A. Wolf, A. Gray, Homogeneous spaces defined by Lie group automorphisms II, J. Differential Geom. 2, 115-159, (1968).
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[^1]: Federal University of Uberlândia, luciana.postigo@gmail.com
[^2]: Federal University of Uberlândia, neitonps@gmail.com
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---
abstract: 'We find a significant anti-correlation between the hard X-ray photon index $\Gamma$ and the Eddington ratio $L_{\rm Bol}/L_{\rm Edd}$ for a sample of Low-Ionization Nuclear Emission-line Regions (LINERs) and local Seyfert galaxies, compiled from literatures with Chandra or XMM-Newton observations. This result is in contrast with the positive correlation found in luminous active galactic nuclei (AGNs), while it is similar to that of X-ray binaries (XRBs) in low/hard state. Our result is qualitatively consistent with the spectra produced from advection dominated accretion flows (ADAFs). It implies that the X-ray emission of low-luminosity active galactic nuclei (LLAGNs) may originate from the Comptonization process in ADAF, and the accretion process in LLAGNs may be similar to that of XRBs in the low/hard state, which is different from that in luminous AGNs.'
author:
- |
Minfeng Gu $^{1}$[^1], Xinwu Cao$^{1}$\
$^{1}$ Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80\
Nandan Road, Shanghai 200030, China
title: 'The anti-correlation between the hard X-ray photon index and the Eddington ratio in LLAGNs'
---
\[firstpage\]
accretion, accretion disks — galaxies: active — galaxies: nuclei — X-rays: galaxies
Introduction
============
Low-luminosity active galactic nuclei [LLAGNs; see @ho08 for recent reviews] are believed to be powered by accretion of matter onto the central supermassive black hole, similarly to powerful active galactic nuclei (AGNs). The intrinsic faintness, i.e. $L_{\rm Bol}<10^{44} \rm erg~s^{-1}$, and the low level of activity are the distinctive characteristics of LLAGNs, which mainly consists of local Seyfert galaxies and Low-Ionization Nuclear Emission-line Regions (LINERs). In terms of Eddington luminosity, most of LLAGNs have $L_{\rm Bol}/L_{\rm
Edd}<10^{-2}$, compared to $L_{\rm Bol}/L_{\rm Edd}\sim1$ of luminous AGNs [@pan06]. It still remains unclear whether LLAGNs are a scaled-down luminosity version of classical AGNs or objects powered by different physical mechanisms. Alternatively, LLAGNs could represent a scaled up version of black hole binaries in the steady-jet, hard X-ray state, as pointed out by the scaling relations reported in @fal04. The accretion processes of LLAGNs are still not well understood. For a sample of 47 local Seyfert galaixes selected from the Palomar optical spectroscopic survey of nearby galaxies [@ho97], @pan06 found a strong linear correlations between the nuclear 2-10 keV, \[OIII\]$\lambda5007$, $\rm H_{\alpha}$ luminosities with the same slope as quasars and luminous Seyfert galaxies, independent of the level of nuclear activity displayed. The authors thus argued that the local Seyfert galaxies are consistent with being a scaled-down version of more luminous AGNs. Most recently, through revisiting the Spectral Energy Distributions (SED) of 13 nearby LINERs using recently published, high-angular-resolution data at radio, ultraviolet (UV), and X-ray wavelengths, @mao07 found that the SEDs of LINERs are quite similar to those of Seyferts up to luminosities of $\rm \sim10^{44}~ergs~s^{-1}$, and there is no evidence for a sharp change in the SED of AGNs at the lowest luminosities. This result indicates that the thin AGN accretion disks may persist at low accretion rates, which is also recently claimed by [@pap08] from the fact that the near-IR to X-ray spectrum of four low-luminosity Seyfert 1 galaxies has the same shape as that of luminous quasars. However, the radiatively inefficient accretion flows [RIAFs: an updated version of ADAF; see, e.g. @yua07a references therein] have been invoked to explain some of the spectral properties observed in many LLAGNs, such as the lack of the “big blue bump" [@ho99], and can be used to explain the SED of several LLAGNs [e.g. @chi06].
X-rays are one of the most direct evidence of nuclear activity and are, therefore, fundamental to study the accretion processes. The nature of the X-ray emission has been recently explored for the samples of LINER galaxies and local Seyfert galaxies from the observations with Chandra and XMM-Newton telescopes [e.g. @pan04; @gon06; @pan06]. The high spatial resolution of Chandra telescope allows the detection of the emission from the genuine active nuclei, not contaminated by off-nuclear sources and/or diffuse emisison. Usually, the AGN nature can be defined as the appearance of nuclei emission in hard X-ray bands. Although the lack of this feature may likely imply non-AGN contribution, such as the emission from the diffuse hot gas and/or the starbursts, it does not necessarily rule out the AGN source nature, since the non-detection can be due to the heavy absorption. It has been pointed out that the lack of UV bump in the LINER SED may not be intrinsic, but simply the effect of obscuration [e.g. @lew03; @dud05; @mao07]. While the spatial resolution of XMM-Newton telescope is not enough to separate the off-nuclear sources and/or diffuse emission, its superior sensitivity and high throughput provide higher signal-to-noise spectra and the presence of diffuse thermal emission can be indirectly inferred from the spectral analysis [e.g. @cap06; @pan04]. Indeed, the Chandra and XMM-Newtwon images and spectra of the local Seyfert galaxies produced in the 0.3-10 keV band have showed a high detection rate ($\sim95$%) of active nuclei, charcterized, in $\sim60$% of the objects, also by the presence of nearby off-nuclear sources and/or, in $\sim35$% of the objects, diffuse emission [@pan04]. From an homogeneous analysis of the Chandra X-ray data available for a sample of 51 LINER galaxies, @gon06 claimed that a high percentage of LINER galaxies, at least $\approx60\%$, could host AGN nuclei, and $\sim40\%$ sources lacking a hard nuclear source were regarded as starburst candidates, because they either lack an energetically significant AGN or contain highly obscured AGN. Consistently, both from the Chandra observations, [@flo06] concluded that in about 60% of 19 LINER sources in their sample an AGN is present, and [@sat05] found that at least 50% of 82 LINERs harbor a central hard X-ray source consistent with an AGN.
The hard X-ray nuclei emission can well trace the AGN nature in LLAGNs, however, the origin of the X-ray emission of LLAGNs is still largely debated. The X-ray emission can be either from the jet [@fal04] or from the RIAF [@mer03]. It could even be from the accretion-flow/hot-corona system of radiative efficient accretion, as suggested by [@pan06] and [@mao07] that the thin AGN accretion disks may still persist at low accretion rates. From an experimental investigation on the correlation between the hard X-ray photon index $\Gamma$ with the Eddington ratio $L_{\rm
Bol}/L_{\rm Edd}$ for a sample of luminous AGNs, [@she06] suggested that the hard X-ray photon index can be a good indicator of accretion rate. In fact, this relation has been explored in various works, mostly for luminous AGNs [e.g. @lu99; @wan04; @bia05; @gre07] and occasionally for X-ray binaries (XRBs) [@yam05; @yua07b], through which the X-ray origin and accretion processes were investigated. However, this has never been explored for LLAGNs. In this paper, we present the relationship between the hard X-ray photon index $\Gamma$ and the Eddington ratio $L_{\rm Bol}/L_{\rm Edd}$ for a sample of LLAGNs, with the aim to constrain the origin of X-ray emission and the accretion processes in these objects. This paper is organized as follows: The sample is described in Section 2, the results are given in Section 3 and Section 4 presents discussions.
The sample
==========
We started from the samples of LINERs and local Seyfert galaxies of the Palomar sample [@ho97; @ho08] and the multiwavelength catalogue of LINERs (MCL) complied by [@car99]. From the Palomar optical spectroscopic survey of nearby galaxies [@ho95], high-quality spectra of 486 bright ($B_{T}\leq12.5$ mag), northern ($\delta>0^{\circ}$) galaxies have been taken and a comprehensive, homogeneous catalog of spectral classifications of all galaxies have been obtained [@ho97]. The Palomar survey is complete to $B_{T}=12.0$ mag and $80\%$ complete to $B_{T}=12.5$ mag. Based on the relative strength of the optical forbidden lines (\[O III\] $\lambda5007$, \[O I\] $\lambda6300$, \[N II\] $\lambda6583$ and \[S II\] $\lambda\lambda6716,6731$) compared to the hydrogen Balmer lines, H II nuclei and AGNs were distinguished. Moreover, the ratio \[O III\] $\lambda5007/\rm H\beta$ was used to distinguish LINERs ($\rm [O
III]/H\beta<3$) from Seyferts ($\rm [O III]/H\beta\geq3$). From this survey it has been shown that AGNs are very common ($\sim40\%$) in nearby galaxies with Seyfert galaxies being $11\%$, pure LINERs $\sim20\%$ and transition objects (with \[O I\] strengths intermediate between those of H II nuclei and LINERs; composite systems having both an HII region and a LINER component [@ho93]) $\sim13\%$ of all galaxies, respectively. The MCL includes most of the LINERs know until 1999, providing information on broad band and monochromatic emission from radio frequencies to X-rays for 476 objects classified as LINERs. The optical classification was reanalyzed by using the line-ratio diagrams by [@vei87], and most of the galaxies can be considered as ‘pure’ LINERs. We searched the literatures for all the available observations with Chandra or XMM-Newton telescopes. Due to the fact that the X-ray emission are complex in these objects, and to ensure that we are exploring the properties of genuine emission from nuclei AGNs, the source selection were carefully performed. Thanks to the high spatial resolution of Chandra, the genuine AGNs are selected from the appearance of the nuclei at hard X-ray band ($>$2 keV). Those sources without the nuclei at hard X-ray band are excluded in this work, though some of them may have central active nuclei, which can be heavily obscured [@gon06]. To get the hard X-ray photon index $\Gamma$, the sufficient count rates to perform the spectral fitting in 2-10 keV band are required. Compton-thick sources are identified by [@pan06] using flux diagnostic diagrams, which are based on measuring the X-ray luminosity against an isotropic indicator of the intrinsic brightness of the source to evaluate the true amount of absorption. Specifically, the flux ratio of X-ray to \[O III\]$\rm \lambda5007$ $F_{\rm X}/F_{\rm [O III]}$ and \[O III\]$\rm \lambda5007$ to far-infrared $F_{\rm [O III]}/F_{\rm
IR}$ is used to identify the Compton thick sources with $F_{\rm
X}/F_{\rm [O III]}<1$, in addition to identify the AGN-dominant (non-starburst) emission in sources with $F_{\rm [O III]}/F_{\rm
IR}>10^{-4}$ [@pan02; @pan06]. In this work, we exclude the Compton-thick sources identified in [@pan06], since the Compton-thick absorption may cause the flatter X-ray photon index and lower absorption when doing spectral fitting in 2-10 keV band, which may not be real. Moreover, the available measurements of black hole mass or stellar velocity dispersion are also required.
Finally, we compiled a sample of 55 sources, of which 27 are LINERs and 28 are local Seyfert galaxies. All the local Seyfert galaxies are from [@ho97] sample, and the LINERs are from either [@ho97] sample or MCL. The photon index from the spectral fitting and the resulting luminosity in 2-10 keV band of nuclei AGNs were collected from literatures, of which XMM-Newton data for 20 sources (all are Seyferts) and Chandra data for 33 sources (6 Seyferts and 27 LINERs). In addition, the ASCA data are collected for 2 Seyfert galaxies (NGC 2639 and NGC 2655) from [@pan04] and [@pan06]. Our sample is listed in Table \[tbl-1\], in which column (1) source name, column (2) redshift, column (3) X-ray satellite used: “C”=Chandra, “X”=XMM-Newton, “A”=ASCA, column (4) the photon index between 2 and 10 keV, column (5) the 2-10 keV luminosity in units of $\rm ergs ~s^{-1}$, column (6) the references of X-ray data, column (7) the mass of central black hole in the units of solar mass, and column (8) is the references of black hole masses.
Results
=======
We collected the black hole masses $M_{\rm BH}$ available from literatures, which were estimated in different ways from gas, stellar and maser kinematics to reverberation mapping or inferred from the mass-velocity dispersion correlations [e.g. @fer00; @tre02]. For those $M_{\rm BH}$ estimated with the mass-velocity dispersion relation different from the [@tre02] relation, we re-calculated $M_{\rm BH}$ with the [@tre02] relation. For those sources without measured $M_{\rm
BH}$ from literatures, we estimate $M_{\rm BH}$ from the stellar velocity dispersion provided by the Hypercat database[^2] using the [@tre02] relation. In Table \[tbl-1\], we report all these $M_{\rm BH}$ estimates, the method used to calculate them and the corresponding references. To calculate the bolometric luminosity, we have assumed that $L_{\rm
Bol}/L_{\rm 2-10 keV}\sim30$, which is typical of luminous AGN, being normally in the range of 25 - 30 [@elv94]. It can be seen from Table \[tbl-1\] that the X-ray luminosity $L_{\rm 2-10 keV}$ of our sample ranges from $10^{38.6}\rm~erg~s^{-1}$ to $10^{43.3}~\rm erg~s^{-1}$. Correspondingly, the bolometric luminosity $L_{\rm Bol}$ varies from $10^{40.1}~\rm erg~s^{-1}$ to $10^{44.8}~\rm erg~s^{-1}$ with most sources (53 of 55 sources) lower than $10^{44}~\rm erg~s^{-1}$.
In Fig. \[fig1\], we present the relationship between the photon index $\Gamma$ at 2-10 keV band and the Eddington ratio $L_{\rm
Bol}/L_{\rm Edd}$ for our sample. To compare with luminous AGNs, we also plot the PG quasars of [@she06]. For better comparisons, we calculated $L_{\rm Bol}=30L_{\rm2-10 keV}$ also for PG quasars, instead of using optical continuum luminosity in [@she06]. It can be clearly seen that the Eddington ratio $L_{\rm Bol}/L_{\rm
Edd}$ of our sample covers a wide range from $\sim10^{-1}$ going down to $10^{-6}$, compared to those of most PG quasars $L_{\rm
Bol}/L_{\rm Edd}>10^{-1}$. For most of our sources, the hard X-ray photon index $\Gamma$ lies in the range of 1.0 - 2.0. We find a significant anti-correlation between the photon index $\Gamma$ at 2-10 keV band and the Eddington ratio $L_{\rm Bol}/L_{\rm Edd}$ with a Spearman correlation coefficient of $r=-0.328$ at $\sim99\%$ confidence level for all our LLAGNs. This significant anti-correlation is found to be slightly improved ($r=-0.517$ at $\sim99.5\%$ confidence level) when only local Seyfert galaxies are considered. However, we find no strong correlation ($r=-0.127$ at $\sim47\%$ confidence level) if we consider LINERs only. This may be caused by the fact that large scatter exist in photon index of LINERs, especially those with $\Gamma>2.5$, and LINERs cover relatively narrower range in $L_{\rm Bol}/L_{\rm Edd}$ than that of local Seyfert galaxies, as can be clearly seen in Fig. \[fig1\]. The significant anti-correlation we found for our LLAGNs is contrast with the positive correlation for luminous AGNs as clearly shown by the PG quasars in Fig. \[fig1\]. However, our result is consistent with that of XRBs at low state [@yam05; @yua07b].
For our sample of 55 LLAGNs, a standard $\chi^2$ minimization method weighted by the errors on $\Gamma$ yields the following linear relation: $$\Gamma=(-0.09\pm0.03)~ \rm log~ (\it L_{\rm Bol}/L_{\rm Edd})+ \rm
(1.55\pm0.07)$$ which is shown in Fig. \[fig1\]. To quantitatively compare with luminous AGNs, we plot our linear fit to [@she06] PG quasars, i.e. $\Gamma=(0.41\pm0.09)~\rm log~ (\it L_{\rm Bol}/L_{\rm
Edd})+\rm (2.17\pm0.07)$. As an updated version of [@she06] work, we also plot the recent linear fit $\Gamma=(0.31\pm0.01)~\rm
log~ (\it L_{\rm Bol}/L_{\rm Edd})+\rm (2.11\pm0.01)$ presented by [@she08] for a sample of 35 moderate- to high-luminosity radio quiet AGNs. It should be noted that the [@she08] fit is based on $L_{\rm Bol}$ derived from optical continuum luminosity for a larger sample. This may cause the differences in two linear fits of luminous AGNs, which however is consistent with each other within errors.
Discussions
===========
In this work, instead of performing a rigorous and homogeneous spectral analysis, we simply collected the hard X-ray photon index from the literatures for our sample sources, especially for LINERs (see Table 1). This may potentially affect the conclusions, since different authors may use different models to fit the X-ray spectra (e.g. absorbed power-law, broken power-law, partially absorbed power-law, power-law plus thermal component, etc.). In principle, for consistent reasons, the same baseline model should be used. However, the commonly used method in literatures, e.g., as shown in [@flo06] and [@gon06], is that the different models are tried to fit the X-ray spectra, and the best fit is usually obtained with the acceptable $\chi^2$ minimization. For a sample of 51 LINERs in [@gon06], two models were used to fit Chandra data, i.e. a single power-law and a single-temperature optically-thin plasma emission (MEKAL or Raymond Smith (Raymond & Smith 1977) model). For each object, five models were attempted: power-law (PL), Raymond-Smith (RS), MEKAL (ME), PL+RS, and PL+ME. The best model was chosen as that with the best $\chi^2$-reduced statistic [see @gon06 for details]. As a result, the best fit can be PL for some sources (e.g. NGC 4374), however it can be PL+ME for other sources (e.g. NGC 2681). In contrast, six models (blackbody, Comptonized blackbody, accretion disk, hot diffuse gas, power-law and XRB Comptonization) and their linear combinations were tested in [@flo06] in order to find the most appropriate model for the X-ray spectra of LINERs. Their spectral fits show that a same model is difficult to well fit all the LINERs X-ray spectra. On the other hand, the different spectral fits can be obtained in different works for same Chandra data in same source. Indeed, seven LINERs in our sample (i.e. NGC 315, NGC 2681, NGC 4374, NGC 4410A, NGC 4457, NGC 4494 and NGC 4552), can be found with same Chandra data reduced in [@gon06] and/or [@flo06], [@sat05]. In these sources, the X-ray spectra can be well fitted by either single power-law or power-law plus thermal component, however, the photon index from different works are consistent with each other within the errors. As a good example, for NGC 4374, the best fit of [@gon06] is power-law model with $\Gamma=2.07_{-0.20}^{+0.24}$, while [@flo06] shows a best fit of power-law plus MEKAL model with $\Gamma=2.0_{-0.1}^{+0.2}$. For these seven LINERs, we choose the results with the smallest error in $\Gamma$. It should be noted that the X-ray data of Seyfert galaxies are all from [@pan04] and [@pan06], in which the homogeneous analysis was committed. In contrast, LINERs are from various works. Compared to a significant anti-correlation between $\Gamma$ and $L_{\rm bol}/L_{\rm Edd}$ in Seyferts, no correlation is found in LINERs. It is not clear whether the inhomogeneous spectral analysis smear the correlation in LINERs. Nevertheless, it would be important to revisit our results through a rigorous and homogeneous spectral analysis. Since our sample data were collected from Chandra and XMM-Newton telescopes, it is thus interesting to see whether the anti-correlation still holds using Chandra and XMM-Newton data separately. We found a significant anti-correlation between $\Gamma$ and $L_{\rm bol}/L_{\rm Edd}$ with $r=-0.553$ at $\sim 99$% confidence level for 20 XMM-Newton sources. However, no strong correlation is found for 33 Chandra sources ($r=-0.177$ at $\sim
68$% confidence level). Clearly, this result is mainly due to the fact that 20 XMM-Newton sources are all Seyferts, and 33 Chandra sources are dominated by LINERs (27 of 33).
The classification of Seyfert/LINER was investigated in many occasions based on four optical line ratios $\rm [O III]/H\beta$, $\rm [N II]/H\alpha$, $\rm [S II]/H\alpha$, and $\rm [O I]/H\alpha$ [e.g. @hec80; @vei87; @ho97; @kew06], and the comparisons between the various classification schemes have also been made, which shown that the source definition may be different in different classification scheme [e.g. @kew06]. Recently based on a large number of emission-line galaxies from Sloan Digital Sky Survey, [@kew06] found that Seyferts and LINERs form clearly separated branches on the optical diagnostic diagrams. They further derived a new empirical classification scheme, which cleanly separates Seyferts and LINERs [@kew06]. To verify the classification of our sample, we have tried to re-classify our sources using their scheme and the line ratios collected from either [@ho97] or [@car99]. For our original 27 LINERs and 28 Seyfert galaxies, the re-classification shows that 32 sources are now LINERs and 23 sources are Seyfert galaxies. Specifically, we find that all original LINERs are still LINERs except for UGC 08696, which was regarded as a LINER in [@car99] using [@vei87] classification scheme, now is a Seyfert galaxy. This high fraction of LINERs remaining as LINERs is actually consistent with the comparisons made by [@kew06] between their scheme and others. Most of LINERs in our sample were defined by [@ho97] scheme, and others by [@vei87] scheme in [@car99]. It should be noted that the [@ho97] scheme actually is very similar to [@vei87] one. As [@kew06] stated that most of galaxies classed as LINERs in the [@ho97] scheme are also LINERs in their scheme. On the other hand, we find that six original Seyfert galaxies (NGC 2639, NGC 2655, NGC 3031, NGC 4168, NGC 4579 and NGC 4639) in our sample may be identified as LINERs with [@kew06] scheme. With these new identifications, we then re-investigate our results. Clearly, the new identification will not change the anti-correlation for whole sample of 55 sources. We find that the anti-correlation is still significant ($r=-0.478$ at $\sim98\%$ confidence level) if we only consider the 23 Seyfert galaxies newly defined with [@kew06] scheme, and again there is no correlation ($r=-0.127$ at $\sim51\%$ confidence level) for newly classified 32 LINERs. We thus conclude that the re-classification does not change our results.
Usually, the bolometric luminosity can be estimated from the luminosity at a given band by applying a suitable bolometric correction. The commonly applied bolometric corrections have been determined from a mean energy distribution calculated from 47 luminous, mostly nearby quasars [@elv94], in which $L_{\rm
Bol}/L_{\rm 2-10 keV}\sim30$. However, as pointed out by [@mar04], the inclusion of IR emission, which is known to be re-processed from the ultraviolet (UV), would overestimate the bolometric luminosity. Moreover, the [@elv94] bolometric corrections were calculated from an X-ray selected sample of quasars, thus may underestimate the bolometric corrections in the X-ray band. In addition, the bolometric corrections were derived from an average quasar template spectrum, thus the dependence on luminosity of the bolometric corrections cannot be obtained [@mar04]. In calculating the bolometric luminosity $L_{\rm
Bol}$, we have simply assumed $L_{\rm Bol}/L_{\rm 2-10 keV}\sim30$. Therefore, the bolometric luminosity $L_{\rm Bol}$ in Fig. \[fig1\] is just a multiple of the X-ray luminosity. As a matter of fact, $L_{\rm Bol}$ depends on the shape of the SED, which could differ among high and low luminosity AGN [@ho99; @mar04]. Indeed, the observed $L_{\rm Bol}/L_{\rm 2-10 keV}$ ratio ranges from 3 to 16 [@ho99] in LLAGNs, and a lower value of $L_{\rm Bol}/L_{\rm
2-10 keV}\approx10$ was suggested at lower luminosities (typical of Seyfert galaxies, i.e., $\rm 10^{42}-10^{44} erg~s^{-1}$) [e.g. @fabi04]. The variations of bolometric corrections on the luminosity was taken into account by [@mar04], in which the AGN template SED were constructed by using the well-known anticorrelation between the optical-to-X-ray spectral index and $\rm
2500~\AA$ luminosity. [@hop07] also investigated the dependence of bolometric corrections on the luminosity with similar approach. Both works shown that the bolometric correction varies among high and low luminosity AGN. More recently, taking the variations in the disk emission in the ultraviolet (UV) into account, [@vas07] found evidence for very significant spread in the bolometric corrections, with no simple dependence on luminosity being evident. Instead, a more well-defined relationship between the bolometric correction and Eddington ratio in AGN is suggested, with a transitional region at an Eddington ratio of $\sim0.1$, below which the bolometric correction is typically 15-25, and above which it is typically 40-70. Most recently, the increase in bolometric correction with Eddington ratio is confirmed and refined by [@vas09] based on simultaneous optical to X-ray SEDs for 29 AGN from the [@pet04] reverberation-mapped sample, with hard X-ray bolometric corrections of $\kappa_{\rm 2-10 keV} \sim 15-30$ for $L_{\rm Bol}/L_{\rm Edd}\lesssim 0.1$, $\kappa_{\rm 2-10 keV}
\thickapprox 20-70$ for $0.1\lesssim L_{\rm Bol}/L_{\rm Edd}\lesssim
0.2$ and $\kappa_{\rm 2-10 keV} \sim 70-150$ for $L_{\rm Bol}/L_{\rm
Edd}\gtrsim 0.2$. Since the X-ray luminosity $L_{\rm 2-10keV}$ in our sample covers over four orders of magnitude, clearly the dependence of bolometric correction on the luminosity can not be ignored if it does exist. We then calculated the bolometric correction using the expression of [@hop07] in 2-10 keV band (see their Equation 2). We find that the bolometric correction $L_{\rm Bol}/L_{\rm 2-10keV}$ of most sources (51 of 55 sources) range from 8 to 16, and the average value of whole sample is about 11. Although the new calculated bolometric correction varies from source to source, we find our results are not changed, i.e. the anti-correlation between $\Gamma$ and $L_{\rm Bol}/L_{\rm Edd}$ is still significant. Alternatively, if a $L_{\rm Bol}/L_{\rm 2-10
keV}$ ratio of 10 is considered for all our LLAGN sources, then the Eddington ratio would be a factor of 3 lower than those in Fig. \[fig1\]. Again, this will not change our correlation results. It should be noted that the more accurate bolometric correction can be performed when the detailed SED of individual source is constructed. However, this is not available now. The adoption of a uniform value of $L_{\rm Bol}/L_{\rm 2-10 keV}$ ratio will introduce dispersion of $\Gamma - L_{\rm Bol}/L_{\rm
Edd}$ relation in Fig. \[fig1\], and the influence on the correlation caused from a uniform bolometric correction needs further exploration.
While the X-ray emission may be from the Compton process in corona in luminous AGNs, it is unclear in LLAGNs. The fundamental plane of supermassive black hole activity derived from the combination of X-ray, radio luminosity and black hole mass, were used to constrain the origin of X-ray emission in LLAGNs [@mer03; @fal04]. However, the different scenarios were proposed. [@fal04] suggested that the radio through X-ray emission of LLAGNs is attributed to synchrotron emission from a relativistic jet, similarly to the scenario proposed for XRBs in their low/hard state. In contrast, [@mer03] suggested a rediatively inefficient accretion flow as the origin for X-ray emission, and a relativistic jet for the radio emission. Recently, it is proposed that in some individual sources the emission from a jet can be responsible for the X-ray emission [e.g. @yua02; @fab03; @fab04]. While several pieces of evidence seem to favor an accretion-related X-ray origin for four LINERs in radio galaxies, a significant (or even dominant) contribution from the jet in the X-ray regime cannot be excluded for some objects [@gli08].
The correlation between the X-ray photon index and the Eddington ratio has been found in various occasions [e.g. @lu99; @wan04; @she06; @gre07]. [@she08] updated the [@she06] work by including more highly luminous AGNs, and they confirmed that the hard X-ray photon index can be a good indicator of accretion rate. Moreover, the authors claimed that a measurement of $\Gamma$ and $L_{\rm X}$ can provide an estimate of $L/L_{\rm
Edd}$ and black hole mass with a mean uncertainty of a factor of $\lesssim3$. There are various explanations for the correlation found in luminous AGNs. The X-ray emission of luminous Seyfert galaxies are usually explained by that the emission is produced by a disk-corona system, where UV soft photon from the accretion disk are comptonized and up-scattered into the hard X-ray band by a hot corona above the accretion disk [@haa91; @haa93]. As the accretion rate increases, the disk flux to irradiate the corona increases. This will cause the corona to cool more efficiently because of Compton cooling, and so the hard X-ray slope will steepen with the increasing accretion rate [@haa91; @haa93; @wan04]. Alternatively, [@wan04] suggested that the hot corona may become weak with increasing accretion rate. [@lu99] found two different accretion classes in Seyfert galaxies and QSOs through investigating the correlation between the X-ray photon index and the Eddington ratio. The authors argued that two classes may correspond to ADAF and thin disk accretion, respectively. While two distinct classes are apparent at the soft X-ray photon index and Eddington ratio panel, it is not clear for the case of the hard X-ray photon index due to the lack of data. Nevertheless, there are strong positive correlations between the hard X-ray photon index and the Eddington ratio for the objects with ASCA data, and for only ADAF class objects as well. To explain the trend of spectrum steepening with increasing accretion rate in ADAF-class objects, they invoked the two-zone accretion disk, i.e. an outer standard thin disk and an inner ADAF. However, we note that all of their ADAF class objects have $\rm
log~(\it L_{\rm Bol}/L_{\rm Edd})> \rm -2.66$, which represent the luminous ADAF objects compared to our objects. Most recently from a large sample of 153 radio quiet quasars, [@kel08] confirmed the correlation between $\Gamma_{\rm X}$ and both $L_{\rm UV}/L_{\rm Edd}$ and $L_{\rm X}/L_{\rm Edd}$, however they found the dependence is not monotonic. The change in the disk/corona structure was proposed by the authors as a possible explanation for the nonmonotonic dependence.
Our study differs from previous work in that we expand the Eddington ratio down to very low region, $\leq10^{-6}$, which is unexplored before. The striking result we find is that the anti-correlation between the hard X-ray photon index and the Eddington ratio of LLAGNs is apparently contrast with the positive one of luminous AGNs. This implies that the X-ray emission mechanism of LLAGNs may be different from that of luminous AGNs. On the other hand, the anti-correlation is consistent with that of XRBs at low/hard state [@yam05; @yua07b]. Through a systematic spectral analysis for 10 XRBs in the low/hard states, using $RXTE$ and $Beppo-SAX$ data, [@yam05] explored the correlation between the hard X-ray photon index and luminosity. They found a clear anti-correlation both for the most frequent observed source XTE J1550-564 and for the compiled data of all 10 XRBs when the 2-200 keV luminosity $L_{2-200}$ is less than $2\times10^{37}\rm ~erg~s^{-1}$, corresponding to about 1% of Eddington limit for 10 solar mass black hole. Recently, the similar anti-correlation was also found at X-ray luminosity below $\sim 2\%L_{\rm Edd}$ from the combined data of two stellar mass black hole X-ray sources, XTE J1118+480 and XTE J1550-564 [@yua07b].
Though there are still debates, LLAGNs are usually believed to be powered by radiatively inefficient accretion flows, such as ADAFs and their variants [@nar94], instead of the standard geometrically thin optically thick accretion disk typically proposed as the accretion mechanism acting in the central regions of luminous AGN [@sha73]. We find that the qualitative behavior of the correlation of LLAGNs is consistent with the prediction of the spectra produced from ADAF model for XRBs in low state [@esi97]. At these low accretion rates, the Comptonization of synchrotron photons by the hot gas in the ADAF constitutes the dominant cooling mechanism. As the accretion rate increases, the optical depth of the ADAF goes up causing a corresponding increase in the Compton $y$-parameter. Consequently, the high-energy part of the spectrum becomes harder and smoother, and the photon index in the hard X-ray band hardens [see Fig. 3a of @esi97]. In the self-similar solution of ADAFs [@nar95; @nar98], the electron scattering optical depth $\tau_{\rm es}$ can be expressed as $$\tau_{\rm es}\simeq24~\alpha^{-1}\dot{m}r^{-1/2}$$ where $\alpha$ represents the viscosity parameter [@sha73], $\dot{m}$ is the accretion rate in Eddington units, and $r$ is the radius in Schwarzschild radii. The inverse Compton spectra of soft photons by the thermal electrons can be expressed as [@ryb79] $$I_{\nu}\propto\nu^{-\alpha}$$ with the spectral index $\alpha=-\rm ln \it \tau_{es}/\rm ln \it A$, where $I_{\nu}$ is intensity, $A$ is the mean amplification of photon energy per scattering by the electrons with temperature $T_{e}$ and is dominantly determined by $T_{e}$. Applying the $\tau_{es}$ expression in equation (2), we can get the spectral index $\alpha\propto-\rm log~ \it \dot{m}$, since the electron temperature $T_{e}$ is only weakly dependent of accretion rate. Our results are thus qualitatively consistent with the theoretical expectations of inverse Compton scattering in ADAFs. Similarly, @yua07b also predicted an anti-correlation between the X-ray spectral slope and the Eddington-scaled luminosity based on ADAF model for objects with low Eddington ratio, in which the X-ray spectral slope is determined by the Compton $y$-parameter. In view of the consistence of our result with the ADAF model predictions, our result seems to suggest that the X-ray emission of LLAGNs may originate from the Comptonization process in ADAFs.
The electron temperature $T_{e}$ in ADAFs is generally in the range $10^{9}-10^{10}$ K, and depends only weakly on the accretion rate [@nar98]. Therefore, the X-ray emission of LLAGNs in our concerned energy band 2-10 keV can be from both the Comptonization and bremsstrahlung emission. The thermal bremsstrahlung spectrum is cut off at $\nu_{\rm cutoff}\sim kT_{e}/h$, thus, $\nu_{\rm cutoff}$ is in the range from several tens to hundreds keV for the plasma with electron temperature $T_{e}\sim10^{9}-10^{10}$ K. This implies that the spectral index of bremsstrahlung emission from ADAFs is around $\alpha\sim0~ (\Gamma=\alpha+1\sim1)$ in 2-10 keV, which however is in contrast with the observations (see Fig. \[fig1\]). Therefore, we speculate that Comptonization is dominant over the bremsstrahlung emission in 2-10 keV for our LLAGNs. While our sample is optically defined and selected, some sources can be defined as radio galaxies based on the radio structure, such as NGC 1275, NGC 315, NGC 4261 and NGC 6251 being regarded as FR I radio galaxies [@fan74]. It thus conceivable that the X-ray emission can be from jet, which previously has been proposed for general LLAGNs [@fal04]. Based on the analysis of time-averaged spectra combined with model-independent information from X-ray temporal and spectral variability, and with inter-band information for four LINERs hosted by radio galaxies, [@gli08] claimed that LINERs represent a very heterogeneous class, which may range from intrinsically low-luminosity objects (in terms of Eddington ratio), to objects potentially strongly absorbed, to brighter objects with properties more in line with Seyfert galaxies. From an investigation on the X-ray emission mechanism using the coupled jet-ADAF model for a sample of eight FR I radio galaxies, @wu07 claimed that the X-ray emission of FR Is are complex from source to source, and X-ray can be dominantly from ADAF for luminous sources, and mainly from jet for the least luminous sources, however, can be a mixture of ADAF and jet for moderate luminosity sources. More generally for LLAGNs, [@yua05] predicted that the X-ray emission is dominated by the emission from the ADAF when the X-ray luminosity of a system is above some critical value, however it will be dominated by the jet when it is below the critical value. The critical value of $L_{\rm 2-10 keV}$ is claimed to be $\sim 10^{-6} L_{\rm Edd}$ to $10^{-7} L_{\rm Edd}$, which varies from source to source. Indeed, the X-ray emission of FR I galaxies NGC 315 [@wu07] and NGC 4261 [@gli03] were claimed to be dominated by ADAFs since the X-ray luminosity is relatively high. On the other hand, the jet origin X-ray emission was proposed for NGC 4374 [@wu07] and NGC 4594 [@pel03] due to the low X-ray luminosity $<10^{-7}L_{\rm
Edd}$. We find that five sources (one Seyfert galaxies and four LINERs) in our sample have $L_{\rm 2-10~ keV}<10^{-7}L_{\rm Edd}$, therefore, the jet contribution in these sources probably can not be ignored based on the predictions of [@yua05]. However, these five sources well follow the general trend of $\Gamma - L_{\rm
Bol}/L_{\rm Edd}$ in Fig. \[fig1\]. This implies that the X-ray emission in these sources may also be dominated by ADAFs as other sources in our sample, though the jet emission can not be excluded. Alternatively, to exclude the possible contamination of jet-origin X-ray emission in low Eddington ratio LINERs and due to the fact that LINERs represent a very heterogeneous class [e.g. @gli08], we retry the correlation analysis for LINERs after excluding four LINERs with $L_{\rm 2-10~ keV}<10^{-7}L_{\rm
Edd}$, of which a jet contribution in X-ray emission is expected [@yua05; @wu07]. However, the correlation between $\Gamma$ and $L_{\rm bol}/L_{\rm Edd}$ is still lack for remaining 23 LINERs ($r=-0.129$ at $\sim44\%$ confidence level). The RIAF model is an updated version of the ADAF model, in which the winds are considered and a relatively high $\delta$ (the fraction of the energy directly heating the electrons to the total viscously dissipated energy in the accretion flow) is adopted compared with the ADAF model. The detailed physics of the putative winds is still quite unclear for RIAF model, and therefore, our discussions is mainly based on the well studied ADAF model. Since the general observational features and structures of the ADAFs and RIAFs are quite similar, the main conclusions of this work will not be altered with the RIAF model.
In summary, we propose that the X-ray emission of LLAGNs may originate from the Comptonization process in ADAFs, and LLAGNs may be similar to XRBs at the low/hard state, where the accretion rate is low, and radiation from accretion flow is inefficient. The detailed fitting of the spectra of individual LLAGNs with ADAF-jet models should further clarify the mechanism of X-ray emission, which is beyond the scope of this work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous referee for insightful comments and constructive suggestions. We thank Feng Yuan, Qingwen Wu and Ramesh Narayan for the helpful comments and discussions. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr). This work is supported by National Science Foundation of China (grants 10633010, 10703009, 10833002, 10773020 and 10821302), 973 Program (No. 2009CB824800), and the CAS (KJCX2-YW-T03).
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-----------------------------------------------------------------------------------------------------------------
Name $z$ Sat $\Gamma$ (2-10 keV) log $L_{\rm refs. log $M_{\rm BH}^{*}$ refs.
2-10~ keV}$
----------- ----------------- ----- ------------------------ ------------- ------- ---------------------- -------
Seyfert galaxies
NGC 1275 0.017559 X 1.95$\pm0.01$ 42.83 1 8.51 2
NGC 2639 0.011128 A 1.64$\pm1.00$ 40.82 1 8.02 2
NGC 2655 0.004670 A 1.20$\pm0.60$ 41.85 1 7.77 2
NGC 2685 0.002945 X 0.50$\pm0.20$ 39.94 1 7.15 2
NGC 3031 -0.000113$^{a}$ X 1.90$\pm0.10$ 40.25 1 7.80$^{b}$ 2
NGC 3147 0.009407 C 1.88$_{-0.12}^{+0.18}$ 41.89 1 8.79 2
NGC 3227 0.003859 X 1.50$\pm0.10$ 41.74 1 7.59$^{c}$ 2
NGC 3486 0.002272 X 0.90$\pm0.20$ 38.86 1 6.14 2
NGC 3516 0.008836 C 1.31$\pm0.10$ 42.29 1 7.36$^{c}$ 2
NGC 3941 0.003095 X 2.10$\pm0.30$ 38.88 1 8.15 2
NGC 4051 0.002336 C 1.33$_{-0.03}^{+0.70}$ 41.31 1 6.11$^{c}$ 2
NGC 4138 0.002962 X 1.50$\pm0.10$ 41.29 1 7.75 2
NGC 4151 0.003319 X 1.30$\pm0.02$ 42.47 1 7.18$^{c}$ 2
NGC 4168 0.007388 X 2.00$\pm0.20$ 39.87 1 7.95 2
NGC 4258 0.001494 C 1.41$_{-0.07}^{+0.18}$ 40.86 1 7.61$^{d}$ 2
NGC 4388 0.008419 C 1.70$_{-0.30}^{+0.40}$ 41.72 1 6.80 2
NGC 4395 0.001064 X 1.30$\pm0.10$ 39.81 1 5.04$^{b}$ 2
NGC 4477 0.004520 X 1.90$\pm0.30$ 39.65 1 7.92 2
NGC 4501 0.007609 X 1.50$\pm0.30$ 39.59 1 7.90 2
NGC 4565 0.004103 X 1.80$\pm0.20$ 39.43 1 7.70 2
NGC 4579 0.005067 C 1.88$\pm0.03$ 41.03 1 7.78 2
NGC 4639 0.003395 X 1.81$\pm0.05$ 40.22 1 6.85 2
NGC 4698 0.003342 X 2.00$\pm0.20$ 39.16 1 7.84 2
NGC 4725 0.004023 X 1.90$\pm0.50$ 38.89 1 7.49 2
NGC 5033 0.002919 X 1.70$\pm0.10$ 41.08 1 7.30 2
NGC 5273 0.003549 X 1.46$\pm0.04$ 41.36 1 6.51 2
NGC 5548 0.017175 X 1.69$\pm0.01$ 43.25 1 8.03 2
NGC 7479 0.007942 X 1.80$\pm1.00$ 41.12 1 7.07 2
LINERs
NGC 0315 0.016485 C 1.48$_{-0.19}^{+0.07}$ 41.64 3 9.24 4
UGC 08696 0.037780 C 1.74$_{-0.61}^{+0.81}$ 42.18 3 7.74 5
IC 1459 0.006011 C 1.89$_{-0.11}^{+0.10}$ 40.56 3 9.00$^{b}$ 6
NGC 1553 0.003602 C 1.20$_{-0.10}^{+0.20}$ 40.18 7 8.00 7
3C 218 0.054878 C 1.17$_{-0.23}^{+0.24}$ 42.15 4 8.99 4
NGC 266 0.015547 C 1.40$_{-0.98}^{+1.80}$ 40.88 8 7.90$^{e}$ 9
NGC 2681 0.002308 C 1.74$_{-0.47}^{+0.52}$ 38.94 3 7.20 7
NGC 3125 0.003712 C 2.00$_{-0.30}^{+0.40}$ 39.67 10 5.65 10
NGC 3169 0.004130 C 2.60$_{-1.00}^{+1.20}$ 41.41 8 7.95 6
NGC 3226 0.003839 C 2.21$_{-0.55}^{+0.59}$ 40.74 8 8.24 6
NGC 3718 0.003312 C 1.48$_{-0.35}^{+0.42}$ 40.06 4 7.97 4
NGC 4143 0.003196 C 1.66$_{-0.27}^{+0.47}$ 40.04 8 8.31 6
NGC 4261 0.007465 C 0.71$_{-0.71}^{+0.80}$ 41.15 4 8.94 4
NGC 4278 0.002165 C 1.64$_{-0.14}^{+0.28}$ 39.96 8 9.20$^{b}$ 6
NGC 4374 0.003536 C 2.00$_{-0.10}^{+0.20}$ 39.60 7 8.80 7
NGC 4410A 0.024190 C 1.73$\pm0.14$ 41.24 4 8.83 4
NGC 4457 0.002942 C 1.70$\pm0.30$ 38.99 7 7.00 7
NGC 4494 0.004483 C 1.80$\pm0.30$ 39.00 7 7.60 7
NGC 4548 0.001621 C 1.70$_{-1.60}^{+1.90}$ 39.79 8 7.51 6
NGC 4552 0.001071 C 1.81$_{-0.10}^{+0.24}$ 39.41 3 8.50 7
NGC 4594 0.003639 C 1.41$_{-0.10}^{+0.11}$ 40.07 3 9.04$^{b}$ 6
NGC 4736 0.001027 C 2.00$_{-0.06}^{+0.23}$ 38.65 3 7.42 6
NGC 5746 0.005751 C 1.22$_{-0.39}^{+0.14}$ 40.07 3 7.49 5
NGC 6240 0.024480 C 1.03$_{-0.15}^{+0.14}$ 42.04 3 9.11 5
NGC 6251 0.024710 C 1.60$_{-0.22}^{+0.29}$ 41.84 3 8.73$^{f}$ 6
NGC 6500 0.010017 C 3.10$_{-1.70}^{+1.10}$ 39.74 8 8.28 6
NGC 7130 0.016151 C 2.51$_{-0.22}^{+0.41}$ 40.49 3 7.54 11
-----------------------------------------------------------------------------------------------------------------
> $^{a}$: distance of 3.5 Mpc [@pan06]. $^{*}$: Calculated using the [@tre02] relation, otherwise stated: – $^{b}$ stellar kinematics; $^{c}$ reverberation mapping; $^{d}$ maser kinematics; $^{e}$ $M_{\rm BH}-L_{\rm bulge}$ relation; $^{f}$ gas kinematics. refs: – (1) [@pan04]; (2) [@pan06]; (3) [@gon06]; (4) [@sat05]; (5) Hypercat database at http://leda.univ-lyon1.fr/; (6) [@mer03]; (7) [@flo06]; (8) [@ter03]; (9) [@don06]; (10) [@dud05]; (11) [@gu06].
![The 2-10 keV photon index versus the Eddington ratio. The solid circles are for local Seyfert galaxies, while the triangles are for LINERs. For comparison, PG quasars of Shemmer et al. (2006) are presented with the rectangles. The solid line is the weighted linear fit to our LLAGNs sample, while the dashed line is the weighted linear fit to Shemmer et al. (2006) PG quasars, and the dotted line is the linear fit to the sample of 35 moderate- to high-luminosity radio quiet AGNs (Shemmer et al. 2008).[]{data-label="fig1"}](f1.eps){width="80.00000%"}
[^1]: E-mail: gumf@shao.ac.cn
[^2]: Available on-line at http://www-obs.univ-lyon1.fr/hypercat/
|
---
abstract: 'Gamma rays at rest frame energies as high as 90 GeV have been reported from gamma-ray bursts (GRBs) by the [*Fermi*]{} Large Area Telescope (LAT). There is considerable hope that a confirmed GRB detection will be possible with the upcoming Cherenkov Telescope Array (CTA), which will have a larger effective area and better low-energy sensitivity than current-generation imaging atmospheric Cherenkov telescopes (IACTs). To estimate the likelihood of such a detection, we have developed a phenomenological model for GRB emission between 1 GeV and 1 TeV that is motivated by the high-energy GRB detections of [*Fermi*]{}-LAT, and allows us to extrapolate the statistics of GRBs seen by lower energy instruments such as the [*Swift*]{}-BAT and BATSE on the [*Compton Gamma-ray Observatory*]{}. We show a number of statistics for detected GRBs, and describe how the detectability of GRBs with CTA could vary based on a number of parameters, such as the typical observation delay between the burst onset and the start of ground observations. We also consider the possibility of using GBM on [*Fermi*]{} as a finder of GRBs for rapid ground follow-up. While the uncertainty of GBM localization is problematic, the small field-of-view for IACTs can potentially be overcome by scanning over the GBM error region. Overall, our results indicate that CTA should be able to detect one GRB every 20 to 30 months with our baseline instrument model, assuming consistently rapid pursuit of GRB alerts, and provided that spectral breaks below $\sim 100$ GeV are not a common feature of the bright GRB population. With a more optimistic instrument model, the detection rate can be as high as 1 to 2 GRBs per year.'
author:
- 'Rudy C. Gilmore'
- Aurelien Bouvier
- Valerie Connaughton
- Adam Goldstein
- Nepomuk Otte
- 'Joel R. Primack'
- 'David A. Williams'
title: 'IACT observations of gamma-ray bursts: prospects for the Cherenkov Telescope Array'
---
Introduction {#sec:intro}
============
The observation of gamma-ray bursts (GRBs) with ground-based atmospheric Cherenkov telescopes has been a tantalizing possibility in recent years. Powerful $>$10-meter telescope arrays such as H.E.S.S., MAGIC, and VERITAS have come online in the last decade, and satellite detectors such as the [*Swift*]{} Burst Alert Telescope (BAT) are capable of providing the necessary localization of GRB events within seconds over the Gamma-ray burst Coordinates Network (GCN[^1]). Despite major campaigns to respond to satellite burst alerts at all three of these instruments [@aharonian09; @albert07d; @garczarczyk08; @aune11], and dozens of follow-up attempts, no conclusive detection of a GRB with an IACT has yet been made. Air shower arrays have also played a complementary role in the search for GRBs. A hint of emission was detected by the Milagrito air-shower array [@atkins03]; however no detections were found by the later Milagro experiment [@abdo07].
Prior to the launch of [*Fermi*]{} on June 11, 2008, knowledge about the emission of GRBs above 100 MeV was limited to a small number of events observed simultaneously in the EGRET and BATSE instruments on the [*Compton Gamma-ray Observatory (CGRO)*]{} . One fascinating finding by EGRET was the discovery of an 18 GeV photon associated with GRB 940217, 1.5 hours after the event. This was a much longer time than the duration of the burst as measured at lower energies by BATSE, which determined a T90 of 150 seconds[^2] (defined as the time between the arrival of 5 percent and 95 percent of the observed fluence). Though the statistics of these EGRET observations were quite limited, they suggested that high energy emission in GRBs did occur in some fraction of events, and that it could last longer than the lower energy emission.
[*Fermi*]{}-LAT, covering the energy range of 20 MeV – 300 GeV [@atwood09], has now detected emission from over 20 GRBs, of the some 800 detected by GBM at 8 keV – 40 MeV energy range [@meegan09]. Photons from four of these LAT GRBs were detected above 10 GeV, and 2 above 30 GeV. At present, the highest energy photons that have been associated with any GRB are a 33.4 GeV photon from long-duration GRB 090902B, and a 31 GeV photon from short-duration GRB 090510. With the redshift of GRB 090902B determined to be $z=1.822$ [@abdo09c], this implies a rest-frame energy of 94 GeV. The LAT therefore confirms that emission in the 10 to 100 GeV decade occurs in at least a small fraction of both short- and long-duration GRBs. However it is not clear how these findings for bright sources extrapolate to the rest of the population, and whether suppression of GeV-scale emission might also happen in a substantial number of cases.
The other major feature of high-energy gamma-ray emission seen by the LAT is the verification of a timescale for the VHE emission that is often longer than that seen by GBM or other experiments sensitive to soft gamma-rays. An unexpected finding is the delayed onset of emission above 100 MeV, typically by $\sim$10 percent of the GBM T90 duration [@dermer10]. As discussed in @ghisellini10, emission at high energy is seen to then continue well beyond this time, with a lightcurve described by a powerlaw with slope -1.5. Understanding the source of this emission, which begins well within the prompt phase of the burst but continues into the afterglow time period, is challenging for models of the high-energy production mechanism.
GeV-scale emission could arise through several mechanisms, and understanding the impact of each on the cumulative spectra will require multiwavelength observations over many orders of magnitude in energy, combined with high event statistics. High energy emission during the prompt phase of the GRB can be most simply explained by a spectral extension of the internal shock processes (inverse Compton (IC) and synchrotron) that produce the keV–MeV flux . The observed spectrum of GRB 080916C, seen over $\sim 7$ orders of magnitude by the [*Fermi*]{}-LAT and GBM instruments, could be explained by a constant synchrotron origin [@abdo09a]. Other possibilities include emission of GeV photons from external shocks in the early afterglow of the GRB [@fan08] or from the reverse shock formed when the GRB ejecta encounter the interstellar medium [@wang05]. The former can explain the delayed onset of high-energy emission seen in most LAT-detected GRBs. Several authors have preferred a purely synchrotron origin in the external shock to describe the LAT GRBs , in contrast to one invoking inverse Compton from the shock electrons [@zou09]. Finally, hadronic processes have been proposed as a source of the high energy component, an idea that connects this radiation to the production of ultra-high energy cosmic rays [@razzaque09a; @asano10].
The redshifts at which GRBs have been detected span from the local universe to $z=8.2$ [@salvaterra09], corresponding to $\sim$95 percent of the age of the universe. Confirmed redshifts for LAT-detected GRBs range span a wide range of this distribution from $z=0.736$ (GRB 090328) to 4.35 (GRB 080916C). This suggests that an IACT-detected GRB could occur at essentially any redshift where star-formation has been observed.
The cosmological UV-IR background radiation produces a barrier to high-energy photons at extragalactic distances . Moreover, the large majority of GRBs are believed to exist at high redshift, where the background flux is highly uncertain, and gamma-rays at observed energies as low as 10 GeV can be impacted [@gilmoreUV]. The effective area, angular resolution, background rejection capabilities of IACTs are strongly energy-dependent, and all of these properties decline in quality below a few hundred GeV for current-generation instruments. Ground-based GRB observations therefore take place in an energy regime where both the low-energy instrument sensitivity and the impact of cosmological background radiation must be carefully taken into account if realistic predictions are to be made.
Our goal in this work is not to comment on the preferred emission mechanism for high-energy GRB photons, but rather to build a phenomenological model that best describes this part of the spectrum, based on the limited set of GeV burst detections to date and the much larger body of data available from lower energy experiments. As the weight of the evidence in the brightest LAT GRBs does indicate the presence of emission mechanisms beyond those producing the prompt flux, we have included allowances for both extended temporal components and separate spectral components from the Band spectrum in this work. In the following sections we show our predictions for the rate at which CTA will detect gamma-ray bursts, using the information available to us from [*Fermi*]{}-LAT, lower energy satellite experiments, and attempted IACT observations with the current generation of instruments. In §\[sec:grbdet\], we describe the model used in this work. In §\[sec:results\], we show results for properties of detected GRBs, calculating both the rates at which detections occur in our model and the photon statistics that can be expected from a detection. We use the detection rates by the currently operating [*Swift*]{}-BAT and [*Fermi*]{}-GBM instruments as the basis for the calculation. We also investigate the effect of varying critical input parameters in our model, such as CTA energy threshold, background rate, and telescope response time delay. In §\[sec:specgrbs\] we show the spectra that could be available to CTA from a few sample GRBs with various properties. The topic of §\[sec:upperlims\] is a comparison of our model with the upper limits that have been set by observations to date with IACT instruments, and with the GRB rate observed with [*Fermi*]{}-LAT. In §\[sec:disc\] we summarize and discuss our findings.
Modeling the Detection of GRBs {#sec:grbdet}
==============================
To a large extent, the challenge of modeling gamma-ray bursts arises out of the large variance in properties seen between events, and the lack of a simple model describing the radiative mechanism. In particular, each of the 4 bright GRBs seen by LAT above 10 GeV shows differing behavior. The GeV-scale emission in GRB 080916C, observed by [*Fermi*]{} three months after launch, was found to be well-described by an extension of the Band function seen at keV and MeV energies [@abdo09a]. Separate spectral components from the Band function were found to be required to match the GeV-scale emission of the three other brightest GRBs in the LAT catalog. Short-duration GRB 090510 was found to be dominated by a hard spectral component of index -1.62 above $\sim 100$ MeV, compared to the average Band high-energy index of -2.4 [@abdo09d; @ackermann10]. Long GRB 090902B was dominated above 100 MeV by emission with a spectral index determined by LAT to be -1.93, compared to an upper Band index of -3.8 [@abdo10a]. The high energy emission also extended in time well past the prompt phase as determined at lower energies, with an only slightly softer spectrum of -2.1 on a timescale of $10^3$ s. Finally, GRB 090926A, detected up to 19.6 GeV with the LAT, was best fit with a high energy hard component of spectral index of -1.72 and an exponential cutoff at 1.4 GeV, with a high-energy Band index of -2.63 [@bregeon11].
An earlier work, @gilmoreGRB, addressed the question of detection of GRB photons with the MAGIC telescope and the [*Fermi*]{}-LAT and the impact of the UV-optical background light. The basis for this model was the population of GRBs seen by [*Swift*]{}-BAT, for which redshifts have been confidently determined, and the high-energy statistics of GRBs detected with [*CGRO*]{}-EGRET. The flux at the MAGIC energy range was calculated by assuming a power-law spectrum continuing to $\sim 200$ GeV. Due to the absorption by the extragalactic background light (EBL), photons at higher energy were greatly attenuated and insignificant in number, and the UV-optical EBL therefore has a large impact on GRB detectability. This calculation showed that MAGIC was capable of detecting tens to thousands of photons from a bright GRB at high redshift, provided that the burst could be observed with a sufficiently low energy threshold.
In this work, we attempt to improve on this previous calculation to make predictions for CTA. To construct a model for the distribution of GRB properties, we draw upon keV and MeV data from two satellite experiments: [*CGRO*]{}-BATSE and [*Swift*]{}-BAT, plus data from the high-energy GRB detections with [*Fermi*]{}-LAT. While [*Swift*]{} has provided us with a large number of GRBs with determined redshifts, the BAT instrument is generally not capable of resolving the Band function peak of the GRB spectra. We therefore combine the [*Swift*]{} redshift distribution with the distribution of Band function parameters seen by the BATSE experiment on the [*CGRO*]{} satellite, at observed energies between 20 keV and 2 MeV[^3]. Band function fits for BATSE GRBs are taken from the BATSE 5B catalog[^4] (Goldstein et al. [*in prep*]{}). To extrapolate the spectra of these bursts to VHE energies, we use the statistics of GRBs seen by the [*Fermi*]{}-LAT instrument above 100 MeV within its first two years of observations.
High energy extrapolation {#sec:highengext}
-------------------------
Predicting the GeV-scale emission of GRBs from the well-sampled statistics of lower-energy instruments requires a considerable amount of extrapolation. Some 4 logarithmic decades in energy lie between the upper extent of the BATSE energy range ($\sim 2$ MeV) and the energy threshold of CTA, which we consider to be between 10 and 25 GeV. We describe our two different approaches to performing this extrapolation in this section.
### Band-function extension model {#sec:bandexdesc}
As a minimal model, we consider the fluence predicted for GRBs at high energy without any significant deviation from the Band fit [@band93]: $$\begin{aligned}
\frac{dN}{dE} & \propto E^\alpha e^{-E(2+\alpha)/E_p}; \; \; E \leq \frac{\alpha-\beta}{2+\alpha}E_p, \nonumber \\
\frac{dN}{dE} & \propto E^\beta(\frac{\alpha-\beta}{2+\alpha}E_p)^{\alpha-\beta}e^{\beta-\alpha}; \; \; E > \frac{\alpha-\beta}{2+\alpha}E_p.\end{aligned}$$ Here $\alpha$ and $\beta$ are low- and high-energy spectral indices, and $E_p$ is the “peak energy” describing the location of the turnover. In this extended Band-function model, termed “bandex” in subsequent plots and discussions, the high energy spectrum is assumed to merge seamlessly with the spectral fit determined at lower energy. A similar model was used in estimating the detectability of GRBs with [*Fermi*]{} LAT in @band09. The high energy normalization is therefore determined by the Band function peak energy and normalization and the upper energy index $\beta$, which continues to GeV energies. In Figure \[fig:bandind\], we show the distribution of values for $\beta$ against the BATSE fluence. A minority of the GRBs in the sample, about 13 percent, have a hard spectrum with $\beta > -2$. We have enforced the requirement that the total fluence per logarithmic decade not be higher in the GeV range than at BATSE energies, and thus we reset these cases to have $\beta = -2$. This requirement is consistent with the LAT–GBM fluence relation for long GRBs observed by LAT, but not the short bursts (see Figure \[fig:flucomp\]), where fluence ratios greater than 1 have been observed.
### Fixed parameter model {#sec:fixeddesc}
In this fixed-parameter (“fixed”) model, we make the assumption that the relative fluence between BATSE energies and GeV energies can be described by a single ratio, which we set here to 0.1. The choice of this parameter is based upon the corresponding ratios found for simultaneously-observed BATSE–EGRET GRBs and for GBM–LAT long-duration GRBs. In Figure \[fig:flucomp\], we reproduce figure 1 in @dermer10, showing this relation for several [*Fermi*]{} and [*CGRO*]{} GRBs. The spectral index at high energies is set to $-2$, consistent with the mean value for EGRET GRBs of $-1.95$ , and near the center of the distribution for LAT-detected events [@ghisellini10]. In general, this model requires a significant departure from the extrapolated Band function, and implies the appearance of a separate high-energy spectral component. As discussed above, such components were seen in GRB 090902B and GRB 090510. A separate spectral component was also the preferred model in describing the total time-integrated emission from GRB 090926A, albeit with a spectral cutoff of this component at 1.4 GeV [@bregeon11]. Though this cutoff component for GRB 090926A was found with high significance in the integrated fluence, in time-resolved analysis this fit was only preferred over simpler power laws within a single narrow time window. As GRB 090902B was found to have a fluence ratio of nearly 10 percent and a time-integrated high energy index 1.93 [@abdo09c; @dermer10], it can be considered the prototypical GRB in motivating this extrapolation scheme. While the GeV-BATSE ratio observed for short GRB 090510 was considerably higher, $\sim 1$, we have not included a separate account of the short population because the emission of these GRBs at multi-GeV energies remains very poorly understood, and they are a small part of the [*Swift*]{} GRB sample, $\sim 9$ percent. Additionally, the time delay to see GRBs from the ground makes short GRBs very difficult to detect even with a flux factor of unity, and including such an possibility is found to have little effect on our findings.
### High-energy lightcurve {#sec:lci}
As we shall see, the lightcurve and emission duration at GeV energies are critical variables in determining the detectability of GRBs from the ground, where the response time of the telescope to transient alerts from satellite instruments limits observations to $\gtrsim 1$ min after the start of the event, and the background can obscure low-luminosity emission occurring over long timescales. Motivated by the finding of @ghisellini10, we will assume that the GRB lightcurve in the early afterglow phase can be described as a power-law falloff. The prompt phase of the GRB can be demarcated by T90 as determined by lower energy gamma-ray instruments. Luminosity during this phase is often seen to fluctuate rapidly and unpredictably, with spiked emission features that undergo rapid exponential time decay [@piran04]. Since only considering time-averaged behavior (where the typical erratic prompt emission of GRBs is neglected) will not affect our results in term of detection rate and photon statistics, we describe this phase as having constant flux. Our total modeled GeV lightcurve then is a plateau from the burst onset ($t=0$) to $t=\mbox{T90}$, followed by a powerlaw falloff: $$\begin{aligned}
\label{eq:lci}
F(t) & = F_0 ; \; \; \; \; & t< \mbox{T90} \\
F(t) & = F_0 [\frac{t}{\mbox{\scriptsize T90}}]^{-\gamma}; & t \geq \mbox{T90} \nonumber\end{aligned}$$ Here $\gamma$ is the power-law index of the afterglow lightcurve. We will use 1.5 as a fiducial value, but will also explore the impact of other possibilities in §\[sec:parvar\]. Note that for this value, two-thirds of the total emission emerges after T90. We assume no spectral evolution between the prompt and afterglow phases.
Redshift distribution {#reddist}
---------------------
The observed fluence distribution for GRBs is not assumed to be directly dependent on redshift. However, redshift is a crucial factor in determining GRB detectability because the cosmological opacity due to EBL is determined by the GRB redshift. We make the assumption in this calculation that the redshift distribution of GRBs to which CTA responds will be similar to that seen by the [*Swift*]{}-BAT experiment, which is the only large sample of GRB redshifts available. Approximately one-third of the GRBs in the [*Swift*]{} population have well-determined redshifts. In Figure \[fig:swiftzs\], we show the distribution of [*Swift*]{} redshifts for 167 GRBs, taken from the online Swift GRB Lookup Table[^5], along with our fit to the distribution which is used in most of this analysis. In §\[sec:gbm\], we will make a speculative alteration to this distribution to describe the redshift distribution of GBM GRBs.
In this work, we make the assumption that redshifts and observed fluence are uncorrelated. In Figure \[fig:fludistbyz\], we show the distribution in fluence for [*Swift*]{} GRBs divided into tertiles in redshift. The distribution is not found to evolve strongly in redshift, and the lowest redshift bin actually has the lowest median fluence. It has been suggested [@lloydronning02; @salvaterra09a] that luminosity evolution in redshift (e.g., by a factor $(1+z)^{\alpha}$ with $\alpha \gtrsim 1$) is required to best fit the redshift-luminosity relation seen in GRBs. While the existence and possible origins of such a factor remain controversial, such an evolutionary term could account for our findings in Figure \[fig:fludistbyz\].
### [*CGRO*]{}–[*Swift*]{} fluence matching {#sec:batseswiftflumatch}
The [*Swift*]{}-BAT population of GRBs is found to have a lower average fluence distribution than the Band-function fits [*CGRO*]{}-BATSE 5B population that is sampled to determine GRB fluence and spectral properties for our burst samples, when the latter is integrated over the BAT energy range. We have adjusted the total fluence of the BATSE burst population by a global factor to better match that of [*Swift*]{}, which we consider as the GRB trigger instrument in §\[sec:de\] and in §\[sec:swiftdetrate\], by using a Kolmogorov–Smirnov test to minimize the difference between the distributions for the brightest 50 percent of bursts in the BATSE sample and the brightest 50 percent in the BAT sample. The fact that we have restricted our fit to the brightest 50 percent of GRBs is motivated by our finding that GRBs with less than median fluence are not detectable by CTA even under optimal conditions. The multiplier applied to BATSE fluences in §\[sec:de\] is . The distribution of the adjusted BATSE population after this correction is shown alongside the [*Swift*]{}-BAT fluence distributions in Figure \[fig:fludistsb\].
### EBL attenuation {#sec:eblatt}
The EBL, specifically at UV-optical wavelengths, is responsible for attenuating the signal of high energy gamma-rays. In some EBL models, this attenuation can affect gamma rays at observed energies as low as 20 to 25 GeV for high redshift sources. The effect of the EBL is to reduce the number of gamma-rays received at high energy, and to reduce the detectability of high-redshift GRBs. We use as a standard assumption in this work the EBL model and opacities of @gspd11 (GSPD11), based on the semi-analytic modeling of @sgpd11 and @somerville08. However, large uncertainties in the EBL normalization are unavoidable at high redshift, and it is useful to see exactly how this uncertainty can influence our predictions. In §\[sec:eblimpact\] we will look at our results assuming a few different models for the background light.
Telescope properties
--------------------
The second step in constructing our model is a parametrization of the performance of the CTA. As many of the array properties are indeterminate at the time of writing, we have relied on the design concept for the array described in @ctaconcept10, as well as reasonable extrapolations from the current generation of IACTs, particularly the MAGIC and VERITAS telescopes.
### Effective area
Our assumptions about the effective area of CTA are based on Configuration E, which assumes a central cluster of four 24-meter class large-size telescopes (LSTs) that provide sensitivity to the lowest energy gamma-rays, and an additional 23 medium-size telescopes of the 12-meter class (MSTs) providing sensitivity at higher energies, $\gtrsim 100$ GeV. Sensitivity at energies above 1 TeV, which is provided by more dispersed arrays of 7-meter class small-size instruments (SSTs), is not crucial to our results here, as most GRBs will occur at redshifts for which emission at these energies is strongly attenuated by the EBL.
The effective area function of the instrument, after all analysis cuts have been performed, is used to determine the counts per GRB and the significance of the detection presented in the next section. As the actual function is unknown at this point in time, we assume two functions for the effective area, which are shown in Figure \[fig:ea\]. Each of these functions includes contributions from the LST and MST arrays, which dominate the total effective area below and above $\sim 100$ GeV, respectively. For the more conservative of the two, labeled “CTA baseline”, the LST contribution is created by shifting the standard VERITAS area function to the 25 GeV threshold that is expected for CTA [@teshima11]. The MSTs use a function that is unmodified in the energy dimension, but has a normalization factor that assumes a linear scaling in effective area with telescope number; we adopt a factor 25/4 = 6.25 for this case. We believe these numbers to be a reasonable estimate of the capabilities of the array. As an alternative, we also present results that incorporate several enhancements to the baseline assumption, and are intended to represent the best possible performance that can reasonably be expected from the instrument. For this case, labeled “CTA optimistic”, the LSTs have the same normalization as the CTA baseline array, but with an energy threshold of 10 GeV, which might be achieved through improved trigger and background rejection techniques. The MSTs are given a normalization 3 times that of the baseline (75/4 = 18.75). This could either be taken to represent the coverage of a 75 telescope array, or a smaller number of telescopes if the effective area increase with telescope number is found to scale at a faster than linear rate. The energy shifts for the LSTs in each case are assumed to take place multiplicatively, i.e., $A_{eff}^{CTA}(E)=A_{eff}^{VER}(k*E)$, with $k=4$ for the baseline area function and 10 for the optimistic.
These parametrizations of the telescope effective area refer to a source at zenith. The changes in telescope performance away from zenith, i.e. at an angle $\theta_z$, are considered using adjustments to the effective area function. Viewing sources at increasing $\theta_z$ generally introduces a higher energy threshold to the observation, due to the increasing distance between the shower core and the telescope array. The increased area of the light pool is also the reason why the effective area increases at higher energies. The first effect is parametrized as a multiplicative energy shift in the effective area function; this is considered as a shift in the telescope energy threshold by a factor $\cos(\theta_z)^{-3}$ away from zenith. The choice of this index, and the use of a multiplicative shift rather than an additive shift, are both motivated by a fit to the VERITAS effective area at various elevations, with cuts optimized to soft sources (VERITAS Collaboration, private communication). The second effect is treated by enhancing the effective area at all energies by a multiplier $\cos(\theta_z)^{-2}$, which represents the geometrical increase in the area of the Cherenkov light pool. The off-zenith effective area is then $$A_{eff}^{CTA}(E,\theta_z)=\eta^{-2}A_{eff}^{CTA}(\eta^3 E,\theta_z=0),
\label{eq:zenshift}$$ where $\eta=\cos(\theta_z)$ and $A_{eff}^{CTA}(E,\theta_z=0)$ are the at-zenith functions shown in Fig. \[fig:ea\].
### Instrument background {#sec:bkg}
Understanding the instrument backgrounds that will impact GRB observations is critical to predicting the detection rate. Unfortunately, while published rates from present-day telescopes can be used to predict the background that will impact observations at $\gtrsim 100$ GeV, little is known about how these rates will extend to lower energies.
We base the background rate in our analysis on that of typical VERITAS observations, using the assumption that the four LSTs will achieve a similar rejection fraction as the four VERITAS instruments above each of their respective energy thresholds, after assuming a power law scaling of the background from 100 GeV to lower energies. Meanwhile, it is assumed that the MSTs will, at a maximum, have the same background as the VERITAS array multiplied by the scaling in effective area normalization. The MSTs contribute only a minority of the total background rate in our analyses, so an overestimate of their contribution would have little effect on our results. The background spectrum is assumed to be a power law of index -2.7. We then take the background at a given energy to scale in proportion to the effective area at that energy. The total background rate for a telescope set (LST or MST) is then $$\left(\frac{dN}{dt}\right)_{bkg} = A\int E^{-2.7}\: A_{eff}^{CTA}(E) \; dE .
\label{eq:bkg}$$ The normalizing factor $A$ is chosen so as to produce the rate for the VERITAS instrument, when this integral is performed over the VERITAS effective area function. A rate of 6 counts per minute (0.1 Hz) is assumed for VERITAS in the case of the baseline effective area, and this is reduced to 2 counts per minute for the optimistic case.
Using the baseline effective area curve and background rate, together with the scaling described, we find a differential sensitivity above 100 GeV similar to that shown for configuration E in Figure 24 of @ctaconcept10. Below 100 GeV, we find that assuming a $\sim$35 GeV threshold for the LST array in our model produces differential sensitivity similar to configuration E, while the 25 GeV threshold we adopt based @teshima11 gives a sensitivity as much as three times better at some energies.
### Response time {#sec:response}
The transient and random nature of GRB emission represents the main difficulty in detecting emission from these sources. The onboard satellite localization time of the event, transmission of the data to the ground, the observer’s response time, and slew time for the IACT all contribute to a total delay time for the commencement of observation, which we quantify in this work as $T_{delay}$. The localization time is dependent on the instrument and brightness of the GRB, but times of $<$ 15 sec are typical. The transmission time of GRB coordinates is expected to be nearly instantaneous [@bastieri05]. The LSTs, which provide coverage at the crucial low energies, are expected to have a slew time of 20 to 30 seconds, while the MSTs may be somewhat slower. As a standard assumption, we assume a total response time of 60 seconds in this work for the LSTs and 100 seconds for the MSTs, but will also discuss in the next section the effect of varying this parameter. To date, most observations with the MAGIC telescopes have commenced after considerably longer times despite the instrument’s rapid slew capabilities, with only a minority occurring with total delay times of $<$ 100 sec [@garczarczyk08]. It may be that the longer delay times are due to reasons other than the mechanical capabilities of the instrument. While the inner telescopes of CTA are expected to have generally the same slewing capabilities as those of MAGIC, we will allow for the possibility that future improvements to the GCN and telescope alert procedures and observer response time could lower the typical delay time from current values.
Limits from current experiments {#sec:upperlims}
===============================
Before discussing our results for simulated CTA observations, it is useful to compare the predictions of our models to the findings of current-generation GeV-scale experiments. This enables a test of our model beyond the bright events seen by the [*Fermi*]{}-LAT.
Analysis of VERITAS GRBs {#sec:verupper}
------------------------
While upper limits on GRBs have been published for all of the major current-generation IACT experiments, a problem arising when comparing to these limits is the dependence on a particular set of assumptions about the high-energy spectrum, lightcurve, and EBL model. These must be equivalent for a meaningful comparison to be made between any two sets of observations and/or predictions. A confirmed GRB redshift is also necessary to determine the impact of EBL on attenuated flux, which can change flux predictions by orders of magnitude.
Here we consider the GRB limits from VERITAS presented in @aune11. This work considered 16 GRBs observed by VERITAS over a 30-month period, 9 of which have spectroscopically-confirmed redshifts. The analysis assumed a characteristic afterglow decay of $t^{-1.5}$ to find the optimal integration timescale for detection, as we have applied throughout this work. The effect of the EBL was compensated by using the model of @gspd11, as used in this paper. Results are presented for gamma-ray spectra of $dN/dE \sim E^{-2.5}$ and $E^{-3.5}$.
To compare our predictions with these results, we have performed a calculation of the high energy emission for the 7 GRBs from @aune11 that have both redshift determinations and have been analyzed assuming a $t^{-1.5}$ afterglow decay. To model these GRBs in such a way that a direct comparison is possible, we use a modified version of our fixed model. Unfortunately, the flux information provided by [*Swift*]{}-BAT is generally not sufficient to resolve the spectral peak of the GRB emission and determine the Band function spectral fit. In our fixed model, we assume that the flux seen in the [*Swift*]{}-BAT bandpass (15–150 keV) is related to the flux in the 20 keV –2 MeV band by applying the common Band function used in @gilmoreGRB, which leads to a ratio $\sim 5$ between flux in the 20 keV –2 MeV and [*Swift*]{} bands. While this factor is intermediate to the range seen in BATSE GRBs [@preece00], the considerable variations present in the spectral indices and peak energy could change this ratio by a large factor. To match the spectral index of -2.5 used in the standard analysis, we assume a spectral index of -2, as used in our work, with a spectral turnover to -2.5 at the energy threshold of each GRB observation.
GRB ID F$_{\mbox{\scriptsize BAT}}$ Redshift VERITAS $<$F$>_{\mbox{\scriptsize UL}}$ $<$F$>_{\mbox{\scriptsize fixed}}$ $<$F$>_{\mbox{\scriptsize fixed}}$/$<$F$>_{\mbox{\scriptsize UL}}$
--------- ------------------------------ ---------- ----------------------------------------- ------------------------------------ --------------------------------------------------------------------
070419A 5.58 0.97 $2.3 \times 10^{-11}$ $9.36 \times 10^{-16}$ $4.07 \times 10^{-5}$
070521 80.10 0.553 $7.6 \times 10^{-12}$ $1.13 \times 10^{-11}$ [**1.48**]{}
080310 23 2.43 $ 2.8 \times 10^{-11}$ $3.27 \times 10^{-17}$ $1.17 \times 10^{-6}$
080330 3.4 1.51 $3.8 \times 10^{-11}$ $2.16 \times 10^{-14}$ $5.69 \times 10^{-4}$
080604 8.0 1.416 $3.1 \times 10^{-11}$ $8.76 \times 10^{-13}$ 0.028
080607 240 3.036 $9.8 \times 10^{-11}$ $5.50 \times 10^{-16}$ $5.61 \times 10^{-6}$
090418A 46 1.608 $6.9 \times 10^{-11}$ $5.31 \times 10^{-12}$ 0.077
In Table \[tab:verlimits\], we compare predictions from our model for GRBs with upper limits placed by VERITAS. In one instance, GRB 070521, highlighted in the rightmost column, the predicted flux exceeds the upper limit set by VERITAS by a factor of about 1.5. We do not believe that this case alone poses a problem for our fixed model, as this factor is much smaller than the scatter seen in MeV-GeV fluence ratios for bright GRBs (Fig. \[fig:flucomp\]), and there is an additional degree of uncertainty in extrapolating the [*Swift*]{}-BAT bandwidth to the BATSE energy range. We also note that the $T_{delay}$ and optimal $T_{obs}$ reported in this case are both quite high, 1118 s, and 1809 s, compared to the T90 duration of 38 s. The predicted high energy emission for this GRB in our model is therefore reliant on the extended-duration lightcurve, up to a timescale of $\sim1$ hour. Finally, in reporting the redshift for GRB 070521 listed in Table \[tab:verlimits\], @hattori07 noted that the detection could be spurious due to a faint afterglow from the supposed host. If the GRB were at higher redshift, then our predicted flux would be lower and the disagreement lessened or removed entirely.
GRB detection with [*Fermi*]{}-LAT
----------------------------------
[*Fermi*]{}-LAT provides the largest set of high-energy GRB detections, and we have compared the rate of detections with this instrument with those predicted in our models. To describe the LAT, we have used the P6 parameters described in @rando09 and on the [*Fermi*]{}-LAT performance website[^6]. Our assumed effective area is based on the “transient” event type, and we assume a background rate of 0.05 Hz within the point-spread function of the instrument. Both the background rate and the effective area at all energies are assumed to evolve as a function of boresight angle uniformly in proportion to the background at 10 GeV. This analysis uses the same parameters assumed previously, the only change being that there is no delay time in LAT observations, and $T_{obs}$ (the time over which the signal is integrated) is considered on timescales as short as 0.1 s.
We find detection efficiencies (fraction of GRBs that are detected) above 100 MeV of 12.6 and 5.1 percent for the bandex and fixed models, respectively, for GBM bursts occurring within 70 degrees of the LAT boresight. The specifics of the calculation used to find detection efficiency will be described in detail in §\[sec:de\]. The fact that these results are inverted from the pattern seen in CTA results in the following sections, where the fixed model generally has a higher detection efficiency, can be explained by the lower energy range covered by the LAT, which favors the softer GRBs in the bandex sample. These fractions can be compared to the 2-year results of @bissaldi11, which report 270 GBM GRBs within this angle, and 18 LAT detections; an overall ratio of 6.7 percent. We have not accounted for autonomous repoints done by the telescope, which have occurred in 45 cases, and could potentially have the effect of raising the detection rate. GRBs occurring at the center of the LAT field of view are found to have a detection efficiency of about 1.3 times that of all GRBs within 70 degrees of boresight. As spacecraft repointings have only occurred in a relatively small fraction of cases (45 out of $\sim$540 GRBs), we conclude that the overall impact of these repoints on the detection efficiency is expected to be minor, even before the observational time delay introduced by the telescope slew is taken into consideration.
As discussed in the Introduction, 4 of these detected GRBs have had detected emission above 10 GeV. In both of our models we find that about 30 percent of LAT-detected GRBs have at least 1 detected photon above 10 GeV, within 1000 sec of the event onset. This is only slightly higher than the corresponding 2-year ratio of $4/18 \approx 22$ percent. Due to the long timescale assumed, the model may overestimate the detection rate in some cases due to practical observing constraints, such as instances in which the GRB happens to vanish below the horizon.
Results {#sec:results}
=======
[ ]{}
The GRB detection capabilities of CTA can be described as the product of two independent factors: $$\label{eq:dr}
\mbox{Detection Rate} = \mbox{DE} \times \mbox{TR}. $$ Here DE denotes the detection efficiency, or probability that a randomly-selected GRB for which CTA is able to take data will be detected with a significance of more than 5 standard deviations, and TR is the trigger rate at which the telescope is able to successfully respond to triggers from satellite instruments. The product of the two is the rate at which confirmed, statistically-significant detections of GRBs will take place. The factor TR can be decomposed into several independent parameters, which are addressed in §\[sec:detrate\].
In the following two sections, we show the results of our modeling of the detection efficiency, and its dependence on various instrumental properties. In §\[sec:de\], we will assume GRBs follow the statistics seen in [*Swift*]{} detections, and use the redshift distribution and [*CGRO–Swift*]{} flux multiplier of that was motivated in §\[sec:grbdet\]. In §\[sec:gbm\], we will consider the alerts provided by the GBM instrument on [*Fermi*]{}, and we will update the redshift distribution and fluence multiplier to better fit this data set. A particular difficulty arising from GBM alerts is the large positional uncertainty, which is in many cases larger than the CTA field of view, and we devote most of this section to addressing some possible strategies to maximize the usefulness of GBM alerts.
Simulated observations and detection efficiency for [*Swift*]{}-like GRBs {#sec:de}
-------------------------------------------------------------------------
For each of the two spectral models considered in §\[sec:highengext\], we consider observations of GRBs randomly placed within a disk of around zenith. Once the spectrum, lightcurve, and telescope effective area and threshold energy have been determined, we calculate the total integrated counts and background counts over 17 observation timescales ($T_{obs}$), with equal logarithmic spacing from 1 to $10^4$ seconds. Observations with the telescope are assumed to commence at a time $T_{delay}$ after the beginning of the burst, and end at $T_{delay}+T_{obs}$. Observed energies are considered from 1 GeV to 1 TeV, although most GRBs experience a spectral cutoff at energies lower than 1 TeV due to the EBL. For each timescale, we calculate the significance $\sigma$ of the GRB detection, using the method described in equation (17) of . For the purposes of this analysis, an on target – off target time ratio of $1/3$ is assumed. The calculated values for $\sigma$ for each timescale are compared, and the highest significance for the bins that have more than 10 photon counts is chosen as the significance for detection of the GRB, and the corresponding $T_{obs}$ is designated the optimal timescale. The GRB is then assumed to be detected if the significance is more than 5 sigma.
### Distributions in $\sigma$ and $N_\gamma$
In Figure \[fig:ctasigma\] we show the basic statistical results of simulated CTA observations for a calculation using the effective area curves of Figure \[fig:ea\], $T_{delay}$ = 60 seconds, and a maximum angle from zenith of . A majority of observed GRBs in both models ($\sim$ 90 percent in the bandex model; $\sim$ 80 percent in the fixed) do not lead to a signal of any appreciable significance ($\sigma<1$) for the baseline effective area. The detection efficiency in the bandex and fixed models is found to be 7.3 percent and 11.4 percent, respectively, for the baseline effective area, and 16 and 33 percent for the optimistic effective area. Overall, the bandex model shows a flatter distribution of $\sigma$-values than the fixed model. This is due to the additional degree of freedom introduced in this model by considering the upper Band index in determining the high-energy GRB output in addition to the BATSE fluence, leading to a wider range of values for the overall high-energy normalization. For the same reason, while the bandex model is more pessimistic in its predictions for detection efficiency, the detected GRBs in this model do often produce more photon counts than the fixed model. This can be seen in the right-hand panel of Figure \[fig:ctasigma\], which shows the distribution of photon counts for detected GRBs.
Fig. \[fig:photcnt\] shows the integral distribution for both total source photon count and integral counts above several energy thresholds for the bandex model with the baseline effective area function. The majority of photons for most GRBs are seen to arrive below 100 GeV, with a significant fraction below 50 GeV, despite the much larger effective area provided by the MSTs at higher energies. Also, for a majority of detected GRBs the expected number of source counts above 300 GeV is less than 1.
Table \[tab:deinstsum\] summarizes the detection efficiencies found for a variety of different possibilities in instrument configuration and assumed GRB population model. In §\[sec:detrate\] we discuss how these results for detection efficiency (fraction of GRBs viewed by the instrument that will be detected) can be converted into a detection rate. We also show results for the VERITAS array in the table, for comparison. In this case we follow the same analysis procedure as for CTA, using the VERITAS effective area function shown in Fig. \[fig:ea\], and assuming a delay time of 100 s, rather than 60 s.
Instrument DE (bandex) DE (fixed)
------------------------------ ------------- ------------
CTA (baseline) 0.0744 0.115
CTA (optimistic) 0.163 0.328
CTA (baseline; LST only) 0.0732 0.110
CTA (baseline; MST only) 0.0231 0.0310
VERITAS ($E_{th} = 65$ GeV) 0.0241 0.0281
VERITAS ($E_{th} = 100$ GeV) 0.0216 0.0235
: Summary of detection efficiencies for several instrumental arrangements. In the ‘LST only’ and ‘MST only’, the effective area and background contributions of the MST and LST components are respectively set to zero. We also show results for the VERITAS effective area, assuming two different energy thresholds. []{data-label="tab:deinstsum"}
Figure \[fig:zdist\] shows the expected distribution of redshifts for detected GRBs, compared to the whole population. The CTA effective area, with sensitivity below 50 GeV, potentially allows GRB detections at high redshift, though those at lower redshift will generally have better photon statistics and will therefore be favored. Assuming the baseline effective area, few GRBs are detected above redshift 2, due to the strong impact at higher redshift of the UV- optical EBL at energies above 50 GeV. When the optimistic effective area is assumed, a subset of GRBs ($\sim 0.1$) are bright enough from 10 to 50 GeV to be detectable even at very high redshift. These detections are still a minority of the full set of detected GRBs, however, and are entirely dependent on the low energy performance of the LST array. In all cases, the distribution is significantly biased towards lower redshifts relative to the [*Swift*]{} distribution as a whole, with median redshifts of $z=0.9$ and 1.2 for the baseline and optimistic effective area functions, respectively.
[ ]{}
In Appendix \[sec:otherprops\], we show a number of other properties for GRBs that pass the detection criteria. This provides some insight into the properties that could be expected of typical IACT GRB detection. In Appendix \[sec:promptonly\], we discuss how our results are affected if only the prompt phase of the burst emission is considered, and the fading afterglow signal is ignored.
### Variation of model parameters {#sec:parvar}
In this section, we discuss the impact that variations in instrument properties and other general assumptions could have on the GRB detection efficiency. This demonstrates the effect of variations from our baseline models discussed in the last section. A summary of results is shown in Fig. \[fig:parvar\].
The impact of VHE observation delay time due to GRB localization and telescope slew time, as discussed in Section \[sec:response\], is dependent on the assumed model for the GRB lightcurve at these energies. The upper left panel of Figure \[fig:parvar\] shows the overall impact of parameter $T_{delay}$ on the detection efficiency, with other modeled parameters held constant.
[ ]{} [ ]{}
Next, in the upper right panel, we show how a higher or lower value of the telescope energy threshold than the $\sim 25$ GeV assumed in the baseline model would influence the detection efficiency. The effective area function is assumed here to have the same shape as presented in Figure \[fig:ea\], but with a shift in energy by a constant multiplicative factor. As discussed in the introduction, GRB observations are strongly affected by spectral cutoffs due to EBL, and raising the telescope threshold energy reduces the redshift range over which GRBs are detectable. Detection efficiency is seen here to vary strongly with energy threshold, for both spectral extrapolation models. Note that setting the energy threshold here to 100 GeV is essentially the same as removing the LSTs from the telescope array (see Table \[tab:deinstsum\]), since at these energies the effective area function is dominated by the MSTs. The large decline in detection efficiency with increasing energy threshold demonstrates the importance of having an LST array with low energy threshold to GRB detection, even though the LSTs may only contribute a fraction of the effective area of the total array at higher energy.
The bottom-left quadrant of Fig. \[fig:parvar\] addresses how altering the afterglow light curve index $\gamma$ in Eq. \[eq:lci\] affects results. As discussed in §\[sec:lci\], we have implemented a lightcurve in this work based on the T90 time of a given GRB at Band peak energies, in which VHE emission is flat for this period and then decays as $t^{-3/2}$. In such a model, $2/3$ of the total VHE energy emerges after the end of the T90 period, leading to a substantial afterglow flux that enables detection of GRBs after the lower energy emission has subsided. A faster or slower falloff of afterglow flux in time will change the optimal integration time for GRBs in our simulation, as well as the distribution in detection significance and therefore the detection efficiency. The effect is found to be relatively minor.
Finally, we show in the bottom-right panel of the figure how altering the normalization of the background rate changes detection efficiency. As discussed in §\[sec:bkg\], the background rate assumed in this work is based on a rate of 6 photons per min over the VERITAS effective area (2 per min for the optimistic effective area), with extrapolation to lower energies achieved with a power law of index -2.7. This figure shows the effect of variations in this base rate.
### Spectral cutoffs and the impact of the EBL {#sec:eblimpact}
As mentioned in §\[sec:eblatt\] and the introduction, the EBL introduces a spectral cutoff in extragalactic gamma-ray observations that affects lower energies at higher redshift. In Table \[tab:eblimpact\], we show how assuming different EBL models can change the overall detection efficiency in our calculation. In general, the magnitude of UV/optical emission will determine the strength of the spectral cutoffs for GRBs at most redshifts. However, at $z \gtrsim 2$ this UV emissivity of galaxies is highly uncertain by a factor of several [@gilmoreUV] (G09). The first two rows in Table \[tab:eblimpact\] are the results for the GSPD11 EBL, which we assume elsewhere in this work. This EBL includes a UV-optical contribution that is nearly maximal in terms of the range allowed by high-redshift measured luminosity functions. The other rows in the table include models with less UV light; the fiducial model of G09 has a similar star-formation rate to GSPD11, but with more dust attenuation in high-redshift star-forming galaxies that reduces the UV emission. The low model in G09 has a smaller amount of star formation, in addition to the larger dust extinction. The model of @franceschini08 (F08) predicts a similar UV flux to GSPD11 at low redshift, but a smaller amount at high redshift. Running our analysis with the low model of G09 and the F08 model, we find detection rates of GRBs that are about 30 to 40 percent higher than the GSPD11 case. However, the low model was disfavored in G09 on the basis of IGM ionization data, in favor of the fiducial model, which only increases detection by 5 to 15 percent over GSPD11. The F08 model only provided gamma-ray opacities for $z \leq 3$, so opacities for high redshift GRBs may be artificially low in this case, and the detection efficiency inflated to some degree.
EBL Model eff. area DE (bandex) DE (fixed)
----------- ------------ ------------- ------------
GSPD11 baseline 0.0744 0.115
GSPD11 optimistic 0.163 0.328
G09 (low) baseline 0.101 0.160
G09 (low) optimistic 0.189 0.425
G09 (fid) baseline 0.0803 0.130
G09 (fid) optimistic 0.171 0.365
F08 baseline 0.105 0.165
F08 optimistic 0.192 0.423
: Detection efficiencies found for a few different EBL models, including the @gspd11 (GSPD11) model used elsewhere in this work, the fiducial and low models of @gilmoreUV (G09), and the observational model of @franceschini08 (F08). For the F08 case, gamma-ray optical depths are only reported for $z \leq 3$ and we have used the $z=3$ result for higher redshifts; this result is therefore higher than it might be were the model extrapolated to higher redshift. The second column shows the effective area function assumed. []{data-label="tab:eblimpact"}
Another caveat in our analysis is the possibility that GRBs typically have a spectral cutoff or turnover at some characteristic energy. Due to our lack of knowledge about the GeV-scale properties of GRBs beyond the few bright events that have been detected by [*Fermi*]{} and [*CGRO*]{}, it is difficult to explore this possibility in detail. However, we can perform a simple test, and examine how our results change if a sharp spectral cutoff is assumed to exist at some characteristic observed energy. Obviously, if this energy is below the sensitivity region of CTA, then the detection efficiency must fall to zero, while if it is above the energy where the EBL has a strong impact on the spectra for the majority of detected GRB, then the effect on results will be minimal. In Figure \[fig:scut\], the impact of a universal step-function cutoff at a given observed energy is shown on the results for the total detection efficiency.
This result suggests that our results will remain sound as long as emission continues unaffected to $\gtrsim 100$ GeV. In general, GRBs in the bandex model are found to be less strongly affected by this spectral cut, as detected bursts in this scenario include those with softer spectra than the universal spectral of -2 used in the fixed model. It is worth emphasizing that only GRBs with fluence greater than the median are generally detectable in our simulation (Fig. \[fig:ctaflu\]), and so the existence of a fluence-dependent cutoff energy that affects only the fainter population of GRBs below 100 GeV would have little effect on our results.
Detection of [*Fermi*]{}-GBM bursts {#sec:gbm}
-----------------------------------
The GLAST Burst Monitor (GBM, [@meegan09])) on [*Fermi*]{} presents several unique challenges as a triggering instrument for ground-based follow-up. GBM is a potent source of GRB alerts, about 250 /yr [@paciesas12], and if the [*Fermi*]{} mission is extended to a 10-year period, ending in 2018 or later, then there would be significant overlap with CTA operations and many alerts provided under optimal viewing conditions for the array.
Unfortunately, GBM is only able to provide approximate coordinates for the GRB in real time, and the substantial uncertainties are typically similar to or larger than the field-of-view (FoV) of the LSTs, in contrast to the arcminute localizations of the [*Swift*]{}-BAT. However, while the analysis of Section \[sec:de\] assumed a static observation centered on the target, one could also imagine strategies to compensate for the limited field-of-view, at the cost of exposure depth. After discussing some specifics of our modeled observation of GBM alerts in the following subsection, we will explain one possible strategy for enhancing the DE of GBM bursts.
### Modeling GBM bursts
Our properties for the population of GBM bursts are taken from a subset of 346 events from the upcoming GBM 2-year catalog (Paciesas et al. [*in prep*]{}). Data for these includes the statistical error on the position for each event, and the fluence in the 50 to 300 keV band. The relationship between the statistical error on the automated localization, produced in ground-processing and distributed within $\sim$10 sec of the GRB trigger time, and the fluence in the 50 to 300 keV band is shown in Figure \[fig:gbmbursts\]. The 1 degree lower limit is due to the current grid size of the localization algorithm.
For small statistical errors, a clear negative trend is seen between burst fluence and positional error. As the brightest GRBs are also the most easily detectable by CTA, this will work to our advantage in the detection efficiency. GRB positional errors are also subject to considerable systematic uncertainty, which is independent of the statistical error and shows no clear trend in brightness or other GRB properties at this time. We assume that 70 percent of the GRBs have a systematic error of 3.2 degrees, and 30 percent have a considerably higher value of 9.5 degrees (Connaughton et al. in prep). The total positional error then follows a 2-dimensional gaussian function, with total RMS angular uncertainty given by $$\sigma_{tot} = \sqrt{\sigma_{sys}^2+\sigma_{stat}^2}.$$
The fluence values available for this population cover a different energy range (50 to 300 keV) than the 20 keV to 2 MeV range covered by BATSE. Our model is based on fluences in the latter energy range. To compensate, we have calculated the ratios between the fluence over the full BATSE energy range and the 50–300 keV range for the sample of BATSE GRBs. We then multiply the GBM fluences in the sample by the median of this collection of ratios, to produce a reasonable distribution of fluences over the BATSE energy range for the GBM population. The ratio found here is .
A summary of fluence distributions in BATSE and GBM is shown in Fig. \[fig:fludistbg\]. Recall that in Section \[sec:de\], we applied a multiplier of to account for the differences between the BATSE and [*Swift*]{}-BAT GRB populations. Our findings here suggest that GBM bursts are considerably brighter, on average, than those of BATSE, which are in turn brighter than [*Swift*]{} detections (even the sub-set of [*Swift*]{} GRBs with known redshifts).
As GBM has substantially different energy coverage from [*Swift*]{}-BAT, and detects GRBs that are considerably brighter, one might expect the redshift distribution of GBM-detected GRBs to differ from the [*Swift*]{}-BAT population. Unfortunately, there are not enough known redshifts within the GBM population to do a comprehensive analysis of the differences. This is largely due to the positional errors on GBM detections which make follow-up observations impossible without more accurate data from another experiment. We can, however, look at the handful of GBM bursts that are also listed with redshifts in the [*Swift*]{} catalogue. A plot of redshifts for these 23 GRBs is shown compared to the redshift distribution of all [*Swift*]{} GRBs in Fig. \[fig:swiftgbmzcomp\]. GBM bursts are found to have a somewhat lower distribution of redshifts overall, and we have made a modification (dotted green line in the plot) to the fit for the [*Swift*]{} distribution that we will use in the analysis that follows.
A final factor we must consider is an account of the camera sensitivity lost in off-axis CTA observations. We account for this by including a simple sensitivity factor that is a function of radius ($r$) from the FoV center: $$\begin{aligned}
S(r) & = 1; r < R_{fov}-1.5 \nonumber \\
S(r) & = 0.3(R_{fov}-0.5-r)+0.7; \nonumber \\ & \hspace{2cm} R_{fov}-1.5 < r \leq R_{fov}-0.5 \nonumber \\
S(r) & = 0.4(R_{fov}-r)+0.5; R_{fov}-0.5 < r \leq R_{fov} \end{aligned}$$ Here $R_{fov}$ denotes the maximum extent at which any observation is possible, and is equal to half the FoV value. This radial dependence gives full sensitivity up to $R_{fov}-1.5$, 70 percent sensitivity at $R_{fov}-0.5$ and 50 percent sensitivity at the edge of the camera. This factor is applied to both expected number counts from the GRB and the background rate, and is used for both LST and MST observations. We set $R_{fov}$ to 4.25 degrees for the LSTs and 8 degrees for the MSTs.
### Static observations of GBM bursts {#sec:statmode}
As a first step, we calculate the detection efficiency of this population without applying any search mechanism in the GRB observation. This analysis, and that of the next section, will only utilize the fixed model, as spectral details for this burst population are not available at the time of writing. Results from the previous sections indicate that detection efficiencies of GRBs are typically a factor of 1.5-2 times higher in the fixed model than in the bandex model; there is no reason to believe a similar relationship would not hold true as well here.
The limiting factor in these observations is the field of view of the telescope. As most GRBs in the sample have a minimum uncertainty of $\sim 4$ degrees from combined statistical and systematic effects, a 3 to 5 degree diameter FoV is insufficient to catch more than a minority of events. Figure \[fig:gbmerrred\] shows the current detection efficiencies calculated, and how these could be increased by future reductions in the amount of uncertainty affecting GBM burst positions as reported in real time.
### Scanning mode observations {#sec:scanmode}
A possible solution to the FoV limitations of CTA is to attempt to rapidly scan over some portion of the GRB error box after the burst alert, rather than simply observing the coordinates of the best GRB localization. We consider in this section the possibility of increasing the GRB detection efficiency using “scanning mode” observations, and address the question of how the search box should be chosen to optimize this rate. A larger search box increases the probability that the GRB will be observed, but at the cost of exposure time. We make the assumption here that the data from such a scanning observation over minutes or hours could be compared with a later determination of the actual burst position, and the significance of the GRB detection would then be computed in an after-the-fact analysis taking into account the photon counts and expected background within the source PSF. We do not consider here the possibility that VHE emission could be identified in real time, i.e., for a “stop-on-target” type of scan.
Our calculation of the detection efficiency follows that of §\[sec:de\] above, with a few modifications. We assume that the time required for the LST array to search a region of 15 degrees radius is about 120 seconds. We therefore append the delay time used in the last section with another term that describes the delay between the commencement of the scan and when the telescope first passes over the GRB’s true location. The total delay is then $$T_{delay}=60 + R_N(0,120) \frac{(R_{srch}-R_{fov})^2}{(15^\circ-R_{fov})^2} \: \; \mbox{sec},$$ where $R_{srch}$ is the radial extent of the search box, $R_{fov}$ is one half the field-of-view for the LSTs, and $R_N(0,120)$ is a random variable between 0 and 120. Afterwards, it is assumed that the telescopes make many successive passes over the GRB, and that the integrated exposure after $T_{delay}$ is approximately equal to that of the standard calculation multiplied by a factor $(R_{fov}/R_{srch})^2$ if the GRB is in the search box, zero otherwise. The received background is adjusted by the same factor. The MST array is assumed to search in the same pattern as the LSTs, and their total exposure is multiplied by an analogous factor.
We consider several possible ways that one might determine the extent the scanned region, $R_{srch}$. The simplest possibility is to use a constant for all GRBs. In a scan over possible values in steps of 0.5 degrees, we find that detection efficiency is maximized at $R_{srch}$ = 6 degrees for the baseline effective area, and 7.5 degrees for the optimistic function. For the baseline effective area, the detection efficiency is increased to 0.057 for $R_{srch} = 6$ degrees, compared to 0.037 in the static case; an increase of over 50 percent. In the optimistic case, the detection efficiency can be more than doubled, from the static value of 0.07 to 0.16 with an optimized scan box. These results show that a scanning strategy for GRB follow-up can raise the detection probability by a large factor, and that it is especially powerful for a telescope with a very low energy threshold.
Total detection rate {#sec:detrate}
--------------------
The second part of our calculation of the detection rate, summarized in Eq. \[eq:dr\], is an estimation of the trigger rate from satellite instruments. This factor accounts for the sky coverage and duty cycle of the instrument, and is the rate at which CTA can respond to and observe GRBs. All other factors influencing GRB detectability are incorporated into the detection efficiency parameter discussed above. Our calculations in the last section allow for observations at a maximum angle from zenith of 75 degrees, covering 37 percent of the sky. The calculations of detection efficiency include the effect of increasing energy threshold for observations far from zenith. The duty cycle of Cherenkov telescopes is limited by the requirement that these telescopes operate mostly on clear, moonless nights, which has generally produced realistic values of about 10 percent, and we expect this factor to remain valid for CTA. Operation of CTA during moonlight may increase the duty cycle to $\sim$13 percent or more, but at the cost of a higher energy threshold. We therefore do not expect a significant change of our predictions for the overall detection rate.
The total trigger rate calculated from a given satellite alert rate $$TR = DC \times SC \times SR \times BF.
\label{eq:tr}$$ Where the CTA duty cycle $DC$ is $\sim 0.1$ and the sky coverage factor $SC$ is 0.37, for a 75 degree radius area around zenith, as we have assumed throughout this work. The satellite rate $SR$ is the number of GRB alerts produced by a given satellite detector per unit time. The bias factor ($BF$) includes the effect of any correlation or anti-correlation between the location of GRB satellite alerts and the sky area covered by CTA. This includes factors such as the bias present in [*Swift*]{} GRB alerts [@gilmoreGRB], which leads to GRB discovery preferentially in the anti-solar direction, working to the advantage of IACTs which are limited to nighttime operations. A factor of 1 indicates no departure from a random distribution of GRB alerts on the sky. We assume a factor of 1.4 for [*Swift*]{}-BAT GRBs and 1.0 for GBM.
### [*Swift*]{}-like GRBs {#sec:swiftdetrate}
The [*Swift*]{} satellite, launched in late 2004, has detected GRBs at a rate of about 95/yr over its first 70 months of operation, and is expected to have an orbital life of $>$15 years [@romano10]. If the science lifetime of [*Swift*]{} overlaps with that of CTA, then [*Swift*]{} will provide a constant source of well-localized GRB alerts. Using Eq. \[eq:tr\], we estimate a detection rate for [*Swift*]{} alerts of $$DR_{\mbox{\tiny Swift}} = DE \times 4.92 \;\frac{\mbox{\small GRB}}{\mbox{\small yr}}.$$ If we assume the ‘best-guess’ instrument parameter of $T_{delay}=60$ sec for the LSTs and use the detection efficiencies from the first two rows of Table \[tab:deinstsum\], we find detection rates of 0.37 and 0.57 yr$^{-1}$ for the bandex and fixed models with the baseline effective area function, and 0.80 and 1.61 with the optimistic effective area function. These correspond to timescales of 32 and 21 months between GRB detections with the baseline area, and 15 and 7.5 months with the optimistic effective area functions.
Another upcoming mission that could provide timely GRB localizations is the [*SVOM*]{} satellite. As described in @gotz09, [*SVOM*]{} will consist of an orbiting gamma-ray telescope covering an energy range similar to that of [*Swift*]{} that is expected to detect $\sim$70 GRBs/yr, as well as ground-based telescopes for follow-up observations. Because the ECLAIRs/CXG instrument on [*SVOM*]{} is intended to cover a similar energy range to [*Swift*]{}-BAT, the population of GRBs detected with this satellite should be similar to the [*Swift*]{} population, and therefore have a similar detection efficiency to that which we have calculated here.
### [*Fermi*]{}-like GRBs
GBM on [*Fermi*]{} has detected GRBs at a rate of about 250 per year (Paciesas et al. [*in prep*]{}). Because these GRBs are detected at all points above the horizon, the anti-solar bias factor affecting [*Swift*]{} (and presumably [*SVOM*]{}) does not apply here. We can therefore write the detection rate for GBM GRBs as $$DR_{\mbox{\tiny GBM}} = DE \times 9.25 \;\frac{\mbox{\small GRB}}{\mbox{\small yr}},$$ meaning that GBM should provide around 10 alerts per year that can be investigated by CTA. While these GRBs are brighter on average than [*Swift*]{} GRBs, ground-based followup is hampered by the large uncertainty in burst location, which is generally several degrees. In §\[sec:scanmode\] we showed that the detection efficiency of GBM alerts can be boosted by executing a rapid scan over some portion of the error box. The optimal values found in our case for a fixed-model type of extrapolation, 0.057 and 0.161 for the baseline and optimistic effective areas, respectively, lead to typical detection timescales of 23 and 8 months. The values are similar to those found for the fixed model with [*Swift*]{}-BAT alerts. It is not possible to do a bandex-type analysis for these GRBs, as spectral information is not available at the time of writing. However, we can speculate that such a calculation would likely lead to detection efficiencies that are a factor of 1.5 to 2 lower than for the fixed model, which was the general finding for the BATSE GRB population. Therefore, we conclude that [*Fermi*]{}-GBM and [*Swift*]{}-BAT could give rise to detection rates that are generally the same. However, we point out that in the case of GBM alerts, the scanning mode necessarily means intermittent exposure on the source with therefore only a partial coverage of the temporal emission of the detected GRBs.
It is also possible that future improvements to real-time trigger analysis of GBM bursts could lead to better localization information. Significant improvements in this area could eliminate the need for scanning or other means to compensate for position uncertainty. As a simple test, we can take this possibility to an extreme and examine a case in which all positional uncertainty is removed from GBM alerts. In this case, the detection efficiency values are 0.19 and 0.45, which give detection rates of 1.8 and 4.2 GRBs yr$^{-1}$, $\sim 3$ times greater than the optimal rates from our scan mode simulation.
Results for specific GRBs {#sec:specgrbs}
=========================
It is useful to consider the spectrum that might be provided by an actual GRB detection. In this section we show sample spectra from a few different possible GRBs, which are modeled using the parameters summarized in Table \[tab:080916Cmod\].
080916C {#sec:080916C}
-------
GRB 080916C was seen on September 16, 2008, by [*Fermi*]{}-LAT and GBM, soon after the beginning of science operations with the instrument. This GRB is notable both for its high redshift ($z=4.35$; [@greiner09]) and its extremely high isotropic-equivalent luminosity, $8.8 \times 10^{54}$ ergs, or 4.9 $c^2$ [@abdo09a]. The finding of $>10 $ GeV emission from this burst can be used to set upper limits on the amount of UV light emitted from star-forming galaxies at high redshift [@fermiEBL; @gilmorePDR]. We can therefore consider this GRB as an archetypical example of a bright, high-redshift GRB with a hard spectrum known to extend into the multi-GeV energy range.
We model the high-energy emission from GRB 080916C using the parameters shown in Table \[tab:080916Cmod\]. As in previous sections, an unbroken intrinsic power law extending to 1 TeV is assumed. The time-integrated flux and high energy spectrum are found using a time-weighted average of the spectra over GBM and LAT energy ranges as presented in Table 1 of @abdo09a.
Parameter Value (080916C) Value (BHLZ) Value (VA)
----------- ----------------------- ---------------------- ----------------------
Flux $4.88 \times 10^{-3}$ $1.0 \times 10^{-2}$ $8.8 \times 10^{-5}$
$ \Gamma$ -2.16 -2.1 -2.0
T90 66 s 100 s 50 s
Redshift 4.35 0.5 2.14
: Parameters assumed in modeling the three GRBs described in §\[sec:specgrbs\]. These include the time-integrated flux normalization at 1 GeV (with units of GeV$^{-1}$ cm$^{-2}$), the spectral index in $dN/dE$, the T90 duration, and redshift. Columns show parameters for GRB 080916C (§\[sec:080916C\]), a “bright, hard, low-$z$” GRB (§\[sec:bhlz\]), and a “very average” GRB with parameters selected from the medians of the fixed sample used in this paper (§\[sec:va\]). []{data-label="tab:080916Cmod"}
As in §\[sec:results\], the GeV lightcurve is assumed to decay as $t^{-1.5}$ after the T90 period, with no spectral evolution. Following our analysis with these assumptions, we find that GRB 080916C could be detected at an angle from zenith as high as 39 degrees with the baseline effective area, or 58 degrees with the optimistic area function. In Figure \[fig:080916Cspect\], we show the spectrum that could be expected from an observation of the burst at a zenith angle of 20 degrees.
In this figure, the spectrum is shown with a bin size of 0.1 dex, along with Poisson error bars for the total number of received photons (signal and noise) in each bin. For the baseline effective area, we find a total of 682 signal photons received over the optimal integration timescale of 178 seconds. A signal is seen up to an energy of 90 GeV in each case, beyond which the signal to noise per bin is well below 1. For the optimistic effective area, the signal extends conclusively down to 10 GeV, and a total of 17950 photons are detected over an optimal timescale of 562 sec. Note that the normalizations of the points in the two cases do not appear quite the same, because integrated flux over $T_{obs}$ is being shown, and the timescales of integration are different. The effect of the EBL is easily seen in a comparison between the attenuated and unattenuated spectra, and the GRB signal is discernible even at energies where an attenuation factor $e^{-\tau} \sim 0.01$ affects observations.
Bright, hard, low redshift (BHLZ) {#sec:bhlz}
---------------------------------
[ ]{}
We now consider the observed spectrum for a bright GRB observed at low redshift. For this “BHLZ” burst, we assume the parameters in the appropriate column of Table \[tab:080916Cmod\]. The very high GeV normalization of this GRB (about the 99th percentile in our bandex sample) and its low redshift mean that it can be conclusively detected even at very large angles from zenith. In Fig. \[fig:bhlzspect\], we show an observation of this GRB at an intermediate zenith angle (30$^\circ$) and a very large angle (70$^\circ$). In the second case, the energy threshold of the telescope is increased by a factor $\cos(70)^{-3} \approx 25$, and therefore the observation is limited to energies above 100 GeV.
A “very average” (VA) GRB {#sec:va}
-------------------------
Finally, we repeat our analysis for a GRB that has a redshift, T90 duration, and high energy fluence chosen from the mean values of the fixed sample of GRBs (see Fig. \[fig:ctaflu\] in Appendix \[sec:otherprops\]). Properties are summarized the last column of Table \[tab:080916Cmod\]. Such a GRB is found to be only marginally detectable even occurring near zenith with a baseline telescope effective area. In Fig. \[fig:ea10spect\], we show the spectra from such a GRB.
Conclusions {#sec:disc}
===========
In this work, we have attempted to make realistic predictions of the GRB detectability with the CTA experiment, in particular by considering a reasonable range of possibilities for the GRB emission and the CTA response functions. The basic conclusions of this work can be summarized as follows:
- CTA can be expected to conclusively detect one GRB every 20 to 30 months assuming a baseline effective area and background rate, or 1 to 2 GRBs per year with the optimistic instrument model.
- Detected GRBs will be at a median redshift $\sim$1, and a typical GRB detected with CTA will provide hundreds or thousands of signal events, mostly appearing below 100 GeV.
- Intrinsic spectral cutoffs will have little effect on our results provided that they are above an observed energy of 100 GeV ($\sim 200$ GeV intrinsic).
- The detection rate of GRBs is a strong function of the instrument energy threshold, and a somewhat weaker function of the typical response time.
- Follow-up of GRB alerts from [*Fermi*]{}-GBM could benefit greatly from a scanning-type observation over the GBM error box, which can boost the detection rate by 1.5 to 2 compared to a static observation. With such a strategy, the detection rate of [*Fermi*]{}-GBM bursts can be comparable to that of [*Swift*]{}-detected GRBs.
- Bright GRBs will provide well-determined spectral information over at least a decade in energy, and will be a valuable source of information about the VHE emission mechanism and intervening cosmological radiation fields.
The detection rates we have determined are roughly in agreement with the independent estimate by @kakuwa11 that has been performed concurrently with our own work. Our findings are contingent on certain assumptions: namely that a satellite instrument (i.e., [*Swift, SVOM,*]{} or [*Fermi*]{}-GBM) will be available to provide burst alerts during CTA operations at a rate similar to that seen in recent experience, and that these alerts will be promptly transmitted and followed up with an instrument slew, when possible. These rates would approximately double in the case that two such satellites are available.
Our predictions rely on a number of very uncertain assumptions about gamma-ray bursts that must be determined from limited data. These include the extrapolation of the spectrum to high energy, the typical lightcurve of the high-energy component, and the amount of extragalactic background light which impedes observations of extragalactic sources in the GeV band. In the case of the first two, we have been guided by the observation by [*Fermi*]{}-LAT of $\gtrsim 10$ GeV photons from 4 bright GRBs (080916C, 090510, 090902B, and 090926A). A constant danger in this model is that these GRBs may not be representative of the population as a whole. If, in actuality, only a small fraction of GRBs have spectra that continue into the multi-GeV range, then our results here would overestimate the detection rate with CTA by a large factor. The aforementioned bright LAT-detected GRBs comprise less than 1 percent of the total observed by GBM and it is quite possible that spectral cutoffs routinely exist between the GBM/BATSE energy ranges and the 20–100 GeV band where most GRB photons would be detected (Figure \[fig:photcnt\]). Our bandex model incorporates some aspects of a cutoff for a significant fraction of GRBs; those bursts with $\beta$ parameter $\lesssim -3.0$ have much less power in the GeV band than near the Band function peak, and are generally not detectable (see Fig. \[fig:ctabeta\] in Appendix \[sec:otherprops\]). Over one-third (36 percent) of GRBs in our model fall into this soft category. But given the limited energy range of BATSE, this number is may not represent the full number of GRBs with spectral turnovers or cutoffs that are below the CTA energy coverage.
Spectral turnovers or cutoffs could exist in GRB spectra due to internal absorption of gamma rays by source photons, or Klein-Nishina suppression of high-energy inverse-Compton emission that could be the basis for GeV-scale emission. As discussed in @baring06, internal absorption will lead to a spectral cutoff above an energy determined by the source bulk Lorentz factor ($\Gamma$). For $\Gamma \gtrsim 1000$ our results are likely unaffected. In general, only lower limits on $\Gamma$ are available for GRBs; one possible exception being the bright LAT-detected GRB 090926A, where the claimed turnover in the GRB spectrum [@bregeon11] can be interpreted as the effect of internal pair opacity [@ackermann11], and used to set limits on the bulk Lorentz factor: $200 \lesssim \Gamma \lesssim 700$. Lower limits inferred from other bright LAT-detected GRBs are $\Gamma \gtrsim 900$, 1200, and 1000 for GRBs 080916C, 090510, and 090202B, respectively [@ackermann11]. The distribution in $\Gamma$ is generally unknown for dimmer GRBs, and it is possible that many bursts have factors in the hundreds, rather than thousands. However, if it is the case that the brightest $\sim 20$ percent of GRBs tend to have unattenuated emission up to $>100$ GeV, then our results would not be strongly affected, as these are the events most likely to be detected in our model.
It is unlikely that our results underestimate the CTA detection rate. An underestimate of brightness in the GeV band would entail an increase in the typical MeV–GeV brightness ratio (Fig. \[fig:flucomp\]) above that seen for bright, hard GRBs with a LAT detection. The lack of LAT detections for most GRBs would seem to disfavor the possibility of large GeV-MeV ratios in fainter GRBs. Another possibility is that the EBL attenuation of the gamma-ray signal due to UV light for high-redshift GRBs is overestimated in the fiducial model of @gspd11. We find that reducing the EBL flux to that of a minimal model, such as G09 Low in Table \[tab:eblimpact\], could increase detection rates by as much as 30 to 40 percent, however disagreements between the G09 Low model and the bulk of the high-redshift data make this an unlikely possibility. Finally, we could have underestimated the performance of CTA itself, though the inclusion of our optimistic effective area model, with its sensitivity at energies as low at $\sim 10$ GeV and reduced background rate, is intended to be as hopeful as reasonably possible about the instrument capabilities. It is worth noting that CTA will produce a number of marginal detections, with $2 < \sigma <5$. These are produced at roughly half the rate of bona fide ($>5\sigma$) detections in our models (Fig. \[fig:ctasigma\]). These marginal detections would have higher significance than any of the GRB observations in the VERITAS analysis (Section \[sec:verupper\]; [@aune11]), where the highest significance quoted is 1.8.
A basic assumption that we have made in this work is that GRBs to which the telescope is triggered, presumably by a satellite instrument, will provide detections at a much higher rate than serendipitous detections, in which a GRB occurs in the field of view of the instrument during other observations. Simple geometry shows that this is always the case. The sky coverage fraction of the LSTs, assuming a 4.25 degree field of view and, neglecting edge effects, is about 0.14 percent. If we extrapolate from the GBM burst rate, and assume $\sim 500$ GRBs occurring per year over all sky, and include the duty cycle factor of 0.1, then we find 1 GRB inside the LST field of view during operations every 13 years. For the MSTs, with an 8 degree field of view, we have one GRB every 4 years. Even with the enhanced detection efficiency that is possible with a delay time of zero (Fig. \[fig:parvar\]; upper left panel), the detection rate for GRBs is still much less for the serendipitous case than for triggered operations. However, a serendipitous detection over the lifetime of CTA is possible, and would certainly be of great value in constraining the prompt VHE emission of the GRB.
In §\[sec:gbm\], we discuss the prospects for detecting GRB afterglows from [*Fermi*]{}-GBM alerts. GBM bursts are found to be more than twice as bright on average than those from [*Swift*]{}, and a cursory look at the small number of redshifts available for this population suggests that they are typically slightly closer as well. The primary difficulty of detecting GBM bursts lies in the large positional uncertainty of the instrument; only about 10 percent of GRBs will fall in view of the LST for an observation at the center of the GBM error box, for a 4.25 degree field of view. We show in §\[sec:scanmode\] how a scanning mode observation over the error box can improve the detection rate by $\gtrsim 50$ percent. With this change, we find that the detection rates of GBM bursts with CTA are nearly the same as [*Swift*]{} GRBs. If improvements to the GBM angular resolution are possible before the onset of CTA, then our results could be enhanced by a significant factor, as described in Fig. \[fig:gbmerrred\]. We have considered the possibility of improved localization with the use of a simple linear scaling of the positional error in this figure, as it is difficult to quantify to what extent such an error reduction could take place over the next several years. It is worth emphasizing that because the brightest GRBs are generally those with the highest probability of detection, reductions in the errors for especially bright and/or hard GRBs will be the most advantageous in increasing the detection efficiency for GBM alerts.
Our results show that GRB detection with CTA will rely heavily on the sensitivity achievable in the 20 to 100 GeV band. As shown in Fig. \[fig:photcnt\], only about 20 percent of the gamma rays found for typical detected GRB are at energies $> 100$ GeV; this is despite the large upturn of the effective area function in both of our assumed telescope models (Fig. \[fig:ea\]) at this energy. It should be emphasized that detection of GRBs with CTA is therefore heavily reliant on the performance of the LSTs. It should also be noted that without an LST component, we find that the detection rate for CTA is only marginally higher than for the VERITAS instrument (Table \[tab:deinstsum\]), and is less than one-third the rate for the complete CTA instrument. The importance of the LSTs is even greater if there exist intrinsic spectral cutoffs in GRB spectra at energies between 20 and 100 GeV. We note, however, that MSTs could play a significant role in providing high statistics data above 100 GeV to perform time-resolved spectroscopy which could reveal much about the GRB physics.
An exciting prospect is to use the spectrum of a GRB seen by CTA to constrain the EBL. While gamma-ray sources have been used in many cases to help constrain the EBL (see references in [@gspd11]), these attempts have mainly focused on relatively low redshift blazars. A high-statistics GRB detection by CTA at $z>1$ could greatly improve our understanding of how the EBL evolves with redshift. Because the EBL impact is significant below 100 GeV at these distances, the LSTs will be crucial for such science. One great advantage of CTA is its ability to potentially detect simultaneously both the attenuated and unabsorbed portions of a gamma-ray spectrum, which allows much more robust limits on the EBL than if only the attenuated spectrum is seen and the intrinsic spectrum must be derived theoretically . For GRBs at and above the median redshift of $z=1.7$, an energy threshold of $\lesssim 20$ GeV will be needed to effectively capture the unattenuated slope.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by a SISSA postdoctoral fellowship (RCG) and grants from the Fermi Guest Investigator Program and the US National Science Foundation. The authors thank the VERITAS Collaboration for the use of unpublished results from the detector Monte Carlo simulation. They also thank Taylor Aune for providing early access to the VERITAS limits on GRB fluence, and acknowledge the GBM operations team for continued access to prompt burst locations and for the GRB catalog information used in the predictive calculations presented here. RCG also thanks Vladimir Vassiliev for a useful discussion related to CTA performance, and Susumu Inoue, Jun Kakuwa, and Ryo Yamazaki for helpful discussions concerning this calculation.
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Other properties of detected GRBs {#sec:otherprops}
=================================
[ ]{}
[ ]{}
In this Appendix, we examine in more detail how the population of GRBs that pass our detection criteria compares to the entire population of simulated [*Swift*]{}-like GRBs. This will give us some insight as to the properties that might be expected of a burst with a confident CTA detection. It will also be useful to look at how the assumption of different effective area functions can affect results.
The distribution of integration timescales that maximize detection significance is shown in Figure \[fig:timescales\]. The two spectral models produce similar results in this distribution. This result suggests that a integration timescale of 100 to 500 seconds after the commencement of ground-based observation will be favored for GRB detection in most cases, assuming a universal $t^{-1.5}$ falloff in the afterglow lightcurve. A small subset of bright GRBs however are still visible against the background some hours after the event trigger ($10^4$ seconds in the longest timescale considered here), and nonzero results are found for all bins in Fig. \[fig:timescales\]. On the right hand side of this plot, we show how detection efficiency varies with GRB T90 duration, as determined by BATSE. Not surprisingly, longer bursts always have a better chance of being detected, but the majority of detected GRBs have T90 values from 30 to 100 sec, due to the scarcity of bursts with T90 $>$ 100 sec.
The differences between the fixed and bandex models become most apparent when we consider the distribution in high energy fluences predicted by each, as are shown in Figure \[fig:ctaflu\]. In general, the bandex model has a much wider distribution in high energy fluence, because the beta parameter introduces another degree of freedom into the extrapolation, and steep beta indices lead to a subset of the bursts in the sample having extremely low levels of high energy emission. Conversely, the brightest bandex GRBs are brighter than the brightest bursts in the fixed model, as the latter are limited to a fluence ratio of 0.1 between $\sim$1 MeV and $\sim$1 GeV, while the corresponding ratio in the bandex model can be as high as 1.0, with $\beta =-2$. This accounts for our somewhat unexpected result that while overall detection rates predicted by the bandex model are lower, detected GRBs in this model tend to be brighter than for the fixed model. Figure \[fig:ctabeta\] shows the distribution of $\beta$ indices for detected GRBs in the bandex model. Only GRBs with fairly hard extrapolated spectra, $\beta \gtrsim -2.5$, are capable of being detected.
Figure \[fig:ctatheta\] shows how the probability of detecting a GRB varies with the zenith angle $\theta_{zen}$ at which it is observed. GRBs in our model are assumed to be observed at a single instantaneous point relative to zenith. While motion on the sky over the observation period $T_{obs}$ will change $\theta_{zen}$ over the course of longer integrations, the effect is small enough that we ignore it here. Detection efficiency is a weak function of $\theta_{zen}$ out to $\sim$40 degrees, where increasing energy threshold is compensated by increasing solid angles. At higher angles the detection efficiency declines more quickly. However, GRBs can in principle be detected out to angles as large as 70 degrees, where the energy threshold is raised by a factor of 25 (Eq. \[eq:zenshift\]). These would have to be at low redshift, so as not to be completely obscured by EBL opacity combined with the elevated energy threshold of the telescope.
Prompt phase observations {#sec:promptonly}
=========================
[ ]{}
GRB detection in our calculation is heavily reliant on emission during the early afterglow phase. Only about 21 percent of GRBs in our sample have prompt emission (T90) phases longer than 60 sec, which we assume as a typical delay time for observations with the LSTs. The majority of GRBs are therefore completely inaccessible during the prompt phase for the standard assumption of a 60 sec time delay. As shown in Figure \[fig:timescales\], there is a definite bias toward longer duration GRBs in the detected portion of the population. Figure \[fig:ctaprompt\] summarizes the amount of fluence in detected GRBs that arises from $t<\mbox{T90}$. About 57 percent of bursts detected with a baseline effective area have no prompt phase fluence, while only about 10 percent have more than half the detected fluence arising from emission during the prompt period. With an optimistic effective area function, the fraction of GRBs seen purely in the afterglow period is slightly higher.
We can also consider an extreme possibility in our detection efficiency calculation: one in which no high energy emission emerges after the prompt phase, or equivalently, the light curve index $\gamma$ in Eq. \[eq:lci\] is taken to $+\infty$. This is found to reduce detection efficiencies to about one-third their values in the standard calculation: 0.027 and 0.040 for the bandex and fixed model with baseline effective area (0.057 and 0.11 in the optimistic case). Fig. \[fig:ctapromptgamcnt\] shows the distribution of sigma values and counts for detected GRBs in such a case. These can be compared to those predicted in Fig. \[fig:ctasigma\].
[^1]: <http://gcn.gsfc.nasa.gov/>
[^2]: <http://gammaray.nsstc.nasa.gov/batse/grb/catalog/4b/>
[^3]: We preferred using the BATSE catalog instead of the GBM catalog because of its better instrument sensitivity and the much larger number of bursts detected.
[^4]: <http://gammaray.nsstc.nasa.gov/~goldstein/>
[^5]: <http://swift.gsfc.nasa.gov/docs/swift/archive/grb_table/>
[^6]: <http://www-glast.slac.stanford.edu/software/IS/> [glast\_lat\_performance.htm](glast_lat_performance.htm)
|
---
abstract: 'The proton-proton momentum correlation functions ($C_{pp}(q)$) for kinematically complete decay channels of $^{23}$Al $\rightarrow$ p + p + $^{21}$Na and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne have been measured at the RIKEN RI Beam Factory. From the very different correlation strength of $C_{pp}(q)$ for $^{23}$Al and $^{22}$Mg, the source size and emission time information were extracted from the $C_{pp}(q)$ data by assuming a Gaussian source profile in the correlation function calculation code (CRAB). The results indicated that the mechanism of two-proton emission from $^{23}$Al was mainly sequential emission, while that of $^{22}$Mg was mainly three-body simultaneous emission. By combining our earlier results of the two-proton relative momentum and the opening angle, it is pointed out that the mechanism of two-proton emission could be distinguished clearly.'
author:
- 'D. Q. Fang'
- 'Y. G. Ma'
- 'X. Y. Sun'
- 'P. Zhou'
- 'Y. Togano'
- 'N. Aoi'
- 'H. Baba'
- 'X. Z. Cai'
- 'X. G. Cao'
- 'J. G. Chen'
- 'Y. Fu'
- 'W. Guo'
- 'Y. Hara'
- 'T. Honda'
- 'Z. G. Hu'
- 'K. Ieki'
- 'Y. Ishibashi'
- 'Y. Ito'
- 'N. Iwasa'
- 'S. Kanno'
- 'T. Kawabata'
- 'H. Kimura'
- 'Y. Kondo'
- 'K. Kurita'
- 'M. Kurokawa'
- 'T. Moriguchi'
- 'H. Murakami'
- 'H. Ooishi'
- 'K. Okada'
- 'S. Ota'
- 'A. Ozawa'
- 'H. Sakurai'
- 'S. Shimoura'
- 'R. Shioda'
- 'E. Takeshita'
- 'S. Takeuchi'
- 'W. D. Tian'
- 'H. W. Wang'
- 'J. S. Wang'
- 'M. Wang'
- 'K. Yamada'
- 'Y. Yamada'
- 'Y. Yasuda'
- 'K. Yoneda'
- 'G. Q. Zhang'
- 'T. Motobayashi'
title: 'Proton-proton correlations in distinguishing the two-proton emission mechanism of $^{23}$Al and $^{22}$Mg'
---
[^1]
[^2]
Introduction
============
The two-particle intensity interferometry has been extensively utilized to determine the space-time extension of particle emitting sources in nuclear and particle physics over the past several decades [@hbt; @goldhar; @Koonin; @Lynch; @Bertulani; @Led; @Lisa; @Hu; @Verde; @ma06]. In heavy ion collisions, the two-particle interferometry is a well-recognized and powerful method to characterize the source of particle emission and to probe and disentangle different reaction mechanisms. In particular, this method can provide information on the space-time evolution of hot nuclei which usually decay by evaporation and/or (multi-)fragmentation. Even though a large number of experiments have been carried out to measure the two-proton correlation in nuclear fragmentation, almost no proton-proton momentum correlation function measurement was reported for a kinematically complete decay channel. In contrast, there have been several measurements of the neutron-neutron momentum correlation function in kinematically complete decay channels for halo nuclei, such as $^{11}$Li [@ieki; @orr; @Marq] and $^{14}$Be [@orr], which were believed to be useful for studying the so-called di-neutron structure of neutron-rich nuclei [@neut].
The phenomenon of two-proton emission is a very interesting but complicated process existing in the nucleus close to the proton-drip line [@gold60; @pfutzner; @blank08; @Olsen; @brown]. The proton-proton correlation plays an important role in the emission mechanism. There is a distinct difference in the spectra of the two-proton relative momentum ($q_{pp}$) and opening angle ($\theta_{pp}$) between the diproton emission and two-body sequential or three-body simultaneous emission. This characteristic has been used to investigate the mechanism of two-proton emissions [@kryger; @raci].
The proton-rich nuclei $^{23}$Al and $^{22}$Mg are very important in determining some astrophysical reaction rates and have attracted a lot of attention in both astrophysics and nuclear structure studies [@iacob; @gade; @wies88; @cai; @fang; @ozawa; @wies99; @sewe]. Recently, we have reported the experimental results for kinematically complete measurements of two-proton emissions from two excited proton-rich nuclei, namely $^{23}$Al $\rightarrow$ p + p + $^{21}$Na and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne [@ma2015]. Based on the analysis of $q_{pp}$ and $\theta_{pp}$ distributions of the two emitted protons, a favorable diproton emission component from the excited states around 14.044 MeV of $^{22}$Mg was observed. But no such signal exhibited in the two-proton emission processes of the excited $^{23}$Al.
As pointed out in Ref. [@ma2015], it is difficult to distinguish between the two-body sequential and three-body simultaneous emission mechanism using the above analysis. In these two mechanisms, the emission time of the two protons is quite different. For three-body simultaneous emission, the two protons are emitted almost at the same time, while the two protons are emitted one by one in sequential emission. The two-particle intensity interferometry method has been demonstrated to be a good way to extract the source size and particle emission time [@lisa93; @verde06]. In this paper, the proton-proton momentum correlation function will be studied for the three-body decay channels $^{23}$Al $\rightarrow$ p + p + $^{21}$Na and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne. The size and emission time information of the source will be extracted. The possibility of distinguishing sequential and three-body simultaneous emission mechanism will be investigated.
Experiment Description
======================
The experiment was performed using the RIPS beamline at the RI Beam Factory (RIBF) operated by RIKEN Nishina Center and Center for Nuclear Study, University of Tokyo. A primary beam of $135A$ MeV $^{28}$Si was used to produce secondary $^{23}$Al and $^{22}$Mg beams with incident energy of 57.4$A$ MeV and 53.5$A$ MeV in the center of the carbon reaction target, respectively. After the reaction target, there were five layers of silicon detectors and three layers of plastic hodoscopes as shown in Fig. \[setup\]. The first two layers of Si-strip detectors located around 50 cm downstream of the target were used to measure the emitting angle of the fragment and protons. Three layers of 3$\times$3 single-electrode Si were used as the $\Delta E$-$E$ detectors for the fragment. The three layers of plastic hodoscopes located around 3 m downstream of the target were used as $\Delta E$ and $E$ detectors for protons. TOF of proton was measured by the first layer. Clear particle identifications were obtained by this setup for the kinematically complete three-body decay reactions. The momentum and emission angle for protons and the residue are determined by analyzing the detector signals. The excitation energy ($E^*$) of the incident nucleus was reconstructed by the difference between the invariant mass of three-body system and mass of the mother nucleus in the ground state. More detailed description of the experiment could be found in Ref. [@ma2015].
![(Color online) The layout of detector setup. For details see the text.[]{data-label="setup"}](fig_1.eps){width="7.5cm"}
Results and discussion
======================
In the present study, the momentum correlation functions between two protons emitted from $^{23}$Al and $^{22}$Mg are studied. Experimentally, the two-proton correlation function is constructed through dividing the coincidence yield $N_c$ by the yield of non-correlated events $N_{nc}$, namely $C_{pp}(q) = K\frac{N_c(q)}{N_{nc}(q)}$, where the relative momentum is given by $q=\frac{1}{2}|{\bf p}_1-{\bf p}_2|$, with ${\bf p}_1$ and ${\bf p}_2$ being the momenta of the two coincident protons. The normalization constant $K$ is determined so that the correlation function goes to unity at large values of $q$, where no correlation is expected.
![(Color online) The proton-proton momentum correlation function ($C_{pp}(q)$) for the reaction channel of $^{23}$Al $\rightarrow$ p + p + X (a) and $^{22}$Mg $\rightarrow$ p + p + X (b). The dots are experimental data. The lines are the calculations by the CRAB code with a Gaussian source.[]{data-label="cppx"}](fig_cppx.eps){width="7.5cm"}
The event-mixing technique [@hbt] was applied to construct the background yield, i.e. by pairing a proton with a randomly chosen uncorrelated proton from different events and then normalized to the number of two-proton correlation events in $N_c$. This method ensures that the uncorrelated distribution includes the same class of collisions and kinematical constraints as the numerator. It has, however, a potential problem: it may attenuate the slight correlations one wishes to measure [@goldhar] due to the existence of possible “residue correlation” from initial two-proton physical correlation, which will overestimate the denominator. To eliminate this “residue correlation”, an iterative calculation method for the $C_{pp}(q)$ was applied and intrinsic correlation was extracted [@Zajc]. A similar method has been first applied to two-neutron momentum correlation function measurement for neutron-halo nuclei [@Marq]. Here it is the first attempt to apply the correlation function analysis on two-proton emission data.
Firstly, we looked at the two-proton correlation in the inclusive reaction channel which is similar to the proton-proton momentum correlation function for hot nuclei. Fig. \[cppx\] showed our measurements of $C_{pp}(q)$ which were obtained by the event-mixing method with an iterative calculation for the two emitted protons, without identifying the residue from the mother nucleus and using any specific $E^*$ window. Fig. \[cppx\](a) and Fig. \[cppx\](b) were the results for the inclusive channel of $^{23}$Al $\rightarrow$ p + p + X and $^{22}$Mg $\rightarrow$ p + p + X, respectively. In this work, the normalization factor $K$ in calculating $C_{pp}(q)$ was determined by the $50<q_{pp}<100$ MeV/c data. The peak height around $q_{pp}=20$ MeV/c reflected the strength of correlation function. To extract the source size, theoretical calculation for $C_{pp}(q)$ was performed by using the correlation function calculation code (CRAB) [@pratt]. A Gaussian profile was assumed for the source and the space distribution was simulated according to the function $S(r) \sim \exp(-r^2/2r_0^2)$, with $r_0$ being the source size parameter. The calculations were compared with the experimental $C_{pp}(q)$ data. The source size was determined by finding the minimum of the reduced chi-square ($\chi^2/\nu$, $\nu$ is the degree of freedom). The fit gave a source size range of $r_0=3.15\sim3.25$ fm for $^{23}$Al (corresponding to the rms radius of $R_{rms}=5.46\sim5.63$ fm). The uncertainty was determined from the minimum chi-square $\chi_0^2$ to $\chi_0^2+1$. The best fit of the calculation was plotted in the figure. The obtained source size could give us information of the average distance between the two emitted protons. This size is much larger than the expected radius of the $^{23}$Al nucleus. Of course, a caution needs to be taken for the value of $r_0$, which should be considered as the apparent size for the source since the emission time between two protons is another folded ingredient. For $^{22}$Mg, the source size was extracted to be $r_0=2.9\sim3.0$ fm which is a little bit smaller than that of $^{23}$Al.
![(Color online) Same as Fig. \[cppx\] but for the reaction channel of $^{23}$Al $\rightarrow$ p + p + $^{21}$Na (a); $^{23}$Al $\rightarrow$ p + p + $^{20}$Ne (b); and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne (c). Inserts show sketch maps for the most probable emission mechanism.[]{data-label="cpp"}](fig_cpp.eps){width="7.5cm"}
Secondly, we checked the kinematically complete three-body channels for both $^{23}$Al and $^{22}$Mg. Fig. \[cpp\] showed the $C_{pp}(q)$ of the two emitted protons which were coincident with the residues from the mother nuclei. For the channel of $^{23}$Al $\rightarrow$ p + p + $^{21}$Na, Fig. \[cpp\](a) showed an almost flat correlation function except for the Coulomb dip in low $q_{pp}$ region, indicating that emission of both protons from $^{23}$Al three-body break-up was uncorrelated in phase space except for the Coulomb interaction. This was consistent with the result of no clear observation of diproton emission from the relative momentum and opening angle spectra of $^{23}$Al at any excited states [@ma2015]. Generally speaking, a flat proton-proton momentum correlation function indicates a very large source size and very weak two-proton correlation. The $C_{pp}(q)$ data was also compared with the Gaussian source calculations. The fit gave a source size $r_0=3.9\sim4.7$ fm which was larger than that of the inclusive channel of $^{23}$Al, indicating a more loose two-proton emission. Since the effect of emission time was not considered, it is difficult to explain the results only by the geometric size of the source.
In contrast, the $C_{pp}$ data for $^{22}$Mg nucleus was very different from that of $^{23}$Al as shown in Fig. \[cpp\](c). In this figure, a strong correlation emerged in $C_{pp}$ spectra for the process of $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne, which indicated a compact source size of two-proton emission. The fit gave $r_0=2.35\sim2.45$ fm, which was smaller than that of the inclusive channel of $^{22}$Mg.
Between the two very different two-proton correlation pattern of $^{23}$Al and $^{22}$Mg, we had checked the intermediate situation. If we looked at the decay process of $^{23}$Al $\rightarrow$ p + p + $^{20}$Ne, where one proton was not detected by the experimental setup, a moderate correlation appeared at $q_{pp}\sim20$ MeV/c as shown in Fig. \[cpp\](b), which could be understood by assuming the following two-step proton emission process of $^{23}$Al. One proton was emitted from $^{23}$Al and its corresponding residue nucleus was $^{22}$Mg; Then the other two protons were ejected from $^{22}$Mg and its corresponding residue nucleus was $^{20}$Ne. Among the three emitted protons, only two protons were detected by the detectors. Because of a strong two-proton correlation in the second step, a moderate two-proton correlation could be eventually observed in the process of $^{23}$Al $\rightarrow$ p + p + $^{20}$Ne. The peak height of $C_{pp}$ in Fig. \[cpp\](b) could be explained by a mixture of Fig. \[cpp\](a) and Fig. \[cpp\](c). The Gaussian source fit gave a size of $r_0=3.1\sim3.3$ fm, which was between the cases of Fig. \[cpp\](a) and Fig. \[cpp\](c). In addition, to give a visual impression of two-proton emission, the sketch maps were plotted as inserts in Fig. \[cpp\] to illustrate the most probable emission mechanism for each channel.
![(Color online) Contour plot of the reduced chi-square ($\chi^2/\nu$) obtained from fitting the proton-proton momentum correlation function by the CRAB calculation for the reaction channel of $^{23}$Al $\rightarrow$ p + p + $^{21}$Na (a); $^{23}$Al $\rightarrow$ p + p + $^{20}$Ne (b); and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne (c).[]{data-label="chi3"}](fig_chi.eps){width="7.2cm"}
The effective source size was extracted from the above analysis. However, it was not sure how the two protons were emitted, i.e. the two protons were emitted sequentially or simultaneously. In these two cases, opposite values of the effective source size were observed for $^{23}$Al and $^{22}$Mg. It indicated that the emission time of protons might be different for these two nuclei. Since time information could also be extracted from the correlation function, it will be very interesting to extract both source size and emission time information. Thus a more general analysis was done for the experimental $C_{pp}(q)$ data. For the different mechanism of two-proton emission, the emission time difference between the two protons is important. Assuming the first proton being emitted at time $t=0$ and the second proton being emitted at time $t$, the space and time profile of the Gaussian source was simulated according to the function $S(r,t) \sim \exp(-r^2/2r_0^2-t/\tau)$ in the CRAB code. $\tau$ refers to the lifetime for the emission of the second proton, which starts from the emission time of the first proton. The agreement between the calculation and the $C_{pp}(q)$ data was evaluated by determining the value of the reduced chi-square. The results were shown in Fig. \[chi3\] by a contour plot of $\chi^2/\nu$ as a function of $r_0$ and $\tau$. For the reaction channel of $^{23}$Al $\rightarrow$ p + p + $^{21}$Na as shown in Fig. \[chi3\](a), the ranges of source parameters were obtain to be $r_0=1.2\sim2.8$ fm and $\tau=600\sim2450$ fm/c based on the best chi-square fit. While for the reaction channel of $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne as shown in Fig. \[chi3\](c), the ranges of source parameters were $r_0=2.2\sim2.4$ fm and $\tau=0\sim50$ fm/c. It means that the emission time difference between two protons for $^{23}$Al and $^{22}$Mg was quite different. For $^{23}$Al, the two protons were emitted at very different time ($\tau>600$ fm/c), i.e. the mechanism is a sequential emission. For $^{22}$Mg, the two protons were emitted almost at the same time ($\tau<50$ fm/c), i.e. the mechanism was essentially simultaneous. Based on the above results and the $q_{pp}$ and $\theta_{pp}$ analysis in Ref. [@ma2015], all observables indicate the three-body simultaneous decay mechanism for $^{22}$Mg. Moreover, for the excitations around the $14.044$ MeV, T = 2 state a strong diproton-like component was observed. For the reaction channel of $^{23}$Al $\rightarrow$ p + p + $^{20}$Ne as shown in Fig. \[chi3\](c), the determined source parameters were $r_0=0.4\sim2.0$ fm and $\tau=350\sim1950$ fm/c, which were also between the above two cases and could be explained by the two-step proton emission process of $^{23}$Al.
Conclusions
===========
In summary, measurement on the proton-proton momentum correlation function was applied to kinematically complete decay of two reaction channels $^{23}$Al $\rightarrow$ p + p + $^{21}$Na and $^{22}$Mg $\rightarrow$ p + p + $^{20}$Ne in this paper. The experiment was performed at the RIKEN RI Beam Factory. The proton-proton momentum correlation function $C_{pp}$ was obtained by the event-mixing method with an iterative calculation. By assuming a simple Gaussian emission source, the effective source sizes were extracted by comparing the CRAB calculation with the experimental $C_{pp}$ data. Different effective source size was obtained for $^{23}$Al and $^{22}$Mg, which comes from the different mechanism of two-proton emission. In a more general analysis including source size and emission time information, a reasonable source size but completely different emission time for the two protons were extracted. The results indicated that the mechanism of two-proton emission from $^{23}$Al was dominately sequential, while that for $^{22}$Mg was mainly three-body simultaneous emission with a strong diproton-like component at excited states around $14.044$ MeV. Based on the previous results [@ma2015] and this work, it is possible to distinguish clearly the mechanism of two-proton emission by investigating on the proton-proton momentum correlation function, the two-proton relative momentum and opening angle distributions. The method presented in this work was applied for the first time to two-proton emitters, and was shown to provide new and valuable information on the mechanism of two-proton emission.
Acknowledgements {#acknowledgements .unnumbered}
================
We are very grateful to all of the staffs at the RIKEN accelerator for providing beams during the experiment. The Chinese collaborators greatly appreciate the hospitality from the RIKEN-RIBS laboratory. This work is supported by the Major State Basic Research Development Program of China under contract No. 2013CB834405, National Natural Science Foundation of China under contract Nos. 11421505, 11475244 and 11175231.
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[^1]: dqfang@sinap.ac.cn
[^2]: ygma@sinap.ac.cn
|
---
abstract: 'We study the early stages of viral infection, and the distribution of times to obtain a persistent infection. The virus population proliferates by entering and reproducing inside a target cell until a sufficient number of new virus particles are released via a burst, with a given burst size distribution, which results in the death of the infected cell. Starting with a 2D model describing the joint dynamics of the virus and infected cell populations, we analyze the corresponding master equation using the probability generating function formalism. Exploiting time-scale separation between the virus and infected cell dynamics, the 2D model can be cast into an effective 1D model. To this end, we solve the 1D model analytically for a particular choice of burst size distribution. In the general case, we solve the model numerically by performing extensive Monte-Carlo simulations, and demonstrate the equivalence between the 2D and 1D models by measuring the Kullback-Leibler divergence between the corresponding distributions. Importantly, we find that the distribution of infection times is highly skewed with a “fat" exponential right tail. This indicates that there is non-negligible portion of individuals with an infection time, significantly longer than the mean, which may have implications on when HIV tests should be performed.'
author:
- Carmel Sagi and Michael Assaf
title: Time Distribution for Persistent Viral Infection
---
Introduction
============
Dynamics of infectious diseases, and in particular, the early stages of infection, have attracted much attention in recent years, see *e.g.*, Refs. [@anderson1992infectious; @nowak2000virus; @helbing2015saving; @wang2016statistical]. An important subcategory of such diseases are those that are transmitted via a viral infection. This work focuses on the investigation of a prototypical stochastic model, which describes the early infection stages of various viral infections such as the Human immunodeficiency virus type 1 (HIV-1 or simply HIV), or the hepatitis C virus (HCV) [@perelson1996hiv; @pearson2011stochastic].
The HIV attacks CD4$^+$ cells which belong to the immune system. An HIV virion (virus particle, the infective extracellular form of the virus) attaches to a healthy CD4$^+$ cell, and enters it. Inside the cell the virion uses reverse transcriptase and changes its RNA to DNA, allowing it to penetrate the cell’s nucleus. Once integrated, the virion uses the cell’s nucleus to produce new virions via bursts (or in a continuous manner) [@pearson2011stochastic]. This process eventually kills the infected cell and releases the virions. In the first stages of infection, before the immune system responds, the dynamics of this process is highly noisy. On the one hand, the virions have a nonzero degradation rate which may lead to extinction of the virus, even without an intervention of the immune system. The exact reasons for virion degradation are unknown; the conjecture is that it mainly results from binding to cells or non specific immune elimination [@perelson1996hiv; @li2009visualizing]. On the other hand, given that the virions have not gone extinct, the noisy dynamics yields a high variability in the infection time – the time it takes the virions to infect a sufficient number of cells such that viral extinction is highly unlikely. Thus, in many cases the *mean* time to infection, which can be calculated using standard techniques, becomes inadequate for reliably predicting the fraction of infected individuals within a population up to a given time.
The dynamics of infected cells and virions can be described by the following reaction set $$V\overset{\beta}{\rightarrow}I,\quad I\overset{\alpha}{\rightarrow}kV,\quad
V\overset{1}{\rightarrow}\emptyset.\label{1}$$ Here $I$ represents the number of infected cells, $V$ represents the number of virions and $\emptyset$ denotes an empty set. The first reaction describes penetration of a virion into a cell and its infection, which occurs at a rate $\beta$. The second reaction, which occurs at a rate of $\alpha$, describes a burst of virions which releases $k$ new virions (see below), and kills the host cell. The third reaction describes degradation of virions at a rate of 1 per individual. Note, that here we rescale all rates by the virion degradation rate, which is ${\cal O}(10)$ per day [@ramratnam1999rapid], and we neglect spontaneous degradation of infected cells, whose rate is negligible [@selliah2003t].
Model (\[1\]) describes the so-called “exposure phase”, which corresponds to the linear regime – the early stages of infection. That is, we assume that (i) the infected cell number is undetectable to the immune system, and (ii) the population of uninfected cells, which are susceptible to virion infection, is constant [@footnote1]. Furthermore, the model is generic and can describe any viral infection that behaves similarly as the HIV, such as the HCV [@ramratnam1999rapid; @martinez2015similarities] or even the influenza virus [@baccam2006kinetics; @footnote2]. Hence, the model can predict the distribution of infection times (DIT) for a wide variety of viruses that proliferate in this manner.
The stochastic process, described by Eq. (\[1\]), has two possible outcomes: extinction of the virions and infected cells, or unlimited growth (or proliferation) of the number of virions and infected cells which is ultimately arrested by non-linear terms. In the latter case, the infection is called persistent if the number of infected cells reaches a given threshold $N_0\gg 1$, from which the extinction probability is vanishingly small. In this work we omit the non-linear terms [@footnote3] focusing on the linear regime, and demonstrate that the stochastic nature of the viral dynamics plays a crucial role in determining the statistics of infection times.
The number of released virions per cell depends on many factors including the different phases of infection [@rusert2004quantification]. Chen et al. have shown that in simian immunodeficiency virus (SIV) infection – a monkey-infecting virus similiar to HIV – about 50,000 virions are released upon a death of a cell [@chen2007determination]. However, a large fraction of these virions are not sufficiently mature and thus are non infectious, while others die before they infect new healthy cells [@briggs2009structure]. As a result, the fraction of released *infectious* virions during the first stages of infection are estimated to be 1 in $10^3$ to $10^4$ virions [@bourinbaiar1994ratio; @marozsan2004relationships; @dimitrov1993quantitation]. In our model, since we are interested in the dynamics of infected cells, we only account for the infectious virions and thus, we consider $k$ values which are on the order of $1-100$.
In recent years, the early stages of HIV dynamics have been extensively investigated using different methods. Tan and Wu [@tan1998stochastic] derived a stochastic infection model and used Monte-Carlo simulations to study time-dependent distributions of the numbers of virions and infected cells. They have also shown that viral extinction probability is finite, see also Refs. [@pearson2011stochastic; @kamina2001stochastic; @heffernan2005monte]. Pearson et al. [@pearson2011stochastic] also computed the DIT numerically. Tuckwell and Le Corfec [@tuckwell1998stochastic] introduced a modified infection model by incorporating white noise into the deterministic model. They have shown that the virion growth does not depend on the initial viral population, while Lee et al. [@lee2009modeling] determined the effect of the initial conditions on the probability to reach a viable infection. Chaudhury et al. [@chaudhury2012spontaneous] investigated the role of demographic noise in early stages of viral infection by numerically solving the master equation. Furthermore, Noecker et al. [@noecker2015simple] studied early phases of infection, by accounting for an additional phase in which infected cells do not release virions; they also computed the DIT numerically. However, despite the large body of work in this field, to the best of our knowledge, the DIT to reach a persistent infection has not been studied analytically so far.
In this work we study the DIT to reach persistent infection in a 2D model, which accounts for the interplay between the dynamics of infected cells and virions. To this end we use extensive Monte-Carlo simulations as well as analytical methods such as the probability generating function formalism. In the limit where the viral dynamics is fast compared to that of the infected cells, we derive an effective 1D model which includes the dynamics of the infected cells only. We demonstrate, both analytically and numerically, that the 1D model can reproduce the DIT of the 2D model with a high accuracy, for a generic bursty scheme.
The paper is organized as follows. In Sec. \[Theoretical Analysis\] we present the theoretical analysis of the model. In Sec. \[Determinstic Approach\] we present the deterministic approach. The stochastic approach is described in Sec. \[Stochastic Approach: 2D model\] by using the probability generation function formalism. In Sec. \[Subsection 1D model\] we find an effective 1D model, while in Sec. \[BLimit\] the bifurcation limit is considered. Section \[Numerical Analysis\] is dedicated to presenting the numerical Monte-Carlo simulation results. In Sec. \[Comparison 1D 2D Models\] we show comparisons between the 1D and 2D models, while in Sec. \[Statistics Infection\] we find the DIT semi-empirically. Finally, in Sec. \[Implications Realistic Population\] we discuss the implications of our model on realistic populations and employ our model to HCV. We conclude our study in Sec. \[Conclusion Discussion\].
Theoretical Analysis {#Theoretical Analysis}
====================
Deterministic approach {#Determinstic Approach}
----------------------
Our starting point is model (\[1\]) describing the dynamics of virions ($V$) and infected cells ($I$). When the populations of infected cells and virions are large, one can neglect demographic noise and write down the dynamics in terms of a set of coupled ordinary differential equations. These describe the time evolution of the *mean* number of virions and infected cells. Using model (\[1\]) we arrive at $$\dot{I}=-\alpha I + \beta V,\quad\dot{V}= -\left(\beta+1\right) V + \alpha k I,\label{2}$$ where $I$ represents the mean number of infected cells, $V$ represents the mean number of virions, and $k$ represents the number of released infectious virions. As a reminder, these equations account for three reactions: infection of a cell at a rate of $\beta$, death of an infected cell and release of $k$ virions at rate $\alpha$, and virion degradation at a rate of 1 per individual.
In realistic systems, $\alpha$ is small compared to the other rates, as the infected cells die at a relatively slow rate compared to other processes, see e.g. Refs. [@pearson2011stochastic; @nowak2000virus]. As a result, we assume henceforth that $\alpha\ll 1$, while $\beta={\cal O}(1)$. In the literature, one is interested in the infection dynamics starting from a single virion at $t=0$ [@pearson2011stochastic; @tuckwell1998stochastic]. Using the initial condition, $V(t=0)=1$ and $I(t=0)=0$, and for $\alpha\ll 1$, the solution to (\[2\]) reads $$I\simeq \frac{\beta}{\beta+1}\left[e^{\lambda_{1} t}-e^{\lambda_{2} t}\right],\;\;\;\;\;\;V\simeq\frac{\beta k \alpha}{\left(\beta +1\right) ^2}e^{\lambda_{1} t}+\left[1-\frac{\beta k \alpha}{\left(\beta +1\right) ^2}\right]e^{\lambda_{2} t},$$ where $\lambda_{1}$, $\lambda_{2}$ are the eigenvalues, satisfying $$\lambda_{1}\simeq \alpha\left(\frac{\beta k }{\beta +1}-1\right),\quad\lambda_{2}\simeq-\left(1+\beta+\frac{\beta k \alpha}{\beta +1 }\right).$$ Let us define the basic reproductive ratio, $R\equiv\beta k/\left(\beta+1\right)$ [@pearson2011stochastic]. In epidemiology, $R$ is used to study the rate of spread of an infectious disease [@anderson1992infectious], while here $R$ represents the number of new cells that a single infected cell will infect during its lifespan. One can show (also in the epidemiological context) that $R>1$ describes a persistent infection, while for $R\le1$ the infection dies out deterministically. Indeed in the latter case both eigenvalues are negative, while for $R>1$, $\lambda_{1}>0$, and the populations of $I$ and $V$ grow exponentially. Furthermore, since $\alpha\ll1$, one has $\lambda_{1}\ll\left|\lambda_{2}\right|$, and thus, at long times $t\gtrsim \mathcal{O}\left(1/\lambda_{1}\right)$ the dynamics is solely governed by $\lambda_1$. At this point it is convenient to define $\lambda\equiv\lambda_{1}$ as the (small) positive eigenvalue. By further rescaling time $\tilde{t}=\lambda t$, the solution to Eq. (\[2\]) at long times satisfies $$I=\frac{\beta}{\beta+1}e^{\tilde{t}},\quad V=\frac{\beta k\alpha}{\left(\beta+1\right)^2}e^{\tilde{t}}.\label{3}$$ Henceforth, we omit the $\sim$ above $t$ and measure time in units of the inverse of $\lambda$ times the virus degradation rate.
An important feature of Eqs. (\[2\]) is that for $\lambda\sim\alpha\ll1$, a timescale separation occurs between the rapidly-varying $V$ and the slowly-varying $I$ [@haken1983synergetics; @constable2014fast]. Indeed, defining $\epsilon=\lambda/\left(\beta+1\right)\ll1$, and using the normalized time, rate equations (\[2\]) become $$\frac{\mathrm{d}I}{\mathrm{d}t}=-\frac{\alpha}{\lambda}I+\frac{\beta}{\lambda}V,\quad
\epsilon\frac{\mathrm{d}V}{\mathrm{d}t}=-V+\frac{\alpha k}{\beta+1}I.\label{4}$$ This indicates that in the limit of $\epsilon\rightarrow0$, $\epsilon dV/dt$ vanishes, and $V$ becomes enslaved to $I$. As a result, $V$ is expected to rapidly fluctuate around a slowly varying trend, $V(I)\simeq\alpha k I/\left(\beta+1\right)$, see Fig. \[fig:1\] and below.
Stochastic approach: 2D model {#Stochastic Approach: 2D model}
-----------------------------
The deterministic approach predicts an exponential growth of $I$ and $V$ for $R>1$, but it neglects demographic noise due to the discreteness of individuals and stochastic nature of the reactions. This noise brings about qualitative changes in the dynamics. For example it allows for the virion and infected cell populations to go extinct. On the other hand, if extinction does not occur, the time to reach $N_0$ infected cells (where $N_0$ defines a persistent infection, see Sec. \[introduction\]) is strongly varying across the different stochastic realizations, which results in a non-trivial DIT, see below. Figure \[fig:1\] shows an example of three stochastic realizations of process (\[1\]) and the deterministic solution \[Eq. (\[3\])\]. One can see the significant variability in the infection times across these realizations.
![(a) Heuristic plot of $I(t)$ (on a semi-log scale) for three realizations of process (\[1\]): The left, middle and right solid lines represent a fast, an average, and a slow realization, respectively. The dashed line denotes the deterministic solution given by Eq. (\[3\]). In (b) shown is $I(t)$ along the average realization (dashed line), and $V(t)$ multiplied by $\left(\beta+1\right)/\left(\alpha k\right)$ (solid line), demonstrating that $V$ is an enslaved variable to $I$. In both panels $\beta=2$, $k=5$, $\alpha=0.01$, $N_0=10^4$.[]{data-label="fig:1"}](fig1.eps){width=".7\linewidth"}
Apart from demographic noise, our model includes an *additional* source of uncertainty – the number of virions produced per burst. To account for the stochastic dynamics, we write down the master equation which describes the evolution of the probability $P_{n,m}\left(t\right)$ to find $n$ infected cells and $m$ virions at time $t$ [@be2016effect; @be2016rare; @be2017enhancing] $$\begin{aligned}
\dot{P}_{n,m}\left(t\right)\!&=&\!\frac{\alpha}{\lambda}\!\left[\!\left(n\!+\!1\right)\!\sum_{k\!=0}^{\infty}D\left(k\right)P_{n+1,m-k}\left(t\right)\!-nP_{n,m}\left(t\right)\!\right]\nonumber\\
&+&\frac{\beta}{\lambda}\left[\left(m+1\right)P_{n-1,m+1}\left(t\right)-mP_{n,m}\left(t\right)\right]+\frac{1}{\lambda}\left[\left(m+1\right)P_{n,m+1}\left(t\right)-mP_{n,m}\left(t\right)\right].\label{5}\end{aligned}$$ Here $D\left(k\right)$ denotes the burst size distribution (BSD) [@be2016effect; @be2016rare; @be2017enhancing] – the probability distribution for $k$ virions to be released upon the death of the cell. Multiplying Eq. (\[5\]) once by $n$ and summing over all $n$’s, and once by $m$ and summing over all $m$’s, and recalling that $\langle n\rangle=\sum_{n,m=0}^{\infty}nP_{n,m}\left(t\right)$, and $\langle m\rangle=\sum_{n,m=0}^{\infty}mP_{n,m}\left(t\right)$, we arrive at the deterministic rate equations \[Eqs. (\[2\])\] for the mean number of infected cells and virions.
Throughout this work we will focus on two types of BSDs: the $K$-step distribution, $D\left(k\right)=\delta_{k, K}$, and the geometric distribution, $D\left(k\right)=q^k\left(1-q\right)$, where $k=0,1,2,\dots$, $q=\langle k\rangle/\left(\langle k\rangle+1\right)$ is the probability to release a single virion, and $\langle k\rangle$ is the average number of virions released. In the latter case, for which the number of released virions is a-priori unknown, the effective reproductive ratio equals $R=\beta\langle k\rangle/\left(\beta+1\right)$. A special case for the $K$-step distribution is the so-called single-step reaction (SSR) for $K=2$, which will be analyzed in detail. Since the typical number of released infectious virions is at most ${\cal O}(100)$, we have taken $K$ and $\langle k\rangle$ to be up to 100.
Before computing $P_{n,m}\left(t\right)$ and the DIT, let us first study the extinction scenario, by computing the quantity $\Pi_{n,m}$ – the probability that starting with $n$ infected cells and $m$ virions, both populations will eventually undergo extinction. It can be shown that $\Pi_{n,m}$ satisfies the following recursive equation [@gardiner2004handbook; @feller1968introduction; @redner2001guide; @parzen1960modern; @mosteller1965fifty; @pearson2011stochastic] $$\Pi_{n,m}=\frac{\alpha n}{ z}\sum_{k=0}^{\infty}D\left(k\right)\Pi_{n-1,m+k}+\frac{m\beta}{z}\Pi_{n+1,m-1}
+\frac{m}{z}\Pi_{n,m-1},\label{6}$$ where $z=\beta m+\alpha n+m$ is the sum of rates to leave a state with $n$ infected cells and $m$ virions. Here, starting from $n$ infected cells and $m$ virions, the probability of extinction equals the sum of three terms: (i) probability of extinction starting from $n-1$ infected cells and $m+k$ virions times the probability that an infected cell has died and released $k$ new virions; (ii) probability of extinction starting from $n+1$ infected cells and $m-1$ virions times the probability that a virion has been absorbed by a cell resulting in its infection, and (iii) probability of extinction starting from $n$ infected cells and $m-1$ virions times the probability that a virion died.
Equation (\[6\]) is a 2D recursive equation, and solving it is equivalent to solving a partial differential equation. Yet, the fact that the transition rates are linear in $n$ and $m$ allows one to find a solution in a straightforward manner, by noting that the probability of extinction starting from $n$ infected cells is equivalent to the probability of extinction of $n$ separate realizations, each starting with one infected cell. The same goes for the virion population. As a result, we can look for the solution as $\Pi_{n,m}=\rho_I^n\rho_V^m$, where $\rho_I$ and $\rho_V$ are the probabilities that an infection initiated with a single infected cell and a single virion, respectively, will result in extinction. The solution has to satisfy the boundary conditions $\Pi_{\infty,\infty}=0$, since $\rho_I,\rho_V<1$, and $\Pi_{0,0}=1$, as for $m=n=0$ extinction has already occurred. Substituting this solution into Eq. (\[6\]), one obtains a characteristic polynomial equation for $\rho_I$ and $\rho_V$, whose solution, for arbitrary $D\left(k\right)$, yields [@pearson2011stochastic] $$\begin{aligned}
\rho_I=\sum_{k=0}^{\infty}D\left(k\right)\rho_V^k,\;\;\;\;\;\;\;\;\sum_{k=0}^{\infty}\frac{\beta}{\beta+1}D\left(k\right)\rho_V^k-\rho_V+\frac{1}{\beta+1}=0.\label{7}\end{aligned}$$ For the $K$-step BSD, $\rho_I$ and $\rho_V$ satisfy [@pearson2011stochastic]: $$\begin{aligned}
\rho_I=\rho_V^{K},\quad0=\rho_V^{K}-\frac{\beta+1}{\beta}\rho_V+\frac{1}{\beta},\label{8}\end{aligned}$$ where these equations can be solved numerically for any $K$. Analytical solutions are available for $K\le4$; For example, for $K=2$, $\rho_V=1/\beta$, while for $K=3$, $\rho_V=\sqrt\beta\sqrt{\beta+4}/\left(2\beta\right)-1/2$. For the geometric BSD we find $$\rho_I=\frac{1+\beta}{\beta \langle k\rangle },\;\;\;\;\rho_V=\frac{1+\beta + \langle k\rangle }{\langle k\rangle\left(1+\beta\right)}.$$
Note that for $R<1$, one can show that $\Pi_{n,m}=1$, namely, virion extinction is guaranteed [@pearson2011stochastic].
![The extinction probability starting with a single virion (and zero infected cells), $\rho_V$, as a function of the cell infection rate $\beta$. (a) The case of $K$-step BSD with $K$=2, 5, 50 (top to bottom). (b) The case of geometric BSD for $\langle k\rangle=2,5,50$ (top to bottom). In both panels, for large $K$, one can see that $\rho_V$ approaches $1/\left(1+\beta\right)$ (denoted by the dashed lines). One can also see that for a given $\langle k\rangle=K$, $\rho_V$ for the geometric BSD is higher than that of the $K$-step BSD, as in the former case there is a nonzero probability to release zero virions in a burst event, see text.[]{data-label="fig:2"}](fig2.eps){width=".7\linewidth"}
In Fig. \[fig:2\] we present $\rho_V$ for both the $K$-step and geometric BSDs as a function of $\beta$ for different values of $K$ and $\langle k\rangle$, respectively. In both cases, $\rho_V$ decreases (and the infection probability increases) with increasing $K$ and $\langle k\rangle$, as more virions are released per burst, and more cells are infected due to the higher number of virions. $\rho_V$ also decreases as $\beta$ increases. For the $K$-step BSD, in the limit of large $\beta$, one can show that the solution for $\rho_V$ approaches $1/\beta$ for any $K\ge 2$. Alternatively, for any $\beta$, when $K$ is large, the second of Eqs. (\[8\]) can be reduced to $\rho_V=1/\left(\beta+1\right)$ as $\rho^K$ is negligible, see Fig. (\[fig:2\]). Indeed, if infection starts with a single virion there are two possibilities: the virion can infect a cell with probability $\beta/\left(\beta+1\right)$ or die with probability $1/\left(\beta+1\right)$. Thus, for $K\gg 1$, if extinction does not occur initially, persistent infection will almost certainly occur. Moreover, for any $\langle k\rangle = K$, the extinction probability for the geometric BSD is higher than that of the $K$-step BSD, as there is a possibility for zero virions to be released in the former case. Note that when $\langle k\rangle\gg \beta$ the geometric BSD solution also approaches $1/\left(\beta+1\right)$.
### Probability generating function formalism {#PG function formalism}
In this subsection we use the probability generating function (PGF) formalism [@gardiner2004handbook] to analyze master equation (\[5\]) and to find the DIT. We do so by calculating the probability distribution of finding $n$ infected cells at time $t$ regardless of the number of virions, which is then used to obtain the DIT to reach $N_0$ cells. Previous works [@pearson2011stochastic; @noecker2015simple] have shown numerical calculations for the DIT, and here we find the distribution analytically.
The PGF for a two-variable probability distribution function has the form $$G\left(x,y,t\right)=\sum_{n,m=0}^{\infty}x^ny^mP_{n,m}\left(t\right),$$ where $x$ and $y$ are auxiliary variables representing the infected cells and virions respectively. Importantly, the PGF encodes the probability distribution, which can be found by Taylor-expanding $G$ about $x=y=0$ [@gardiner2004handbook]: $$P_{n,m}\left(t\right)=\frac{1}{n!m!}\left.\frac{\partial^{n+m} G\left(x,y,t\right)}{\partial x^n\partial y^m}\right|_{x=0, y=0}.\label{9}$$
To find the PGF, we multiply master equation (\[5\]) by $x^ny^m$ and sum over all $n'$s and $m'$s, arriving at a first-order partial differential equation $$\frac{\partial G}{\partial t}=\frac{\partial G}{\partial y}\frac{\left(\beta x-\beta y+1-y \right)}{\lambda}\\
+\frac{\partial G}{\partial x}\frac{\alpha}{\lambda}\left[\sum_{k=0}^{\infty}y^kD\left(k\right)- x\right]\!.\label{10}$$ This equation cannot be solved analytically due to the coupling between $x$ and $y$. To proceed, we exploit the smallness of the parameter $\alpha$ which gives rise to time-scale separation between $V$ and $I$. Indeed, we notice that the coefficient multiplying $\partial G/\partial y$ diverges when $\lambda\sim\alpha\rightarrow0$, while the other coefficients are $\mathcal{O}\left(1\right)$. As a result, to regularize Eq. (\[10\]) we demand that the numerator in the coefficient of $\partial G/\partial y$ vanish. Thus, we demand that $\beta x-\beta y+1-y=0$, which yields $y=\left(1+\beta x\right)/\left(1+\beta\right)$. This is the analogous relation in the stochastic picture between $V$ and $I$, to the deterministic relation between these variables, see below Eq. (\[4\]) [@footnote4]. Using this relation between $y$ and $x$, Eq. (\[10\]) becomes $$\frac{\partial G}{\partial t}=\frac{\partial G}{\partial x}\frac{\alpha}{\lambda}\left[\sum_{k=0}^{\infty}\left(\frac{1+\beta x }{1+\beta }\right)^kD\left(k\right)- x\right].\label{11}$$ Note, that $G$ is now only a function of $x$. In fact, by eliminating the $y$ variable, finding $G(x,t)$ yields the probability to find $n$ infected cells at time $t$ regardless of the number of virions, which is exactly what we are after.
### The case of single-step reaction {#The Case SSR}
We now solve Eq. (\[11\]) for the SSR case, for which $D(k)=\delta_{k,2}$. Here, Eq. (\[11\]) becomes $$\frac{\partial G}{\partial t}=\frac{\partial G}{\partial x}\frac{\alpha}{\lambda}{\left[\left(\frac{1+\beta x }{1+\beta}\right)^2-x\right]}.\label{12}$$ This equation has to be solved with the initial condition $P_n\left(t=0\right)=\delta_{n,1}$, namely, starting from a single infected cell. Using the definition of $G$ in 1D, $G\left(x,t\right)=\sum_{n=0}^{\infty}x^n P_n\left(t\right)$, the initial condition becomes $G\left(x,t=0\right)=x$. In addition, conservation of probability yields the boundary condition $G\left(x=1,t\right)=\sum_{n=0}^{\infty}P_n\left(t\right)=1$. With these initial and boundary conditions, employing the method of characteristics [@gardiner2004handbook], the solution of Eq. (\[12\]) reads $$G\left(x,t\right)=\frac{1-Q_{2D}-e^t (x-1)\left(Q_{2D}-1\right)-x}{1-Q_{2D}+e^t (x-1)-x}.\label{13}$$ Here we have defined the survival probability starting from one cell, $Q_{2D}=1-\rho_I=1-1/\beta^2$, where the subscript 2D denotes the two dimensional model (\[1\]). To compute $P_n\left(t\right)$ – the probability to find $n$ infected cells at time $t$ – we use the 1D version of (\[9\]), which yields $$P_n\left(t\right)=\frac{Q_{2D}^2e^t}{\left(e^t-1+Q_{2D}\right)^2}\,\left(\frac{e^t-1}{e^t-1+Q_{2D}}\right)^{n-1}.$$ Being interested in the long time behavior such that $t\gg1$, we finally arrive at $$P_n\left(t\right)\simeq Q_{2D}^2 e^{-n Q_{2D} e^{-t}
-t}.\label{PNT}$$ We now use this quantity to compute the DIT to reach a given $N_0$, $P_t\left(N_0\right)$. Using the fact that the probability contained in a differential area is invariant under change of variables, we have $\left|P_n\left(t\right)\mathrm{d}n\right|=\left|P_t\left(n\right)\mathrm{d}t\right|$, or $P_n(t)|dn/dt|=P_t(n)$, where at long times, we can estimate $dn/dt\simeq n$, since $n(t\gg 1)\simeq e^{t}$. Normalizing the resulting distribution such that $\int_0^{\infty} P_t(N_0)dt=1$, we find the DIT to be $$P_t\left(N_0\right)\simeq\frac{Q_{2D} \left(N_0-1\right) }{1-e^{Q_{2D} (1-N_0)}}e^{-Q_{2D}N_0e^{-t}-t}.\label{Time Distribution 2D}$$ This expression is one of our main results. It is also interesting to compute the probability to have at least $N_0$ infected cells at time $t$, i.e., the cumulative distribution of Eq. (\[PNT\]). Summing $P_n\left(t\right)$ over $n$ from $N_0$ to $\infty$ we find: $$\label{cumdist}
P_{n\geq N_0}\left(t\right)=Q_{2D} e^{-Q_{2D} N_0
e^{-t}}.$$ At $t\rightarrow\infty$, its value approaches the infection probability, $Q_{2D}$. The analytical expression for the cumulative distribution excellently agrees with numerical simulations, see Fig. \[fig:3\].
![Shown is the cumulative distribution – the probability to reach at least $N_0$ infected cells – as a function of the rescaled time, in the case of the SSR. Here, the analytical result given by Eq. (\[cumdist\]) (solid line) is compared with results of numerical simulations (triangles). At long times, $t\gg 1$, the cumulative distribution approaches the infection probability, $Q_{2D}$. Parameter values are $N_0=10^3$, $\alpha=0.1$ and $\beta=2$. []{data-label="fig:3"}](fig3.eps){width=".6\linewidth"}
Our numerical simulations of model (\[1\]) were carried out by using the Gillespie algorithm [@gillespie1977exact]. A comparison between the analytical and numerical DITs for the case of SSR is shown in Fig. \[fig:4\], and excellent agreement is observed. In order to obtain the numerical DIT, we have determined an infection threshold and binned the infection times from all realizations that reached infection, using bins of size $\Delta t$. The corresponding error in each bin is approximately given by $1/\sqrt{MQ_{2D}P_t(N_0)\Delta t}$, where $M$ is the total number of realizations such that $MQ_{2D}$ is the number of realizations that reached infection, and $P_t(N_0)\Delta t$ is the probability to get infected between time $t$ and $t+\Delta t$. We made sure that in all our plots this error was at most 10$\%$, and the size of the symbols in all figures accounts for this error.
![Shown is the distribution of infection times to reach $N_0$ infected cells, as a function of the rescaled time, in the case of the SSR. Analytical results \[Eq. (\[Time Distribution 2D\])\] (solid line) are compared with numerical simulations (triangles), for $\beta=2$, $\alpha=0.01$ and $N_0=10^4$.[]{data-label="fig:4"}](fig4.eps){width=".65\linewidth"}
Effective 1D model {#Subsection 1D model}
------------------
So far, we have seen that for $\alpha\ll 1$, the dynamics of the virions is enslaved to that of the infected cells. We now exploit this property, which is generic and holds for any BSD, to reduce our 2D model into an effective 1D model, which can be analyzed, both analytically and numerically, in a simpler manner. Indeed, in the limit of $\alpha\rightarrow0$, the dynamics of the virions is instantaneous compared to that of the infected cells. As a result, one can write $I\rightarrow kV\rightarrow kI$. Assuming that the cells also degrade at some rate due to the degradation of virions, to be determined below, the effective 1D dynamics of the infected cells can be described by the following two reactions $$I\overset{\gamma}{\rightarrow}kI,\quad I\overset{\delta}{\rightarrow}\emptyset.\label{1Dmodel}$$ Here $\gamma$ and $\delta$ are the effective burst and degradation rates, respectively, yet to be determined, while $k$ is the number of infected cells created per burst event, drawn from the same BSD, $D\left(k\right)$, as in the 2D model. To find $\gamma$ and $\delta$, we demand that the growth rate of the infected cell population, and its survival probability, coincide between the 1D and 2D models.
Equation (\[1Dmodel\]) yields the following rate equation: $\dot{I}=\left[\left(k-1\right)\gamma-\delta\right]I$, whose solution is $I=e^{\left(\left[k-1\right]\gamma-\delta\right)t}$. Thus, to match between the 1D and 2D growth rates, we demand that $\left(k-1\right)\gamma-\delta$ be equal to $\lambda=\alpha\left[\beta k/\left(\beta+1\right)-1\right]$. As before, upon rescaling time $\lambda t\rightarrow t$, we have $I_{1D}=e^t$, where the subscript 1D stands for the solution of the 1D model.
To find the second constraint on the rates of the 1D model, we calculate $\Pi_n$ – the extinction probability starting from $n$ infected cells – in the framework of the 1D model. $\Pi_n$ satisfies the following recursive equation [@Redner2010] $$\Pi_n=\frac{ n\gamma}{z}\sum_{k=0}^{\infty}D\left(k\right) \Pi _ {n+\left(k-1\right)}+\frac{ n\delta}{z} \Pi _ {n-1},\label{1D extinction probability}$$ where $z=n\left(\gamma+\delta\right)$ is the sum of all rates. As before, since the reaction rates are linear in $n$, the solution has the form $\Pi_n=\eta^n$, where $\eta$ is the probability that an infection initiated with a single infected cell will result in extinction. The solution must also satisfy the boundary conditions $\Pi_\infty=0$ and $\Pi_0=1$. Plugging the solution into Eq. (\[1D extinction probability\]) we arrive at $$0=\sum_{k=0}^{\infty}\frac{\gamma}{\gamma+\delta}D\left(k\right)\eta^k-\eta+\frac{\delta}{\gamma+\delta}.\label{rho_1}$$ For the case of the $K$-step BSD we find [@Redner2010] $$0=\eta^{K}-\frac{\gamma+\delta}{\gamma}\eta+\frac{\delta}{\gamma},\label{extinction probability k const}$$ while for the geometric BSD the result is $\eta=[\gamma+\delta\left(\langle k\rangle+1\right)]/[\langle k\rangle\left(\gamma+\delta\right)]$.
We now demand that the extinction probability starting with one infected cell in the 1D model be equal to the extinction probability starting with one virion in the 2D model. The demand comes from the equivalency of the virion dynamics in 2D and the infected cell dynamics in 1D. Indeed, in 2D, a virion can either die or become an infected cell, which then turns with probability 1 to $k$ virions. This is identical to the behavior of an infected cell in 1D. Therefore, since the equation for $\eta$ in the 1D case and the equation for $\rho_V$ in the 2D case are identical, comparing the coefficients in Eqs. (\[7\]) and (\[rho\_1\]), valid for generic $D(k)$, yields $\delta/\gamma=1/\beta$. Combined with the demand on the effective growth rate, we find $$\gamma=\frac{\beta\alpha}{\beta+1},\quad\delta=\frac{\alpha}{\beta+1}.$$ Having found $\gamma$ and $\delta$, we can now analyze the master equation for $P_n\left(t\right)$ in the 1D case: $$\dot{P_n}(t)=\frac{\alpha}{\lambda\left(\beta\!+\!1\right)}\left[\beta\left(\sum_{k=0}^{\infty}D\left(k\right)\left[n\!-\!k\!+1\right]P_{n-k+1}(t)-n P_n(t)\right)+(n\!+\!1) P_{n+1}(t)- n P_n(t)\right]\!.$$ As before, multiplying this equation by $x^n$, and summing over all $n$’s, we arrive at the following evolution equation for the PGF: $$\label{PG1}
\frac{\partial G}{\partial t}=\frac{\alpha}{\lambda\left(\beta+1\right)}\left[ \beta\sum_{k=0}^{\infty}D\left(k\right)x^k-\beta x+1- x\right]\frac{\partial G}{\partial x}.$$
Similarly as in the 2D case, by employing the method of characteristics this equation is solvable only for the SSR case, namely $D\left(k\right)=\delta_{k, 2}$. In this case, taking the initial condition $G(x,0)=x$ and boundary condition $G(1,t)=1$, and defining the survival probability starting from one infected cell in the 1D model as $Q_{1D}=1-\eta=1-1/\beta$, we have $$G\left(x,t\right)=\frac{1-Q_{1D}-e^t (x-1)(Q_{1D}-1)- x}{1-Q_{1D}+e^t (x-1)-x}.\label{GF1D}$$ This equation coincides with Eq. (\[13\]). Therefore the DIT in 1D coincides with Eq. (\[Time Distribution 2D\]) up to the value of the survival probability, which differs between the 2D and 1D models. This discrepancy, however, can be remedied by noticing that the threshold for infection, $N_0$, also varies between the 2D and 1D models. Indeed, in the 2D model $I_{2D}(t)$ grows on average as $\beta/\left(\beta+1\right)e^{t}$ while in the 1D model, $I_{1D}(t)$ grows as $e^{t}$. Therefore, the threshold in the 1D model has to be multiplied by $(\beta+1)/\beta$, in order for the DITs to fully coincide.
We have simulated the 1D model using the Gillespie algorithm, and compared the analytical and numerical distributions. As can be seen in Fig. \[fig:5\], both the 1D and 2D numerical distributions excellently agree with the analytical result \[Eq. (\[Time Distribution 2D\])\]. Here and in all other figures showing comparisons between the 1D and 2D models, given a threshold of $N_0$ in the 2D model, we have taken $(\beta+1)N_0/\beta$ as the threshold for the 1D model.
![Shown is the distribution of infection times to reach $N_0$ infected cells as a function of the rescaled time, for the SSR. Analytical results (solid line) are compared with results of numerical simulations of the 1D (squares) and 2D (triangles) models, for $\beta=2$, $\alpha=0.01$, and $N_0=10^4$. The fact that the curves are almost indistinguishable indicates that the effective 1D model is an excellent approximation of the 2D model for $\alpha\ll 1$.[]{data-label="fig:5"}](fig5.eps){width=".65\linewidth"}
Bifurcation limit of the 1D model {#BLimit}
---------------------------------
Equation (\[PG1\]) is exactly solvable only for the SSR case. Yet, close to the bifurcation limit, where $R$ approaches 1, the equation is also approximately solvable, for any $K$. To find the solution, we notice that the extinction probability $\eta$ approaches $1$ for $R-1\ll 1$. Thus, substituting $\eta=1-\delta\eta$ in Eq. (\[extinction probability k const\]), where $\delta\eta\ll 1$, and solving the resulting equation in the leading order in $\delta\eta$, yields the survival probability $Q_{1D}=\delta\eta=2(R-1)/(K-1)$.
To solve Eq. (\[PG1\]) close to the bifurcation limit, we substitute $D(k)=\delta_{k,K}$, and expand the coefficients of $\partial G/\partial x$ around $x=1$ up to second order in $\left|1-x\right|\ll1$, which yields $$\label{Gbif}
\frac{\partial G}{\partial t}\simeq \left[\left(x-1\right)+\frac{1}{Q_{1D}}\left(x-1\right)^2\right]\frac{\partial G}{\partial x}.$$ It can be shown that this approximation holds as long as $K(R-1)\ll 1$. Note, that it is justified to expand Eq. (\[PG1\]) in the vicinity of $x=1$, as close to the bifurcation limit, the probability distribution at long times is determined by a narrow region of $G$ in the vicinity of $x=1$ [@assaf2006spectral; @assaf2007]. In fact, what we are doing here is approximating the $K$-step process by a single-step process $I\rightarrow2I$ \[for which Eq. (\[Gbif\]) is exact\], which turns out to be a good approximation when $R\rightarrow 1$. Since Eq. (\[Gbif\]) coincides with Eq. (\[PG1\]) upon substituting $D(k)=\delta_{k,2}$ and using the definition of $Q_{1D}$ close to the bifurcation limit, we find that in this limit, the DIT coincides with Eq. (\[Time Distribution 2D\]) up to the value of the survival probability $Q_{1D}$, given above Eq. (\[Gbif\]). In Fig. \[fig:6\] we compare the analytical result close to bifurcation with numerical simulations for $K=3$ and $K=4$. For each $K$, we have taken two $R$ values: $R=1.1$ and $R=2$. As shown in the figure, the DIT close to bifurcation is accurately given by the SSR result, as long as $K(R-1)\ll 1$.
![Shown are distributions of infection times to reach $N_0$ infected cells as a function of the rescaled time, for the $K$-step BSD. The analytical result close to bifurcation (solid line) is compared with numerical simulations (triangles) for different $K$ and $R$ values: (a) $K=3$ and $R=1.1$; (b) $K=3$ and $R=2$; (c) $K=4$ and $R=1.1$; (d) $K=4$ and $R=2$. One can see that the analytical and numerical results agree well as long as $K\left(R-1\right)\ll 1$, see text. Here $\alpha=0.01$, $N_0=10^4$, and $\beta$ is determined by the $K$ and $R$ values. []{data-label="fig:6"}](fig6.eps){width=".9\linewidth"}
Numerical Analysis {#Numerical Analysis}
==================
Monte-Carlo simulation results {#Comparison 1D 2D Models}
------------------------------
After having compared the 1D and 2D models, both analytically and numerically, for the SSR case and close to bifurcation, we now compare the models numerically for generic BSDs. An example can be seen in Fig. \[fig:7\], where we have compared the models for the case of $K$-step BSD with four different $K$’s, demonstrating an excellent agreement between the models.
![Shown are distributions of infection times to reach $N_0$ infected cells as a function of the rescaled time, for the $K$-step BSD. Here $\beta=2$, $N_0=100$, $\alpha=0.01$ and (a) $K=10$, (b) $K=20$, (c) $K=50$, (d) $K=100$. The numerical results of the 2D model (triangles) agree well with those of the effective 1D model (squares) for all $K$ values. []{data-label="fig:7"}](fig7.eps){width=".9\linewidth"}
A method that is commonly used to measure the similarity between two distributions is called the Kullback-Leibler (KL) divergence [@kullback1951information], defined by $$D_{KL}\left(p\|q\right)=\sum_{i=1}^{N}p\left(x_i\right)\log{\frac{p\left(x_i\right)}{q\left(x_i\right)}}.$$ Here $p$ is a probability distribution and $q$ is an approximated probability distribution to $p$. In general, as $D_{KL}$ approaches zero, the probability distribution of $q$ approaches $p$, whereas when $D_{KL}=\mathcal{O}\left(1\right)$, $q$ is a poor approximation of $p$. The results of the KL divergence between the 1D and 2D models are shown in Fig. \[fig:8\]. We have seen that the time-scale separation leading to the effective 1D model requires that $\alpha\ll 1$. As a result, we expect the 1D approximation to break down as $\alpha$ becomes ${\cal O}(1)$. In Fig. \[fig:8\] the KL divergence is shown as a function of $\alpha$ for both the $K$-step and the geometric BSDs. It is evident that as $\alpha$ increases the approximation deteriorates, whereas for $\alpha=\mathcal{O}(1)$, $D_{KL}$ becomes $\mathcal{O}(1)$, which indicates that the 1D approximation breaks down in this regime.
![Shown is the Kullback-Leibler divergence (see text) between the numerically-calculated distributions of infection times, of the 1D and 2D models, for the $K$-step (triangles) and geometric (squares) BSDs, versus $\alpha$. One can see that, as long as $\alpha\ll 1$, the 1D model remains a good approximation of the 2D model. Here $\beta=0.3$ and $K=\langle k\rangle=50$. []{data-label="fig:8"}](fig8.eps){width=".7\linewidth"}
Inferring the statistics of infection times {#Statistics Infection}
-------------------------------------------
Previously we have analytically computed the DIT for the SSR case. We now present a semi-empirical argument that allows finding the DIT for a generic choice of BSD. To do so, let us observe how the mean and variance of the DIT behave as a function of the BSD. In the SSR case, computing the mean and standard deviation (STD) is straightforward. The mean time to infection is given by $\langle t\rangle=\int_0^{\infty}tP_t\left(N_0\right)dt\simeq \ln (Q_{2D}N_0)$, where $P_t(N_0)$ was given from Eq. (\[Time Distribution 2D\]), and the approximation holds when $\langle t\rangle\gg1$. Furthermore, the variance of the DIT in this case satisfies, $\sigma^2=\int_0^{\infty}t^2P_t(N_0)\mathrm{d}t - \langle t\rangle^2\simeq\pi^2/6$.
For $K>2$, $P_t(N_0)$ is unknown analytically, and thus the mean and variance cannot be found in this way. However, we have empirically observed that for large $K$, the right tail of the DIT decays exponentially with a slope that equals $1/(R-1)$. At the same time, these distributions have a bulk region, around the maximum, obtained approximately at $t=\ln N_0$. Thus, when $K$ is large, both the mean and variance of the DIT are governed by the exponentially decaying right tail, namely, they are solely determined by $R$, and are independent on other parameters. Using the slope of the right tail which we found empirically, the mean becomes $\langle t_{K-step}\rangle\simeq\ln N_0+R-1$, while the STD satisfies $\sigma_{K-step}\simeq R-1$. When the BSD is geometrically distributed, we have similarly found that $\langle t_{geometric}\rangle\simeq\ln N_0+R$ and $\sigma_{geometric}\simeq R$.
![(a) The relative error (see text) between the theoretical expression for the mean time to infection, $\langle t\rangle_{theo}$, and the numerical result, $\langle t\rangle_{num}$, as a function of $K$, for $\beta=2$. (b) The relative error between the theoretical expression for the STD of the distribution of infection times, $\sigma_{theo}$, and the numerical result, $\sigma_{num}$, as a function of $K$, for $\beta=2$. In both panels the triangles represent the $K$-step BSD and the squares represent the geometric BSD, with $\langle k\rangle=K$. []{data-label="fig:9"}](fig9.eps){width=".6\linewidth"}
Figure \[fig:9\] presents the relative error between the estimated expressions for the mean and STD, and the numerical results. We denote by $\langle t\rangle_{theo}$ and $\sigma_{theo}$ the theoretical mean and STD respectively, while $\langle t\rangle_{num}$ and $\sigma_{num}$ respectively denote the numerical mean and STD of the DIT. The relative error was calculated by computing $\left|\langle t\rangle_{theo}-\langle t\rangle_{num}\right|/\langle t\rangle_{num}$, and $\left|\sigma_{theo}-\sigma_{num}\right|/\sigma_{num}$. One can see a good agreement between the semi-empirical and numerical results, which improves as $K$ (and correspondingly $R$) is increased, where the tail of the DIT becomes closer to an exponential distribution.
To verify our empirical result that for large $K$ the DITs depend only on $R$, we plotted in Fig. \[fig:10\] several DITs by varying both $K$ and $\beta$ such that $R$ remained constant. This figure clearly demonstrates that the DITs are governed by a single parameter when $K$ is large.
![Shown are distributions of infection times to reach $N_0$ infected cells, as a function of the rescaled time, for the $K$-step BSD. Here $\alpha=0.01$, and the different symbols represent $K=10$ $(\triangle)$, $K=15$ $(*)$, $K=20$ $(\diamond)$ and $K=50$ $(\square)$, while we chose values of $\beta$ such that $R=1.9$ and $\lambda=0.009$ for all the curves. One can see that despite changing $K$ by a factor of $5$, the curves are almost indistinguishable, indicating that the distribution is solely governed by $R$, see text.[]{data-label="fig:10"}](fig10.eps){width=".65\linewidth"}
Having shown that for the $K$-step BSD the DIT is solely governed by $R$ (as long as $K\gg1$), we wanted to check whether the *width* of the BSD affects the DIT. To do so, we took a bi-modal BSD with $D(k)=1/2\delta_{k,K-\Delta}+1/2\delta_{k,K+\Delta}$, where $\Delta$ is the width of the BSD. For each BSD used, we kept the mean $K$ constant while changing its width (by changing $\Delta$). The results were then compared with the DIT in the case of the $K$-step BSD, and we found that the BSDs were independent on $\Delta$, see Fig. \[fig:11\]. These results indicate that when $K$ is large, the DIT only depends on the BSD’s mean (through $R$). That is, the DIT is independent on the higher moments of the BSD such as its width. This result also holds for the geometric BSD as discussed above.
![Shown are distributions of infection times to reach $N_0$ infected cells, as a function of the rescaled time. Here the result for the $K$-step BSD $(*)$ is compared with results of bimodal BSDs with a width of $5$ $(+)$, $10$ $(\square)$, and $50$ $(\bigtriangledown)$, for $R=16.66$, $K=50$, $\beta=0.5$, $\alpha=0.01$, and $N_0=10^3$. The figure indicates that at large values of $K$, the distribution of infection times is insensitive to the higher moments of the BSD (such as its width). []{data-label="fig:11"}](fig11.eps){width=".6\linewidth"}
Implications on Realistic Populations {#Implications Realistic Population}
-------------------------------------
We now show which possible implications our semi-empirical results have on the estimation of the DIT, using realistic biological parameters, for both the HIV and HCV.
As a first example, let us take a set of realistic rates that has been studied in [@pearson2011stochastic] for the HIV, where $\beta=0.15$, $\alpha=0.05$ and $K=20$, for which $R\simeq 2.6$. Pearson et al. [@pearson2011stochastic] have shown that for this set of parameters taking $N_0=32$ guarantees persistent infection, with a vanishingly low extinction probability. We have numerically computed the DIT for this set of parameters, for the cases of $K$-step and geometric BSDs, with $\langle k \rangle=K$, see Fig. \[fig:12\] [@footnote5].
![Shown are distributions of infection times to reach $N_0$ infected cells, versus the number of infection days. (a) HIV with $\beta=0.15$, $\alpha=0.05$, $\langle k\rangle=K=20$ and $N_0=32$, such that $N_0^{1D}=245$. (b) HCV with $\beta=0.125$, $\alpha=0.0175$, $\langle k\rangle=K=50 $ and $N_0^{1D}=110$. In both panels the triangles represent the $K$-step BSD and the squares represent the geometric BSD. Note, that in both cases the distribution displays a fat right tail, see text.[]{data-label="fig:12"}](fig12.eps){width=".9\linewidth"}
With these parameters the mean time to infection is $\langle t\rangle\simeq 3$ days for both BSDs, while the STDs are $\sigma_{K-step}\simeq 1.5$ and $\sigma_{geometric}\simeq 1.9$ days. Notably, while 3 days is the mean infection time, in the case of the $K$-step BSD, approximately 5$\%$ of the population will only be infected after 6 days, and 0.2$\%$ after 10 days. For the geometric BSD, approximately 8$\%$ of the population will be infected after 6 days, and 0.7$\%$ after 10 days. These numbers are much higher than those obtained for a Gaussian DIT with the same mean and STD.
As a second example, we took HCV (hepatitis C virus), with realistic parameters of $\alpha=0.0175$, $K=50$ and $\beta=0.125$ [@neumann1998hepatitis]. In Fig. \[fig:12\] we also plotted the DIT for the HCV and found that the mean time for infection satisfies $\langle t_{K-step}\rangle\simeq 9.5$ and $\langle t_{geometric}\rangle\simeq 11.9$ days, while the STDs are $\sigma_{K-step}\simeq 7.6$ and $\sigma_{geometric}\simeq 9.5$ days. Importantly, our results indicate that there is a non-negligible fraction of the population whose infection time is significantly longer than the typical infection time, by many standard deviations, and thus, it is imperative to know the entire DIT. That is, the fact that the DIT is highly-skewed, for both the HIV and HCV, clearly demonstrates that knowing the bulk of the DIT, described by the mean infection time and its standard deviation, is insufficient for assessing the infection times of a significant portion of the population.
Conclusions and Discussion {#Conclusion Discussion}
==========================
We have investigated the dynamics of the early infection stage of viral diseases such as HIV. The model we have considered included the dynamics of both the virions and infected cells. By employing the probability generating function formalism we were able to analytically find the distribution of infection times (DIT) for a particular choice of virion burst size distribution (BSD), while for other choices of BSD we have computed the DIT numerically and semi-empirically. Furthermore, by exploiting the time-scale separation between the dynamics of the virions and infected cells, we were able to reduce the 2D model into an effective 1D model for the infected cells only. Using the measure of Kullback-Leibler divergence, we have shown that the results for the DIT in the 1D model coincide with those of the 2D model, for any choice of BSD, in the limit of fast virion dynamics. We have also considered the bifurcation limit where the infected cell population grows slowly, and found the DIT for any arbitrary BSD.
Our results for the DIT indicate that for a realistic choice of parameters, the right tail of the DIT is exponential and thus, it is skewed towards the right. Therefore, the bulk of the DIT, described by the mean infection time and its standard deviation, is a poor measure for assessing whether an individual has been infected or not. Importantly, this may have implications on when HIV (or any other viral infection that behaves similarly) tests should be performed.
Notably, while our model is generic and holds for a wide variety of viruses, it is valid only for the early stages of infection, as it neglects immune system responses and medical treatment, such as vaccinations or medications. It also neglects other types of interactions between viruses and healthy T cells via apoptosis (self-destruction) [@selliah2003t], and cell to cell transmission of the virus, which may be significantly more efficient than infection by virions [@dimitrov1993quantitation; @galloway2015cell; @murooka2012hiv]. It would be interesting to study the implications of these various factors on the statistics of infection times.
Acknowledgements
================
This work was supported through the Israel Science Foundation Grant No. 300/14 and the United States-Israel Binational Science Foundation grant No. 2016-655.
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|
---
abstract: |
There is strong evidence in favor of the idea that dark matter is self interacting, with the cross section-to-mass ratio $\sigma / m \sim
1~{\text{cm}}^2/{\text{g}}\sim 1~\text{barn}/{\text{GeV}}$. We show that viable models of dark matter with this large cross section are straightforwardly realized with non-Abelian hidden sectors. In the simplest of such models, the hidden sector is a pure gauge theory, and the dark matter is composed of hidden glueballs with a mass around 100 MeV. Alternatively, the hidden sector may be a supersymmetric pure gauge theory with a $\sim
10~{\text{TeV}}$ gluino thermal relic. In this case, the dark matter is largely composed of glueballinos that strongly self interact through the exchange of light glueballs. We present a unified framework that realizes both of these possibilities in anomaly-mediated supersymmetry breaking, where, depending on a few model parameters, the dark matter may be composed of hidden glueballinos, hidden glueballs, or a mixture of the two. These models provide simple examples of multicomponent dark matter, have interesting implications for particle physics and cosmology, and include cases where a subdominant component of dark matter may be extremely strongly self interacting, with interesting astrophysical consequences.
author:
- 'Kimberly K. Boddy'
- 'Jonathan L. Feng'
- Manoj Kaplinghat
- 'Tim M. P. Tait'
bibliography:
- 'bibsidm.bib'
title: ' Self-interacting dark matter from a non-Abelian hidden sector '
---
Introduction
============
The standard model of cosmology describes a universe that is dominated by the vacuum energy $\Lambda$ and collisionless cold dark matter (CDM). The success of the $\Lambda$CDM model is based on its well-established record of describing the features of the large-scale structure observed in the Universe.
On smaller scales, however, the picture is much less clear. $N$-body simulations of collisionless CDM appear to disagree with observations on small scales, motivating dark matter properties that differ significantly from the standard paradigm. In particular, if dark matter is self interacting (able to scatter elastically with itself), simulations show that the core sizes and central densities of dwarf spheroidal galaxies, low-surface-brightness spirals, and galaxy clusters can all be brought in line with observations [@Rocha:2012jg; @Peter:2012jh; @Vogelsberger:2012ku; @Zavala:2012us]. Modifying $\Lambda$CDM to incorporate self-interacting dark matter (SIDM), sometimes called the $\Lambda$SIDM model, is consistent with constraints from the Bullet Cluster, measurements of dark matter halo shapes, and subhalo survival requirements. To make the simulations and observations consistent, the ratio of the dark matter’s self-interaction cross section to its mass should be in the range $\sigma/m \sim 0.1 - 10~{\mathrm{cm^2/g}}$. The requirement of such strong self interactions eliminates from consideration all of the most studied dark matter candidates, including weakly interacting massive particles (WIMPs), axions, and sterile neutrinos.
At the same time, such large cross sections are on par with nuclear-scale cross sections ($1~{\mathrm{cm^2/g}} \simeq
1.78~{\mathrm{barn/GeV}}$). The possibility that dark matter has color and interacts through the strong interactions of the standard model (SM) is highly constrained, for example, by searches for anomalous isotopes in sea water [@Hemmick:1989ns; @Verkerk:1991jf; @Yamagata:1993jq]. However, dark matter may self interact through non-Abelian forces (such as a dark analogue of QCD) in a hidden sector. As we will show below, this is straightforwardly realized in even the simplest such hidden sectors, with SU($N$) gauge symmetry and no additional matter content. For the confinement scales $\Lambda \sim 100~{\text{MeV}}$, the hidden gluons confine to form glueballs, and the resulting glueball dark matter has the required self interactions. For hidden sectors that are roughly the same temperature as the visible sector, the glueball relic density is generically too large, but the desired relic density may be obtained by adjusting the relative temperatures of the hidden sector and visible sector, as we discuss below.
This hidden glueball scenario for self-interacting dark matter is remarkably simple, but it is decoupled from the visible sector, both in the technical sense and in the sense that it is not motivated by any of the well-known problems of the SM. In addition, the correct relic density is arranged by tuning a free parameter, the ratio of hidden to visible sector temperatures, and so the scenario cannot be claimed to naturally produce the right thermal relic density, as in the case of WIMPs. At first sight, it might appear to be difficult to enhance the model to accommodate all of these desirable features, especially since the WIMP miracle requires weak-scale annihilation cross sections, whereas the required self interactions naturally suggest strong interactions.
In fact, however, we will show that all of these features are present in a supersymmetric version of the hidden glueball scenario, in which the hidden sector is a supersymmetric pure gauge theory. In this model, the dark matter is a $\sim 10~{\text{TeV}}$ hidden gluino, which freezes out in the early Universe when the temperature is high. At freeze-out, the theory is weakly coupled, but, as the Universe cools and expands, the theory confines, forming hidden glueballinos and glueballs. The glueballinos strongly interact via exchange of the hidden glueballs with the required self-interaction cross section. This scenario is straightforwardly accommodated in anomaly-mediated supersymmetry breaking (AMSB) scenarios [@Randall:1998uk; @Giudice:1998xp], which provide a connection to the problems of the SM and also allow the glueballinos to naturally inherit the correct relic density through the WIMPless miracle [@Feng:2008ya; @Feng:2008mu], a possibility discussed previously in Refs. [@Feng:2011ik; @Feng:2011uf; @Feng:2011in]. For related work on strongly interacting hidden sectors and dark matter, see Refs. [@Kribs:2009fy; @Alves:2009nf; @Falkowski:2009yz; @Lisanti:2009am; @Alves:2010dd; @Kumar:2011iy; @Cline:2013zca; @Kang:2006yd; @Higaki:2013vuv; @Heikinheimo:2013fta; @Bai:2013xga].
Of course, the supersymmetric models also contain glueballs, which, as in the nonsupersymmetric case, may be dark matter. As we will see, in different regions of the AMSB parameter space, the dark matter may be composed of mostly glueballinos, mostly glueballs, or a mixture of the two. For the case where the dark matter is composed of mostly glueballinos, we detail two possibilities. In the first case, the hidden sector is coupled to the visible sector only indirectly through the supersymmetry breaking mechanism. Since this coupling is extremely weak, the sectors can have different temperatures, and the glueball relic density may be very small for cold hidden sectors. An example cosmological timeline of events in this case is given in [Fig. \[fig:timeline1\]]{}.
![Example timeline of events in the supersymmetric pure SU($N$) theory without connectors, in terms of the hidden- and visible-sector temperatures $T^h$ and $T$. The hidden-sector coupling $\alpha_h$ is shown as a function of these temperatures. It is weak at gluino freeze-out but grows as the temperature drops, leading to confinement and the formation of glueballino and glueball dark matter at a temperature $\sim \Lambda$. The scenario is described in detail in [Sec. \[sec:noconnectors\]]{}, and the chosen parameters are represented by the yellow dot in [Fig. \[fig:AMSB-pure-relic\]]{}. []{data-label="fig:timeline1"}](running)
In the second case, the hidden sector is coupled to the visible sector through connector fields. The visible and hidden sectors, therefore, have the same temperature at early times, leading, [*a priori*]{}, to too-large glueball relic densities. Decays of glueballs are in conflict with big bang nucleosynthesis (BBN) and other astrophysical and particle constraints. Instead, we rely on a novel nonthermal process in the early Universe to deplete the gluon density, thereby suppressing the glueball density after confinement. In this case, the gluons annihilate into singlet right-handed neutrinos with $\sim
1~{\mathrm{GeV}}$ mass, and we reduce the hidden gluon density by forcing the right-handed neutrinos to decay into SM particles more quickly than they can annihilate back into hidden gluons. A representative timeline for this case is shown in [Fig. \[fig:timeline2\]]{}.
![As in [Fig. \[fig:timeline1\]]{}, but for supersymmetric pure SU($N$) theory [*with*]{} connectors. Since the hidden and visible sectors communicate efficiently in the early Universe, they share a temperature, $T$. The gluon population is depleted through annihilations to and the subsequent decays of the $\nu_R$ in the visible sector, and the resulting scenario has pure glueballino dark matter. The scenario is described in detail in [Sec. \[sec:connectors\]]{}, and the chosen parameters are represented by the yellow dot in [Fig. \[fig:AMSB-pureConn-relic\]]{}. []{data-label="fig:timeline2"}](running-withConn)
This work is organized as follows. In [Sec. \[sec:evidence\]]{}, we review the astrophysical evidence for self-interacting dark matter. In [Sec. \[sec:glueball\]]{}, we begin with the simplest possible case: nonsupersymmetric pure gauge hidden sectors and glueball dark matter. We discuss glueball self interactions and relic densities and determine the preferred parameters for this simple model. We then move to supersymmetric models with pure gauge hidden sectors and glueballino dark matter. In [Sec. \[sec:glueballinoscattering\]]{}, we review the calculation of the glueballino self-interaction cross section, and in [Sec. \[sec:glueballinorelic\]]{}, we discuss the glueballino relic density and the realization of the WIMPless miracle in the AMSB framework. Finally, with this groundwork, we present full AMSB models of glueballino/glueball dark matter without and with connectors in [Secs. \[sec:noconnectors\] and \[sec:connectors\]]{}, respectively. We conclude in [Sec. \[sec:conclusions\]]{}.
Last, we make a note on naming conventions. In the rest of this work, we follow the literature: glueballinos denote gluino-gluon bound states, while gluinoballs denote gluino-gluino bound states. In addition, unless otherwise stated, gluon, gluino, glueball, and glueballino refer to hidden sector particles and are denoted by $g$, $\tilde{g}$, ${\text{gb}}$, and ${\text{gbino}}$, respectively.
Astrophysical evidence for self-interacting dark matter {#sec:evidence}
=======================================================
The $\Lambda$CDM model is quite successful in describing the large-scale structure. The predictions of the standard six-parameter $\Lambda$CDM cosmology match remarkably well to the latest measurements of the cosmic microwave background (CMB) by the WMAP [@Hinshaw:2012aka] and Planck [@Ade:2013zuv] at large multipoles of the power spectrum. Additionally, CDM fits the dark matter power spectrum very well [@Reid:2009xm], using observations of luminous red galaxy clustering in the Sloan Digital Sky Survey [@Abazajian:2008wr].
Despite these agreements on large scales, observations of small-scale structures indicate that CDM is insufficient. Challenges to the $\Lambda$CDM paradigm arise largely from tensions between observation and cosmological simulations. Simulations of CDM create dark matter halos with density profiles that have steep, inverse-power-law behaviors (cusps) towards the center of the halo [@Navarro:1994hg; @Navarro:1995iw; @Navarro:1996gj; @Bullock:1999he; @Wechsler:2001cs]. Conversely, observations show that low-surface-brightness spiral galaxies (LSBs) [@Flores:1994gz; @Simon:2004sr; @KuziodeNaray:2007qi; @Dutton:2010dw; @deNaray:2011hy; @Oh:2010ea; @Salucci:2011ee; @Castignani:2012sr], satellite dwarf galaxies [@Walker:2011zu; @Salucci:2011ee], and galaxy clusters [@Sand:2003ng; @Sand:2007hm; @Newman:2009qm; @Newman:2011ip; @Coe:2012kj; @Umetsu:2012nj] exhibit constant-density cores. In addition to the core-cusp discrepancy [@deBlok:2009sp], the simulated central densities of halos are too high. By matching the luminosity function of the Milky Way to the Aquarius simulations [@Springel:2008cc], the brightest subhalos in the Milky Way are a factor of 5 less massive than predicted [@BoylanKolchin:2011de; @BoylanKolchin:2011dk]. If $\Lambda$CDM is correct, we are left to explain this “too-big-to-fail" problem in which the largest subhalos of the Milky Way do not luminesce; otherwise, some additional physics is needed in simulations to decrease the central densities of these overly massive halos.
To address these concerns with $\Lambda$CDM, there are a few generic possibilities to consider [@Weinberg:2013aya]: adding feedback from baryons in simulations [@Oh:2010mc; @Governato:2012fa; @Newman:2012nw], warm dark matter (WDM) [@Tremaine:1979we; @Bond:1980ha; @Olive:1981ak], and self-interacting dark matter [@Spergel:1999mh; @Firmani:2000ce; @Firmani:2000qe]. Feedback exists and should be included in simulations, but there may not be enough energy to eject a sufficient amount of mass from the halo center to solve the too-big-to-fail problem [@BoylanKolchin:2011dk]. WDM tends to be too efficient in wiping out structure, leaving too few subhalos in the Milky Way [@Polisensky:2010rw]. Additionally, lower bounds on WDM masses from Lyman-$\alpha$ forest measurements constrain the ability of WDM to solve the core-cusp problem over the full range of astrophysical scales needed [@Seljak:2006qw; @Viel:2013fqw]. Even with its mass unconstrained, WDM still leaves dwarf halos cuspy, though it does lower the central densities [@deNaray:2009xj].
On the other hand, self-interacting dark matter can soften halo cores and lower central densities, while preserving large-scale structure [@Spergel:1999mh] and satisfying bounds of $\sigma/m
\lesssim 1~{\mathrm{cm^2/g}}$ from the Bullet Cluster [@Randall:2007ph]. Indeed, simulations with constant dark matter cross section-to-mass ratios in the range $0.1 -
1~{\mathrm{cm^2/g}}$ show that self interactions can bring theory in line with observations of both halo profiles and shapes [@Rocha:2012jg; @Peter:2012jh]. Velocity-dependent self interactions widen this range to $0.1 - 10~{\mathrm{cm^2/g}}$ and can also soften cores and reduce the density of the brightest satellites to solve the too-big-to-fail problem [@Vogelsberger:2012ku; @Zavala:2012us].
With these results from simulation, dark matter with self interactions is a strong contender within particle physics to be a solution to the small-scale formation woes in astrophysics. From a particle physics perspective, we will see that self scattering is a quite reasonable and perhaps even generic property for dark matter to possess.
Glueball dark matter {#sec:glueball}
====================
The simplest construction resulting in dark matter that is a composite of a strongly interacting hidden sector is a pure Yang-Mills gauge theory. At large-energy scales, the theory consists of a weakly coupled set of massless gluons whose couplings are described by the gauge coupling. The theory is expected to confine at low energies at a scale $\Lambda$, where the gauge coupling becomes strong enough that perturbation theory breaks down [@Dietrich:2006cm; @Appelquist:1998rb; @Miransky:1996pd; @Cohen:1988sq; @Appelquist:1988yc; @Poppitz:2009tw; @Poppitz:2009uq]. At this point it develops a mass gap, and the low energy physics is described by a set of glueball states whose masses are characterized by $\Lambda$ through dimensional transmutation.
At very low energies $\ll \Lambda$, the physics is described by an effective field theory composed of the low-lying glueball states. In the absence of any coupling to the SM, the lightest of these states will be effectively stable.[^1] The detailed mass spectrum (and spins) of these states depends on the underlying choice of theory and is further clouded by strong coupling, which leaves results based on perturbation theory suspect. Generically, one expects the glueball spectrum to have a lowest-lying element whose mass is ${\cal O}(\Lambda)$, which, following the guidance of QCD, we take to be a $J^{CP}=0^{++}$ state [@Narison:1997nw; @Morningstar:1999rf]. There will also be a collection of excited states with various spins and whose mass splittings are roughly multiples of $\Lambda$.
Glueball self interactions {#sec:glueballscattering}
--------------------------
The various glueball states will interact with one another as a residual of the strong dynamics that bind them. Dimensional analysis dictates that the interactions among them will be proportional to $\Lambda$ to an appropriate power, with dimensionless coefficients characterized by naïve dimensional analysis (NDA) [@Manohar:1983md; @Cohen:1997rt]. For example, a description of a scalar glueball state, $\phi_0$, would look like $$\begin{aligned}
{\cal L}_{{\text{gb}}} & = &
\frac{1}{2} \partial_\mu \phi_0 \partial^\mu \phi_0 - \frac{1}{2} m^2 \phi_0^2
+ \frac{A}{3!} \phi_0^3 + \frac{\lambda}{4!} \phi_0^4 + \ldots \ ,\end{aligned}$$ where NDA would suggest that for the lowest-lying state $m \simeq
\Lambda$, and $A \simeq (4 \pi) \Lambda$, $\lambda \simeq (4 \pi)^2$, and the $+...$ indicates interaction terms in the form of higher dimensional operators that are suppressed by powers of $\Lambda$. Interactions involving the various glueball excited states can be formulated in a similar way.
For energies $\ll \Lambda$, the physics should be well described by an effective field theory composed of the lightest glueball. At kinetic energies of the order $\Lambda$, more of the lowest lying states become accessible and need to be included in the effective theory. At energies $\gg \Lambda$, the physics is described by the interactions of the gluons together with the structure functions that describe their distribution inside of the glueballs.
Although it is clear that glueballs are strongly self interacting, it is very difficult to make precise predictions for the scattering rate, given our general ignorance concerning strongly coupled theories. The expected cross section will be characterized by the confinement scale and strong coupling, $$\begin{aligned}
\sigma \left( {\text{gb}}~{\text{gb}}\rightarrow {\text{gb}}~{\text{gb}}\right) \simeq \frac{4 \pi}{\Lambda^2}~,\end{aligned}$$ which can also be understood from the geometric size of the glueballs, whose radius is $\sim 1 / \Lambda$.
Glueball relic density {#sec:glueballrelic}
----------------------
If the glueballs are stable on the scale of the age of the Universe, they will contribute to the total observed dark matter relic density. At early times, when their temperature is $T_h \gg \Lambda$, the hidden sector is represented by a plasma of gluons whose comoving relativistic number density is given by $$Y_\infty = \frac{n_{g}}{s}
= \left. \frac{[\zeta(3)/\pi^2]g_\textrm{eff} {T^h}^3}{(2\pi^2/45)g_{*S}T^3}
\right|_{t_f}
= \frac{45\zeta(3)}{2\pi^4} \xi_f^3 \frac{g_\textrm{eff}}{g_{*S}(t_f)} \ ,
\label{eq:Yinf}$$ where $s$ is the entropy in the visible sector, $\xi_f \equiv T^h / T$ is the ratio of temperatures in the hidden and visible sectors, $\zeta(3)\approx 1.202$ is the zeta function, and $g_\textrm{eff}=2(N^2-1)$ for an SU($N$) gauge theory. We use an early time, $t_f$ (which we will identify with the time of gluino freeze-out in the supersymmetric models discussed below), as a reference point. The quantity $Y_\infty$ remains constant as the Universe expands.
As the hidden sector temperature $T^h$ cools below the critical temperature $T^c \sim \Lambda$ [@Lucini:2005vg], there is a transition to the confined phase, and the energy density of the gluon plasma is converted into glueballs. The result is that after confinement the Universe is filled with nonrelativistic glueballs whose comoving number density is the same as that of the gluons up to factors of ${\mathcal{O}(1)}$. Consequently, today the glueballs are nonrelativistic with a relic density: $$\Omega_{{\text{gb}}} \sim \frac{Y_\infty s_0 \Lambda}{\rho_{c0}} \ .
\label{eq:gbRelic}$$ This expression assumes that there are no number-changing processes, but the glueballs may interact through a dimension-5 operator to give $3 \to 2$ scatterings [@Carlson:1992fn]. We ignore this possibility here and leave it for future work.
Viable glueball parameters {#sec:glueballmodels}
--------------------------
![Glueball dark matter in the case of a nonsupersymmetric pure gauge SU($N$) hidden sector. The self-interaction cross section and relic density are given in the $(\xi_{\Lambda}, \Lambda)$ plane, where $\Lambda$ is the confinement scale in the hidden sector, and $\xi_{\Lambda} \equiv
T^h/T$ is the ratio of hidden to visible sector temperatures at the time that $T^h = \Lambda$. The self-interaction cross section is in the range $\langle \sigma_T \rangle / m_X=0.1 -
1~{\mathrm{cm^2/g}}$ in the shaded region. The glueball relic density is $\Omega_{{\text{gb}}} = \Omega_{\text{DM}} \simeq 0.23$ on the diagonal contours for the number of colors $N$ indicated. \[fig:glueball\]](glueball-only)
Glueball dark matter is thus primarily characterized by two quantities: the confinement scale $\Lambda$, which simultaneously controls the dark matter mass and its self-interaction cross section, and $\xi_\Lambda$, the ratio of temperatures of the hidden and visible sectors at the time of confinement. Also relevant is the number of gluon degrees of freedom; for an SU($N$) gauge theory this is specified by $N$ through $g_\textrm{eff}=2(N^2-1)$. In [Fig. \[fig:glueball\]]{}, we show the parameter space in the $(\xi_{\Lambda},
\Lambda)$ plane. The scattering cross section is independent of $\xi_\Lambda$, which together with the choice of $N$ controls the relic density of glueballs. The scattering cross sections of interest suggest $\Lambda \sim 100$ MeV, amusingly close to $\Lambda_{\rm QCD}
\approx 300$ MeV. Note that since the cross section is constant, the acceptable upper limit from simulations is $1~{\mathrm{cm^2/g}}$, in particular, to stay within cluster constraints. This limit will increase to $10~{\mathrm{cm^2/g}}$ for velocity-dependent cross sections, which we begin discussing in [Sec. \[sec:glueballinoscattering\]]{}. The relic density requires the hidden-sector temperature to be a few orders of magnitude below the visible temperature at the time of confinement.
Glueballino self interactions {#sec:glueballinoscattering}
=============================
The simplest extension to the pure gauge hidden sector discussed in [Sec. \[sec:glueball\]]{} is to add a massive (mass $m_X \gg \Lambda$) gauge adjoint Majorana fermion to the theory, resulting in a spectrum with two types of composites: the bosonic glueballs with a mass $\sim \Lambda$ and the fermionic states with masses $\sim
m_X$ [@Farrar:1995ew; @Raby:1997pb; @Raby:1998xr; @Kauth:2009ud]. Each sector contains excited states whose mass splittings are again characterized by $\Lambda$. In the absence of further ingredients, the massive fermionic states are stable because of Lorentz invariance, and this construction allows one to realize a situation where the dark matter is (mostly) composed of the heavy composite fermions that self interact via exchange of the much lighter glueballs, naturally realizing two widely separated energy scales. This dark sector is identical to a softly broken $N=1$ supersymmetric gauge theory and can be considered the supersymmetric version of the model of [Sec. \[sec:glueball\]]{}. In that language, the composite fermions are glueballino states.
The self interactions of glueballinos are dominated by the exchange of glueballs. At low energies, when the kinetic energy available is $\lesssim \Lambda$, the scattering will be elastic. If there is sufficient kinetic energy, $$\begin{aligned}
\label{eq:inelastic}
\frac{1}{2} m_X v^2 \geq \Lambda \ ,\end{aligned}$$ inelastic scattering into excited states and glueball emission becomes possible, leading to novel effects, such as additional rapid halo cooling. The inelastic effects are not modeled in the $\Lambda$SIDM simulations and so are not well understood. For the remainder of this work, we focus on the elastic scattering regime and comment later in this section on systems where this approximation breaks down.
NDA suggests that the coupling between glueballs and glueballinos is $\alpha \sim 1$. Even for elastic scattering, there will be a large number of distinct glueball states, which are capable of mediating self interactions of the glueballinos, but the dominant contribution arises from the lightest glueball states, which mediate the longest range interactions. Thus, we model the induced potential between two glueballinos as an attractive Yukawa interaction with a range $\Lambda$ and strength $\alpha \sim 1$: $$V(r)=-\frac{\alpha}{r}\exp(-\Lambda r) \ .$$
It is common to use the transfer cross section $$\sigma_T = \int d\Omega (1-\cos\theta) \frac{d\sigma}{d\Omega}$$ to compare predictions to observations and simulations. We have numerically solved the Schrödinger equation to calculate $\sigma_T$, following the methods of Ref. [@Tulin:2013teo]. For the astrophysical systems of interest, to achieve the desired cross sections of $0.1 - 10 ~{\mathrm{cm^2/g}}$ with $m_X \gtrsim {\mathrm{TeV}}$, the parameters must be in the classical scattering regime, $m_X v \gg
\Lambda$. Scattering from Yukawa potentials has been studied in this regime in the context of classical, complex plasmas [@Khrapak2003:yukawa; @Khrapak2004:yukawa; @Khrapak:2004transfer], and simple analytic fits to numerical results for the transfer cross section have been derived. These plasma physics results may be applied directly to the present dark matter case [@Feng:2009hw] (in fact, they describe the dark matter model exactly, whereas the Yukawa potentials are only an approximation to screened Coulomb interactions in the plasma physics context), and we have checked that these agree well with our numerical results.
Within a galactic halo or cluster, the dark matter particles have a velocity distribution that we take to be Maxwell-Boltzmann, and so $$f(v_i) = \left(\pi v_0^2 \right)^{-3/2} e^{-v_i^2/v_0^2} \ ,$$ where $v_0$ is the mode and $\langle v^2_i \rangle =(3/2)v_0^2$ is the square of the three-dimensional velocity dispersion. This distribution is expected for cross sections of $\sigma/m=1.0~{\mathrm{cm^2/g}}$ and above [@Vogelsberger:2012sa]; for the slightly lower cross sections that are still of interest to us, the distribution may be modified, but we do not expect this to impact our results significantly. Simulations [@Rocha:2012jg] show that $\langle v^2_i \rangle \approx (1.2\, V_\textrm{max})^2$, where $V_\textrm{max}$ is the peak circular velocity of a given system, and thus $v_0 \approx 0.98\, V_\textrm{max}$. The astrophysical systems of interest have values of $V_\textrm{max}$ in the ranges $20 -
50~{\mathrm{km/s}}$ for dwarfs, $50-130~{\mathrm{km/s}}$ for LSBs, and $700-
1000~{\mathrm{km/s}}$ for clusters. We make a simplistic estimate for the dark matter escape velocity, $v_\textrm{esc}^2=2v_0^2$, so that the largest relative velocity between particles is $2\sqrt{2}v_0$. For two scattering dark matter particles with the velocities $\vec{v}_1$ and $\vec{v}_2$, the velocity-averaged transfer cross section is $$\begin{aligned}
\langle \sigma_T \rangle
&=& \int \frac{d^3v_1\ d^3v_2}{(\pi v_0^2)^3}
e^{-v_1^2/v_0^2} e^{-v_2^2/v_0^2} \sigma_T(|\vec{v}_1-\vec{v}_2|) \nonumber\\
&=& \int_0^{2\sqrt{2}v_0} \frac{d^3v}{(2\pi v_0^2)^{3/2}}
e^{-v^2/2v_0^2} \sigma_T(v) \ .
\label{eq:sigmaT-avg}\end{aligned}$$ Note that although the escape velocity may be an underestimate here, increasing it by a factor of 10 changes $\langle \sigma_T \rangle $ only at the 1% level.
The thermally averaged transfer cross section, then, depends on four parameters: $m_X$, $\Lambda$, $\alpha$, and $V_{\text{max}}$. In [Fig. \[fig:AMSB-pure-scattering-params\]]{}, we plot the ratio $\langle
\sigma_T \rangle /m_X$ in the $(m_X, \Lambda)$ plane for $\alpha = 1$ and three representative characteristic velocities: $V_\textrm{max} =
40~{\mathrm{km/s}}$ for dwarfs, $V_\textrm{max} = 100~{\mathrm{km/s}}$ for LSBs, and $V_\textrm{max} = 1000~{\mathrm{km/s}}$ for clusters. For masses $m_X \sim 1~{\mathrm{TeV}}$ and $\Lambda\sim 10~{\mathrm{MeV}}$, we achieve transfer cross sections around the targeted range between $0.1~{\mathrm{cm^2/g}}$ and $1.0~{\mathrm{cm^2/g}}$ for all three systems under consideration. The transfer cross section decreases as a function of $v$ in the classical regime; thus, systems with larger characteristic velocities have smaller cross sections, all else being equal. The LSB line at $0.1~{\mathrm{cm^2/g}}$, for instance, lies below that for dwarfs, because a larger interaction range (corresponding to a smaller $\Lambda$) is needed to counter its larger velocity to give the same $\sigma_T$ as the dwarfs. Toward the lower values of $m_X$, the scattering exhibits resonant behavior due to the formation of quasibound states [@Tulin:2013teo], analogous to Sommerfeld enhancements in annihilations.
The region below the straight magenta lines in [Fig. \[fig:AMSB-pure-scattering-params\]]{} is where the dark matter typically has $(1/2) m_X v^2 > \Lambda$, and modifications from inelastic scattering processes can be important. We urge the reader to keep in mind that while in this region the classical elastic scattering cross section (for our assumed Yukawa potential) falls below about $3\pi/\Lambda^2$, we expect other energy-exchange mechanisms to become important in dark matter halos. Note that for clusters ($v \sim 3 \times 10^{-3}$), this is a substantial region of the interesting parameter space: $(m_X/{\rm TeV}) \gtrsim
(\Lambda/ 10~{\rm MeV})$. This suggests that the elastic glueballino scattering curves plotted for clusters in [Fig. \[fig:AMSB-pure-scattering-params\]]{} and other figures are far from the whole story. We expect new astrophysical phenomenology, especially in clusters of galaxies, and this deserves separate consideration.
![The ratio of the thermally averaged transfer cross section to the dark matter mass $\langle \sigma_T \rangle /m_X$ in the $(m_X,
\Lambda)$ plane for $\alpha = 1$ and three different astrophysical systems: dwarf galaxies ($V_\textrm{max} = 40~{\mathrm{km/s}}$, solid), LSBs ($V_\textrm{max} = 100~{\mathrm{km/s}}$, dashed), and clusters ($V_\textrm{max} = 1000~{\mathrm{km/s}}$, dotted). For each system, three values of the cross section are shown: $0.1~{\mathrm{cm^2/g}}$ (top), $1~{\mathrm{cm^2/g}}$ (middle), and $10~{\mathrm{cm^2/g}}$ (bottom). The region below the straight magenta lines shows where inelastic processes may modify the picture based on elastic scattering for each type of system.[]{data-label="fig:AMSB-pure-scattering-params"}](scattering)
Glueballino relic density {#sec:glueballinorelic}
=========================
One goal of supersymmetrizing the pure gauge hidden sectors considered in [Sec. \[sec:glueball\]]{} is to revive the possibility of dark matter with naturally the right relic density, as in the case of WIMPs, but now for self-interacting dark matter. In this section, we first review the machinery required to calculate a glueballino relic density from the freeze-out of thermal relic gluinos. We then discuss the possibility of realizing the correct thermal relic density through the WIMPless miracle in AMSB models [@Feng:2011ik].
Gluino freeze-out {#sec:freezeout}
-----------------
In a supersymmetric pure gauge hidden sector, the gluinos are initially in equilibrium with a thermal bath of gluons at a hidden-sector temperature $T^h$. As the Universe cools below the gluino mass $m_X$, however, the gluinos freeze out. The gluino is the lightest supersymmetric particle in the hidden sector, and we will assume it is stable. In the absence of couplings to the visible sector, stability is guaranteed by Lorentz symmetry, as the gluino is the only fermion in the hidden sector.
The gluino relic density is determined by the usual thermal freeze-out analysis, but with the slight extra complication that it occurs in a hidden sector with a temperature that may differ from the visible sector. For $S$-wave annihilation, the relic density of a thermal relic in a hidden sector is [@Feng:2008mu] $$\Omega_X \approx \frac{s_0}{\rho_{c0}}
\frac{3.79 x_f}{(g_{*S}/\sqrt{g_*^\textrm{tot}}) {M_{\text{Pl}}}\langle \sigma v \rangle } \ ,
\label{glueballinoRelic}$$ where $s_0$ is the entropy of the visible sector today, $x_f \equiv
m_X/T_f$, $\rho_{c0}$ is the critical density today, and $g_*^\textrm{tot}=g_* + \xi^4_f g_*^h$ at freeze-out.
We now discuss the various quantities entering [Eq. (\[glueballinoRelic\])]{}. For the annihilation process $\tilde{g}
\tilde{g} \to g g$, we use an $S$-wave cross section, $$\langle \sigma v \rangle = k_N \frac{\pi\alpha_X^2}{m_X^2} \ ,$$ where $\alpha_X = g_h^2(m_X)/4\pi$ is the fine-structure constant with a corresponding hidden-gauge coupling $g_h$ evaluated at the scale $m_X$, and $k_N$ is an ${\mathcal{O}(1)}$ $N$-dependent coefficient, which we simply set to $1$. Additionally, we set $x_f=25\xi_f$, which is a good approximation for a large set of parameters [@Feng:2008mu]. Given this, $\Omega_X$ scales approximately linearly with $\xi_f$. The latest Planck results give a value of $\Omega_\textrm{DM} h^2 =
0.1196 \pm 0.0031$ from a six-parameter fit to the $\Lambda$CDM model [@Ade:2013zuv].
To determine the number of relativistic degrees of freedom in the visible and hidden sectors, note that, although the hidden and visible sectors need to interact gravitationally, they do not necessarily have to communicate otherwise, even at high energies. Thus, the sector temperatures are generically unrelated, and the ratio $\xi = T^h/T$ parametrizes this difference. The comoving entropies in the visible and hidden sector are independently conserved, and the values of $\xi$ at the different times $t_i$ and $t_f$ are related by $$\frac{g_{*S}^h(t_i)}{g_{*S}(t_i)} \xi_i^3
= \frac{g_{*S}^h(t_f)}{g_{*S}(t_f)} \xi_f^3 \ .
\label{eq:xiRelation}$$ The effective numbers of relativistic degrees of freedom associated with the entropy (energy) density in the visible and hidden sectors are $g_{*S}$ and $g_{*S}^h$ ($g_*$ and $g_*^h$), respectively. As we will see, for most of the parameter space of interest, the gluinos freeze out at visible-sector temperatures at or above $T_\textrm{SM}\approx
300~{\mathrm{GeV}}$, so that all SM particles are relativistic and $g_{*S}=g_*=106.75$. Although there may be minimal supersymmetric standard model (MSSM) superpartners with low enough masses to contribute to $g_*$ at freeze-out, we assume the contribution is negligible, with most of the visible supersymmetric-partner spectrum being above $m_X$. For the gluons and gluinos, $$g_*^h = g_{*S}^h =
\begin{cases}
4(N^2-1) & T^h \gtrsim m_X \\
2(N^2-1) & m_X \gtrsim T^h > \Lambda \ .
\end{cases}$$
The WIMPless miracle and AMSB {#sec:wimpless}
-----------------------------
As noted above, the gluino thermal relic density has the parametric dependence $$\Omega_X \sim \frac{1}{\langle \sigma v \rangle }
\sim \frac{m_X^2}{\alpha_X^2} \ .$$ For weak-scale masses and weak interaction coupling strengths, $\Omega_X$ is of the desired size; this is the essence of the WIMP miracle. For the hidden sector, we have great freedom in choosing the parameters $m_X$ and $\alpha_X$, and may simply choose them to yield the correct relic density. However, it is preferable if the correct mass-to-coupling ratio is set in a noncontrived way. This is a property of models that realize the WIMPless miracle [@Feng:2008ya; @Feng:2008mu], where the dark matter mass and coupling are not fixed individually, but the ratio $m_X^2/\alpha_X^2$ is fixed to the desired value by the model framework.
Supersymmetric models with AMSB [@Giudice:1998xp; @Randall:1998uk] provide a particularly nice realization of the WIMPless miracle [@Feng:2011ik; @Feng:2011uf; @Feng:2011in]. In AMSB, the MSSM is sequestered from the supersymmetry-breaking sector, so the gaugino masses in the visible sector do not receive any tree-level contributions and are instead generated by the Weyl anomaly, leading to $$m_v \sim \frac{\alpha_v}{4\pi} m_{3/2} \ ,$$ where $m_{3/2}$ is the gravitino mass, $\alpha_v$ is a SM fine-structure constant, and $m_v$ is of the order of the weak scale, if these models are to address the gauge hierarchy problem. In any additional sequestered hidden sector of the theory, the hidden-sector superpartner masses will be given by a similar relation, $$m_X \sim \frac{\alpha_X}{4\pi} m_{3/2} \ ,$$ where $\alpha_X$ is the hidden sector’s fine-structure constant. Since there is only one gravitino mass, $m_X / \alpha_X \sim m_v /
\alpha_v$, and any hidden sector thermal relic in AMSB can be expected to have the desired relic density, even if $m_X$ and $\alpha_X$ differ, perhaps greatly, from the SM values.
The visible sector of AMSB models contains a stable thermal relic, the lightest neutralino. However, the standard AMSB relations imply that this is the wino. Winos annihilate very efficiently, and must have masses around $2.7-3.0~{\text{TeV}}$ to be all of dark matter [@Hisano:2006nn; @Hryczuk:2010zi]. The thermal relic density scales as $\sim m_{\tilde{W}}^{-2}$, and so for lighter and more natural values closer to the Large Electron-Positron (LEP2) collider experimental limit $m_{\tilde{W}}
\agt 100~{\text{GeV}}$ [@lep2-01-03.1; @lep2-02-04.1], the wino thermal relic density is completely negligible. We will therefore neglect it below, and take this as additional motivation to develop AMSB models with viable hidden-sector dark matter candidates.
The particle spectrum in AMSB models is completely specified by quantum numbers, dimensionless couplings, and the gravitino mass. In the visible sector, the wino mass limit $m_{\tilde{W}} \gtrsim
100~{\mathrm{GeV}}$ implies $$m_{3/2} \gtrsim 37~{\mathrm{TeV}} \ .
\label{eq:LEP2-bound}$$ In the hidden sector, at scales above $m_X$, the one-loop $\beta$-function coefficient is $b_H=-3N$, and the theory is asymptotically free. The gluino mass is $$m_X = -b_H \frac{\alpha_X}{4\pi} m_{3/2} =
3N \frac{\alpha_X}{4\pi} m_{3/2} \ .$$ Below $m_X$, we have a nonsupersymmetric SU($N$) gauge theory with a $\beta$-function coefficient: $b_L=-(11/3)N$. The theory is expected to confine at the scale $$\Lambda \sim m_X \exp\left(\frac{-6\pi}{11 N\alpha_X}\right)
= m_X \exp\left(\frac{-9m_{3/2}}{22 m_X}\right) \ .
\label{Lambda}$$ With this relationship, the relic density in [Eq. (\[glueballinoRelic\])]{} becomes $$\Omega_X \approx \frac{s_0}{\rho_{c0}}
\frac{\left[g_* + 2(N^2-1)\xi_f^4\right]^{1/2}}{g_{*S}}
\frac{3.79 \cdot 25 \xi_f}{{M_{\text{Pl}}}}
\frac{9 N^2}{16 \pi^3} m_{3/2}^2 \ .
\label{glueballinoRelic-reduced}$$
Glueballino/Glueball dark matter without connectors {#sec:noconnectors}
===================================================
Given the results above, we can now present simple AMSB models of self-interacting dark matter. We begin by considering the simple case without connector fields, in which the visible and hidden sectors are decoupled. The visible sector is the MSSM; the tachyonic slepton problem is assumed to be solved in a way that does not impact the masses of the MSSM gauginos, and the wino is assumed to be the visible lightest supersymmetric particle (LSP), with a negligible thermal relic density. The hidden sector is a pure SU($N$) gauge theory, consisting of gluinos and gluons, which confine to form glueballino and glueball dark matter.
There are only four independent parameters in the theory, which may be taken to be $$m_X, ~\Lambda, ~N, ~\xi_f \ .$$ These determine $\alpha_X$ and $m_{3/2}$ through [Eq. (\[Lambda\])]{}. In contrast to the model-independent discussion of [Sec. \[sec:glueballinoscattering\]]{}, in AMSB models, renormalization group equations relate the high-scale parameters $m_X$ and $\alpha_X$ to the low-scale parameter $\Lambda$. In terms of these parameters, the glueball self-interaction cross section and relic density are determined as described in [Secs. \[sec:glueballscattering\] and \[sec:glueballrelic\]]{}, and the glueballino self-interaction cross section and relic density are determined as described in [Secs. \[sec:glueballinoscattering\] and \[sec:glueballinorelic\]]{}.
We first present results for models with mostly glueballino dark matter in [Fig. \[fig:AMSB-pure-relic\]]{}. We scan over the $(m_X, \Lambda)$ plane. At every point in this plane, we require that glueballinos make up 90% (top panel) or 99.99% (bottom panel) of the dark matter, and glueballs make up the remaining 10% or 0.01%. The constraints on $\Omega_{{\text{gbino}}}$ and $\Omega_{{\text{gb}}}$ determine $N$ and $\xi_f$; contours of constant $N$ and $\xi_f$ are shown. The lower bound of [Eq. (\[eq:LEP2-bound\])]{} excludes parameter space with a low $m_X$. In the remaining parameter space, $m_X/\Lambda \gtrsim 10^3$, which is more than sufficient to ensure $T_f^h > \Lambda$, so gluino freeze-out occurs in the weakly interacting theory, and the thermal freeze-out calculation is valid.
\
These relic density results for a particular glueballino density may be understood as follows. On a given curve of constant $N$, larger dark matter masses imply larger thermal relic densities and so require smaller values of $\xi_f$ to keep $\Omega_{{\text{gbino}}}$ fixed. Once $\xi_f$ decreases, a larger $\Lambda$ is required to keep $\Omega_{{\text{gb}}}$ constant. Note also that for $\Lambda \sim {\text{MeV}}$ and $\xi_f \sim 1$, glueballs overclose the Universe. To avoid this, $\xi_f$ must be lowered, and, to have mostly glueballino dark matter, $m_X$ must be a bit larger than the weak scale. In short, the presence of glueballs forces the model away from the [*a priori*]{} most natural parameter space with a low $m_X$ and $\xi_f \sim 1$. In the context of AMSB, however, it is rather natural to assume that the hidden and visible sectors are separated at high scales and $\xi_f \ll
1$. Given this, the WIMPless miracle is nicely realized in regions of parameter space with $\xi_f \sim 0.01$ and $N \sim {\mathcal{O}(1)}$ for $\Omega_{{\text{gbino}}} = 0.9 \Omega_\text{DM}$.
There are also differences between the 90% and 99.99% glueballino cases. The curves of constant $N$ and constant $\xi_f$ shift as the relative amounts of glueball and glueballino dark matter change. By focusing on a particular point in the $(m_X,\Lambda)$ plane and comparing [Eq. (\[eq:gbRelic\])]{} and [Eq. (\[glueballinoRelic-reduced\])]{}, we find $$\xi_f \sim \frac{\Omega_{\text{gb}}^{1/2}}{\Omega_{\text{gbino}}^{1/2}}
\qquad\textrm{and}\qquad
N \sim \frac{\Omega_{\text{gbino}}^{3/4}}{\Omega_{\text{gb}}^{1/4}}$$ for $N^2 \gg 1$. When the glueball density is reduced by three orders of magnitude, we expect $N$ to increase by a factor of $10^{3/4} \sim 6$ and $\xi_f$ to decrease by a factor of $10^{3/2} \sim 30$; this can be seen in [Fig. \[fig:AMSB-pure-relic\]]{}.
Of course, the goal is not simply to obtain a multicomponent model of dark matter with the correct relic densities, but to obtain self-interacting dark matter. The regions with the preferred self-interaction cross sections are also shown in [Fig. \[fig:AMSB-pure-relic\]]{}. For values of $m_X \sim 10~{\text{TeV}}$, $\Lambda
\sim 1~{\text{MeV}}$, $2 \leq N \alt 10$, and $10^{-3} \alt \xi_f \alt
10^{-2}$, we find models that satisfy the relic density constraints and also satisfy the scattering constraints for dwarfs and LSBs. Viable models also exist for the lower values of $m_X$ down to the LEP2 limit for larger values of $N$ and lower values of $\xi_f$. A representative model is one with $m_X \simeq 14~{\mathrm{TeV}}$, $\Lambda \simeq 0.35~{\mathrm{MeV}}$, $N=2$, and $\xi_f \simeq 0.02$; this is shown as a yellow dot in [Fig. \[fig:AMSB-pure-relic\]]{}. For these parameters, [Fig. \[fig:timeline1\]]{} shows how the dark matter coupling behaves from the scale $m_X$ down to confinement.
Measurements of nuclei abundances and of the CMB place restrictions on the number of light degrees of freedom $\Neff$ around the time of BBN that contribute to the expansion of the Universe. Results from Planck give $\Neff=3.30 \pm 0.27$ [@Ade:2013zuv]. An interesting question, then, is whether these models also imply nonstandard values of $\Neff$. Once the hidden-sector temperature drops below the confinement scale, glueballinos and glueballs form. This occurs when the visible sector’s temperature is $$T_\Lambda = \frac{T_\Lambda^h}{\xi_\Lambda} \sim \frac{\Lambda}{\xi_\Lambda}
= \frac{\Lambda}{\xi_f} \left(\frac{g_{*S}(t_f)}{g_{*S}(t_\Lambda)}\right)^{1/3} \ ,
\label{eq:confinementTemp}$$ using [Eq. (\[eq:xiRelation\])]{} with $g_{*S}^h(t_\Lambda)=g_{*S}^h(t_f)$. For the representative example parameters given above, the confinement scale is $T_\Lambda \sim 90~{\mathrm{MeV}}$; confinement occurs well before BBN and structure formation, as expected. There is therefore no relativistic, massless species to act as the hidden-sector bath during BBN. At the time of BBN, the hidden-sector temperature is not well defined, and its contribution to $\Neff$ is essentially zero.
We next consider the case of mostly glueball dark matter. To be concrete, we present the case of $\Omega_{{\text{gb}}} = 0.9 \,
\Omega_{\text{DM}}$ and $\Omega_{{\text{gbino}}} = 0.1 \, \Omega_{\text{DM}}$ in [Fig. \[fig:AMSB-pure-relic-glueball\]]{}. Once again, we show contours of constant $N$ and $\xi_f$, but now we include the glueball scattering constraints from [Fig. \[fig:glueball\]]{}, since glueballs are the dominant component of dark matter. The values of $m_X$ that satisfy relic and scattering constraints for a given $N$ are fairly similar to those in the case of mostly glueballino dark matter; however, the corresponding values of $\Lambda$ are a few orders of magnitude larger than the mostly glueballino case.
\
In [Figs. \[fig:AMSB-pure-relic\] and \[fig:AMSB-pure-relic-glueball\]]{}, the fraction of glueballino to glueball dark matter is fixed. Of course, different values are possible. In [Fig. \[fig:AMSB-pure-relic-Omega\]]{}, we fix $N=2$ and vary $m_X$ and $\Lambda$; $\xi_f$ is set by the requirement that $\Omega_{{\text{gbino}}} + \Omega_{{\text{gb}}} = \Omega_{\text{DM}}$. The results are presented in the $(\langle \sigma_T \rangle_{{\text{gbino}}} / m_X,
\langle \sigma_T \rangle_{{\text{gb}}} / \Lambda)$ plane, where $V_{\text{max}} = 40~{\text{km}}/{\text{s}}$, and contours of constant $\Omega_{{\text{gbino}}} / \Omega_{{\text{gb}}}$ are shown. Regions excluded by LEP2 and by cluster bounds are shaded.
[Figure \[fig:AMSB-pure-relic-Omega\]]{} shows that the fraction of dark matter that is composed of glueballinos may take almost any value in the parameter space. Of course, regions of parameter space that are overwhelmingly glueballino dominated and have too-large glueballino self interactions are excluded, as are regions that are overwhelmingly glueball dominated with too-large glueball self interactions. The parts of parameter space that are certainly excluded by these considerations are indicated, but the position of this boundary is somewhat uncertain and requires detailed $N$-body simulations (modeling both components of dark matter) to determine. The cluster constraints [@Randall:2007ph; @Peter:2012jh] are relevant here because glueballs have a velocity-independent scattering cross section and these constraints dictate that glueballs must be the subdominant component of dark matter in all of the parameter space shown in [Fig. \[fig:AMSB-pure-relic-Omega\]]{}.
Especially interesting, however, are the regions of parameter space with a subdominant component of dark matter that self interacts very strongly. For example, the dark matter may be 99% glueballinos and 1% glueballs, but the glueballs may have $\langle \sigma_T
\rangle_{{\text{gb}}} / \Lambda \sim 10^5 - 10^{11}~{\text{cm}}^2/{\text{g}}$. Such possibilities are not ruled out by the constraints discussed so far but may have very interesting astrophysical implications.
It has been pointed out that, at early times before the halo has had time to form a core through self interactions, seed black holes can grow by accreting self-interacting dark matter [@Hennawi:2001be]. In the mixed self-interacting dark matter scenario where one of the components has $\langle
\sigma_T\rangle/m \gg 1{\text{cm}}^2/{\text{g}}$, this accretion can be highly enhanced. The possibility that supermassive black hole growth is seeded by the self interactions of either the glueballs or glueballinos is an exciting prospect. There is not yet a clear picture of how $10^9~{\rm M}_\odot$ quasars are assembled already by $z
\gtrsim 6$ within the standard $\Lambda$CDM cosmology. Models starting with the expected $100~{\rm M}_\odot$ seeds require special assumptions about the mass accretion histories of these quasars [@oai:arXiv.org:0807.4702], which become more strained as higher redshift quasars are found [@oai:arXiv.org:1106.6088]. Self interactions within the dark matter sector may have a big role to play in this story, as they generically enhance the early black hole accretion rate.
![Mixed dark matter without connectors to the standard model. We show curves of constant $\Omega_{{\text{gbino}}}/\Omega_{{\text{gb}}}$ in the $(\langle \sigma_T
\rangle_{{\text{gbino}}} /m_X, \langle \sigma_T \rangle_{{\text{gb}}} /\Lambda)$ plane, for $N=2$ and considering dwarf systems with $V_\textrm{max}=40~{\mathrm{km/s}}$. The black curves have $\Omega_{{\text{gbino}}}/ \Omega_{{\text{gb}}} = 0.1, 1, 10, 100$, as indicated. The bound from LEP2 is shown in the red shaded region. A stronger bound from clusters is shown in the lower hatched magenta region; since the glueball scattering cross section is the same on all scales, its value is limited for the dwarf systems to avoid violating bounds from cluster scales. We caution the reader that the bound may be stronger and it is certainly not as sharp as indicated by the hatched region. The hatched magenta wedge near the upper right-hand portion of the graph represents an upper limit of $10~{\mathrm{cm^2/g}}$ for the case of mostly glueballino dark matter, which will have important implications for cores in dwarfs galaxies and may be excluded by a comparison to the observed core sizes and densities (e.g., Ref. [@Rocha:2012jg; @zavala13]). []{data-label="fig:AMSB-pure-relic-Omega"}](mixed-sidm)
There is a tight correlation between the mass of supermassive black holes in the centers of galaxies and the velocity dispersion or luminosity of the bulge [@oai:arXiv.org:0903.4897]. By requiring that the predicted masses of supermassive black holes are not overly large, it should also be possible to constrain the ratio $\Omega_{\text{gb}}/\Omega_{\text{gbino}}$ in mixed self-interacting dark matter models where $\langle \sigma_T \rangle_{\text{gb}}/\Lambda$ is large. To correctly implement this constraint, many new features of our simple model and their astrophysical consequences will have to be worked out. We highlight a few of these below.
The details of the capture of glueballs by a seed black hole will differ significantly from the treatment in Ref. [@Hennawi:2001be]. The black hole capture depends sensitively on the density profile of glueballs, and this is tightly correlated with the potential well of the galaxy, which is dominated by glueballinos. In particular, although an isolated strongly self-interacting dark matter halo will undergo core collapse, this is not true when the strongly self-interacting component (glueballs) is a small fraction of the dark matter.
A complicating factor is that the glueballs and glueballinos will scatter off of each other. Each collision will change the velocity of glueballs by ${\cal O}(1)$, but the velocity of glueballinos will only change by $\Lambda/m_X \ll 1$. The glueballino-glueball scattering cross section should be of the order of the geometric cross section ($\sim 1/\Lambda^2$), and, thus, this could be an important effect if the number density of glueballinos is much larger than that of glueballs (either because of a small $\Omega_{\text{gb}}/\Omega_{\text{gbino}}$ or as glueballs are depleted due to accretion by the black hole). Conversely, this scattering could also have an impact on the glueballino density profile if the number density of glueballs is large enough to overcome the small momentum transfer.
Another important effect, relevant for halo properties as well as black hole growth, is cooling. We have focused on elastic collisions in this paper, but as mentioned previously there are also inelastic processes leading to cooling through the emission of glueballs. Cooling will funnel more glueballs into the inner regions (modulo angular momentum constraints) and increase the black hole accretion rate. Note that, unlike the baryons, competing effects from star formation and subsequent heating by UV photons are not relevant for glueballs.
As an extreme example, one could assume that all of the glueballs are bound up in the central supermassive black hole. In this case, we can use measured ratios of the black hole masses to halo masses to put an upper limit on $\Omega_{\text{gb}}/\Omega_{\text{gbino}}$. For the Milky Way, this ratio is $\sim 10^{-5}$, while for Andromeda the ratio is more like $10^{-4}$. (It should be kept in mind that the black hole will also accrete baryons and grow, so this is a lenient upper bound.) Rather than focus on the Local Group, one could look more generally at the black hole mass–virial mass relation for all galaxies, but as expected there is a lot of scatter in this relation [@2012MNRAS.419.2497B].
To illustrate the effect of these constraints on the model parameter space, we have shown two possibilities in [Fig. \[fig:AMSB-pure-relic\]]{}: one with $\Omega_{\text{gb}}/\Omega_{\text{gbino}}= 0.1$ (which may not be viable given the arguments above) and a second with $\Omega_{\text{gb}}/\Omega_{\text{gbino}}= 10^{-4}$. There is no impediment in making this ratio even smaller, although there is no natural reason to do so. In addition, as $\Omega_{\text{gb}}/\Omega_{\text{gbino}}$ is reduced, the regions with small $N$ move into the regime where the inelastic process will be important for all relevant velocities (dwarfs to clusters).
Glueballino/Glueball dark matter with connectors {#sec:connectors}
================================================
Although a pure SU($N$) hidden sector with no connectors can accommodate both early Universe and structure formation constraints, it is interesting to consider the possibility of connector fields that allow communication between the hidden and visible sectors. Such scenarios may have, of course, a larger number of testable implications. In addition, as we will see, if the connectors mediate annihilation or decays to the visible sector, the viable parameter space may be significantly altered.
If the hidden and visible sectors communicate, we expect the temperatures of the two sectors to coincide nearly until kinetic decoupling at confinement. If glueballs are stable, they will generically overclose the Universe, and so there must be a mechanism to reduce the glueball density. Let us assume that this mechanism exists and reduces the glueball relic density to a negligible level. We can then immediately determine the consequences for the parameter space. For a given point in the $(m_X, \Lambda)$ plane with $\xi_f =
1$, there are contours of constant $N$ on which $\Omega_{{\text{gbino}}} =
\Omega_{\text{DM}}$. These are shown in [Fig. \[fig:AMSB-pureConn-relic\]]{}, along with the self-interaction constraints. We see that the LEP2 bound excludes all but the $N \leq 4$ possibilities, but now, for small $N$, the allowed values of $m_X$ are much reduced and more natural relative to the case without connectors.
![Glueballino dark matter in AMSB models with pure SU($N$) hidden sectors and connectors to the SM. Glueballinos are assumed to make up all of the dark matter. The relic density constraints are given in the $(m_X,\Lambda)$ plane with $\xi_f=1$; contours of constant $N$ are shown. The gray shaded bands are from [Fig. \[fig:AMSB-pure-scattering-params\]]{} and give the regions where the glueballino self-interaction cross sections are in the preferred range. The red shaded region is excluded by null searches for visible-sector winos at LEP2. The yellow dot defines a representative model with $m_X \simeq 2.5~{\mathrm{TeV}}$, $\Lambda
\simeq 1.4~{\mathrm{MeV}}$, and $N=2$. []{data-label="fig:AMSB-pureConn-relic"}](glueballino-withConn)
A straightforward way to eliminate glueballs is through decays, but other constraints render this scenario unacceptable. The glueballs have a mass around $1$ to $10~{\mathrm{MeV}}$, so possible decay products will be photons, electrons, and neutrinos. Decays to photons will typically take too long and happen well after BBN. If too much energy and entropy is injected into the visible sector at $T \lesssim 1~{\mathrm{MeV}}$, then there is an unacceptably large contribution to $\Neff$. Decays to electrons after $1~{\mathrm{MeV}}$ face a similar problem, and, in addition, they can break up deuterium and ruin its BBN abundance (if the glueball is heavy enough). Decays to light neutrinos are problematic because glueballs can be produced in supernovae, escape the neutrino sphere, and cool the supernovae too efficiently. If we attempt to adjust parameters to get around the difficulties with either electrons or neutrinos, then we encounter problems with $e^+ e^-$ collider constraints. We are led to consider alternative processes to eliminate the glueball density.
Since decays after confinement are highly constrained, we investigate reducing the glueballino density by depleting the gluon density before confinement. The gluons may annihilate to SM particles via loop diagrams, but the reverse process needs to be suppressed. Let us introduce a right-handed neutrino $\nu_R$. The $\nu_R$ is a SM gauge singlet with a mass, $m_R\sim{\mathrm{GeV}}$, and it could be one of the sterile states in a seesaw mechanism to produce neutrino masses. Our goal is for the gluons to annihilate into right-handed neutrinos, which then decay quickly into SM particles before they can annihilate back into gluons.
To implement this scenario, we postulate that there is a connector field, $C$, with a mass $m_C$ that allows communication between the hidden and visible sector. The connector has a Yukawa interaction, $\lambda_R
C \bar{\nu}_R \nu_R$, in the visible sector and a gauge interaction with the gluons with a strength $g_h$ in the hidden sector. Integrating out the connector produces the effective interaction $$\mathcal{L} \sim \frac{1}{16\pi^2} \frac{\lambda_R^2 g_h^2}{m_C^3}
G^h_{\mu \nu} G^{h\, \mu \nu} \bar{\nu}_R \nu_R \ .$$ This interaction leads to an annihilation cross section, $$\langle \sigma v \rangle_{g g \to \bar{\nu}_R \nu_R} \sim
\frac{\lambda_R^4 g_h^4}{8\pi (16\pi^2)}
\frac{T^4}{m_C^6} \equiv \sigma_0 z^{-4} \ ,$$ where $z=m_R/T$. Note that the annihilation of gluons into right-handed neutrinos is subdominant to the annihilation rate of gluons into gluinos and can be ignored in the gluino freeze-out calculations. The right-handed neutrino decays with a rate of $$\Gamma_R \sim \frac{g_\nu^2}{4\pi} \frac{m_R^2}{T} \equiv \Gamma_0 z$$ into SM particles at the tree level with a coupling strength $g_\nu$. As long as the neutrino decay rate is much faster than the gluon annihilation into neutrinos (and both are faster than the Hubble expansion), the gluons cannot maintain their equilibrium density, and their energy is transferred to SM particles. The depletion terminates no later than $\sim m_R$, when any surviving right-handed neutrinos freeze out, and the gluon density decreases subsequently only due to Hubble expansion.
To give a concrete example, consider the following parameters: $N=2$, $m_X=2.5~{\mathrm{TeV}}$, $\Lambda\simeq 1.4~{\mathrm{MeV}}$, $m_C =
0.5~{\mathrm{TeV}}$, $m_R = 1~{\mathrm{GeV}}$, $g_h=1.1$, $\lambda_R=1.6$, and $g_\nu=0.1$. The output glueball relic density is $\sim$ 5% of the total dark matter abundance. We find this result by numerically solving the coupled Boltzmann equations for the gluons and right-handed neutrinos: $$\begin{aligned}
Y_g'(z) &=& -z^{-6} \sigma_0 \frac{s(m_R)}{H(m_R)}
\left(Y_g^2 - Y_R^2 \right) \\
Y_R'(z) &=& -z^{-6} \sigma_0 \frac{s(m_R)}{H(m_R)}
\left(Y_R^2 - Y_g^2 \right)
- z^2 \frac{\Gamma_0}{H(m_R)} Y_R \ ,\end{aligned}$$ where $s(m_R)$ and $H(m_R)$ are the entropy and Hubble rate at $T=m_R$. The initial conditions $Y_R(z_f)$ and $Y_g(z_f)$ are given by [Eq. (\[eq:Yinf\])]{} at the dark matter freeze-out: $z_f=25 m_R/m_X$. These differential equations tend to be fairly stiff, so, in certain regions of parameter space, it is beneficial to decouple the equations. We may do so if the neutrino decay term dominates, allowing us to approximate $Y_R$ as exponentially decaying. Solving the decoupled differential equation yields results that are numerically similar (typically within 10%) to solving the full set of coupled equations when the decay term dominates.
There are few constraints on this mechanism. Prior to confinement, a large amount of entropy is transferred from the gluons to light SM particles. Since the right-handed neutrinos are still relativistic, there is no entropy nonthermally deposited into the visible sector. All the right-handed neutrino decay products fall into equilibrium with the bath well before BBN. With a nonzero glueball density, a concern might be that the glueballs will be able to decay to SM particles via off-shell right-handed neutrinos and nonthermally deposit entropy into the visible sector. If the right-handed neutrino decays into a left-handed neutrino and the Higgs, then we expect the glueball decay rate into $\bar{\nu}_L \nu_L e^+ e^- e^+ e^-$ to be $$\Gamma_\textrm{gb} \sim y_e^4 g_\nu^4 \frac{\Lambda^{19}}{m_C^6 m_h^8 m_R^4} \ ,$$ where $y_e$ is the electron Yukawa coupling and $m_h$ is the mass of the Higgs. This decay rate is slow enough that the glueballs are essentially stable and, further, will not contribute significantly to $\Neff$, since they are nonrelativistic below $1~{\mathrm{MeV}}$, and they will not have a large impact on the expansion rate of the Universe during BBN, given their small energy density. Our glueball depletion process is robust, and it is consistent with terrestrial and cosmic constraints [@Abazajian:2012ys].
Conclusions {#sec:conclusions}
===========
We have explored the possibility that dark matter may be a composite particle, made up of bound states of a dark analogue of QCD in the hidden sector. Such constructions lead to rich and varied phenomena that are distinct from the WIMP scenario more typically considered. It also naturally leads to large self interactions of the dark matter, which can explain several observational puzzles in the small-scale structure of the Universe.
The simplest scenarios contain only dark gluons, which confine into glueballs with cosmologically interesting scattering cross sections for confinement scales around 100 MeV. Arranging the correct relic density requires one to disconnect the temperatures in the hidden and visible sectors such that their ratio at confinement is $\sim
10^{-3}$.
A richer theory arises when one considers supersymmetric versions, for which the dark gluino mass provides a separate mass scale and (in AMSB) can provide the correct relic density of glueballinos via the WIMPless miracle. The phenomenology depends crucially on how connected the hidden sector is to the visible matter. If there are no light connecting particles, one can dial the balance of dark matter from glueballs to glueballinos by adjusting the relative temperatures of the hidden and visible sectors. These mixed scenarios are strongly interacting analogues of atomic dark matter [@Kaplan:2009de; @CyrRacine:2012fz; @Cline:2013pca; @Kahlhoefer:2013dca; @Fan:2013yva] and inspire further simulation of the galactic dynamics in cases where there are two components of dark matter with naturally very different mass scales and different self-interaction rates. Such simulations would be very helpful to better understand the observational limits on these theories. For clusters, another important issue is the fact that the dark matter may have enough energy to scatter inelastically, bringing the details of the dark composite sector to the forefront of the physics; further work is needed to better understand the implications. We have also pointed out that our models have rich implications for the early growth of supermassive black holes. The mechanism by which $\sim 10^9 \, {\rm M}_\odot$ quasars are assembled as early as redshifts of 6–7 is a mystery, and self-interacting dark matter could have a major role to play in this story.
If the hidden and visible sectors are closely connected such that the temperatures remain comparable even at late times, the hidden glueballs will generically overclose the Universe. We considered a depletion mechanism into right-handed neutrinos and found that it can efficiently remove hidden gluons before confinement. Self-interaction strengths required to explain the astrophysical puzzles on small scales are obtained for glueballino masses $\gtrsim 1$ TeV and confinement scales $\sim$ MeV.
The possibility of strong self interactions in the dark sector is well motivated by observations of lower-than-expected dark matter densities in the centers of galaxies. A strongly interacting hidden sector naturally realizes this possibility. Even in the simple models explored in this paper, we have discovered new features that must be incorporated into numerical simulations to correctly predict the spatial distribution of dark matter in the central parts of structures from dwarf galaxies to clusters of galaxies.
Note Added {#note-added .unnumbered}
==========
After publishing this article, we became aware of the fact that the glueball and glueballino scattering cross sections are proportional to $1/N^2$ for large $N$. The plots present an accurate estimate of the scattering cross sections for small $N$, but the cross sections have been overestimated for large $N$.
We are grateful for helpful conversations with David B. Kaplan, Jared Kaplan, Matthew Reece, Ira Rothstein, Yael Shadmi, Jessie Shelton, Sean Tulin, and Hai-Bo Yu. K.B. thanks the University of California, Irvine, Department of Physics and Astronomy for their hospitality throughout this work. K.B. is supported in part by the U.S. DOE Award No. DE–FG02–92ER40701 and by the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics. J.L.F. and T.M.P.T. are supported in part by U.S. NSF Grant No. PHY–1316792. T.M.P.T. is further supported in part by the University of California, Irvine through a Chancellor’s Fellowship. M.K. is supported in part by NSF Grants No. PHY–1214648 and No. PHY–1316792. We used the LSODA software from LLNL [@Hindmarsh:1983ode; @Radhakrishnan:1993ode] to solve the Schrödinger equation. All other calculations were performed with *Mathematica* 8.0.
[^1]: Note that gravitational interactions will mediate very suppressed decays to light SM particles, but these are irrelevant for $\Lambda \ll {M_{\text{Pl}}}$.
|
---
abstract: 'Introducing a notion of the weighted$\ $mean $\sigma _{r}^{\infty }-$curvature and using the weighted Newton transformations we derive in this paper some integral formulae on weighted manifolds. These formulae generalize the flux formula and some of its examples of applications obtained by Alias, de Lira and Malacarne [@alm].'
address: ' Ecole Supérieure de Management. Tlemcen Ageria.'
author:
- Mohammed Abdelmalek
- Mohammed Benalili
title: Some integral formulae on weighted manifolds
---
Introduction
============
Several works have been done in the past to study the geometric properties of constant $k$-mean curvature hypersurfaces in space forms, where the $k$-mean curvature is defined as the $k$-th elementary symmetric function of the eigenvalues of the second fundamental form see ([@alm]). Motivated in part by connections with the Ricci flow, much works have also been done on geometric properties of manifolds and hypersurfaces when the manifold is endowed with a “weighted” volume element; i.e. one integrates using the smooth measure $e^{-f}dvol_{g}$ for $dvol_{g}$ the Riemannian volume element of the metric $g$ see ( [@C], [@Co], [@cr], [@e], [@m] ). In this work, we follow Case ([@C]) to introduce the notion of weighted $k$-mean curvature and using the weighted Newtons transformations introduced in ([@C]) we obtain an integral formula on weighted manifolds and give some applications. This latter formula was first introduced by Kusner in ([@K]) and nowadays it’s called the flux formula. Then it was extended to $k$-curvature in a nice paper by Alías, de Lira, and Malacarne see ([@alm]). Where they studied the properties of certain geometrical configurations, more particularly they established a flux formula and gave examples of geometric applications. Our paper extends some properties obtained by the authors cited above for weighted manifolds.
Preliminaries
=============
In this section we fix the notations and recall some definitions and properties of the weighted symmetric functions and the weighted Newton transformations: for more details see ([@C],[@m]).
Given a complete $n$-dimensional Riemannian manifold $(M,\left\langle
,\right\rangle )$ and a smooth function $f:M\longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.$ The weighted manifold $M_{f}$ associate to $M$ is the triplet $%
(M,\left\langle ,\right\rangle ,dv_{f}),$where $dv_{f}=e^{-f}dv$ and $dv$ is the standard volume element of $M.$
Consider the tensional connection $$\widetilde{\nabla }_{X}Y=\nabla _{X}Y+\left\langle X,Y\right\rangle
V-\left\langle V,Y\right\rangle X$$where $V=\left\langle \nabla f,\nu \right\rangle \nu $, $\nu $ is a vector field on $M$ orthogonal to $X$ and $\nabla $ stands for the covariant derivative on $M.$ This connection is one of the three basic types of metric connections introduced by Elie Cartan. It was studied by I. Agricola and M. Kraus [@ak].
If $B$ and $\widetilde{B}$ are second forms on $M$ defined by $B\left(
X,Y\right) =\left\langle \nabla _{X}Y,\nu \right\rangle $ and $\widetilde{B}%
\left( X,Y\right) =\left\langle \widetilde{\nabla }_{X}Y,\nu \right\rangle $, we have $$\widetilde{B}\left( X,Y\right) =B\left( X,Y\right) +\left\langle
X,Y\right\rangle \left\langle \nabla f,\nu \right\rangle$$An example of this situation is as follows: let$\ \psi _{t}:M^{n}\rightarrow
(\overline{M}^{n+1}$,$\left\langle ,\right\rangle )$ be a one family of parameter of immersions of an $n-$dimensional manifold $M^{n}$ into an $n+1$- $\overline{M}^{n+1}$ Riemannian manifold where $\nabla $ is the connection induced on $M^{n}$ and $\widetilde{\nabla }$ is the the tonsorial connection on $M^{n}$, so the mean curvatures of $M$ with respective to the connections $\nabla $ and $\widetilde{\nabla }$ are related by$$n\widetilde{H}=nH+\left\langle \nabla f,\nu \right\rangle$$which is the classical weighted mean curvature.
The important point is that hypersurfaces of constant weighted curvature appear as critical points of certain weighted volume functionals. The fundamental analogy with constant mean curvature hypersurfaces draws the attention in this area. In terms of matrices we have $$\widetilde{B}=B+\left\langle \nabla f,\nu \right\rangle I$$so if $\widetilde{\tau }_{i}$ and $\tau _{i}$ are the eigenvalues of $%
\widetilde{B}$ and $B$ respectively, we get$$\widetilde{\tau }_{i}=\tau _{i}+\left\langle \nabla f,\nu \right\rangle .$$Now, putting $\lambda =\left\langle \nabla f,\nu \right\rangle ,$ we obtain [@abn] $$\widetilde{\sigma }_{k}=\sigma _{k}\left( \widetilde{B}\right) =\underset{j=0%
}{\overset{k}{\sum }}\binom{n-k+j}{j}\lambda ^{j}\sigma _{k-j}\left(
B\right) . \label{1}$$where $\widetilde{\sigma }_{k}$ stand for the symmetric functions of $%
\widetilde{B}.$
Let $\overline{M}$ be an $(n+1)$-dimensional Riemannian manifold, and $\psi
:M\longrightarrow \overline{M}$ be an isometrically immersed hypersurface with $\triangledown $ and $\overline{\triangledown }$ the Levi-Civita connections on $M$ and $\overline{M}$ respectively. The Weingarten formula of this immersion is written as follows$$\text{ }AX=-\left( \overline{\triangledown }_{X}N\right) ^{\intercal }
\notag$$where $A$ is the shape operator of the hypersurface $M$ with respect to the Gauss map $N,$ and $^{\top }$ denotes the orthogonal projection on the tangent vector bundle of $M.$ As it is well known $A$ is a linear self adjoint operator and at each point $p\in M,$ its eigenvalues $\mu
_{1},...,\mu _{n}$ are the principal curvatures of $M.$
Associate to the shape operator $A$ are the weighted elementary symmetric functions $\sigma _{k}^{\infty }:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\times
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}\longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ defined recursively ( and introduced by Case in [@C] ) by $$\left\{
\begin{array}{l}
\sigma _{0}^{\infty }(\mu _{0},\mu )=1, \\
k\sigma _{k}^{\infty }(\mu _{0},\mu )=\sigma _{k-1}^{\infty }(\mu _{0},\mu )%
\underset{j=0}{\overset{n}{\dsum }}\mu _{j}+\underset{i=1}{\overset{k-1}{%
\dsum }}\underset{j=1}{\overset{n}{\dsum }}\left( -1\right) ^{i}\sigma
_{k-1-i}^{\infty }(\mu _{0},\mu )\mu _{j}^{i}\text{ \ \ \ \ \ for }k\geq 1%
\end{array}%
\right. \notag$$where $\mu _{0}\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $\mu =\left( \mu _{1},...,\mu _{n}\right) \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$. In particular for $\mu _{0}=0$ we recover $\sigma _{k}^{\infty
}(0,\mu )=\sigma _{k}(\mu )$ the classical elementary symmetric functions defined in [@r].
([@C])The weighted Newton transformations (W.N.T) $T_{k}^{\infty }(\mu
_{0},A)$ are defined inductively from $A$ by :$$\left\{
\begin{array}{l}
T_{k}^{\infty }(\mu _{0},A)=I \\
T_{k}^{\infty }(\mu _{0},A)=\sigma _{k}^{\infty }(\mu
_{0},A)I-AT_{k-1}^{\infty }(\mu _{0},A)\text{\ \ \ \ for }k\geq 1%
\end{array}%
\right. \notag$$or equivalently $$\text{ }T_{k}^{\infty }(\mu _{0},A)=\underset{j=0}{\overset{k}{\dsum }}%
\left( -1\right) ^{j}\sigma _{k-j}^{\infty }(\mu _{0},A)A^{j} \notag$$where $I$ stands for the identity on the Lie algebra of vector fields $%
\varkappa (M)$, $\sigma _{k}^{\infty }(\mu _{0},A)=\sigma _{k}^{\infty }(\mu
_{0},\mu _{1},...,\mu _{n})$ and $\mu _{1},...,\mu _{n}$ are the eigenvalues of $A$.
It should be noted that $T_{k}^{\infty }(0,A)=T_{k}(A)$ is the classical Newton transformations introduced in [@r].
These functions enjoy the nice following properties.
[@C] For $\mu _{0},\mu _{1}\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $\mu \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$, we have$$\sigma _{k}^{\infty }(\mu _{0}+\mu _{1},\mu )=\sum_{j=0}^{k}\frac{\mu
_{1}^{j}}{j!}\sigma _{k-j}^{\infty }(\mu _{0},\mu ). \notag$$In particular,$$\sigma _{k}^{\infty }(\mu _{1},\mu )=\sum_{j=0}^{k}\frac{\mu _{1}^{j}}{j!}%
\sigma _{k-j}\left( \mu \right) \label{2}$$$$\text{trace}(AT_{k}^{\infty }\left( (\mu _{0},\mu )\right) )=(k+1)\sigma
_{k+1}^{\infty }(\mu _{0},\mu )-\mu _{0}\sigma _{k}^{\infty }(\mu _{0},\mu ).
\label{2'}$$For $i\in \left\{ 1,...,n\right\} $ we have$$\sigma _{k,i}^{\infty }(\mu _{0},\mu )=\sigma _{k}^{\infty }(\mu _{0},\mu
)-\mu _{i}\sigma _{k-1,i}^{\infty }\left( \mu _{0},\mu \right) \notag$$and the $i^{th}$ eigenvalue of $T_{k}^{\infty }(\mu _{0},\mu )$ is equal to $%
\sigma _{k,i}^{\infty }(\mu _{0},\mu )$ where $\sigma _{k,i}^{\infty }\left(
\mu _{0},\mu \right) =\sigma _{k}^{\infty }\left( \mu _{0},\mu _{1},...,\mu
_{i-1},\mu _{i+1},...,\mu _{n}\right) .$
We can see by (\[1\]) and (\[2\]) that $\widetilde{\sigma }_{k}$ and $%
\sigma _{k}^{\infty }(\mu _{0},\mu )$ are polynomials of the same degree but with slightly different coefficients, which differ by a multiplicative constant only.
The weighted $k^{th}$ mean curvature $H_{k,f}$ is given by:$$\text{ }\left(
\begin{array}{c}
n \\
k%
\end{array}%
\right) H_{k,f}=\sigma _{k}^{\infty }(\left\langle \nu ,\triangledown
f\right\rangle ,A) \notag$$where $\nu $ is the unit outpointing vector field normal to $M$ in $%
\overline{M}.$
In particular for $k=1$ and in view of formula \[2\], we get $$nH_{1,f}=\sigma _{1}^{\infty }(\left\langle \nu ,\triangledown
f\right\rangle ,A)=\sigma _{1}\left( A\right) +\left\langle \nu
,\triangledown f\right\rangle =nH+\left\langle \nu ,\triangledown
f\right\rangle \notag$$which is the classical definition of the weighted mean curvature of the hypersurface $M$ studied by Gromov [@G].
To clarify this notion of curvature we will study the case $k=2$.
Consider a one family of parameter $\ \psi _{t}:M^{n}\rightarrow \overline{M}%
^{n+1}\left( c\right) $ of immersions of an $n-$dimensional closed manifold $%
M^{n}$ into an $n+1$ space form $\left( \overline{M}^{n+1}\text{,}%
\left\langle ,\right\rangle \right) $ of constant curvature $c$. Denote by $%
X $ the deformation vector field and by $\nu $ the normal vector field to $%
\overline{M}^{n+1}$. Put $\lambda =\left\langle X,\nu \right\rangle .$
Consider the variational problem$$\delta \left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right) =0 \label{3.1}$$that is to say$$\begin{aligned}
\delta \left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right) &=&\frac{d}{dt}%
\left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right) \label{3.2} \\
&=&\frac{d}{dt}\left( \int_{M}\left( \sigma _{1}+\mu \right) dV_{f}\right)
\notag \\
&=&\int_{M}\left( \frac{d\sigma _{1}}{dt}+\frac{d\mu }{dt}\right)
dV_{f}+\int_{M}\left( \sigma _{1}+\mu \right) \frac{d\left( dV_{f}\right) }{%
dt}. \notag\end{aligned}$$Now, by formula (9c) in page 469, we have$$\frac{d\sigma _{1}}{dt}=\lambda \left( \sigma _{1}^{2}-2\sigma _{2}\right)
+\lambda _{,ii}+\sigma _{1,j}X^{j}+nc\lambda$$and by the well known fact $\frac{d}{dt}dV=(-\lambda \sigma
_{1}+X_{,j}^{j})dV$, we infer$$\frac{d\left( dV_{f}\right) }{dt}=(-\lambda \sigma
_{1}-X^{j}f_{,j}+X_{,j}^{j})dV_{f}$$and $$\frac{d\mu }{dt}=\left\langle \nabla \nabla _{X}f,\nu \right\rangle
-\left\langle \nabla f,\left[ \nu ,X\right] \right\rangle$$By the definition of the weighted divergence, we have $%
div_{f}X=-X^{j}f_{,j}+X_{,j}^{j}$ and replacing in (\[3.2\]), we get
$$\begin{aligned}
\frac{d}{dt}\left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right)
&=&\int_{M}\left( \lambda \left( \sigma _{1}^{2}-2\sigma _{2}\right)
+\lambda _{,ii}+\sigma _{1,j}X^{j}++nc\lambda +\left\langle \nabla \nabla
_{X}f,\nu \right\rangle -\left\langle \nabla f,\left[ \nu ,X\right]
\right\rangle \right) dV_{f} \\
&&+\int_{M}\left( \sigma _{1}+\mu \right) (-\lambda \sigma
_{1}+div_{f}\left( X\right) )dV_{f} \\
&=&\int_{M}\left( -2\lambda \sigma _{2}+\lambda _{,ii}+\left\langle \nabla
\nabla _{X}f,\nu \right\rangle -\left\langle \nabla f,\left[ \nu ,X\right]
\right\rangle \right) dV_{f} \\
&&+\int_{M}\left( -\lambda \mu \sigma _{1}+\mu div_{f}\left( X\right)
+nc\lambda \right) dV_{f} \\
&=&\int_{M}\left( -\lambda \left( 2\sigma _{2}^{\infty }-\mu _{1}\sigma
_{1}^{\infty }\right) +\lambda _{,ii}+\left\langle \nabla \nabla _{X}f,\nu
\right\rangle -\left\langle \nabla f,\left[ \nu ,X\right] \right\rangle
\right) dV_{f} \\
&&+\int_{M}\left( \mu div_{f}\left( X\right) +nc\lambda \right) dV_{f}.\end{aligned}$$
we have
The Euler-Lagrange equation corresponding to the problem (\[3.1\]) is $$-\lambda \left( 2\sigma _{2}^{\infty }-\mu _{1}\sigma _{1}^{\infty }\right)
+\lambda _{,ii}+\left\langle \nabla \nabla _{X}f,\nu \right\rangle
-\left\langle \nabla f,\left[ \nu ,X\right] \right\rangle +\mu div_{f}\left(
X\right) +nc\lambda =0.$$
To clarify the idea we will consider simpler cases: put $f(x)=\frac{1}{2}%
\left\Vert x\right\Vert ^{2.}$, and $X=g\nu $, where $g$ is a $C^{2}$-function, so $\lambda =g$ and $\mu =\left\langle x,\nu \right\rangle $ the support function. We get $$\begin{aligned}
div_{f}\left( X\right) &=&-\left( g\nu ^{j}\right) f_{,j}+\left( g\nu
^{j}\right) _{,j} \\
&=&-g\left\langle x,\nu \right\rangle +g_{j}\nu ^{j}+g\nu _{,j}^{j} \\
&=&-g\mu +\nu (g)-g\sigma _{1} \\
&=&g\sigma _{1}^{\infty }+\nu \left( g\right) ,\end{aligned}$$$$\begin{aligned}
\int_{M}g_{,ii}dV_{f} &=&\int_{M}g\left( f_{,ii}-f_{i}f^{,i}\right) dV_{f} \\
&=&\int_{M}g\left( n-2f\right) dV_{f}\end{aligned}$$and$$\left\langle \nabla \nabla _{X}f,\nu \right\rangle -\left\langle \nabla f,
\left[ \nu ,X\right] \right\rangle =g\nu \left( \mu _{1}\right) .$$Hence$$\begin{aligned}
\frac{d}{dt}\left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right)
&=&\int_{M}\left( -g\left( 2\sigma _{2}^{\infty }-2\mu \sigma _{1}^{\infty
}\right) +g\left( n-2f\right) +g\nu \left( \mu \right) \right) dV_{f} \\
&&+\int_{M}\mu \nu \left( g\right) dV_{f}\end{aligned}$$and since $$\begin{aligned}
\int_{M}\mu \nu \left( g\right) dV_{f} &=&-\int_{M}\nu (\mu e^{-f})gdV \\
&=&-\int_{M}g\left( \nu \left( \mu \right) -\mu ^{2}\right) dV_{f}\end{aligned}$$we deduce$$\frac{d}{dt}\left( \int_{M}\sigma _{1}^{\infty }dV_{f}\right) =\int_{M}g%
\left[ -2\sigma _{2}^{\infty }+2\mu \sigma _{1}^{\infty }-2f+\mu
^{2}+n\left( 1+c\right) \right] dV_{f}\text{.}$$
Under the above assumptions the Euler-Langrange equation of the problem ([3.1]{}) is given by $$-2\sigma _{2}^{\infty }+2\mu \sigma _{1}^{\infty }-2f+\mu ^{2}+n\left(
1+c\right) =0.$$
Example: Consider a hypersurface $M_{n}$ of the unit sphere $\ S^{n+1}$. The Euler-Lagrange is then written$$-\sigma _{2}^{\infty }+\sigma _{1}^{\infty }+n=0$$or equivalently$$\left( 2-\underset{j=0}{\dsum^{n}\mu _{j}}\right) \sigma _{1}^{\infty }+2n+%
\underset{j=0}{\overset{n}{\dsum }}\mu _{j}=0$$where $\mu _{j}$ stand for the eigenvalues of the second fundamental form.
Or$$\left( \sigma _{1}^{\infty }\right) ^{2}-4\sigma _{1}^{\infty }-2n-1=0$$which gives $$\sigma _{1}^{\infty }=2\pm \sqrt{2n+3}$$or$$\sigma _{1}=1\pm \sqrt{2n+3}.$$We can cite a candidate to our situation: Cliffor torus in $S^{n+1}$i.e. $%
M^{n}$ is a product of spheres $S^{n_{1}}(r_{1})\times S^{n_{2}}(r_{2})$ , $%
n_{1}+n_{2}=n$ of appropriate radii $r_{1}$, $r_{2}.$ An $H(r)$-torus in $%
S^{n+1}$ is obtained by the canonical immersions $S^{n-1}\left( r\right)
\subset R^{n},S^{1}(\sqrt{1-r^{2}})\subset R^{2}$, $0<r<1$, as $%
S^{n-1}\left( r\right) \times S^{1}(\sqrt{1-r^{2}})\subset S^{n+1}$. The principal curvatures are given, for a chosen orientation, by $$\mu _{1}=...=\mu _{n-1}=\frac{\sqrt{1-r^{2}}}{r}\text{, }\mu _{n}=-\frac{r}{%
\sqrt{1-r^{2}}}$$so for the $H(r)$-torus $$\sigma _{1}=\left( n-1\right) \frac{\sqrt{1-r^{2}}}{r}-\frac{r}{\sqrt{1-r^{2}%
}}=\frac{n(1-r^{2})-1}{r\sqrt{1-r^{2}}}.$$For the torus to be an example we have to show that $$\frac{n(1-r^{2})-1}{r\sqrt{1-r^{2}}}=1+\sqrt{2n+3} \tag{3.3}$$has a root. For this, we consider the continuous function $\phi \left(
r\right) =-(1+\sqrt{2n+3})$ with $0<r<\frac{n-1}{n}$ and $n\geq 2.$ We notice that $\lim_{r\rightarrow 0^{+}}\phi (r)=+\infty $ and $\phi \left(
\frac{n-1}{n}\right) =\frac{n}{\sqrt{2n+1}}-1-\sqrt{2n+3}<0.$ Consequently the equation (\[3.3\]) admits at least one root.
We say that an hypersurface $M$ of $\overline{M}$ is $\sigma _{r}^{\infty }$-minimal, if $H_{r,f}=0.$ In particular $M$ is $f$-minimal if $H_{f}=-\frac{1%
}{n}\left\langle \nu ,\triangledown f\right\rangle .$
Here we need $\mu _{0}=\left\langle \nabla f,\nu \right\rangle $. For the safe of brevity, we put $T_{k}^{\infty }=T_{k}^{\infty }(\mu _{0},A)$ and $%
\sigma _{k}^{\infty }=\sigma _{k}^{\infty }(\mu _{0},A).$
The weighted divergence of the weighted Newton transformations is define by $$\func{div}_{f}T_{k}^{\infty }=e^{f}\func{div}\left( e^{-f}T_{k}^{\infty
}\right) \notag$$where$$\func{div}\left( T_{k}^{\infty }\right) =\text{trace}\left( \triangledown
T_{k}^{\infty }\right) =\underset{j=0}{\overset{k}{\dsum }}\triangledown
_{e_{i}}\left( T_{k}^{\infty }\right) \left( e_{i}\right) \notag$$and $\left\{ e_{1},...,e_{n}\right\} $ is an orthonormal basis of the tangent space of $M.$
\[lem1’\]$$tr\left( T_{k-1}^{\infty }\circ \nabla _{v}A\right) =\left\langle \nabla
\sigma _{k}^{\infty }(\mu _{1},A)-\sigma _{k-1}^{\infty }\left( \mu
_{1},A\right) \nabla \mu _{1},v\right\rangle .$$
The computations will be in a basis that diagonalizes $A$. Let $A=\left(
\begin{array}{ccc}
\lambda _{1} & & \\
& . & \\
& & \lambda _{n}%
\end{array}%
\right) .$ Since the eigenvalues of $T_{k-1}^{\infty }$ are given by $$\begin{aligned}
t_{i} &=&\sigma _{k-1}^{\infty }\left( \mu _{1},\lambda _{1},...,\lambda
_{i-1},\lambda _{i+1},...,\lambda _{n}\right) \\
&=&\dsum\limits_{j=0}^{k-1}\frac{\mu _{1}^{j}}{j!}\sigma _{k-1-j}\left(
\lambda _{1},...,\lambda _{i-1},\lambda _{i+1},...,\lambda _{n}\right) \\
&=&\dsum\limits_{j=0}^{k-1}\frac{\mu _{1}^{j}}{j!}\frac{\partial }{\partial
\lambda _{i}}\sigma _{k-1-j}\left( \lambda _{1},...,\lambda _{n}\right) \\
&=&\dsum\limits_{j=0}^{k-1}\frac{\mu _{1}^{j}}{j!}\dsum\limits_{i\neq
i_{j},\ i_{1}<...<i_{k-1-j}}\lambda _{i_{1}}...\lambda _{i_{k-1-j}}.\end{aligned}$$So$$tr\left( T_{k-1}^{\infty }\circ \nabla _{v}A\right)
=\dsum\limits_{i=1}^{n}t_{i}\nabla _{v}\lambda _{i}$$$$\begin{aligned}
&=&\dsum\limits_{j=0}^{k}\frac{\mu _{1}^{j}}{j!}\nabla
_{v}\dsum\limits_{i_{1}<...<i_{k-j}}\lambda _{i_{1}}...\lambda _{i_{k-j}} \\
&=&\nabla _{v}\sigma _{k}^{\infty }(\mu _{1},A)-\nabla _{v}\mu
_{1}\dsum\limits_{j=0}^{k-1}\frac{\mu _{1}^{j}}{j!}\sigma _{k-1-j}\left(
A\right) \\
&=&\nabla _{v}\sigma _{k}^{\infty }(\mu _{1},A)-\left( \nabla _{v}\mu
_{1}\right) \sigma _{k-1}^{\infty }\left( \mu _{1},A\right) .\end{aligned}$$
\[lem1\] The weighted divergence of the weighted Newton transformations $%
T_{k}^{\infty }$ are inductively given by the following formula $$\func{div}_{f}T_{0}^{\infty }=\triangledown f$$and$$\func{div}_{f}T_{k}^{\infty }=\sigma _{k}^{\infty }\triangledown f+\sigma
_{k-1}^{\infty }\left( \mu _{1},A\right) \nabla \mu _{1}-A\func{div}%
_{f}T_{k-1}^{\infty }-\sum_{i=1}^{n}\left( \overline{R}(N,T_{k-1}^{\infty
}(e_{i}))e_{i}\right) ^{\top }\text{\ \ \ \ \ for }k\geq 1\text{.} \notag$$
We have$$\begin{aligned}
\func{div}_{f}T_{k}^{\infty } &=&e^{f}\func{div}\left( e^{-f}T_{k}^{\infty
}\right) \\
&=&e^{f}\overset{n}{\underset{i=1}{\sum }}\left[ \triangledown
_{e_{i}}(e^{-f}T_{k}^{\infty })(e_{i})\right] \\
&=&e^{f}\overset{n}{\underset{i=1}{\sum }}\left[ \triangledown
_{e_{i}}(e^{-f}T_{k}^{\infty }(e_{i}))-e^{-f}T_{k}^{\infty }(\triangledown
_{e_{i}}e_{i})\right] \\
&=&e^{f}\overset{n}{\underset{i=1}{\sum }}\left[ e^{-f}\triangledown
_{e_{i}}(T_{k}^{\infty }(e_{i}))+e_{i}\left( e^{-f}\right) T_{k}^{\infty
}(e_{i})-e^{-f}T_{k}^{\infty }(\triangledown _{e_{i}}e_{i})\right] \\
&=&e^{f}\overset{n}{\underset{i=1}{\sum }}\left[ e^{-f}\triangledown
_{e_{i}}(T_{k}^{\infty }(e_{i}))-e^{-f}\left\langle \triangledown
f,e_{i}\right\rangle T_{k}^{\infty }(e_{i})-e^{-f}T_{k}^{\infty
}(\triangledown _{e_{i}}e_{i})\right] \\
&=&\func{div}T_{k}^{\infty }-T_{k}^{\infty }\left( \triangledown f\right)\end{aligned}$$It is not difficult to see that $$\text{div}_{M}T_{k}^{\infty }=\nabla \sigma _{k}^{\infty }-A_{k}\text{div}%
_{M}T_{k-1}^{\infty }-\sum_{i=1}^{n}(\nabla _{e_{i}}A_{k})(T_{k-1}^{\infty
}(e_{i}))\text{.}$$Indeed, $$\begin{aligned}
(\nabla _{e_{i}}T_{k}^{\infty })(e_{i}) &=&\nabla _{e_{i}}(T_{k}^{\infty
}(e_{i}))-T_{k}^{\infty }(\nabla _{e_{i}}e_{i}) \\
&=&\nabla _{e_{i}}((\sigma _{k}^{\infty }I-AT_{k-1}^{\infty })e_{i})-(\sigma
_{k}^{\infty }I-AT_{k-1}^{\infty })(\nabla _{e_{i}}e_{i}) \\
&=&\nabla _{e_{i}}(\sigma _{k}^{\infty }e_{i})-\nabla
_{e_{i}}(AT_{k-1}^{\infty }(e_{i}))-\sigma _{k}^{\infty }(\nabla
_{e_{i}}e_{i})+(AT_{k-1}^{\infty })(\nabla _{e_{i}}e_{i}) \\
&=&e_{i}(\sigma _{k}^{\infty })e_{i}+\sigma _{k}^{\infty }(\nabla
_{e_{i}}e_{i})-\nabla _{e_{i}}(AT_{k-1}^{\infty }(e_{i}))-\sigma
_{k}^{\infty }(\nabla _{e_{i}}e_{i})+(AT_{k-1}^{\infty })(\nabla
_{e_{i}}e_{i}) \\
&=&e_{i}(\sigma _{k}^{\infty })e_{i}-(\nabla _{e_{i}}(AT_{k-1}^{\infty
}(e_{i}))-(AT_{k-1}^{\infty })(\nabla _{e_{i}}e_{i})) \\
&=&(d_{e_{i}}\sigma _{k}^{\infty })(e_{i})-(\nabla _{e_{i}}(AT_{k-1}^{\infty
}))e_{i} \\
&=&\langle \nabla \sigma _{k}^{\infty },e_{i}\rangle e_{i}-(\nabla
_{e_{i}}A)(T_{k-1}^{\infty }(e_{i}))-A((\nabla _{e_{i}}T_{k-1}^{\infty
})(e_{i}))\end{aligned}$$Thus, $$\begin{aligned}
\text{div}_{M}T_{k}^{\infty } &=&\sum_{i=1}^{n}(\nabla _{e_{i}}T_{k}^{\infty
})(e_{i}) \\
&=&\sum_{i=1}^{n}\langle \nabla \sigma _{k}^{\infty },e_{i}\rangle
e_{i}-\sum_{i=1}^{n}(\nabla _{e_{i}}A)(T_{k-1}^{\infty
}(e_{i}))-\sum_{i=1}^{n}A((\nabla _{e_{i}}T_{k-1}^{\infty })(e_{i})) \\
&=&\nabla \sigma _{k}^{\infty }-Adiv_{M}T_{k-1}^{\infty
}-\sum_{i=1}^{n}(\nabla _{e_{i}}A)(T_{k-1}^{\infty }(e_{i}))\end{aligned}$$The Godazzi equation and the fact that $\nabla _{e_{i}}A$ is a self-ajoint operator allow us to write, $$\dsum\limits_{i=1}^{n}\langle (\nabla _{e_{i}}A)T_{k-1}^{\infty
}(e_{i}),v\rangle =\langle \sum_{i=1}^{n}\left( \overline{R}%
(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top },v\rangle
+trace(T_{k-1}^{\infty }\circ \nabla _{v}A)$$where $v$ is an arbitrary vector tangent $M$.
Thus,$$\left\langle div_{M}T_{k}^{\infty },v\right\rangle =\langle \nabla \sigma
_{k}^{\infty },v\rangle -\langle Adiv_{M}T_{k-1}^{\infty },v\rangle -\langle
\sum_{i=1}^{n}\left( \overline{R}(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right)
^{\top },v\rangle -tr(T_{k-1}^{\infty }\circ \nabla _{v}A)$$Using now Lemma \[lem1’\]$$tr(T_{k-1}^{\infty }\circ \nabla _{v}A)=\left\langle \nabla \sigma
_{k}^{\infty }(\mu _{1},A)-\left( \nabla \mu _{1}\right) \sigma
_{k-1}^{\infty }\left( \mu _{1},A\right) ,v\right\rangle$$we have,$$\left\langle div_{M}T_{k}^{\infty },v\right\rangle =\sigma _{k-1}^{\infty
}\left( \mu _{1},A\right) \left\langle \nabla \mu _{1},v\right\rangle
-\langle Adiv_{M}T_{k-1}^{\infty },v\rangle -\langle \sum_{i=1}^{n}\left(
\overline{R}(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top },v\rangle$$or equivalently,$$div_{M}T_{k}^{\infty }=\sigma _{k-1}^{\infty }\left( \mu _{1},A\right)
\nabla \mu _{1}-Adiv_{M}T_{k-1}^{\infty }-\sum_{i=1}^{n}\left( \overline{R}%
(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top }.$$Finally$$\begin{aligned}
\func{div}_{f}T_{k}^{\infty } &=&T_{k}^{\infty }\left( \triangledown
f\right) +\sigma _{k-1}^{\infty }\left( \mu _{1},A\right) \nabla \mu
_{1}-Adiv_{M}T_{k-1}^{\infty }-\sum_{i=1}^{n}\left( \overline{R}%
(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top } \\
&=&\left( \sigma _{k}^{\infty }I-AT_{k-1}^{\infty }\right) \left(
\triangledown f\right) +\sigma _{k-1}^{\infty }\left( \mu _{1},A\right)
\nabla \mu _{1}-A\left( \func{div}_{f}T_{k-1}^{\infty }-T_{k-1}^{\infty
}\left( \triangledown f\right) \right) -\sum_{i=1}^{n}\left( \overline{R}%
(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top } \\
&=&\sigma _{k}^{\infty }\triangledown f+\sigma _{k-1}^{\infty }\left( \mu
_{1},A\right) \nabla \mu _{1}-\left( AT_{k-1}^{\infty }\right) \left(
\triangledown f\right) -A\left( \func{div}_{f}T_{k-1}^{\infty }\right)
+\left( AT_{k-1}^{\infty }\right) \left( \triangledown f\right)
-\sum_{i=1}^{n}\left( \overline{R}(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right)
^{\top } \\
&=&\sigma _{k}^{\infty }\triangledown f+\sigma _{k-1}^{\infty }\left( \mu
_{1},A\right) \nabla \mu _{1}-A\func{div}_{f}T_{k-1}^{\infty
}-\sum_{i=1}^{n}\left( \overline{R}(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right)
^{\top }\text{.}\end{aligned}$$Which achieves the proof of Lemma \[lem1\].
\[cor1\] If $\overline{M}$ has constant sectional curvature, then $$\func{div}_{f}T_{k}^{\infty }=T_{k}^{\infty }\left( \triangledown f\right)
+\sigma _{k-1}^{\infty }\left( \mu _{1},A\right) \nabla \mu _{1}.$$
* If* $\overline{M}$ has constant sectional curvature, then $\left(
\overline{R}(N,T_{k-1}^{\infty }(e_{i}))e_{i}\right) ^{\top }=0,$ and we have* *$$\func{div}_{f}T_{k}^{\infty }=\sigma _{k}^{\infty }\triangledown f-A\func{div%
}_{f}T_{k-1}^{\infty }+\sigma _{k-1}^{\infty }\left( \mu _{1},A\right)
\nabla \mu _{1}.$$The desired relation results by a recursive argument.
Main results
============
The aim of this part is to derive an integral formula on weighted manifolds with constant sectional curvature and to give some of its geometric applications. The method is based on the computation of the weighted divergence $\func{div}_{f}\left( T_{k}^{\infty }Y^{\intercal }\right) $ and $%
\left\langle \func{div}_{f}T_{k}^{\infty },Y\right\rangle $, where $Y$ is a conformal vector field. To do so, we first consider the following geometric configuration: let $\overline{M}^{n+1}$ be an oriented Riemannian manifold with metric $\left\langle ,\right\rangle $, $P^{n}\subset \overline{M}^{n+1}$ an oriented connected submanifold of $\overline{M}^{n+1}$and $\Sigma
^{n-1}\subset P^{n}$ a compact hypersurface of $P^{n}.$ Let $\varphi
:M^{n}\longrightarrow \overline{M}^{n+1}$ be a compact oriented hypersurface of boundary $\partial M$.
Let $p\in \Sigma ^{n-1}$ and $\left\{ e_{1},...,e_{n-1}\right\} $ an orthonormal basis of $T_{p}\Sigma ^{n-1}$. We can choose a global vector field $\nu $ such that $\left\{ e_{1},...,e_{n-1},\nu (p)\right\} $ is an orthonormal basis of $T_{p}M^{n}$. Let $N$ be the globally vector normal to $%
M^{n}$, then $\left\{ e_{1},...,e_{n-1},\nu (p),N\right\} $ is an orthonormal basis of $T_{p}\overline{M}^{n+1}$. Suppose now the existence of a closed conformal vector field $Y$ on $\overline{M}^{n+1}$; that is to say there exists a $\phi \in C^{\infty }(\overline{M}^{n+1})$ such that $$\overline{\triangledown }_{V}Y=\phi V \notag$$for every vector fields $V$ over $\overline{M}^{n+1}.$
If $\left\{ e_{1},...,e_{n}\right\} $ is an orthonormal basis of $T_{p}M^{n}$ that diagonalizes $A,$then $$\begin{aligned}
\left\langle \func{div}_{f}T_{k}^{\infty },Y\right\rangle &=&\left\langle
e^{f}\func{div}\left( e^{-f}T_{k}^{\infty }\right) ,Y\right\rangle \\
&=&e^{f}\func{div}\left( e^{-f}T_{k}^{\infty }Y\right) -\underset{i=0}{%
\overset{n}{\dsum }}\left\langle T_{k}^{\infty }\left( e_{i}\right)
,\triangledown _{e_{i}}Y\right\rangle \\
&=&e^{f}\func{div}\left( e^{-f}T_{k}^{\infty }Y\right) -\underset{i=0}{%
\overset{n}{\dsum }}\phi \left\langle T_{k}^{\infty }\left( e_{i}\right)
,e_{i}\right\rangle \\
&=&\func{div}_{f}\left( T_{k}^{\infty }Y\right) -\phi trT_{k}^{\infty }.\end{aligned}$$And in virtue of formula \[2’\] we have $$\begin{aligned}
trT_{k}^{\infty } &=&n\sigma _{k}^{\infty }(\mu _{0},\mu )-tr\left(
AT_{k-1}^{\infty }(\mu _{0},\mu )\right) \\
&=&\left( n-k\right) \sigma _{k}^{\infty }(\mu _{0},\mu )+\mu _{0}\sigma
_{k-1}^{\infty }(\mu _{0},\mu ) \\
&=&\left( n-k\right) \left(
\begin{array}{c}
n \\
k%
\end{array}%
\right) H_{k,f}+\left\langle \nu ,\triangledown f\right\rangle \left(
\begin{array}{c}
n \\
k-1%
\end{array}%
\right) H_{k-1,f}.\end{aligned}$$
So,$$\func{div}_{f}\left( T_{k}^{\infty }Y\right) =\left\langle \func{div}%
_{f}T_{k}^{\infty },Y\right\rangle +\phi \left(
c_{k}H_{k,f}+c_{k-1}\left\langle \nu ,\triangledown f\right\rangle
H_{k-1,f}\right) \notag$$
where $c_{k}=(n-k)\dbinom{n}{k}$ and $c_{k-1}=n\dbinom{n}{k-1}$
Integrating the two sides of this latter equality and applying the divergence theorem, we obtain for $1\leq k\leq n-1$,$$\begin{aligned}
\underset{M^{n}}{\int }\func{div}_{f}\left( T_{k}^{\infty }Y\right) dv_{f}& =%
\underset{M^{n}}{\int }e^{-f}\func{div}_{f}\left( T_{k}^{\infty }Y\right) dv
\\
& =\underset{M^{n}}{\int }\func{div}\left( e^{-f}T_{k}^{\infty }Y\right) dv
\\
& =\underset{\partial M}{\int }e^{-f}\left\langle T_{k}^{\infty }Y,\nu
\right\rangle ds \\
& =\underset{\partial M}{\int }\left\langle T_{k}^{\infty }Y,\nu
\right\rangle ds_{f}.\end{aligned}$$
Hence,$$\underset{\partial M}{\int }\left\langle T_{k}^{\infty }\nu ,Y\right\rangle
ds_{f}=\underset{M^{n}}{\int }\left\langle \func{div}_{f}T_{k}^{\infty
},Y\right\rangle dv_{f}+c_{k}\underset{M^{n}}{\int }\phi
H_{k,f}dv_{f}+c_{k-1}\underset{M^{n}}{\int }\phi \left\langle \nu
,\triangledown f\right\rangle H_{k-1,f}dv_{f}\text{.}$$Consequently, we have the following proposition
\[prop2\] Let $\varphi :M^{n}\longrightarrow \overline{M}^{n+1}$ an immersed compact oriented hypersurface of boundary $\partial M$. Denoting by $N$ the global vector fields normal to $M^{n}$, and $\nu $ the outward pointing conormal vector field to $M^{n}$ along $\partial M$. Then for $%
1\leq k\leq n-1$ and for every closed conformal vector field $Y$ on $%
\overline{M}^{n+1}$, we have :$$\underset{\partial M^{n}}{\int }\left\langle T_{k}^{\infty }\nu
,Y\right\rangle ds_{f}=\underset{M^{n}}{\int }\left\langle \func{div}%
_{f}T_{k}^{\infty },Y\right\rangle dv_{f}+c_{k}\underset{M^{n}}{\int }\phi
H_{k,f}dv_{f}+c_{k-1}\underset{M^{n}}{\int }\phi \left\langle \nu
,\triangledown f\right\rangle H_{k-1,f}dv_{f}.$$
If $\overline{M}^{n+1}$ has constant sectional curvature, we obtain by Corollary \[cor\],
Under the hypothesis of the proposition \[prop2\], if $\overline{M}^{n+1}$ has constant sectional curvature, then $$\underset{\partial M}{\int }\left\langle T_{k}^{\infty }\nu ,Y\right\rangle
ds_{f}=\underset{M^{n}}{\int }\left\langle \nabla f,T_{k}^{\infty
}Y\right\rangle dv_{f}+c_{k}\int \phi H_{k,f}dv_{f}+c_{k-1}\int \left( \phi
\mu _{1}+\frac{1}{n}\left\langle \nabla \mu _{1},Y\right\rangle \right)
H_{k-1,f}dv_{f}$$with $\mu _{1}=\left\langle \nabla f,\nu \right\rangle $.
Under the hypothesis of the proposition \[prop2\], if $\overline{M}^{n+1}$ has constant sectional curvature and $f$ is constant then $$\underset{\partial M}{\int }\left\langle T_{k}\nu ,Y\right\rangle
ds_{f}=c_{k}\underset{M^{n}}{\int }\phi H_{k}dv_{f}. \label{3}$$where $T_{k}$ is the classical Newton transformation.
If $f$ is constant, $H_{k}$ is a non zero constant and $Y$ is an homothetic vector field, we can assume that $\phi =1,$ and get $$c_{k}H_{k}vol(M^{n})=\underset{\partial M}{\int }\left\langle T_{k}^{\infty
}\nu ,Y\right\rangle ds$$or equivalently $$vol\left( M^{n}\right) =\frac{1}{c_{k}H_{k}}\underset{\partial M}{\int }%
\left\langle T_{k}^{\infty }\nu ,Y\right\rangle ds.$$
Let $\varphi :M^{n}\longrightarrow \overline{M}^{n+1}$ be an immersed compact oriented hypersurface of boundary $\partial M$. Denoting by $N$ the global vector fields normal to $M^{n}$, and $\nu $ the outward pointing unit conormal vector field to $M^{n}$ along $\partial M$. Suppose that $\overline{%
M}^{n+1}$ is of constant sectional curvature, $f$ is constant and the $%
k^{th} $ mean curvature $H_{k}$ is non zero constant. Then for $1\leq k\leq
n-1$ and for every homothetic vector field $Y\in \overline{M}^{n+1})$, we have $$vol\left( M^{n}\right) =\frac{1}{c_{k}H_{k}}\underset{\partial M}{\int }%
\left\langle T_{k}^{\infty }\nu ,Y\right\rangle ds. \label{4}$$
An estimate of the integrant in formula \[4\] leads to an estimate of the volume of $M^{n}$ by the area of its boundary$\ \partial M.$
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---
abstract: 'The development of analytic-gradient methodology for excited states within conventional time-dependent density-functional theory (TDDFT) would seem to offer a relatively inexpensive alternative to better established quantum-chemical approaches for the modeling of photochemical reactions. However, even though TDDFT is formally exact, practical calculations involve the use of approximate functionals, in particular the TDDFT adiabatic approximation, whose use in photochemical applications must be further validated. Here, we investigate the prototypical case of the symmetric CC ring opening of oxirane. We demonstrate by direct comparison with the results of high-quality quantum Monte Carlo calculations that, far from being an approximation on TDDFT, the Tamm-Dancoff approximation (TDA) is a practical necessity for avoiding triplet instabilities and singlet near instabilities, thus helping maintain energetically reasonable excited-state potential energy surfaces during bond breaking. Other difficulties one would encounter in modeling oxirane photodynamics are pointed out but none of these is likely to prevent a qualitatively correct TDDFT/TDA description of photochemistry in this prototypical molecule.'
author:
- 'Felipe Cordova, L. , Andrei Ipatov, and Mark E. Casida'
- Claudia Filippi
- Alberto Vela
title: ' Troubleshooting Time-Dependent Density-Functional Theory for Photochemical Applications: Oxirane '
---
Introduction {#sec:intro}
============
A complete understanding of photochemistry often requires photodynamics calculations since photochemical reactions may have excess energy and do not follow the lowest energy pathway, and since dynamic, as much as energetic, considerations often govern how a reaction jumps from one electronic potential energy surface (PES) to another. Unfortunately, full photodynamics calculations are prohibitively expensive for all but the smallest molecules unless simplifying approximations are made. One such approximation is to adopt a Tully-type mixed quantum/classical surface-hopping (SH) trajectory approach [@TP71; @T90] where the electrons are described quantum mechanically and the nuclei classically. Then, since the quantum part of the calculation remains the computational bottleneck, further simplifications are required for an efficient electronic structure computation of the photochemical dynamics.
Time-dependent density-functional theory (TDDFT) [@MUN+06] represents a promising and relatively inexpensive alternative [@DM02] to accurate but more costly quantum chemical approaches for on-the-fly calculations of excitation energies. TDDFT methods for excited-state dynamics are not yet a standard part of the computational chemistry repertory but are coming on line. In particular, the basic methodology for the computation of analytic gradients has now been implemented in a number of codes [@VA99; @VA00; @FA02; @RF05; @H03; @DK05] and important practical progress is being made on the problem of calculating SH integrals within TDDFT [@CM00; @B02; @CDP05; @M06; @HCP06; @TTR07]. Indeed first applications of mixed TDDFT/classical SH dynamics are being reported [@CDP05; @HCP06; @TTR07]. Problems in this new technology can be expected and modifications in the basic TDDFT methods will be necessary to enlarge the class of its potential applications. In particular, practical TDDFT calculations employ approximate functionals whose performance in describing photochemical problems requires careful validation.
Few applications of TDDFT to problems relevant to photochemistry can be found in the literature [@DKZ01a; @DKZ01b; @DKSZ02; @SDK+02; @OGB02; @CSM+06] and the few attempts made to assess the value of TDDFT for photochemical applications report either encouraging [@CCS98; @FMO04; @BGDF06] or discouraging results [@BM00; @CTR00; @RCT04; @WGB+04; @LQM06]. The fundamental reasons for these descrepencies stem from the fact that conventional TDDFT has several problems which may or may not be fatal for modeling a given photochemical reaction. The main difficulties encountered in conventional TDDFT include the underestimation of the ionization threshold [@CCS98], the underestimation of charge transfer excitations [@TAH+99; @CGG+00; @DWH03], and the lack of explicit two- and higher-electron excitations [@C95; @MZCB04; @CZMB04; @C05]. Still other difficulties are discussed in pertinant reviews [@C01; @MBA+01; @ORR02; @D03; @MWB03; @SM05; @MUN+06].
To help TDDFT become a reliable tool for modeling photochemistry, we must first obtain a clear idea of what are its most severe problems. It is the objective of this article to identify the most critical points where improvement needs to be made, at least for one prototypical molecule and one type of reaction path. The reaction chosen for our study is the photochemical ring opening of oxirane. The photochemistry of oxirane and of oxirane derivatives has been much studied both theoretically and experimentally (Appendix \[sec:photochem\]). It is a text-book molecule used to discuss orbital control of stereochemistry in the context of electrocyclic ring opening. Textbook discussions usually focus on CC ring opening (see e.g. Ref. [@MB90] pp. 258-260) but a discussion of CO ring opening may also be found (Ref. [@MB90] pp. 408-411). At first glance, this would seem to be a particularly good test case for TDDFT. That is, oxirane is a simple molecule with a relatively simple photochemistry where TDDFT should work and, if not, where its failures will be particularly easy to analyze. In this paper, we will focus on the symmetric ring opening of this molecule even though the photochemical ring opening does not follow a symmetric pathway. The reason for our choice is the usual reason [@symmetry], namely that the use of symmetry greatly facilitates analysis and hence the construction of and comparison with highly accurate quantum Monte Carlo results. A mixed quantum/classical SH trajectory study of asymmetric ring opening will be reported elsewhere [@T07].
In the next section, we give a brief review of DFT and of TDDFT. In sec. \[sec:QMC\], we review the formalism behind the more exact theory against which we will be comparing our TDDFT results. Computational details are given in Sec. \[sec:details\]. Section \[sec:results\] reports our results and discussion and Sec. \[sec:conclude\] summarizes.
(Time-Dependent) Density-Functional Theory {#sec:TDDFT}
==========================================
The (TD)DFT PES for the $I$th excited state is calculated at each nuclear configuration by adding the $I$th TDDFT excitation energy to the DFT ground state energy, so $E_I = E_0 + \omega_I$. \[Hartree atomic units ($\hbar=m=e=1$) are used throughout the paper.\] Even though the use of parentheses in the expression (TD)DFT better emphasizes the hybrid DFT + TDDFT nature of the calculation, we will usually follow the common practice of simply refering to TDDFT PESs. The purpose of this section is to review those aspects of TDDFT most necessary for understanding the rest of the paper.
The hybrid DFT + TDDFT nature of (TD)DFT calculations indicates that problems with the ground state PES can easily become problems for the excited-state PESs. Thus it is important to begin with a few words about the ground-state problem. The simplest methods for treating the ground state are the Hartree-Fock (HF) method and the Kohn-Sham formulation of DFT. In recent years, the use of hybrid functionals has permitted the two energy expressions to be written in the same well-known form, $$\begin{aligned}
E^{\text{hybrid}}_{xc} = E^{\text{GGA}}_{xc} + c_x ( E^{\text{HF}}_x - E^{\text{GGA}}_x )\end{aligned}$$ where $E^{\text{HF}}_x$ is the HF exchange energy and $E^{\text{GGA}}_x$ and $E^{\text{GGA}}_{xc}$ are generalized gradient approximation (GGA) exchange (x) and exchange-correlation (xc) energies. The coefficient $c_x$ controls the amount of Hartree-Fock exchange, being unity for Hartree-Fock, zero for pure DFT, and fractional (typically around 0.25 [@PEB96]) for hybrid functionals. For more information about DFT, we refer the reader to Refs. [@PY89; @DG90; @KH00].
Time-dependent density-functional theory offers a rigorous approach to calculating excitation energies. Unlike time-dependent Hartree-Fock (TDHF), TDDFT is formally exact. Consequently, although approximate exchange-correlation functionals must be used in practice, we can hope that good approximations will lead to better results than those obtained from TDHF which completely lacks correlation effects. In practice, the use of hybrid functionals means that TDDFT contains TDHF as a particular choice of functional (i.e., the exchange-only hybrid functional with $c_x=1$).
The most common implementation of TDDFT is via a Kohn-Sham formalism using the so-called adiabatic approximation which assumes that the self-consistent field responds instantaneously and without memory to any temporal change in the charge density. This allows the time-dependent exchange-correlation action quantity in TDDFT to be replaced with the more familiar exchange-correlation energy from conventional TDDFT. Excitations may be obtained from the linear response formulation of TDDFT (LR-TDDFT). A key quantity in LR-TDDFT is the exchange-correlation kernel, $$f_{xc}^{\sigma,\tau}({\bf r},{\bf r}')
= \frac{
\delta^2 E_{xc}[\rho_\uparrow,\rho_\downarrow]
}
{
\delta \rho_\sigma({\bf r}) \delta \rho_\tau({\bf r}')
} \, ,
\label{eq:TDDFT.8}$$ which, along with the Hartree kernel, $ f_{H}^{\sigma,\tau}({\bf r},{\bf r}')
= 1/ \left| {\bf r} - {\bf r}' \right| $, and the Hartree-Fock exchange kernel (whose integrals can be written in terms of the Hartree kernel) determines the linear response of the Kohn-Sham self-consistent field in the adiabatic approximation. In Casida’s formulation [@C95], the excitation energies are obtained by solving a random phase approximation (RPA)-like pseudo-eigenvalue equation, $$\left[ \begin{array}{cc} {\bf A} & {\bf B} \\
{\bf B} & {\bf A} \end{array} \right]
\left( \begin{array}{c} \vec{X}_I \\ \vec{Y}_I \end{array} \right)
= \omega_I
\left[ \begin{array}{cc} +{\bf 1} & {\bf 0} \\
{\bf 0} & -{\bf 1} \end{array} \right]
\left( \begin{array}{c} \vec{X}_I \\ \vec{Y}_I \end{array} \right) \, ,
\label{eq:TDDFT.10}$$ where the matrices ${\bf A}$ and ${\bf B}$ are defined by, $$\begin{aligned}
A_{ia\sigma,jb\tau} & = & \delta_{i,j} \delta_{a,b} \delta_{\sigma,\tau}
\left( \epsilon_{a\sigma} - \epsilon_{i\sigma} \right)
+ \left( ia \vert f_H \vert bj \right) +
\left( ia \vert f_{xc}^{\sigma,\tau} \vert bj \right) \nonumber \\
& - & c_x \delta_{\sigma,\tau} \left[
\left( ij \vert f_H \vert ba \right) +
\left( ia \vert f_{x}^{\sigma,\sigma} \vert bj \right) \right] \nonumber \\
B_{ia\sigma,bj\tau} & = & \left( ia \vert f_H \vert jb \right)
+ \left( ia \vert f_{xc}^{\sigma,\tau} \vert jb \right)
\nonumber \\
& - & c_x \delta_{\sigma,\tau} \left[
\left( ib \vert f_H \vert ja\right) +
\left( ia \vert f_{x}^{\sigma,\sigma} \vert jb \right)
\right] \, ,
\label{eq:TDDFT.11}\end{aligned}$$ and the integrals are in Mulliken charge cloud notation, $$\left( pq \vert f \vert rs \right) = \int \int \psi_p^* ({\bf r}) \psi_q({\bf r})
f({\bf r}, {\bf r}') \psi_r^* ({\bf r}') \psi_s({\bf r}') \, d{\bf r} d{\bf r}' \, .
\label{eq:TDDFT.12}$$
It is to be emphasized that the TDDFT adiabatic approximation includes only dressed one-electron excitations [@C95; @C05]. This is particularly easy to see in the context of the Tamm-Dancoff approximation (TDA) to Eq. (\[eq:TDDFT.10\]) whose TDDFT variant is usually attributed to Hirata and Head-Gordon [@HH99]. The TDA simply consists of neglecting the ${\bf B}$ matrices to obtain, $ {\bf A} \vec{X}_I = \omega_I \vec{X}_I $. The number of possible solutions to this equation is the dimensionality of ${\bf A}$ which is just the number of single excitations. In fact, the LR-TDHF TDA (exchange-only hybrid functional with $c_x=1$) is simply the well-known configuration interaction singles (CIS) method.
An issue of great importance in the context of the present work is triplet instabilities. These have been first analyzed by Bauernschmitt and Ahlrichs [@BA96] in the context of DFT, but the explicit association with LR-TDDFT excitation energies was made later [@CGG+00; @C01]. Following Ref. [@C01], we suppose that the ground-state DFT calculation has been performed using a same-orbitals-for-different-spin (SODS) ansatz and we now wish to test to see if releasing the SODS restriction to give a different-orbitals-for-different-spin (DODS) solution will lower the energy. To do so we consider an arbitary unitary tranformation of the orbitals, $$\psi_{r\sigma}^\lambda ({\bf r}) = \exp\left[i\lambda \left(\hat{R}+i\hat{I}\right)\right]
\psi_{r\sigma}({\bf r})\,.$$ where $\hat{R}$ and $\hat{I}$ are real operators. The corresponding energy expression is, $$E_\lambda = E_0 + \lambda^2 \left[ \vec{R}^\dagger \left( {\bf A} - {\bf B} \right)
\vec{R} + \vec{I}^\dagger \left( {\bf A} + {\bf B} \right) \vec{I} \right]
+ {\cal O}(\lambda^3) \, ,
\label{eq:TDDFT.15}$$ where matrix elements of the $\hat{R}$ and $\hat{I}$ operators have been arranged in column vectors and the ${\cal O}(\lambda)$ term disappears because the energy has already been minimized before considering symmetry-breaking. The presence of the terms $({\bf A} \pm {\bf B})$ shows the connection with the pseudoeigenvalue problem (\[eq:TDDFT.10\]) which can be rewritten as the eigenvalue problem, $$\left( {\bf A} + {\bf B} \right) \left( {\bf A} - {\bf B} \right) \vec{Z}_I
= \omega_I^2 \vec{Z}_I\,,
\label{omega2_eq}$$ where $\vec{Z}_I=\vec{X}_I-\vec{Y}_I$. It is an easy consequence of Eq. (\[eq:TDDFT.11\]) that the matrix $\left( {\bf A} - {\bf B} \right)$ is always positive definite for pure functionals ($c_x=0$), as long as the [*aufbau*]{} principle is obeyed. However $\left( {\bf A} + {\bf B} \right)$ may have negative eigenvalues. In that case, the energy $E_\lambda$ will fall below $E_0$ for some value of $\vec{I}$, indicating a lower symmetry-broken solution. At the same time, this will correspond to a negative value of $\omega_I^2$ (i.e., an imaginary value of $\omega_I$.) A further analysis shows the corresponding eigenvector corresponds to a triplet excitation, hence the term “triplet instability” for describing this phenomenon. In principle, when hybrid functionals are used, singlet instabilities are also possible.
Thus imaginary excitation energies in LR-TDDFT are an indication that something is wrong in the description of the ground state whose time-dependent response is being used to obtain those excitation energies. As pointed out in Refs. [@CGG+00] and [@CIC06], the TDA actually acts to [*decouple*]{} the excited-state problem from the ground-state problem so that TDA excitations may actually be [*better*]{} than full LR-TDDFT ones. Typically, full and TDA LR-TDDFT excitation energies are close near a molecule’s equilibrium geometry, where a single-determinantal wave function is a reasonable first approximation and begin to differ as the single-determinantal description breaks down.
Interestingly, there is a very simple argument against symmetry breaking in exact DFT for molecules with a nondegenerate ground state. Since the exact ground-state wave function of these molecules is a singlet belonging to the totally symmetric representation and so with the same symmetry as the molecule, the spin-up and spin-down charge densities will be equal and also have the same symmetry as the molecule. It follows that the same must hold for the spin-up and spin-down components of the exact exchange-correlation potential. Since the potentials are the same, then the molecular orbitals must also be the same for different spins. Thus, no symmetry breaking is expected in exact DFT for molecules with a nondegenerate ground state. Note however that the functional is still dependent on spin so that the spin kernels $ f_{xc}^{\uparrow,\uparrow}({\bf r},{\bf r}')$ and $f_{xc}^{\uparrow,\downarrow}({\bf r},{\bf r}')$ are different.
The interested reader will find additional information about TDDFT in the recent book of Ref. [@MUN+06].
Quantum Monte Carlo {#sec:QMC}
===================
The quality of our (TD)DFT calculations is judged against highly accurate quantum Monte Carlo (QMC) results. For each state of interest, the QMC results are obtained in a three step procedure. First, a conventional complete active space (CAS) self-consistent field (SCF) calculation is performed. The resultant CASSCF wave function is then reoptimized in the presence of a Jastrow factor to include dynamical correlation and used in a variational Monte Carlo (VMC) calculation. Finally, the VMC result is further improved via diffusion Monte Carlo (DMC).
To properly describe a reaction involving bond breaking, it is necessary to adopt a multi-determinantal description of the wave function. In the CASSCF method, a set of active orbitals is selected, whose occupancy is allowed to vary, while all other orbitals are fixed as either doubly occupied or unoccupied. In a CASSCF($n$,$m$) calculation, $n$ electrons are distributed among an active space of $m$ orbitals and all possible resulting space- and spin-symmetry-adapted configuration state functions (CSFs) are constructed. The final CASSCF($n$,$m$) wave function consists of a linear combination of these CSFs, like in a full configuration interaction (CI) calculation for $n$ electrons in $m$ orbitals, except that also the orbitals are now optimized to minimize the total energy. In the present article, we consider 4 electrons in an active space of 6 orbitals which represents the minimal active space for a proper description of all 9 states of interest.
When several states of the same symmetry are requested, there is a danger in optimizing the higher states that their energy is lowered enough to approach and mix with lower states, thus giving an unbalanced description of excitation energies. A well-established solution to this problem is the use of a state averaged (SA) CASSCF approach where the weighted average of the energies of the states under consideration is optimized [@WM81; @HJO00]. The wave functions of the different states depend on their individual sets of CI coefficients using a common set of orbitals. Orthogonality is ensured via the CI coefficients and a generalized variational theorem applies. Obviously, the SA-CASSCF energy of the lowest state will be higher than the CASSCF energy obtained without SA. In the present work, we need to apply the SA procedure only for the $2^1A_1$ state in the $C_{2v}$ ring opening. Although the CASSCF and SA-CASSCF $1^1A_1$ energies are found to be really very close, we calculate the $2^1A_1$ energy as $ E_{\text{CAS}}(1^1A_1)
+ \left[ E_{\text{CAS}}^{\text{SA}}(2^1A_1)- E_{\text{CAS}}^{\text{SA}}(1^1A_1) \right]$. A similar procedure is used in the case of the corresponding QMC calculations.
QMC methods [@FMNR01] offer an efficient alternative to conventional highly-correlated [*ab initio*]{} methods to go beyond CASSCF and provide an accurate description of both dynamical and static electronic correlation. The key ingredient which determines the quality of a QMC calculation is the many-body trial wave function which, in the present work, is chosen of the Jastrow-Slater type with the particular form, $$\Psi^{\text{VMC}}_I = \Psi^{\text{CAS}}_I \prod_{A,i,j} {\cal J}(r_{ij}, r_{iA}, r_{jA}) \, ,
\label{eq:QMC.2}$$ where $r_{ij}$ denotes the distance between electrons $i$ and $j$, and $r_{iA}$ the distance of electron $i$ from nucleus $A$. We use here a Jastrow factor ${\cal J}$ which correlates pairs of electrons and each electron separately with a nucleus, and employ different Jastrow factors to describe the correlation with different atom types. The determinantal component consists of a CAS expansion, $\Psi^{\text{CAS}}_I$, and includes all possible CSFs obtained by placing 4 electrons in the active space of 6 orbitals as explained above. All parameters in both the Jastrow and the determinantal component of the wave function are optimized by minimizing the energy. Since the optimal orbitals and expansion coefficients in $\Psi^{\text{CAS}}_I$ may differ from the CASSCF values obtained in the absence of the Jastrow factor ${\cal J}$, it is important to reoptimize them in the presence of the Jastrow component.
For a wave function corresponding to the lowest state of a given symmetry, we follow the energy-minimization approach of Ref. [@UTFSH07]. If the excited state is not the lowest in its symmetry, we obtain the Jastrow and orbitals parameters which minimize the average energy over the state of interest and the lower states, while the linear coefficients in the CSF expansion ensure that orthogonality is preserved among the states [@F07]. Therefore, the wave functions resulting from the state-average optimization will share the same Jastrow parameters and the same set of orbitals but have different linear coefficients. This scheme represents a generalization of the approach of Ref. [@SF04] where only the orbitals were optimized and orthogonality was only approximately preserved. The present approach is therefore superior to the one of Ref. [@SF04], which was however already giving excellent results when tested on several singlet states of ethylene and a series of prototypical photosensitive molecules [@SF04; @SBF04]. We note that, when a CAS expansion is used in the absence of the Jastrow component, the method is analogous to the CASSCF technique for the lowest state of a given symmetry, and to a SA-CASSCF approach if the excited state is not the lowest in its symmetry.
The trial wave function is then used in diffusion Monte Carlo (DMC), which produces the best energy within the fixed-node approximation \[i.e., the lowest-energy state with the same zeros (nodes) as the trial wave function\]. All QMC results presented are from DMC calculations.
Computational Details {#sec:details}
=====================
SCF and TD Details
------------------
Calculations were carried out with two different computer programs, namely [Gaussian]{} 03 [@Gaussian] and a development version of [deMon2k]{} [@deMon2K]. We use the two programs in a complementary fashion as they differ in ways which are important for this work. In particular, [Gaussian]{} can carry out HF calculations while [deMon2k]{} has no Hartree-Fock exchange. Similarly, [Gaussian]{} can carry out time-dependent HF (TDHF) and configuration interaction singles (CIS) calculations which are not implemented in [deMon2k]{}. In contrast, the Tamm-Dancoff approximation (TDA) for TDDFT may only be performed with [deMon2k]{}. The present version of [deMon2k]{} is limited to the local density approximation for the exchange-correlation TDDFT kernel, while the kernel is more general in [Gaussian]{}. Finally, [Gaussian]{} makes automatic use of symmetry and prints out the irreducible representation for each molecular orbital. The particular version of [deMon2k]{} used in the present work does not have this feature so we rely on comparison with the output of [Gaussian]{} calculations for this aspect of our analysis.
All [Gaussian]{} results reported here are calculated with the extensive 6-311++G\*\*(2d,2p) basis set [@KBSP80; @CCS83]. All calculations used default convergence criteria. The DFT calculations used the default grid for the exchange-correlation integrals.
The same basis set is used for [deMon2k]{} calculations as in the Gaussian calculations. The GEN-A3\* density-fitting basis set is used and density-fitting is carried out without imposing the charge conservation constraint. The SCF convergence cutoff is set at $10^{-7}$. We always use the FIXED FINE option for the grid. Our implementation of TDDFT in [deMon2k]{} is described in Ref. [@IFP+05].
We use two different density functionals, the local density approximation (LDA) in the parameterization of Vosko, Wilk, and Nusair [@VWN80] (referred to as SVWN5 in the [Gaussian]{} input), and the popular B3LYP hybrid functional [@B3LYP]. The corresponding TDDFT calculations are referred to as TDLDA and TDB3LYP respectively.
CAS Details {#CAS_Details}
-----------
All CASSCF calculations are performed with the program [GAMESS(US)]{} [@Gamess]. In all SA-CASSCF calculations, equal weights are employed for the two states.
We use scalar-relativistic energy-consistent Hartree-Fock pseudopotentials [@BFD07] where the carbon and oxygen 1$s$ electrons are replaced by a non-singular $s$-non-local pseudopotential and the hydrogen potential is softened by removing the Coulomb divergence. We employ the Gaussian basis sets [@BFD07] constructed for these pseudopotentials and augment them with diffuse functions. All calculations are performed with the cc-TZV contracted (11$s$11$p$1$d$)/\[3$s$3$p$1$d$\] basis for carbon and oxygen, augmented with two additional diffuse $s$ and $p$ functions with exponents 0.04402 and 0.03569 for carbon, and 0.07376 and 0.05974 for oxygen. The $d$ polarization functions for carbon and oxygen are taken from the cc-DZV set. For hydrogen, the cc-DZV contracted (10$s$9$p$)/\[2$s$1$p$\] basis is augmented with one $s$ diffuse function with exponent 0.02974.
QMC Details
-----------
The program package [CHAMP]{} [@Champ] is used for the QMC calculations. We employ the same pseudopotentials and basis sets as in the CASSCF calculations (see Sec. \[CAS\_Details\]).
Different Jastrow factors are used to describe the correlation with a hydrogen, an oxygen and a carbon atom. For each atom type, the Jastrow factor consists of an exponential of the sum of two fifth-order polynomials of the electron-nuclear and the electron-electron distances, respectively [@FU96]. The parameters in the Jastrow factor and in the determinantal component of the wave function are simultaneously optimized by energy minimization following the scheme of Ref. [@UTFSH07], where we employ the simple choice $\xi=1$. For the excited states with the same symmetry as the ground state, the ground- and excited-state wave functions are optimized in a state-average manner with equal weights for both states [@F07]. An imaginary time step of 0.075 H$^{-1}$ is used in the DMC calculations.
Results {#sec:results}
=======
![a) Woodward-Hoffmann orbital correlation scheme. Symmetry labels are for the $C_{2v}$ point group in the case of reactants and products, for the $C_s$ point group along the disrotatory pathway, and for the $C_2$ point group along the conrotatory pathway. b) Thermal ring opening. c) Photochemical ring opening. Symmetry labels depend upon the labels of the $x$, $y$, and $z$ coordinates. In this article, the COC ring lies in the $(y, z)$-plane and the $z$-axis coincides with the $C_2$ axis, in agreement with the IUPAC convention[@M55]. []{data-label="fig:WHcorrdiag"}](fig1.pdf)
The Woodward-Hoffmann (WH) rules for orbital control of symmetry in electrocyclic reactions were an important motivation in the 1970s and 1980s to seek stereospecific photochemical ring-opening reactions of oxiranes [@H71b; @GP76; @L76; @H77; @HRS+80; @P87]. Since some oxiranes (notably diphenyl oxirane) do appear to follow the WH rules for thermal and photochemical ring opening, the WH rules might seem like the obvious place to begin our study of symmetric ring opening pathways in oxirane. This is somewhat counterbalanced by the fact that oxirane itself is an exception to the WH rules for photochemical ring opening and by the fact that it is now clear that the WH rules do not apply nearly as well to photochemical as to thermal reactions (Appendix \[sec:photochem\]). Nevertheless, we will take the WH model as a first approximation for understanding and begin by describing the model for the particular case of oxirane.
Many models at the time that Woodward and Hoffmann developed their theory [@WH65a; @WH65b; @WH69] were based upon simple Hückel-like $\pi$-electron models, so, not surprisingly, the historical WH model for oxirane uses a three-orbital model consisting of one $p$-orbital on each of the oxygen and two carbon atoms. (See Fig. \[fig:WHcorrdiag\].) Elementary chemical reasoning predicts that the closed cycle should form three molecular orbitals, $\sigma$, $n$, $\sigma^*$, in increasing order of energy. Similarly the open structure has the “particle-in-a-box” orbitals familiar from simple Hückel theory. Woodward and Hoffmann observed that a reflection plane ($\sigma$) of symmetry is preserved along the disrotatory reaction pathway while a $C_2$ rotation symmetry element is preserved along the conrotatory reaction pathway. This observation allows the reactant-product molecular orbital correlation diagram to be completed by using the fact that the symmetry representation of each orbital is preserved within the relevant symmetry group of the molecule along each reaction path (WH principle of conservation of orbital symmetry) and connecting lowest orbitals with lowest orbitals. Figure \[fig:WHcorrdiag\] shows that a conrotatory ([*con*]{}) thermal reaction connects ground-state configurations in the reactant and product while a disrotatory ([*dis*]{}) thermal reaction connects the reactant ground-state configuration with an electronically excited-state configuration. Thus, the [*con*]{} thermal reaction is expected to be prefered over the [*dis*]{} reaction. Also shown in Fig. \[fig:WHcorrdiag\] is the photochemical reaction beginning with the $n \rightarrow \sigma^*$ excited state. In this case, the [*dis*]{} mechanism is expected to be prefered since the [*con*]{} mechanism leads to a still higher level of excitation in the product than in the reactant molecule while the level of excitation is preserved along the [*dis*]{} pathway.
Let us now see whether this orbital model is reflected first in the vertical absorption spectrum of oxirane, then in the $C_{2v}$ ring opening pathway, and finally in the [*con*]{} and [*dis*]{} ring-opening pathways. At each step, the performance of TDDFT will be assessed against experiment and against other levels of theory.
Absorption Spectrum
-------------------
![Comparison of HF/6-311G$^{**}$(2d,2p), B3LYP/6-311G$^{**}$(2d,2p), and LDA/6-311G$^{**}$(2d,2p) optimized geometries with the experimental gas phase geometry from Ref. [@H47]. Note that the structure has $C_{2v}$ symmetry.[]{data-label="fig:optgeom"}](fig2.pdf)
The first step towards calculating the electronic absorption spectrum of oxirane is the optimization of the geometry of the gas phase molecule. This was carried out using the HF method and DFT using the LDA and B3LYP functionals. The calculated results are compared in Fig. \[fig:optgeom\] with the known experimental values. It is seen that electron correlation included in DFT shortens bond lengths, bringing the DFT optimized geometries into considerably better agreement with the experimental geometries than are the HF optimized geometries. This better agreement between DFT and experiment also holds for bond angles. There is not much difference between the LDA and B3LYP geometries with the exception of the COC bond angle which is somewhat better described with the B3LYP than with the LDA functional.
TDHF TDLDA TDB3LYP Expt.
--------------- --------------- --------------- ------------------
9.14 (0.0007) 6.01 (0.0309) 6.69 (0.0266) 7.24(s)
9.26 (0.0050) 6.73 (0.0048) 7.14 (0.0060) 7.45(w)
9.36 (0.0635) 6.78 (0.0252) 7.36 (0.0218) 7.88(s), 7.89(s)
9.56 (0.0635) 7.61 (0.0035) 7.85 (0.0052)
9.90 (0.0478) 7.78 (0.0304) 8.37 (0.0505)
9.93 (0.0935) 8.13 (0.0014) 8.39 (0.0168)
8.15 (0.0405) 8.40 (0.0419)
: Principal oxirane singlet excitation energies (eV) and oscillator strengths (unitless). []{data-label="tab:spect"}
Several experimental studies of the absorption spectrum of oxirane are available [@LD49; @LW58; @FAHP59; @BRK+69; @BBG+92]. The positions of the principal electronic excitations were identified early on but their assignment took a bit longer. The accepted interpretation is based on the study by Basch [*et al*]{}. [@BRK+69] who combined information from vacuum ultraviolet spectra, photoelectron spectra, and quantum chemical computations, and assigned the observed transitions to O($n$) $\rightarrow$ $3s$ and $3p$ Rydberg transitions. Let us see how this is reflected in our calculations.
While theoretical absorption spectra are often shifted with respect to experimental absorption spectra, we expect to find two strong absorptions in the low energy spectrum, separated from each other by about 0.65 eV. Vertical absorption spectra are calculated using TDHF, TDLDA, and TDB3LYP, all at the B3LYP-optimized geometry. The results are shown in Table \[tab:spect\]. TDB3LYP is the method in best agreement with experiments, yielding two strong absorptions red-shifted from experiment by about 0.5 eV and separated by 0.67 eV. The TDLDA method is qualitatively similar, apart from a stronger red shift. In particular, the TDLDA method shows two strong absorptions red-shifted from experiment by about 1.1 eV and separated by 0.78 eV. In contrast, the TDHF spectrum is blue-shifted by about 2 eV and is otherwise of questionable value for interpreting the experimental spectrum. In what follows, we have decided to assign the lowest three absorptions using the results of our TDB3LYP calculations.
![B3LYP MOs. Left: ring structure. Right: open structure.[]{data-label="fig:orbitals"}](fig3.pdf)
This first involves an examination of the B3LYP molecular orbitals (MOs). These orbitals are shown in Fig. \[fig:orbitals\]. The electronic configuration of the ring structure is, $$\cdots [6a_1(\sigma)]^2 [2b_1(n)]^2 [7a_1(3s)]^0 [4b_2(\sigma^*)]^0 \cdots \, .
\label{eq:results.1}$$ The group theoretic MO labels ($a_1$, $a_2$, $b_1$, and $b_2$) correspond to representations of the $C_{2v}$ symmetry group. The additional labels, $\sigma$, $n$, and $\sigma^*$, show our chemical interpretation of the B3LYP orbitals and their correspondence with the MOs in the WH three-orbital model.
![Walsh diagram for $C_{2v}$ ring opening calculated at the B3LYP level. To construct this diagram, the COC bond angle was varied and all other geometric parameters were relaxed within the constraint of $C_{2v}$ symmetry. The HOMO is the $2b_1$ orbital on the left hand side and the $4b_2$ orbital on the right hand side. []{data-label="fig:Walsh_C2v"}](fig4.pdf)
![B3LYP MOs implicated in the principal UV absorptions.[]{data-label="fig:excite_orbs"}](fig5.pdf)
Confirmation of the chemical nature of our B3LYP MOs was obtained by constructing a Walsh diagram for $C_{2v}$ ring opening. This graph of MO energies as a function of ring-opening angle ($\alpha$) is shown in Fig. \[fig:Walsh\_C2v\]. As the ring opens, the C($2p$) $6a_1(\sigma)$ bond breaks and so increases in energy. At the same time, the C($2p$) $4b_2(\sigma^*)$ antibond becomes less antibonding and so decreases in energy. The O($2p$) lone pair $2b_1(n)$ is not involved in bonding and so maintains a roughly constant energy throughout the ring-opening process. Experimental information about occupied MOs is available from electron momentum spectroscopy (EMS) via the target Kohn-Sham approximation [@DCCS94]. The ordering of the B3LYP occupied orbitals is consistent with the results of a recent EMS study of oxirane [@WMWB99]. In particular, the two highest energy occupied orbitals are seen to have a dominant $p$-type character. The interpretation of the unoccupied orbitals is more problematic in that the B3LYP unoccupied orbitals in our calculations are not bound (i.e., have positive orbital energies). We are thus attempting to describe a continuum with a finite basis set. It is thus far from obvious that the unoccupied orbital energies will converge to anything meaningful as the finite basis set becomes increasingly complete. The Walsh diagram shows that the $4b_2(\sigma^*)$ unoccupied orbital becomes bound for ring opening angles beyond about 80$^\circ$. It is also rather localized and hence it makes sense to assign it some physical meaning. This is certainly consistent with previous HF studies using the STO-3G minimal basis set [@BSD79a] and the more extensive 6-31G\*\* basis sets [@HSS01]. Since our 6-311++G\*\*(2d,2p) basis set is even larger, it is not too surprising that we find an additional unoccupied orbital, namely the $7a_1(3s)$ orbital shown in Fig. \[fig:excite\_orbs\]. Although apparently at least partially localized, this orbital remains unbound at all bond angles in the Walsh diagram and care should be taken not to overinterpret its physical nature. Nevertheless this $7a_1(3s)$ orbital intervenes in an important way in the interpretation of our calculated electronic absorption spectra and it is upon analysis of the spectra that we will be able to associate this orbital with the $3s$ Rydberg state.
One should also be conscious in using the calculated TDB3LYP absorption spectrum to assign the experimental gas phase UV spectrum that the TDDFT ionization continuum begins at minus the value of the HOMO orbital energy [@CJCS98]. The value of $-\epsilon_{\text{HOMO}}$ obtained in the different calculations is 6.40 eV with the LDA, 7.68 eV with the B3LYP hybrid functional, and 12.27 eV with HF. As expected from Koopmans’ theorem, the HF value of $-\epsilon_{\text{HOMO}}$ is in reasonable agreement with the experimental ionization potential of 10.57 eV [@NCK77]. The presence of a fraction of HF exchange in the B3LYP hybrid functional helps to explain why its value of $-\epsilon_{\text{HOMO}}$ lies between that of the pure DFT LDA and that of HF. As far as our TDB3LYP calculations are concerned, the value of $-\epsilon_{\text{HOMO}}$ means that assignment of experimental excitation energies higher than 7.7 eV should be avoided. Fortunately this still allows us to assign the first singlet excitation energies. Examination of the TDB3LYP coefficients yields the following assignments for the three principal UV absorption peaks: $$\begin{aligned}
\mbox{6.69 eV} & : & 1^1B_1[2b_1(n) \rightarrow 7a_1(3s)] \\
\mbox{7.14 eV} & : & 2^1B_1[2b_1(n) \rightarrow 8a_1(3p_z)] \\
\mbox{7.36 eV} & : & 2^1A_1[2b_1(n) \rightarrow 3b_1(3p_x)] \, .
\label{eq:results.2}\end{aligned}$$ Comparison with the experimental assignment of Basch [*et al*]{}. justifies the identification of these orbitals with the Rydberg orbitals $3s$, $3p_z$, and $3p_x$.
It is worth pointing out that the expected $^1[2b_1(n),4b_2(\sigma^*)]$ excitation is of $^1A_2$ symmetry and as such corresponds to a spectroscopically forbidden transition. It is in any event fairly high in energy (9.77 eV with TDHF). The $^1[6a_1(\sigma),4b_2(\sigma^*)]$ transition has $^1B_2$ symmetry and so is spectroscopically allowed, but is still found at fairly high energy in our calculations (8.15 eV with TDLDA, 8.37 with TDB3LYP, and 9.93 with TDHF).
{width="\columnwidth"} {width="\columnwidth"}
$C_{2v}$ Ring-Opening
---------------------
We now consider how DFT performs for describing the ground state and how well TDDFT performs for describing the lowest excited state of each symmetry for $C_{2v}$ ring-opening of oxirane. Comparisons are made against CASSCF and against high-quality DMC energies calculated at the same geometries. All geometries along the $C_{2v}$ pathway have been fully optimized at each O - C - O ring-opening angle ($\alpha$) using the B3LYP functional. The orbital energies as a function of ring opening angle have already been given in the Walsh diagram (Fig. \[fig:Walsh\_C2v\]).
Fig. \[fig:all\_TDDFT\_curves\] shows the TDB3LYP and TDLDA curves for the ground ($1 ^1A_1$) state and the lowest excited-states of each symmetry ($2 ^1A_1$, $1 ^3A_1$, $1 ^1B_1$, $1 ^3B_1$, $1 ^1A_2$, $1 ^3A_2$, $1 ^1B_2$, and $1 ^3B_2$). Several things are worth noting here. The [*first*]{} point is that, some of the differences between the TDB3LYP and TDLDA curves are apparent, not real, as the points for the two graphs were not calculated at exactly the same angles. In particular, the TDLDA misses the point at 120$^\circ$ where we encountered serious convergence difficulties due to a quasidegeneracy of $\sigma$ and $\sigma^*$ orbitals (Fig. \[fig:Walsh\_C2v\]). Under these circumstances, the HOMO, which suffers from self-interaction errors, can lie higher than the self-interaction-free LUMO. As the program tries to fill the orbitals according to the usual [*aufbau*]{} principle, electron density sloshes on each iteration between the two orbitals making convergence impossible without special algorithms. The TDB3LYP calculations were found to be easier to converge, presumably because they have less self-interaction error.
The [*second*]{} point is that only [deMon2k]{} TDLDA calculations are reported here, although we have also calculated TDLDA curves using [Gaussian]{} and found similar results when both programs printed out the same information. Unfortunately, [Gaussian]{} did not always print out the lowest triplet excitation energy, so we prefer to report our more complete [deMon2k]{} results.
The [*third*]{} point is the presence of a triplet instability. This means that, going away from the equilibrium geometry, the [*square*]{} of the first triplet excitation energy decreases, becomes zero, and then negative. The exact meaning of triplet instabilities will be discussed further below. For now, note that [*we follow the usual practice for response calculations by indicating an imaginary excitation energy as a negative excitation energy.*]{} However it is important to keep in mind that a negative excitation energy in this context is only a common convention and not a physical reality. Incidentally, we presume that triplet instability is the source of the difficulty with the [Gaussian]{} output mentioned above–the associated coefficients become more complicated and [Gaussian]{} may have difficulty analyzing them. A [*fourth*]{} and final point is that the TDDFT ionization threshold occurs at minus the value of the HOMO energy which is significantly underestimated with ordinary functionals such as the LDA and B3LYP [@CJCS98]. This artificially low ionization threshold is indicated in the two graphs in Fig. \[fig:all\_TDDFT\_curves\]. In the TDB3LYP case, the TDDFT ionization threshold is high enough that it is not a particular worry. In the TDLDA case, the TDDFT ionization threshold is lower by about 1 eV. This may explain some of the quantitative differences between the high-lying TDB3LYP and TDLDA curves, although by and large the two calculations give results in reasonable qualitative, and even semi-quantitative, agreement.
We now wish to interpret the TDDFT curves. TDDFT excitations may be characterized in terms of single electron excitations from occupied to unoccupied MOs [@C95]. Unlike in HF, unoccupied and occupied orbitals in pure DFT (e.g., the LDA) see very similar potentials and hence the same number of electrons. This means that the orbitals are preprepared for describing electron excitations and that simple orbital energy differences are often a good first approximation to describing excitation energies. In the two-orbital model and pure DFT, the singlet, $\omega_S$, and triplet, $\omega_T$, TDDFT excitation energies for the transition from orbital $i$ to orbital $a$ are, $$\begin{aligned}
\omega_S & = & \sqrt{\left(\epsilon_a - \epsilon_i\right)
\left[\left(\epsilon_a - \epsilon_i\right) + 2(ia \vert 2f_H + f_{xc}^{\uparrow,\uparrow}
+ f_{xc}^{\uparrow,\downarrow} \vert ai ) \right] } \nonumber \\
\omega_T & = & \sqrt{\left(\epsilon_a - \epsilon_i\right)
\left[\left(\epsilon_a - \epsilon_i\right) + 2(ia \vert f_{xc}^{\uparrow,\uparrow}
- f_{xc}^{\uparrow,\downarrow} \vert ai ) \right] } \, .
\label{eq:results.3}\end{aligned}$$ In the TDA, this becomes $$\begin{aligned}
\left(\omega_S\right)_{\text{TDA}} & = & \left(\epsilon_a - \epsilon_i\right)
+ (ia \vert 2f_H + f_{xc}^{\uparrow,\uparrow} + f_{xc}^{\uparrow,\downarrow} \vert ai )
\nonumber \\
\left(\omega_T\right)_{\text{TDA}} & = & \left(\epsilon_a - \epsilon_i\right)
+ (ia \vert f_{xc}^{\uparrow,\uparrow} - f_{xc}^{\uparrow,\downarrow} \vert ai ) \, ,
\label{eq:results.4}\end{aligned}$$ from which it is clear from the sizes and signs of the integrals that, $$\left( \omega_T \right)_{\text{TDA}} \leq \epsilon_a -
\epsilon_i \leq \left( \omega_S \right)_{\text{TDA}}\, .
\label{eq:results.5}$$ For Rydberg states the inequalities become near equalities because the overlap of orbitals $i$ and $a$ goes to zero. While no longer strictly valid, these general ideas are still good starting points for understanding results obtained with the hybrid functional B3LYP.
The frontier MOs shown in the Walsh diagram (Fig. \[fig:Walsh\_C2v\]) supplemented with the addition of the two additional Rydberg orbitals, $3b_1(3p_x)$ and $5b_2(3p_y)$, provides a useful model not only for interpreting the results of our TDDFT calculations, but also for constructing the active space necessary for the CASSCF calculations used to construct the QMC wavefunctions. Fig. \[fig:CAS\_and\_DMC\_curves\] shows the results of our best QMC calculation and of the results of the CAS(4,6) calculation on which it is based. For the most part, the CAS and DMC curves differ quantitatively but not qualitatively. The exceptions are the $1 ^3A_1$ and $1 ^1B_2$ curves where the DMC results, which are significantly lower than the corresponding CAS(4,6) results at some geometries, contain important amounts of dynamic correlation not present at the CAS(4,6) level of calculation. In the remaining graphs (except for Fig. \[fig:B2\_curves\]) we will suppress CAS(4,6) curves in favor of presenting only the higher quality DMC curves since these offer the better comparison with TDDFT.
{width="\columnwidth"} {width="\columnwidth"}
We now give a detailed comparison of our TDDFT results with DMC methods, dividing the states into three sets. The first set consists of states where double excitations (lacking in the TDDFT adiabatic approximation) are likely to be important. The second set consists of states which show the effects of triplet instabilities. And the third set consists of Rydberg excitations.
The states of $A_1$ symmetry are those most likely to be affected by the absence of double excitations. Although not shown here, we have constructed the two HF curves obtained by enforcing a double occupancy of either the $6a_1(\sigma)$ or the $4b_2(\sigma^*)$ orbital. Both single determinants have $^1A_1$ symmetry. The two curves thus generated simply cross at about 120$^\circ$. In the absence of configuration mixing, the ground state curve always follows the lower state and shows an important cusp at 120$^\circ$. Introducing configuration mixing via a CASSCF calculation leads to an avoided crossing.
![$C_{2v}$ ring opening curves: $1^1A_1$ and $2^1A_1$ states. Note that the $1^3A_1$ state has also been included at the DMC level of calculation.[]{data-label="fig:1A1_curves"}](fig8.pdf){width="\columnwidth"}
Now, DFT is different from HF because DFT is exact when the functional $E_{xc}$ is exact while HF always remains an approximation. On the other hand, the Kohn-Sham equations of modern DFT resemble the HF equations and tend to inherit some of their faults when approximate functionals are used. Fig. \[fig:1A1\_curves\] shows the comparable curves at the TDDFT and DMC levels. As expected, the unphysical cusp is absent in the $1 ^1A_1$ ground state curve at the DMC level of calculation. The unphysical cusp is present at both the B3LYP and LDA levels where significant divergences between the DFT and DMC calculations occur between about 100$^\circ$ and 150$^\circ$. However, this region is very much reduced compared to what we have observed to happen for the HF curve where significant divergences beginning at about 75$^\circ$. This is consistent with the idea that even DFT with approximate functionals still includes a large degree of electron correlation.
As to the $2 ^1A_1$ excited state curve, only the DMC calculation shows an indication of an avoided crossing. The inclusion of the $1^3A_1$ excited state curve calculated at the DMC level helps to give a more complete understanding of the curve for this state. Below about 100$^\circ$ and above about 150$^\circ$, the $2 ^1A_1$ and $1^3A_1$ states have nearly the same energy, consistent with the idea that these correspond to the $^1[a_1(\sigma),a_1(3s)]$ Rydberg transition below 110$^\circ$ and to the $^1[b_2(\sigma^*),b_2(3p_z)]$ Rydberg transition above 150$^\circ$. In between the $^1[a_1(\sigma)^2,b_2(\sigma^*)^2]$ two-electron valence excitation cuts across the $^1A_1$ manifold. The TDDFT calculations are qualitatively capable of describing these one-electron Rydberg excitations but not of describing the two-electron excitation in the bond-breaking region. As expected, the cusp in the DFT $1^1A_1$ ground state curve is simply reflected as cusps in the TDDFT $2 ^1A_1$ excited-state curve.
![$C_{2v}$ ring opening curves: $1^3B_2$ and $1^1B_2$ states. Note that the ground state ($1^1A_1$) curve has only been shown for the B3LYP calculation since the LDA curve is nearly identical.[]{data-label="fig:B2_curves"}](fig9.pdf){width="\columnwidth"}
The only state which shows triplet instabilities is the $1^3B_2$ state. The TDLDA $1^3B_2$ energies are [*imaginary*]{} (even if designated as negative in Figs. \[fig:all\_TDDFT\_curves\] and \[fig:B2\_curves\]) between about about 100$^\circ$ and about 140$^\circ$ while the TDB3LYP $1^3B_2$ energies are [*imaginary*]{} over a larger range, between about 90$^\circ$ and 160$^\circ$. Certainly, one way to understand how this can happen is through the formulae Eq. (\[eq:results.3\]), but a better way to understand triplet instabilities is in terms of the fact that the quality of response theory energies depends upon having a high-quality ground state. Problems occur in TDDFT excitation energies because the functional $E_{xc}$ is only approximate.
Stability analysis and triplet instabilities have already been discussed in a general way in Sec. \[sec:TDDFT\] where it was seen that no symmetry breaking is expected for a closed-shell singlet ground state when $E_{xc}$ is exact. However, symmetry breaking can occur when $E_{xc}$ is approximate. To deepen our understanding of triplet instabilities, let us look at a two-orbital model for the dissociation of the $\sigma$ bond in H$_2$, which is similar to the dissociation of the CC $\sigma$ bond in oxirane. Here we follow the argument in Ref. [@CGG+00] and consider the wave function, $$\Psi_\lambda = \vert \sqrt{1-\lambda^2} \sigma + \lambda \sigma^* , \sqrt{1-\lambda^2} \bar{\sigma}
- \lambda \bar{\sigma}^* \vert \,,
\label{eq:results.6}$$ which becomes $\vert s_A, \bar{s}_B \vert$ as $\lambda \rightarrow 1/\sqrt{2}$. The corresponding energy expression is, $$E_\lambda = E_0 + 2\lambda^2 \frac{\omega_T^2}{\epsilon_{\sigma^*} - \epsilon_\sigma} + {\cal O}(\lambda^3) \, .
\label{eq:results.7}$$ This result suggests the more general result [@C01] that symmetry breaking will occur if and only if there is an imaginary triplet excitation energy ($\omega_T^2<0$). Certainly, one way to overcome the problem of triplet excitation energies is to improve the quality of the exchange-correlation functional. However, another way is to use the TDA. The reason is clear in HF theory where TDHF/TDA is the same as CIS, whose excited states are a rigorous upper bound to the true excitation energies. Therefore, not only will the excitations remain real but variational collapse will not occur. To our knowledge, there is no way to extend these ideas to justify the use of the TDA in the context of TDDFT calculations except by carrying out explicit calculations to show that the TDA yields improved PESs for TDDFT calculations. This was previously shown for H$_2$ [@CGG+00] and is evidently also true for oxirane judging from Fig. \[fig:B2\_curves\] where the TDLDA/TDA curve is remarkably similar to the DMC curve.
As shown in Fig. \[fig:B2\_curves\], triplet instabilities are often also associated with singlet near instabilities. In this case, the TDDFT $1 ^1B_2$ singlet excitation energies are much too low. The TDA brings the TDLDA curve into the same energetic region of space as the corresponding DMC curve, but still compared with the DMC curve the TDLDA TDA curve appears to be qualitatively incorrect after 120$^\circ$ where it is seen to be decreasing, rather than increasing, in energy as the angle opens. This, in fact, is the behavior observed in the CAS(4,6) singlet calculation but is opposite to what happens in the more accurate DMC calculation.
![$C_{2v}$ ring opening curves: $1^1B_1$ state. The TDLDA and TDLDA/TDA curves are practically superimposed.[]{data-label="fig:1B1_curves"}](fig10.pdf){width="\columnwidth"}
 
The remaining $1^{1,3}B_1$ and $1^{1,3}A_2$ states are primarily Rydberg in nature with the corresponding singlets and triplets being energetically nearly degenerate. The graph for a single one of these states suffices to illustrate the general trend observed in the case of all four states. Fig. \[fig:1B1\_curves\] shows what happens for the $1^1B_1$ states. All the curves have qualitatively the same form. This is particularly true for the TDB3LYP curve which closely resembles the DMC curve but shifted down by a little over one eV in energy. Notice also that the TDLDA and TDLDA/TDA curves are essentially superimposed. This is because, $$\begin{aligned}
\left(\omega_S\right)_{TDA}^2 - \omega_S^2 & = & (ia \vert 2f_H + f_{xc}^{\uparrow,\uparrow}
+ f_{xc}^{\uparrow,\downarrow} \vert ai )^2 \nonumber \\
\left(\omega_T\right)_{TDA}^2 - \omega_T^2 & = & (ia \vert f_{xc}^{\uparrow,\uparrow}
- f_{xc}^{\uparrow,\downarrow} \vert ai )^2 \, .
\label{eq:results.8}\end{aligned}$$ For Rydberg states the overlap between orbitals $i$ and $a$ is very small and so the difference between full and TDA TDLDA calculations can be neglected. From Eq. (\[eq:results.2\]) this also means that the Rydberg excitation energies also reduce to simple orbital energy differences.
Fig. \[fig:DMC\_and\_TDA\_curves\] shows that the TDLDA/TDA yields, at least in this case, a mostly qualitatively reasonable description of photochemically important PESs as compared with high-quality QMC PESs. By this we only mean that corresponding TDLDA/TDA and QMC potential energy curves have typically the same overall form and are found at roughly similar energies. As discussed above, significant differences do remain in the shapes of some potential energy curves ($2 ^1A_1$ and $1 ^1B_2$) and the detailed energy ordering of the curves differs between the TDLDA/TDA and QMC results.
[*Con*]{}- and [*Dis*]{}rotatory Ring Opening
---------------------------------------------
Conrotatory and disrotatory potential energy surfaces were calculated using the simplifying assumption that each set of 4 atoms OCH$_2$ is constrained to lie in a plane. Our coordinate system is defined in Fig. \[fig:coord\]. All other geometrical parameters were relaxed for the thermal reaction preserving $C_s$ and $C_2$ symmetry for the [*con*]{} and the [*dis*]{} path, respectively.
![$(\alpha,\theta)$ coordinate system used in this paper.[]{data-label="fig:coord"}](fig12.pdf)
### Thermal Reaction
Fig. \[fig:TDLDA\_surfaces\] shows the ground state ($S_0$) PES. In accordance with the WH model there appears to be a much lower energy barrier for [*con*]{} ring opening than for [*dis*]{} ring opening. Also shown is the lowest triplet state ($T_1$). It is clear that triplet instabilities occur along the disrotatory pathway as the CC $\sigma$ bond breaks. Remarkably, they do not occur along the conrotatory pathway in the LDA. (They should, of course, be absent when the exchange-correlation functional is exact.) Although triplet instabilities account for about 50% of the LDA surface, this fraction actually increases to 93% for the B3LYP functional where triplet instabilities occur along both the conrotatory and disrotatory reaction pathways. Finally, when the HF method is used triplet instabilities account for 100% of the PES. Perhaps because we forced the OCH$_2$ to lie in a single plane, triplet instabilities appear to be nearly everywhere in these last methods, potentially spelling trouble for mixed TDDFT/classical photodynamics calculations. We would thus like to caution against the use of hybrid functionals for this type of application. At the same time, we reiterate our recommendation to use the TDA because, of course, the TDLDA/TDA excited state PESs (Fig. \[fig:TDA\_surfaces\]) show no triplet instabilities.
![TDLDA $S_0$ (light grey) and $T_1$ (dark grey) PESs. Note that a “negative excitation energy” is just a convenient graphical trick for representing an imaginary excitation energy. Negative excitation energies correspond to triplet instabilities.[]{data-label="fig:TDLDA_surfaces"}](fig13.pdf)
![TDLDA/TDA $S_0$ (light grey) and $T_1$ (dark grey) PESs. In this case negative excitation energies are real, not imaginary, quantities.[]{data-label="fig:TDA_surfaces"}](fig14.pdf)
### Photochemical Reaction
In considering whether the WH model has any validity for describing con- versus disrotatory rotation for photochemical ring opening in oxirane, we are immediately faced with the problem that there are a large number of excited states with similar energies which cross each other (Fig. \[fig:CAS\_and\_DMC\_curves\]). Provisionally, we have decided to assume that state symmetry is conserved and so to look only at one PES, namely the $1 ^1B_1 S_1$ surface (which we shall simply refer to as $S_1$) which begins as the $^1[2b_1(n)\rightarrow 7a_1(3s)]$ excitation for the closed cycle and then evolves as the molecular geometry changes. In so doing, we are following the spirit of simple WH theory which makes maximum use of symmetry. Turning back to this theory (Fig. \[fig:WHcorrdiag\]) and thinking of excitation to a Rydberg state as analogous to electron removal, it suffices to remove an electron from the $n$ orbital of the [*thermal*]{} WH diagram, thereby giving the $\sigma^2 n^1$ configuration, to have an idea of what might be the relative importance of the [*con*]{} and [*dis*]{} mechanisms for the $S_1$ surface. In this case the [*con*]{} mechanism leads to the energetically nearest-neighbor single excitation, $\sigma \rightarrow n$, and the [*dis*]{} mechanism leads to the nearest-neighbor single excitation, $\sigma\rightarrow \sigma^*$. Since both mechanisms correspond to nearest-neighbor excitation, no particular preference is [*a priori*]{} expected for one mechanism over the other. The ground ($S_0$) and excited state singlet ($S_1$) surfaces calculated with the TDLDA/TDA are shown in Fig. \[fig:TDA\_singlet\_surfaces\]. Indeed the simple WH theory appears to be confirmed in that the $S_1$ surface is remarkably flat.
![TDLDA/TDA $S_0$ (light grey) and $S_1$ (white) potential energy surfaces.[]{data-label="fig:TDA_singlet_surfaces"}](fig15.pdf)
Unfortunately this analysis is far too simplistic. In particular, Kasha’s rule [@K50] tells us that the first excited triplet ($T_1$) or singlet ($S_1$) states are the most likely candidates for the initiation of a photochemical reaction, where now $T_1$ and $S_1$ are the [*globally*]{} lowest states and not necessarily the lowest state of a given symmetry. This is based upon the idea that relaxation of higher excited states is rapid. Such relaxation is due to environmental effects or vibronic coupling which need not preserve the symmetry of the electronic state. Hence $S_1$ at one geometry is not necessarily $S_1$ at another geometry. Indeed looking again at the DMC curves in Fig. \[fig:CAS\_and\_DMC\_curves\], it is easy to believe that the molecule will eventually arrive in the $1 ^1B_2(\sigma,\sigma^*)$ state during ring opening and that, because the $\sigma$ orbital is higher in energy than the $\sigma^*$ orbital at this geometry (Fig. \[fig:Walsh\_C2v\]), that the usual WH argument will still predict a preference for the [*dis*]{} mechanism.
Again we are falling into a trap imposed by the use of symmetry. Excitation, for example, from the $2b_1(n)$ orbital (Fig. \[fig:orbitals\]) into the $8a_1(3p_z)$ Rydberg orbital (Fig. \[fig:excite\_orbs\]) might lead to preferential CO, rather than CC, bond breaking to the extent that ring opening augments the valence CO($\sigma*$) character of the target orbital [@T07]. The difficulty of predicting [*a priori*]{} such behavior is part of what motivates us to move towards dynamics as a better tool for photochemical modeling.
Conclusion {#sec:conclude}
==========
In this paper, we have examined the potential energy curves and surfaces for the symmetric ring opening of oxirane to assess possible difficulties which might be encountered when using TDDFT in photodynamics simulations of its photochemistry. Our TDDFT calculations provided useful insight helpful in constructing the active space needed for more accurate CASSCF and still more accurate DMC calculations. It is indeed worth noting that identifying active spaces is probably one of the more common uses of TDDFT in photochemical studies. Here, we summarize our main conclusions obtained by comparing our TDDFT and DMC results.
Oxirane does not seem to be a molecule where charge transfer excitations are important. The artificially low ionization threshold typical in TDDFT for most functionals is still high enough for the B3LYP functional so as not to pose a serious problem. Even for the LDA functional, where the TDDFT ionization threshold is lower, the shapes of the excited-state Rydberg curves seem to be qualitatively correct even if they should not be considered quantitative.
Problems do show up as the CC $\sigma$ bond breaks. By far, the most severe problem encountered is the presence of triplet instabilities and singlet near instabilities, where the excited-state PES takes an unphysical dive in energy towards the ground state. In the case of the triplet, the excitation energy may even become imaginary, indicating that the ground state energy could be further lowered by allowing the Kohn-Sham orbitals to break symmetry. Although this problem is very much diminished compared to that seen in the HF ground state, it is still very much a problem as seen for example in the 2D surfaces of the con and disrotatory ring opening of oxirane.
It is difficult to overstate the gravity of the triplet instability problem for the use of TDDFT in photodynamics simulations. One should not allow symmetry breaking as it is an artifact which should not occur when the exchange-correlation functional is exact. Moreover, there may be more than one way to lower the molecular energy by breaking symmetry, and searching for the lowest energy symmetry broken solution can be time consuming. Finally, the assignment of excited states using broken symmetry orbitals is far from evident. On the whole, it thus seems better to avoid the problem by using the TDA to decouple (at least partially) the quality of the excited-state PES from that of the ground state PES. Our calculations show that this works remarkably well for the $1^3B_2$ curve along the $C_{2v}$ pathway, in comparison with good DMC results. The $1^1B_2$ state, which appeared to collapse in energy without the TDA in the region of bond breaking, is also restored to a more reasonable range of energies. For this reason, we cannot even imagine carrying out photochemical simulations with TDDFT without the use of the TDA. Or, to put it more bluntly, the TDA, which is often regarded as an approximation on conventional TDDFT calculations, gives [*better*]{} results than does conventional TDDFT when it comes to excited-states PESs in situations where bond breaking occurs.
The TDA is unable to solve another problem which occurs as the CC $\sigma$ bond breaks, namely the presence of an unphysical cusp on the ground state curve (or surface). This cusp is there because of the difficulty of approximate density functionals to describe a biradical structure with a single determinantal wave function. The cusp is also translated up onto the excited-state curves because of the way that PESs are constructed in TDDFT. However the problem is very much diminished compared to that seen in HF because of the presence of some correlation in DFT even when the exchange-correlation functional is approximate. The ultimate solution to the cusp problem is most likely some sort of explicit incorporation into TDDFT of two- and higher-electron excitations. Among the methods proposed for doing just this are dressed TDDFT or polarization propagator corrections [@CZMB04; @MZCB04; @C05] and spin-flip TDDFT [@SHK03; @SK03; @WZ04].
Our examination of the con- and disrotatory ring-opening pathways revealed another important point which has nothing to do with TDDFT. This is that the manifold of excited state PESs is too complicated to interpret easily. The best way around this is to move from a PES interpretation to a pathway interpretation of photochemistry, and the best way to find the pathways moving along and between adiabatic PESs seems to be to carry out dynamics. For now, it looks as though the use of TDDFT/TDA in photodynamics simulations may suffice to describe the principal photochemical processes in oxirane. On-going calculations [@T07] do indeed seem to confirm this assertion in so far as trajectories have been found in good agreement with the accepted Gomer-Noyes mechanism [@GN50; @IIT73].
Photochemistry of Oxiranes {#sec:photochem}
==========================
![Typical reactions of alkyl oxiranes.[]{data-label="fig:alkyl_oxirane"}](fig16.pdf)
The generic structure of oxiranes is shown as the starting point in the chemical reactions in Figs. \[fig:alkyl\_oxirane\] and \[fig:aryl\_oxirane\]. When R$_1$, R$_2$, R$_3$, and R$_4$ are hydrogens or alkyl groups, then the prefered reaction is CO cleavage both photochemically and thermally (Step 1, Fig. \[fig:alkyl\_oxirane\]). In particular it is estimated that the CO rupture energy is about 52 kcal/mol while the CC rupture energy is 5-7 kcal/mol higher [@BSD79a]. Since the molecule is not symmetric along the CO ring-opening pathway, the WH model does not apply. Photochemical CO ring opening may be followed by alkyl migration [@AWR+04] (Step 2, Fig. \[fig:alkyl\_oxirane\]). In the particular case of oxirane itself (R$_1$=R$_2$=R$_3$=R$_4$=H), hydrogen migration is followed by breaking of the CC single bond (Step 3, Fig. \[fig:alkyl\_oxirane\]). This is the Gomer-Noyes mechanism [@GN50] which was confirmed experimentally by Ibuki, Inasaki, and Takesaki [@IIT73].
![Typical reactions of aryl oxiranes.[]{data-label="fig:aryl_oxirane"}](fig17.pdf)
In contrast, cyano and aryl substitutions favor CC bond breaking (Step 1, Fig. \[fig:aryl\_oxirane\]) to form what is often refered to as a 1,3-dipolar species or a carbonyl ylide. This is the case where there may be sufficient symmetry that the WH model applies. The photochemistry of phenyl and phenyl substituted oxiranes has been reviewed in Refs. [@H77; @HRS+80; @P87] and on pages 565-566 of Ref. [@T91]. Evidence for carbonyl ylides goes back to at least the 1960s when Ullman and Milks investigated the tautomerization of 2,3-diphenylindenone oxide [@UM62; @UM64] and Linn and Benson investigated the ring-opening reaction of tetracyanoethylene oxide [@LB65; @L65]. Finding cases where the WH model applies and where its predictions can be verified turns out to be not straightforward, particularly in the photochemical case. There are several reasons for this. First of all, too much asymmetry should be avoided in order to assure the applicability of the WH orbital symmetry conservation rule and so avoid passing directly onto carbene formation (Step 2, Fig. \[fig:aryl\_oxirane\]). Secondly, the substituted oxirane should include groups which are bulky enough to confer the structural rigidity needed to avoid premature radiationless relaxation, but not bulky enough to favor carbene formation. The predictions of the WH model have been found to hold for [*cis*]{}- and [*trans*]{}-1,2-diphenyloxirane [@MP84] while carbene formation dominates for tetraphenyl oxirane [@TABG80].
Supplementary Material {#sec:suppl}
======================
Benchmark quality diffusion Monte Carlo (DMC) energies calculated at the geometries given in Table \[tab:B3LYPopt\] are reported here for the ground state ($1 ^1A_1$, Tables \[tab:A1\] and \[tab:SA-A1\]) and the lowest excited state of each symmetry ($2 ^1A_1$, Table \[tab:SA-A1\]; $1 ^3A_1$, Table \[tab:A1\]; $1 ^1B_1$, $1 ^3B_1$, Table \[tab:B1\]; $1 ^1A_2$, $2 ^3A_2$, Table \[tab:A2\]; and $1 ^1B_2$, $1 ^3B_2$, Table \[tab:B2\]). We set the zero of the energy to coincide with the DMC $1 ^1A_1$ energy at 60$^\circ$ and report the statistical error on the energy in parenthesis. Note that the negative value of the $1 ^1A_1$ SA-DMC energy at 60$^\circ$ is statistically compatible with zero. We hope that these DMC data will encourage further developing and testing of improved TDDFT algorithms suitable for addressing the problems mentioned in this article.
[ccc]{}\
\
\
$\angle$ COC ($^\circ$) & $1 ^1A_1$ & $1 ^3A_1$\
60. & 0.0000 (0.0009) & 0.3099 (0.0010)\
75. & 0.0212 (0.0009) & 0.2811 (0.0010)\
90. & 0.0651 (0.0010) & 0.2836 (0.0009)\
105. & 0.0998 (0.0010) & 0.2998 (0.0010)\
120. & 0.1222 (0.0009) & 0.3406 (0.0011)\
135. & 0.1153 (0.0009) & 0.3223 (0.0009)\
150. & 0.1083 (0.0009) & 0.3106 (0.0010)\
165. & 0.1048 (0.0010) & 0.3025 (0.0010)\
179.5 & 0.1017 (0.0010) & 0.3007 (0.0010)\
\
[ccc]{}\
\
\
$\angle$ COC ($^\circ$) & $1 ^1A_1$ & $2 ^1A_1$\
60. & -0.0018 (0.0009) & 0.3246 (0.0009)\
75. & 0.0254 (0.0010) & 0.2904 (0.0009)\
90. & 0.0694 (0.0010) & 0.2865 (0.0010)\
105. & 0.1011 (0.0010) & 0.2962 (0.0009)\
120. & 0.1263 (0.0009) & 0.2699 (0.0010)\
135. & 0.1196 (0.0010) & 0.2902 (0.0010)\
150. & 0.1103 (0.0010) & 0.3109 (0.0011)\
165. & 0.1054 (0.0010) & 0.3063 (0.0010)\
179.5 & 0.1037 (0.0010) & 0.3054 (0.0010)\
\
[ccc]{}\
\
\
$\angle$ COC ($^\circ$) & $1 ^1B_1$ & $1 ^3B_1$\
60. & 0.2788 (0.0010) & 0.2636 (0.0010)\
75. & 0.3083 (0.0009) & 0.3026 (0.0010)\
90. & 0.3138 (0.0010) & 0.3140 (0.0009)\
105. & 0.3329 (0.0010) & 0.3316 (0.0010)\
120. & 0.3058 (0.0010) & 0.2970 (0.0010)\
135. & 0.3023 (0.0010) & 0.2887 (0.0010)\
150. & 0.3113 (0.0010) & 0.2972 (0.0010)\
165. & 0.3274 (0.0010) & 0.3111 (0.0011)\
179.5 & 0.3343 (0.0010) & 0.3155 (0.0010)\
\
[ccc]{}\
\
\
$\angle$ COC ($^\circ$) & $1 ^1A_2$ & $1 ^3A_2$\
60. & 0.2972 (0.0010) & 0.3001 (0.0009)\
75. & 0.3021 (0.0010) & 0.2971 (0.0009)\
90. & 0.2901 (0.0010) & 0.2857 (0.0010)\
105. & 0.2835 (0.0010) & 0.2769 (0.0010)\
120. & 0.2870 (0.0010) & 0.2843 (0.0010)\
135. & 0.3344 (0.0010) & 0.3319 (0.0011)\
150. & 0.3179 (0.0010) & 0.3165 (0.0010)\
165. & 0.3111 (0.0010) & 0.3088 (0.0010)\
179.5 & 0.3086 (0.0010) & 0.3066 (0.0010)\
\
[ccc]{}\
\
\
$\angle$ COC ($^\circ$) & $1 ^1B_2$ & $1 ^3B_2$\
60. & 0.3484 (0.0009) & 0.3301 (0.0010)\
75. & 0.3040 (0.0010) & 0.2395 (0.0010)\
90. & 0.2810 (0.0010) & 0.1657 (0.0010)\
105. & 0.2549 (0.0011) & 0.1229 (0.0009)\
120. & 0.2470 (0.0011) & 0.1165 (0.0010)\
135. & 0.2481 (0.0010) & 0.1232 (0.0009)\
150. & 0.2604 (0.0010) & 0.1410 (0.0010)\
165. & 0.2718 (0.0010) & 0.1611 (0.0009)\
179.5 & 0.2733 (0.0010) & 0.1667 (0.0010)\
\
[ccccc]{}\
\
\
$\angle$ COC ($^\circ$) & $\angle$ HCH ($^\circ$) & $\angle$ HCOC ($^\circ$) & $R(\mbox{CH})$ (Å) & $R(\mbox{CO})$ (Å)\
60. & 115.52 & 111.317 & 1.08375 & 1.44558\
75. & 118.69 & 105.214 & 1.08311 & 1.36333\
90. & 122.04 & 97.339 & 1.08131 & 1.34639\
105. & 122.28 & 84.096 & 1.08311 & 1.35811\
120. & 118.06 & 109.402 & 1.08576 & 1.38032\
135. & 119.88 & 105.829 & 1.08353 & 1.33439\
150. & 121.78 & 101.409 & 1.08154 & 1.30455\
165. & 123.27 & 96.054 & 1.08009 & 1.28694\
179.5 & 123.86 & 90.207 & 1.07954 & 1.28114\
\
This study was carried out in the context of a Franco-Mexican collaboration financed through [*ECOS-Nord*]{} Action M02P03. Financing by [*ECOS-Nord*]{} made possible working visits of AV to the [*Université Joseph Fourier*]{} and of MEC to [*Cinvestav*]{}. FC acknowledges support from the Mexican Ministry of Education via a CONACYT (SFERE 2004) scholarship and from the [*Universidad de las Americas Puebla*]{} (UDLAP).
We like to acknowledge useful discussions with Mathieu Maurin, Enrico Tapavicza, Neepa Maitra, Ivano Tavernelli, Todd Martínez, and Massimo Olivucci.
MEC and FC thank Pierre Vatton, Denis Charapoff, Régis Gras, Sébastien Morin, and Marie-Louise Dheu-Andries for technical support of the [*Département de Chimie Molécularie*]{} and [*Centre d’Expérimentation le Calcul Intensif en Chimie*]{} (CECIC) computers. CF aknowledges the support by the [*Stichting Nationale Computerfaciliteiten*]{} (NCF-NWO) for the use of the SARA supercomputer facilities.
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|
---
abstract: |
For a cardinal of the form $\kappa=\beth_\kappa$, Shelah’s logic $L^1_\kappa$ has a characterisation as the maximal logic above $\bigcup_{\lambda<\kappa} L_{\lambda, \omega}$ satisfying Strong Undefinability of Well Order (SUDWO). SUDWO is a strengthening of the Undefinability of Well Order (UDWO). We prove that if $\kappa$ is singular of countable cofinality, Karp’s chain logic [@Karpintroduceschain] is above $L^1_\kappa$, while it is already known that it satisfies UDWO and Interpolation. Moreover, we show that in these circumstances, the chain logic is—in a sense—maximal among logics with chain models to satisfy UDWO.
We then show that the chain logic gives a partial solution to Problem 1.4. from Shelah’s [@Sh797], which asked whether for $\kappa$ singular of countable cofinality there was a logic strictly between $ L_{\kappa^+, \omega}$ and $L_{\kappa^+, \kappa^+}$ having Interpolation. We show that modulo accepting as the upper bound a model class of $L_{\kappa, \kappa}$, Karp’s chain logic satisfies the required properties. In addition, we show that this chain logic is not $\kappa$-compact, a question that we have asked on various occasions. We contribue to the further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power.
In conclusion, we have shown that the simply defined chain logic emulates the logic $L^1_\kappa$ in satisfying Interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a Completeness Theorem.[^1]
author:
- 'Mirna Džamonja[^2]'
- 'Jouko Väänänen[^3]'
bibliography:
- 'JMbiblio.bib'
title: 'Chain Logic and Shelah’s Infinitary Logic'
---
Introduction {#intro}
============
In 2012 Saharon Shelah [@Sh797] introduced the logic $L^1_\kappa$, defined for $\kappa$ such that $\kappa=\beth_\kappa$. The motivation was to find a strong logic which has many nice properties of first order logic, such as Interpolation, and which moreover, like first order logic, has a model-theoretic characterisation. Lindstr[ö]{}m characterised [@Lindstrom] first order logic as the maximal abstract logic (a notion defined precisely below) satisfying the Downward L[ö]{}wenheim-Skolem theorem to $\aleph_0$ and Compactness. In spite of the abundance of abstract logics discovered in the 1960s and 1970s, rare are those that have a characterisation of this type[^4]. Shelah proved (Theorem 3.4 in [@Sh797]) that $L^1_\kappa$ has such a characterisation, namely it is the maximal logic which is above $\bigcup_{\lambda<\kappa}L_{\lambda, \omega}$ and which satisfies a strong form of the Undefinability of Well Order, called SUDWO.
A feature of Shelah’s logic is that it is defined via its elementary equivalence relation. In fact, no syntax is known for this logic in the sense of the syntax of logics such as $L_{\kappa,\lambda}$. In joint but as yet unpublished work with A. Villaveces the second author has introduced a fragment of $L^1_\kappa$, which does have a simple syntax and which comes within the so called $\Delta$-extension of $L^1_\kappa$. In joint but as yet unpublished work with B. Veli[č]{}kovi[ć]{} the second author has introduced a strengthening of $L^1_\kappa$ which has a syntax in the usual sense and which is equivalent to $L^1_\kappa$ in the case that $\kappa=\beth_\kappa$.
Coming back to the motivation for $L^1_\kappa$, it had became apparent early on that infinitary logics beyond $L_{\omega_1,\omega}$ do not permit a Completeness theorem or an Interpolation theorem in the same sense as first order logic. Already in 1964, Carol Karp introduced the concept of a chain model and showed, in co-operation with her students Ellen Cunningham and Judy Green, see [@Cunnigham] and [@Green], that for $\kappa$ strong limit of cofinality $\omega$, the infinitary logic $L_{\kappa\kappa}$ behaves quite nicely in chain models. In particular, they proved a Completeness theorem and the Interpolation theorem for chain logic. The main point about chain logic which allows it to have many nice properties, is that the notion of a model is changed to the notion of a chain model (see Definition \[chainour\]). The price we pay is that the notion of isomorphism is also changed, being replaced by the notion of chain-isomorphism (see Definition \[chainour\]). However, both the notion of chain model and the chain isomorphism are rather simple and are also well studied in the literature, as is seen by the various references we quote, starting from [@Karpintroduceschain].
The advantage of chain logic is that its syntax is the same as the syntax of the classical $L_{\kappa,\kappa}$. The difference is in the concept of a model. Our purpose in this paper is to establish the mutual relationships between different chain logics and Shelah’s logic. The main difficulty is that chain logic and $L^1_\kappa$ are based on a different concept of a model, chain versus the classical model. However, accepting the notion of a chain model, the logic $L^c_{\kappa, \kappa}$ has syntax, Interpolation, Undefinability of Well Order (UDWO) and an associated $EF$-game. Moreover, in an appropriate sense, at singular cardinals of countable cofinality, it is also characterised as a maximal logic with UDWO. In fact, the only known property of $L^1_\kappa$ for which it is not known if $L^c_{\kappa, \kappa}$ has it, or if it even makes sense to require it, is SUDWO. SUDWO is a strengthening of UDWO by an extra property that Shelah calls an ‘a posteriori’ property, meaning that its definition is motivated by the proof of the characterisation theorem. We now explain some of our findings, which among other things suggest that SUDWO is strictly stronger than UDWO.
For $\kappa$ singular of countable cofinality satisfying $\kappa=\beth_\kappa$ (such as the first fixed point of the $\beth$-hierarchy), we prove that the chain logic $L^c_{\kappa, \kappa}$ of [@Karpintroduceschain] is above $L^1_\kappa$ in the Chu order and that this is in some sense strict. The Chu order extends the one used in the characterisations above (Theorem \[complok\] and Theorem \[chainbigger\]). Our work also gives that chain logic provides a solution to Problem 1.4. from [@Sh797], which was one of the motivations for $L^1_\kappa$:
[**Problem 1.4. from [@Sh797]**]{}: Suppose that $\kappa$ is a singular strong limit cardinal of countable cofinality. Is there a logic between $L_{\kappa^+, \omega}$ and $L_{\kappa^+, \kappa}$ which satisfies Interpolation?
Our Corollary \[solves1.4.\] shows that modulo replacing $L_{\kappa^+, \kappa}= L_{\kappa, \kappa}$ (since $\kappa$ is singular, see Observation \[moreforless\]) by a model class of $L_{\kappa, \kappa}$, the chain logic has the required properties. The logic $L^1_\kappa$, although it has Interpolation, does not solve this problem, because it is not above $L_{\kappa^+, \omega}$ (it is above $\bigcup_{\lambda<\kappa} L_{\lambda, \omega}$).
As one of our main findings, we show that the chain logic has a Lindstr[ö]{}m-style characterisation as the maximal to satisfy UDWO, where UDWO and maximality is interpreted in the appropriate sense, Theorem \[maxudwo\]. We also show (Theorem \[incompactness\]) that chain logic is not $\kappa$-compact, resolving a question about this logic which we have asked on numerous occasions.
The final part of our paper contributes to the further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, which surprisingly still has a lot of expressive power.
Organisation, background and notation
-------------------------------------
The paper is organised as follows. §\[intro\] describes the motivation and the main result, after which it gives the mathematical background and notation. §\[Chudefined\] defines one of the main tools of the paper, which are Chu transforms, to be used in later sections for comparing various abstract logics. §\[sec:chain\] introduces chain logic and its basic properties, and locates it between $L_{\kappa,\omega}$ and a model class of $L_{\kappa,\kappa}$. §\[sec:Shelah\] introduces Shelah’s logic $L^1_\kappa$. The main results on comparing chain logic and $L^1_\kappa$ are found in §\[sec:main\]. The characterisation of chain logic as the maximal chain logic to satisfy UDWO is in §\[maximalitysec\]. Further results about chain logic appear in §\[sec:further\]. §\[fin\] contains some concluding remarks.
Throughout the paper $\tau$ denotes a fixed relational language of countable size which in some contexts may be enriched with up to $\kappa$ constant symbols, for some fixed cardinal $\kappa$ to be specified. The arity of the relation symbols is always assumed to be $<\kappa$. When we wish to discuss different languages at the same time, we may also use the term ‘vocabulary’ to denote each of them. The words ‘model of $\tau$’ refer to non-empty sets equipped with the interpretation of the symbols of $\tau$. The concept of a model, sometimes also called a ‘structure’, refers to models of $\tau$ in which we have also interpreted the satisfaction relation of the logic we are working with.
For simplicity, we work with a singular cardinal $\kappa$ of countable cofinality and a cofinal strictly increasing sequence $\langle \kappa_n:\,n<\omega\rangle$ of cardinals with limit $\kappa$.
We study various logics simultaneously, so it is useful to have an abstract definition of a logic. From [@Joukoablogic]:
\[abslogic\] A [*logic*]{} is a triple of the form ${\mathfrak L}=(L, \models_{\mathfrak L}, S)$ where $\models_{\mathfrak L} \subseteq S\times L$ and $S$ comes with a notion of isomorphism, usually understood from the context. We think of $L$ as the set or class of sentences of ${\mathfrak L}$, $S$ as a set or class of models of ${\mathfrak L}$ and of $\models_{\mathfrak L}$ as the satisfaction relation. The classes $L$ and $S$ can be proper classes. If the rest is clear from the context, we often identify ${\mathfrak L}$ with $L$.
To be able to use the context of [@Sh797], we need to assume some additional properties of the logics in question. Following Shelah’s notation we define:
\[nicelogic\] A logic $(L, \models_{\mathfrak L}, S)$ is [*nice*]{} iff it satisfies the following requirements:
- for any $n$-ary relation symbol $P$ and constant symbols $c_0, \ldots c_{n-1}$ in $\tau$, $P[c_0, \ldots c_{n-1}]$ is a sentence in $L$,
- $L$ is closed under negation, conjunction and disjunction,
- for any $\varphi\in L$ and $M\in S$, $M {\nvDash}_{\mathfrak L} \varphi$ if and only if $M \models_{\mathfrak L} \neg\varphi$,
- $M \models_{\mathfrak L} \varphi_1\wedge \varphi_2$ iff $M \models_{\mathfrak L} \varphi_1$ and $M \models_{\mathfrak L} \varphi_2$, and similarly for disjunction,
- for any $M\in S$, $a\in M$ and a sentence $\psi[a]\in S$ such that $M \models_{\mathfrak L}\psi[a]$, we have that $M\models_{\mathfrak L} (\exists x) \psi(x)$, and conversely, if $M\models_{\mathfrak L} (\exists x) \psi(x)$ then there is $a\in M$ such that $M \models_{\mathfrak L}\psi[a]$,
- if $M_0$ and $M_1$ are isomorphic models of $\tau$, by some isomorphism $f$, and if both $M_0, M_1$ are in $S$, then for every $\varphi\in L$ we have $M_0 \models_{\mathfrak L} \varphi[a_0, \ldots a_{n_1}]$ iff $M_1 \models_{\mathfrak L} \varphi[f(a_0), \ldots f(a_{n_1})]$.
[**Remark.**]{} In this paper we deal with the logics of the form $L_{\lambda, \theta}$, the chain logics $L^c_{\kappa, \kappa}$ and $L^{c,*}_{\kappa, \kappa}$, and Shelah’s logic $L^1_\kappa$. They are all nice logics (a fact that is not trivial for Shelah’s logic but is proved in Claim 2.7(1) of [@Sh797]), except that the closure under isomorphism in $L^{c,*}_{\kappa, \kappa}$ and \*$L^{c}_{\kappa, \kappa}$ has to be replaced by the closure under chain isomorphism (Definition \[chainour\]). We also note that our investigation will include non-standard interpretations of $\models$, see for example equation (\[newtruth\]).
${\mathfrak L}$ is [*$(\theta,\lambda)$-compact*]{} if every set of $\le\theta$ sentences of $L$ such that every subset of size $<\lambda$ has a model (in $S$), itself has a model (in $S$). ${\mathfrak L}$ is [*$\lambda$-compact*]{} if it is $(\theta,\lambda)$-compact for all $\theta$.
Our notation follows that of [@Ebbinghaus].[^5]
We shall often discuss logics of the form $L_{\lambda,\theta}$, in which sentences are formed as in the case of first order logic, but we are allowed to use conjunctions and disjunctions of length $<\lambda$ and strings of $<\theta$ quantifiers. The following is a well-known observation about such logics.
\[moreforless\] If $\lambda$ is a singular cardinal, then every $L_{\lambda^+,\theta}$ sentence is expressible in an equivalent form as an $L_{\lambda,\theta}$ sentence.
The proof is by induction on the complexity of the sentence, where $\bigvee_{i<\lambda} \varphi_i$ is translated into $\bigvee_{j<{{\rm cf}}(\lambda)} \bigvee_{i<\lambda_j} \varphi_i$ for an increasing sequence of cardinals $\langle \lambda_j:\,j<{{\rm cf}}(\lambda)\rangle$ converging to $\lambda$. ${\bigstar}_{\ref{moreforless}}$
Comparison between logics and Chu transforms {#Chudefined}
============================================
We develop a general approach for comparing logics by using the notion of Chu spaces.This notion, studied in category theory (cf. [@BarrChu] and [@ChuonChu]) has proven useful in computer sciences and generalised logic. A similar notion was studied in [@Joukoablogic Definition 2.2]. [^6] The main difference with our definition is in the density condition below. We prove Theorem \[compactnesspreservation\] as a version of Lemma 2.3 of [@Joukoablogic].
\[def:Chu\] A [*Chu space*]{} over a set $K$ is a triple $(A, r, X)$ where $A$ is a set of points, $X$ is a set of states and the function $r:\,A\times X{\rightarrow}K$ is a $K$-valued binary relation between the elements of $A$ and the elements of $X$. When $K=\{0,1\}$ we just speak of Chu spaces and $r$ becomes a 2-valued relation.
A [*Chu transform*]{} between Chu spaces $(A, r, X)$ and $(A', r,' X')$ over the same set $K$ is a pair of functions $(f,g)$ where $f:\, A\to A'$, $g:\,X'\to X$ and the [*adjointness condition*]{} $r'(f(a), x'))= r(a, g(x'))$ holds.
We consider abstract logics $(L,\models, S)$ (see Definition \[abslogic\]) as Chu spaces and we use the following notion of Chu transforms to quasi-order abstract logics.
\[def:lesseq\] We say that $(L,\models, S)\le (L',\models', S') $ if there is a Chu transform $(f,g)$ between $(L,\models, S)$ and $(L',\models', S')$ where $f$ preserves the logical operations in the following sense
[([****]{})]{}\[star\] for any boolean combination $\varphi$ of sentences $\psi_0, \ldots, \psi_n$ of $L$, $f(\varphi)$ of $L$ is $L'$-equivalent to the corresponding boolean combination of $f(\psi_0), \ldots, f(\psi_n)$,
and such that the range of $g$ is [*dense*]{} in the following sense:
- for every $\phi \in L$ for which there is $s\in S$ with $s\models \phi$, there is $s\in {\mbox{\rm ran}}(g)$ with $s\models \phi$.
For example, any $g$ which is onto will satisfy the density condition.
\[partailorder\] The relation $\le$ introduced in Definition \[def:lesseq\] is reflexive and transitive.
\[compactnesspreservation\] Suppose that $(L,\models, S)\le (L',\models', S')$ and $(L',\models', S')$ is compact. Then so is $(L,\models, S)$.
Let $(f,g)$ be the Chu transform which witnesses $(L,\models, S)\le (L',\models', S')$. Suppose that $\Sigma\subseteq L$ is finitely satisfiable and let $\Sigma'=\{ f(\varphi): \varphi\in \Sigma\}$. We now claim that $\Sigma'$ is finitely satisfiable. Any finite $\Gamma'\subseteq \Sigma'$ is of the form $\{ f(\varphi): \varphi\in \Gamma\}$ for some finite $\Gamma\subseteq \Sigma$. Therefore there is $M\in S$ with $M\models \varphi$ for all $\varphi\in \Gamma$. Since $g$ is not necessarily onto, we cannot use it to obtain from $M$ an element of $S'$.
However, $\bigwedge \Gamma$ is a sentence of $L$, by closure under conjunctions. Since $\models$ satisfies Tarski’s definition of truth for quantifier-free formulas, the fact that $M\models \varphi$ for all $\varphi\in \Gamma$ implies $M\models \bigwedge \Gamma$. By the density requirement on $g$, there is $M'\in S'$ such that $g(M') \models \bigwedge \Gamma$ and hence $M'\models' f(\bigwedge \Gamma)$. By the preservation of the logical operations by $f$, we have that $f(\bigwedge \Gamma)$ is $L'$-equivalent to $\bigwedge_{\varphi\in \Gamma} f(\varphi)$, so that $M'\models' f(\varphi)$ for all $\varphi\in \Gamma$ and $M'\models'\Gamma'$. Thus $\Gamma'$ is finitely satisfiable in $S'$, which by the assumption implies that there is $N'\in S'$ with $N'\models \Sigma'$. Therefore $g(N')\models \Sigma$. ${\bigstar}_{\ref{compactnesspreservation}}$
With slight modifications the proof of Theorem \[compactnesspreservation\] goes through for the higher degrees of compactness:
\[kappacompactnesspreservation\] Suppose that $(L,\models, S)\le (L',\models', S')$ as witnessed by a pair $(f,g)$. Suppose that the following conditions are satisfied:
1. $L, L'$ are closed under conjunctions of $<\lambda$ sentences,
2. $\models$ satisfies Tarski’s definition of truth for quantifier-free formulas, including conjunctions and disjunctions of size $<\lambda$,
3. $f$ preserves conjunctions and disjunctions of size $<\lambda$, in the sense of \[star\]([****]{}).
Then, for any $\theta$, if $(L',\models', S')$ is $(\lambda, \theta)$-compact, so is $(L,\models, S)$.
We consider two logics $\mathfrak L$ and $\mathfrak L'$ equivalent if $\mathfrak L\le \mathfrak L'$ and $\mathfrak L'\le \mathfrak L$. For the purpose of studying various properties of logics, which is our main concern, once we prove that the relevant property is preserved by a Chu transform, two equivalent logics are the same with respect to that property. One such property is compactness, as we proved in Theorem \[compactnesspreservation\].
Chain logic {#sec:chain}
===========
In [@singtrees] the authors explored Karp’s notion of a [*chain model*]{} and the associated model theory of $L_{\kappa, \kappa}$ based on that concept. The definition used in [@singtrees] was as follows.
\[chainour\] A [*chain model (of length $\mu$)*]{} consists of a model $\mathfrak A$ and a sequence $\langle A_\alpha:\,\alpha<\mu\rangle$, called a [*chain*]{}. It is assumed that $A=\bigcup_{\alpha<\mu} A_\alpha$, and that $\langle A_\alpha:\,\alpha<\mu\rangle$ is an increasing sequence of sets[^7] satisfying $|A_\alpha|<|A|$ for each $\alpha$. A [*chain isomorphism*]{} between $\mathfrak A$ and $\mathfrak B$ is an isomorphism $f:\,\mathfrak A{\rightarrow}\mathfrak B$ such that for all $\alpha$ the image of $A_\alpha$ is contained in some $B_\beta$ and, conversely, the preimage of any $B_\alpha$ is contained in some $A_\beta$. If there is a chain isomorphism between $\mathfrak A$ and $\mathfrak B$ we write $\mathfrak A\cong^{c}\mathfrak B$.
Notice that the decomposition of a model $\mathfrak A$ into a chain is not required to be elementary. The instance that we study specifically is $\mu=\omega$, when we shall often write $\mathfrak A=(A_n)_{n<\omega}$. The defining feature of chain models is the following modification of the truth definition of $L_{\lambda, \kappa}$, given by the induction on complexity of $\varphi$. For quantifier-free sentences, the notions of $\models$ and $\models^c$ agree, and the quantifier case is covered by the following:
$$\label{newtruth}
(A_n)_{n<\omega}\models^c\exists \bar{x}\varphi( \bar{x})\iff\mbox {there are }n<\omega \mbox{ and } \bar{a}\in A_n \mbox{ with }
(A_n)_{n<\omega}\models \varphi[ \bar{a}],$$
where $\bar{x}$ is a sequence of length $<\kappa$. If we restrict our attention to chain models, the model theory of $L_{\kappa, \kappa}$ (equivalently $L_{\kappa^+, \kappa}$ as $\kappa$ is singular) is very much like that of $L_{\omega_1,\omega}$. Karp proved several theorems about chain logic, and further results by various authors are the undefinability of well order UDWO (Makkai [@Makkaichainlogic]), Craig Interpolation Theorem, Beth Definability Theorem ([@Cunnigham]), and an extension of Scott’s analysis of countable models to chain models of size $\kappa$ (which is the contribution of [@singtrees]).[^8] Various kinds of chain models and their interaction are presented in §\[kindsof\] below.
One can use the concept of a chain model to study other logics besides $L_{\kappa, \kappa}$, adapting the truth definition from (\[newtruth\]) accordingly, and in fact this was Karp’s approach. See [@Cunnigham] for a historical overview.
Various kinds of chain models {#kindsof}
-----------------------------
As mentioned in the introduction, Definition \[chainour\] permits several modifications. We specify them in the following definition.
\[chainournew\] (1) A [*weak chain model*]{} consists of a model $A$ of ${{\cal L}}$ and a sequence $\langle A_n:\,n<\omega\rangle$ of subsets of $A$, called a [*chain*]{}, where the only assumption that is made is that for all $n<\omega$, $A_n\subseteq A_{n+1}$ and $A=\bigcup_{n<\omega}A_n$.
We denote such a weak chain model as $(A_n)_{n<\omega}$. If the meaning is clear from the context, we may just use [*“chain model"*]{} for “weak chain model".
[(2)]{} (equivalent to Definition \[chainour\]) A [*chain model of strict power at most $\kappa$*]{} is a weak chain model $(A_n)_{n<\omega}$ where for each $n$, $|A_n|<|A|$.
[(3)]{} A [*proper chain model*]{} is a weak chain model $(A_n)_{n<\omega}$ where for each $n$, $|A_{n}|<|A_{n+1}|$.
\[variouschainmodels\] (1) Any ordinary model $\mathfrak A$ is a weak chain model, where all elements are the same set, the universe of $\mathfrak A$. In particular, weak chain models exist in all cardinalities.
[(2)]{} The cardinality requirements on proper chain models mean that such models can only exist in cardinals of countable cofinality.
[(3)]{} Every chain model $(A_n)_{n<\omega}$ of strict power at most $\kappa$ gives rise to a chain-isomorphic proper chain model of the form $(A_{k_n})_{n<\omega}$, obtained by a passing to a subsequence of $\langle A_n:\,n<\omega\rangle$.
We give a simple example showing that the truth in chain models depends on the choice of the chain, namely we produce two chain models $(A_n)_{n<\omega}$ and $(B_n)_{n<\omega}$ with $\bigcup_{n<\omega}A_n=\bigcup_{n<\omega}B_n$, and a sentence $\varphi$ true in $(A_n)_{n<\omega}$ but false in $(B_n)_{n<\omega}$.
\[dependenceonchain\] Let ${{\cal L}}=\{<\}$, where $<$ is a binary relation symbol. Let $\mathfrak A$ be the model of ${{\cal L}}$ represented by the lexicographic sum of $\langle \kappa_n:\, n<\omega^\ast\rangle$, so each $\kappa_n$ is ordered as the corresponding ordinal and for $m<n<\omega$ we have that the block $\kappa_m$ is above the block $\kappa_n$. We can see this sum as consisting of pairs $(\alpha, n)$ where $n<\omega$ and $\alpha<\kappa_n$, ordered by $(\alpha, n)<(\beta, m)$ if $m<n$ or if $m=n$ and $\alpha<\beta$.
Let us choose a sequence $\langle \alpha_n:\, n<\omega\rangle$ so that $\alpha_n\in \kappa_n$, hence the sequence $\langle (\alpha_n,n):\,n<\omega\rangle$ is $<$-decreasing in $\mathfrak A$. Let $\varphi$ be the $L_{\kappa,\kappa}$ sentence $$(\exists x_0) (\exists x_1) \ldots (\exists x_n) \ldots \bigwedge_{n<\omega} (x_{n+1} < x_n).$$ Let $A_0=\{(\alpha_n,n):\,n <\omega\}\cup \kappa_0 \times \{0\}$ and for $n>0$ let $A_n= \kappa_n \times \{n\}\cup \bigcup_{k<n} A_k$. For $n<\omega$ let $B_n=\bigcup_{k\le n} \kappa_k \times \{k\}$.
Then $(A_n)_{n<\omega}\models^c \varphi$ while $(B_n)_{n<\omega}\models^c \neg \varphi$.
\(1) Let ${\mathfrak M}^{c}$ stand for the class of proper chain models and let ${\mathfrak M}^{c, \ast}$ stand for the class of all weak chain models.
[(2)]{} The class of ordinary models is denoted by $\mathcal M$ and the class of ordinary models of size $\lambda$ by $\mathcal M_\lambda$ .
[(3)]{} Let $L^c_{\lambda, \kappa}=(L_{\lambda,\kappa}, \models^c, {\mathfrak M}^{c})$ and let $L^{c,\ast}_{\lambda, \kappa}=(L_{\lambda,\kappa}, \models^c, {\mathfrak M}^{c,\ast})$.
[(4)]{} We use the usual notation $L_{\lambda,\theta}$ to denote the logic $(L_{\lambda,\theta}, \models,
\mathcal M)$.
\[wearenice\] Modulo replacing the closure under isomorphisms in the last item of Definition \[nicelogic\] by the closure under chain isomorphisms, the logics $L^c_{\kappa, \kappa}$ and $L^{c,\ast}_{\kappa, \kappa}$ are nice logics.
Comparison with $L_{\kappa, \omega}$ and $L_{\kappa, \kappa}$
-------------------------------------------------------------
We start with the following easy observation.
\[comparison\]
[(1)]{} Every formula of the logic $L_{\lambda, \omega}$ is a formula of $L^c_{\lambda, \kappa}$ and equally of $L^{c,\ast}_{\lambda, \kappa}$ and on such formulas the notions of $\models$ and $\models^c$ agree. Formulas of $L_{\lambda, \kappa}$ are also formulas of $L^c_{\lambda, \kappa}$, but then the notions $\models$ and $\models^c$ do not necessarily agree.
[(2)]{} Suppose that $\varphi$ is a sentence in $L_{\lambda, \omega}$. Let $\mathfrak A$ be a (weak, proper) chain model with a decomposition $(A_n)_{n<\omega}$. Then $\mathfrak A$ is a model of $\varphi$ iff $(A_n)_{n<\omega}$ is a (weak, proper) chain model of $\varphi$.
\(1) By definition.
[(2)]{} By induction on the complexity of $\varphi$. If $\varphi$ is quantifier-free then the definition of $\models^c\varphi$ is the same as that of $\models \varphi$. Suppose that $\phi\equiv(\exists x)\psi(x)$. If $\mathfrak A\models \varphi$ then there is a witness $x$ in some $A_n$ and hence $(A_n)_{n<\omega} \models^c\varphi$. The other direction is similar. ${\bigstar}_{\ref{comparison}}$
The following Theorem \[lowerbound\] is the reason that (weak) chain logic is not $\kappa$-compact, as we shall see in Corollary \[notcompact\].
\[lowerbound\] (1) $L_{\kappa, \omega} \le L^{c,\ast}_{\kappa,\kappa}$.
[(2)]{} If $\kappa$ is a strong limit cardinal then $(L_{\kappa, \omega},\models, {\mathcal M}_{\ge \kappa}) \le L^{c}_{\kappa,\kappa}$.
[(3)]{} $(L^{c,\ast}_{\kappa,\kappa}, \models^c, {\mathcal M}^c_{ \kappa}) \le L^{c}_{\kappa,\kappa}$, where ${\mathcal M}^c_{\ge\kappa}$ stands for weak chain models of size $\ge\kappa$ .
\(1) Let $f$ be the identity function and let $g((M_n)_{n<\omega})=\bigcup_{n<\omega} M_n$. Notice that $g$ is onto. By Observation \[comparison\] we have that $(f,g)$ is a Chu transform witnessing the announced inequality.
[(2)]{} The pair $(f,g)$ as above will still be a Chu transform, but it is not immediate that $g$ satisfies the density condition as it now acts only on proper chain models, so models of size $\kappa$ or some other singular cardinal of countable cofinality. The conclusion uses a downward Lowenheim-Skolem theorem for $L_{\kappa, \omega}$ that holds in the following form and which uses the assumption that $\kappa$ is a strong limit (see Theorem 3.4.1 in [@Dickmann]):
\[DLS\] Assume that $\kappa$ is a strong limit cardinal. Then any sentence $\varphi$ of $L_{\kappa, \omega}$ that has a model of size $\ge\kappa$ also has a model of size $\kappa$.
The lemma immediately implies the density of $g$, since any model of $L_{\kappa, \omega}$ of size $\kappa$ can be represented as a proper chain model and the two will satisfy the same sentences by Observation \[comparison\](2).
[(3)]{} This time we take both $f$ and $g$ to be the identity functions. The adjointness property is easily verified, but the density condition needs an argument. It will be provided by Corollary 2.5 of [@Cunnigham], which is a Downward Lowenheim-Skolem theorem for the chain logic. It states that every $L_{\kappa, \kappa}$ -sentence which has a chain model, must have a strict chain model of power at most $\kappa$.
So suppose that an $L_{\kappa, \kappa}$-sentence $\varphi$ has a chain model of $(A_n)_{n<\omega}$ of power at least $\kappa$, but this model is not necessary a strict chain model of size $\kappa$. Notice that $|A_n|$ must converge at least to $\kappa$, therefore for any sequence of cardinals $\langle \kappa_n:\, n<\omega\rangle$ with supremum $\kappa$, for every $n$, there will be $A_m$ with $|A_m|\ge \kappa_n$. Hence the sentence $$\theta_n\equiv (\exists_{i<\kappa_n} x_i) \forall_{i,j<\omega} [x_i\neq x_j]$$ is true in the chain model $(A_n)_{n<\omega}$ and so is the conjunction $\theta\equiv \bigwedge_{n<\omega} \theta_n$. This shows that the sentence $\varphi\wedge \theta$ has a chain model and hence by the Downward Lowenheim-Skolem theorem for the chain logic, we have that $\varphi\wedge \theta$ must have a strict chain model $(B_n)_{n<\omega}$ of power at most $\kappa$. The definition of of $\theta_m$ implies that for every $n$, the model $(B_n)_{n<\omega}$ must have size at least $\kappa_m$. Putting this together, we get that $(B_n)_{n<\omega}$ has power exactly $\kappa$, and therefore $(B_n)_{n<\omega}=g((B_n)_{n<\omega})$ is in the range of $g$ and a model of $\varphi$. ${\bigstar}_{\ref{lowerbound}}$
Shelah’s $L^1_\kappa$ {#sec:Shelah}
=====================
In this section we recall the definition and basic properties of Shelah’s logic $L^1_\kappa$ as defined in [@Sh797]. Throughout this section we assume that $\kappa=\beth_\kappa$, not necessarily singular.
Before defining $L^1_\kappa$ we define its elementary equivalence by means of a game. The game is called the “borrowing game", $BG$, because the partitions into $\omega$ many blocks given by the functions $h_n$ let us “borrow" the time in which we have to “repay" the promise of constructing a partial isomorphism.
\[DG\] Let $\beta$ be an ordinal and $\theta$ a cardinal. When we define $L^1_\kappa$ we assume $\theta<\kappa$ and $\beta<{\theta}^+$. Let ${\mathfrak A}$ and ${\mathfrak B}$ be models of some fixed vocabulary $\tau$. [^9]
The game $${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$$ is defined as follows:
1. Player I picks $\beta_0<\beta$, $\theta_0\le \theta$ and $A_0\in[A]^{\theta_0}$.
2. Player II picks $h_0:\,A_0\to\omega$ and a partial isomorphism $g_0:h_0^{-1}(0)\to B$.
3. Player I picks $\beta_1<\beta_0$ and $B_1\in[B]^{\theta_0}$.
4. Player II picks $h_1:\, B_1\to\omega$ and a partial isomorphism $g_1\supseteq g_0$ such that $h_1^{-1}(0)\subseteq {\mbox{\rm ran}}(g_1)$ and $h_0^{-1}(1)\subseteq {\mbox{\rm dom}}(g_1)$.
5. Player I picks $\beta_2<\beta_1$ and $A_2\in[A]^{\theta_0}$.
6. Player II picks $h_2:\,A_2\to\omega$ and a partial isomorphism $g_2\supseteq g_1$ such that $h_2^{-1}(0)\subseteq {\mbox{\rm dom}}(g_2)$, $h_1^{-1}(1)\subseteq {\mbox{\rm ran}}(g_2)$ and $h_0^{-1}(2)\subseteq {\mbox{\rm dom}}(g_2)$.
7. ...
8. Eventually $\beta_n=0$ and the game ends. Player II wins if she can play to the end. Otherwise Player I wins.
Let ${\mathfrak A}\sim_{\beta,\theta} {\mathfrak B}$ iff Player II has a winning strategy in ${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$.
We define the relation (of ${\mathfrak A}$ and ${\mathfrak B}$) $${\mathfrak A}\equiv^\beta_\theta{\mathfrak B}$$ to be the transitive closure of $\sim_{\beta,\theta}$.
\[refinement\] (1) For a fixed $\theta$ and $\beta\le\beta'<\theta^+$, the relation $\sim_{\beta',\theta}$ refines the relation $\sim_{\beta,\theta}$, that is ${\mathfrak A}\sim_{\beta',\theta} {\mathfrak B}$ implies that ${\mathfrak A}\sim_{\beta,\theta} {\mathfrak B}$, for any models ${\mathfrak A}$ and ${\mathfrak B}$ of vocabulary $\tau$. The relation $\equiv^{\beta'}_\theta$ refines $\equiv^\beta_\theta$, that is ${\mathfrak A}\equiv^{\beta'}_\theta {\mathfrak B}$ implies that ${\mathfrak A}\equiv^\beta_\theta {\mathfrak B}$ for any models ${\mathfrak A}$ and ${\mathfrak B}$ of vocabulary $\tau$.
[(2)]{} If $\theta\le\theta'$ then for every $\beta<\theta^+$ the relation $\sim_{\beta,\theta'}$ refines the relation $\sim_{\beta,\theta}$, that is ${\mathfrak A}\sim_{\beta,\theta'} {\mathfrak B}$ implies that ${\mathfrak A}\sim_{\beta,\theta} {\mathfrak B}$ for any models ${\mathfrak A}$ and ${\mathfrak B}$ of vocabulary $\tau$. The relation $\equiv^\beta_{\theta'}$ refines $\equiv^\beta_\theta$, that is ${\mathfrak A}\equiv^\beta_{\theta' }{\mathfrak B}$ implies that ${\mathfrak A}\equiv^\beta_\theta {\mathfrak B}$ for any models ${\mathfrak A}$ and ${\mathfrak B}$ of vocabulary $\tau$.
\(1) Any play of ${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$ is a play of ${\rm BG}^{\beta'}_\theta({\mathfrak A},{\mathfrak B})$, so the first conclusion follows. The second conclusion follows from the first. The statement (2) is proved similarly. ${\bigstar}_{\ref{refinement}}$
It is not known whether $\sim_{\beta,\theta}$ is an equivalence relation, although we suspect that it is not.
\[loksatisfaction\] If $\beta<\theta^+$, then any union of equivalence classes of $\equiv^\beta_\theta$ is called a [*sentence*]{} of $L^1_{\theta}$. Let $$L^1_\kappa=\bigcup_{\theta<\kappa}L^1_{\theta}.$$ We define the satisfaction relation in $L^1_\kappa$ by letting ${\mathfrak A}{\vDash}_{L^1_\kappa} K$ if ${\mathfrak A}\in K$.
Note that $L^1_\kappa$ does not (a priori) have a syntax, we just know its semantics. The definition of sentences is such that every sentence $K$ is a proper class of models. The question arises how to define the basic logical operations, especially negation. The following theorem is helpful.
\[smallnumberofclasses\] The equivalence relation $\equiv^\beta_\theta$ has $\le \beth_{\beta+1}(\theta)$ equivalence classes.
Theorem \[smallnumberofclasses\] in particular shows that the family of the $\equiv^\beta_\theta$ equivalence classes is a set, and we have defined sentences as unions of subfamilies of elements of such sets. This allows to define the logical operations $\neg$ and $\vee$ (and hence all the other logical operations), as follows:
\[logoperation\] Suppose that $K$ is a sentence of $L^1_\kappa$, so there are the least $\theta<\kappa$ and the least $\beta<\theta^+$ such that $K$ is a union of equivalence classes of $\equiv^\beta_\theta$. Let ${{\cal E}}^\beta_\theta$ denote the set of all equivalence classes of $\equiv^\beta_\theta$. We define $$\neg K=\bigcup \{e\in {{\cal E}}^\beta_\theta:\, e\notin K\}.$$ For $1\le n<\omega$ and sentences $K_i \,(i<n)$, we find the least $\theta<\kappa$ and least $\beta<\theta^+$ such that each $K_i$ is a union of equivalence classes of $\equiv^\beta_\theta$ (such $\theta$, $\beta$ exist by Observation \[loksatisfaction\]). We define $$\bigvee_{i<n} K=\bigcup K_i=\{e\in {{\cal E}}^\beta_\theta:\, (\exists i<n)\,e\in K_i\}.$$
If $\kappa=\beth_\kappa$ is singular, $L^1_\kappa$ has Craig Interpolation and a Lindstrom-type characterisation involving a strong undefinability of well-order called SUDWO. Moreover, $L^1_\kappa$ is the classically maximal logic stronger than $\bigcup_{\lambda<\kappa} L_{\lambda,\omega}$ which satisfies SUDWO. These facts are all proved in [@Sh797].
Comparison between $L^1_\kappa$ and the chain logic {#sec:main}
===================================================
In addition to the running assumption that $\kappa$ is singular of cofinality $\omega$, we assume that $\kappa=\beth_\kappa$. Based on the EF games for chain logic (see [@singtrees]), we can define:
\[chainbeta\] Suppose that ${\mathfrak A}=(C_n)_{n<\omega}$ and ${\mathfrak B}=(D_n)_{n<\omega}$ are chain models. For a sequence $\xb$ of length $<\kappa$ we say that it is *bounded* if it is contained in some $C_n$ or $D_n$. For simplicity, assume $(\bigcup_nC_n)\cap(\bigcup_n D_n)=\emptyset$.
Let $\beta<\kappa^+$. Define the game $EF^c_{\kappa,\beta}$ in which Player I plays a sequence of bounded sequences $\xb_i$, each of length $<\kappa$, and ordinals $\beta_0>\beta_1>...$, where $\beta_0<\beta$ (so the game is finite) and II responds by an increasing sequence $g_i$ of partial chain isomorphisms of cardinality $<\kappa$, including $\xb_i$ in its domain if $\xb_i$ is in ${\mathfrak A}$ or in its range if $\xb_i$ is in ${\mathfrak B}$. The first player that cannot move loses. We write ${\mathfrak A}\sim^{c}_{\kappa, \beta}{\mathfrak B}$ if Player II has a winning strategy in $EF^c_{\kappa,\beta}$.
The global EF game $EF^c_{\kappa,\infty}$ is defined so that Player I plays a sequence of bounded sequences $\xb_i$, each of length $<\kappa$ and II responds by an increasing sequence $g_i$ of partial chain isomorphisms including $\xb_i$ in its domain if $\xb_i$ is in ${\mathfrak A}$ or in its range if $\xb_i$ is in ${\mathfrak B}$. The first player that cannot move loses and II wins if the game lasts $\omega$ steps.
It is worth noting that the relation $\sim^c_{\kappa, \beta}$ is an equivalence relation.
\[chainEF\] Two chain models ${\mathfrak A}=(C_n)_{n<\omega}$ and ${\mathfrak B}=(D_n)_{n<\omega}$ are in the relation ${\mathfrak A}\sim^c_{\kappa, \beta}{\mathfrak B}$ iff they satisfy the same $L^c_{\infty, \kappa}$-sentences of quantifier rank $<\beta$.
This is a standard proof, going back to [@MR0209132].
\[TFAEequivalence\] Let $\kappa$ be a singular strong limit cardinal of cofinality $\omega$ and $\mathcal K$ a class of chain models of a fixed vocabulary $\tau$ Then the following are equivalent:
1. $\mathcal K$ is definable by a formula of $L^{c}_{\infty,\kappa}$ with the quantifier rank $< \beta$.
2. The model class $\mathcal K$ is closed under $\sim^c_{\kappa, \beta}$.
3. $\mathcal K$ is definable by a formula of $L^{c}_{\lambda,\kappa}$ with the quantifier rank $< \beta$, where $\lambda=\max(\kappa,(\beth_{\beta+2}(|\tau|)^+)$.
We recall that the formulas of $L^c_{\lambda, \kappa}$ are exactly the formulas of $L_{\lambda, \kappa}$.
The direction $(1)\implies (2)$ follows from Lemma \[chainEF\].
To prove that $(2)\implies (3)$, similarly to Lemma 7.55 in [@joukobookgames], we show by induction on $\beta$ that the number of formulas with quantifier rank $<\beta$ over a set $X$ of free variables has size $\le \beth_{\beta+1}(|X|+ |\tau|)$. Each $\sim^c_{\kappa, \beta}$-equivalence class is determined by the set of sentences with the quantifier rank $<\beta$ which are true in one or any model in the class. Therefore, there are $\le \beth_{\beta+2}(|\tau|)$ such classes $e$. Choose for each $e$ such that $\mathcal K\cap e\neq \emptyset$ a representative ${\mathfrak A}_e\in \mathcal K$. Hence $\mathcal K$ is definable by the formula $$\bigvee_{e\cap {{\cal K}}\neq \empty}
\bigwedge \{\varphi:\, \mbox{q.r.}(\varphi)\le\beta\,\,\&\,\, {\mathfrak A}_e\models^c\varphi\}.$$
The direction $(3)\implies (1)$ follows from the definition. ${\bigstar}_{\ref{TFAEequivalence}}$
\[Kstar\] For a class $K$ of models of $\tau$, let ${{\cal C}}(K)$ stand for the class of all possible weak chain representations of models in $K$.
\[betaequiv\] If $\kappa$ is as above, $\beta<\kappa$ and $(C_n)_{n<\omega}$ and $(D_n)_{n<\omega}$ are $L^{c,2\cdot\beta}_{\kappa,\kappa}$-equivalent, then $\bigcup_{n<\omega} C_n \equiv^\beta_\theta
\bigcup_{n<\omega} D_n$ (in the sense of Definition \[DG\]) for any $\theta<\kappa$ for which $\beta<\theta^+$.
Consequently, suppose that $K$ is a class of models of $L^1_\kappa$ closed under $\equiv^\beta_\theta$. Then the class ${{\cal C}}(K)$ is closed under $\sim^c_{\kappa,2\cdot\beta}$ and hence definable by an $L^c_{\kappa, \kappa}$ sentence of quantifier rank $<2\cdot \beta$.
In fact we prove that ${\mathfrak A}\sim_{\beta,\theta}{\mathfrak B}$, by providing a winning strategy to II in the game ${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$. By the assumption ${\mathfrak A}\sim^c_{\kappa,2\cdot\beta}{\mathfrak B}$, we fix a winning strategy $\sigma$ for II in $EF^c_{\kappa,2\cdot\beta}$. The $n$-th move of II in the game ${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$ is supposed to produce a pair of functions $(h_n, g_n)$. To define $h_n$ for $n$ even we let for $a\in A_{n}$: $$h_{n}(a)= \min\{k : a\in C_k\}$$ and for $n$ odd, for $b\in B_{n}$ we let $$h_{n}(b)= \min\{k : b\in D_k\}.$$ That is, player II simply uses the existing chain partitions of ${\mathfrak A}$ and ${\mathfrak B}$.
To find $g_n$ Player II uses $\sigma$ twice. We first assume that $n=2k$ is even. Assume we have defined $g_{n-1}$ (stipulating $g_{-1}=\emptyset$) so that the requirements for II in the game ${\rm BG}^\beta_\theta({\mathfrak A},{\mathfrak B})$ are satisfied. Then II extends $g_{n-1}$ to a partial isomorphism whose domain includes $h_0^{-1}(2k)\cup\ldots \cup h_{2k}^{-1}(0)$ by using $\sigma$ in the play of $EF^c_{\kappa,\beta}$ where $\sigma$ has given $g_{n-1}$ in the $2(n-1)$-th step of II and I has played $h_0^{-1}(2k)\cup\ldots \cup h_{2k}^{-1}(0)\setminus {\mbox{\rm dom}}(g^{n-1})$ in the step $2(n-1)+1$. In this way we extend $g_{n-1}$ to $g'_n$. Then we continue that play of $EF^c_{\kappa,\beta}$ by letting I respond by $g_0^{-1}(2k)\cup\ldots \cup g_{2k}^{-1}(0)\setminus {\mbox{\rm dom}}(g'_n)$ and letting II use $\sigma$ to extend $g'_n$ to $g_n$. The strategy for $n$ odd is similar.
Now suppose that $K$ is closed under $\equiv^\beta_\theta$ and that ${\mathfrak A}=(C_n)_{n<\omega}$ is in ${{\cal C}}(K)$, while ${\mathfrak B}=(D_n)_{n<\omega}$ is $\equiv^{c,2\cdot\beta}_{\kappa, \kappa}$-equivalent to ${\mathfrak A}$. Then, on the one hand, $\bigcup{\mathfrak A}\in K$, by the definition of ${{\cal C}}(K)$ and on the other hand, $\bigcup{\mathfrak A}\equiv^\beta_\theta \bigcup{\mathfrak B}$ by the first part of the lemma. Hence ${\mathfrak B}\in K$, as $K$ is closed under $\equiv^\beta_\theta$-equivalence and therefore $(D_n)_{n<\omega}\in {{\cal C}}(K)$ by the definition of ${{\cal C}}(K)$. ${\bigstar}_{\ref{betaequiv}}$
\[complok\] For $\kappa$ as above and $\tau$ a fixed vocabulary, we have: $$(L^1_\kappa, \models_{L^1_\kappa}, {{\cal K}}) \le (L^{c,\ast}_{\kappa, \kappa}, \models^c, {{\cal K}}^c),$$ where ${{\cal K}}$ is the class of all models of $\tau$ and ${{\cal K}}^c$ the class of all weak chain models of $\tau$.
We define a Chu transform $(f,g)$. To define $f$, suppose that $K$ is a sentence of $L^1_\kappa$. Thus there are $\beta<\theta^+<\kappa$ such that $K$ is the union of certain equivalence classes of ${\mathfrak A}\equiv^\beta_\theta{\mathfrak B}$ and in particular $K$ is closed under the relation $\sim_{\beta, \theta}$. To associate a sentence of $L^{c}_{\kappa,\kappa}$ to $K$, we prove that ${{\cal C}}(K)$ is definable in $L^{c}_{\kappa,\kappa}$. By Lemma \[TFAEequivalence\], since $\kappa=\beth_\kappa$, it is suffices to show that ${{\cal C}}(K)$ is closed under $\sim^c_{\kappa,\beta}$. This follows from Lemma \[betaequiv\].
To define $g$, we let $g((C_n)_{n<\omega})=\bigcup_{n<\omega} C_n$.
To prove that $(f,g)$ satisfies the adjointness condition, suppose that for some chain model $(C_n)_{n<\omega}$ and an $L^1_\kappa$ sentence $K$ we have that ${\mathfrak A}=\bigcup_{n<\omega} C_n\models_{L^1_\kappa} K$. This means that for some relevant $\beta, \theta$, $K$ is a union of $\equiv^\beta_\theta$-equivalence classes and that ${\mathfrak A}\in K$. By Lemma \[betaequiv\], ${{\cal C}}(K)$ is closed under $\equiv_{L^{c,\beta}_{\kappa, \kappa}}$ and by Lemma \[TFAEequivalence\], it is $L^c_{\kappa, \kappa}$-definable by a sentence $\varphi$ of quantifier rank $<\beta$. Moreover, we have chosen $f(K)$ to be some $L^c_{\kappa, \kappa}$-equivalent representation of $\varphi$. Since $(C_n)_{n<\omega}\in {{\cal C}}(K)$, we have that $(C_n)_{n<\omega}\models^c \varphi$ and hence $(C_n)_{n<\omega}\models^c f(K)$. In the other direction, suppose that $(C_n)_{n<\omega}\models^c f(K)$ for some $K$. We can find the corresponding $\beta, \theta$ and $\varphi$, so that we now have that $(C_n)_{n<\omega}\models^c \varphi$ and by the choice of $\varphi$ it follows that $\bigcup_{n<\omega} C_n\models_{L^1_\kappa} K$.
We check that $f$ preserves negation. Suppose that for a class $K$ of $L^1_\kappa$-models, the class ${{\cal C}}(K)$ is characterised by an $L^c_{\kappa, \kappa}$-sentence $\varphi$. Then $\neg K$ contains exactly those models ${\mathfrak A}$ which are not in any equivalence class contained in $K$ and therefore ${{\cal C}}(\neg K)$ contains exactly those chain models whose union is not in $K$. These are exactly those chain models that are not in ${{\cal C}}(K)$ and hence by the choice of $\varphi$, exactly those chain models that do not satisfy $\varphi$, which hence satisfy $\neg \varphi$. So $f(\neg K)$ is $L^c_{\kappa, \kappa}$-equivalent to $\neg(f(K))$.
To show that $f$ preserves disjunction, suppose that we are given classes $K_i(i<n)$ of $L^1_\kappa$-models such that for any $i$ the class ${{\cal C}}(K_i)$ is characterised by an $L^c_{\kappa, \kappa}$-sentence $\varphi_i$. Then ${{\cal C}}( \bigvee_{i<n} K_i)=\bigcup_{i<n} {{\cal C}}(K_i)$ and is hence characterised by $\bigvee_{i<n} \varphi_i$.
The density condition is easy to prove since $g$ is onto: any model is the union of a weak chain model. ${\bigstar}_{\ref{complok}}$
We now show that $L^{c,\ast}_{\kappa, \kappa}$ is strictly stronger than $L^1_\kappa$ in Shelah’s sense. Namely, Shelah [@Sh797] only considers the classically studied order between logics, where ${{\cal L}}_1\le {{\cal L}}_2$ means that for every $\tau$ and $\varphi\in {{\cal L}}_1(\tau)$ there is $\psi\in {{\cal L}}_2(\tau)$ such that for every model, $M\models_{{{\cal L}}_1} \varphi$ iff $M\models_{{{\cal L}}_2} \psi$. In this definition, the models are fixed, so in the language of Chu transforms, $g$ is the identity, whereas we wish to compare two logics for which the notions of a model are different, such as chain logic and $L^1_\kappa$. Then non-trivial Chu transforms are needed. In the spirit of the classical definition, our definition of a Chu transform says that $g$ is ‘practically’ the identity. We shall use similar technology to show that $L^{c,\ast}_{\kappa, \kappa}$ is not in this ‘Chu identitary’ sense weaker than $L_{\kappa,\kappa}$, in contrast to our results in Theorem \[PCtheorem\], which say that $L^{c,\ast}_{\kappa, \kappa}$ is Chu below a model class of $L_{\kappa,\kappa}$.
\[\] Suppose that $(f,g)$ is a Chu transform between logics ${{\cal L}}_1=(L_1, \models_1, S_1)$ and ${{\cal L}}_2=(L_2, \models_2, S_2)$ is in the same vocabulary $\tau$, where the elements of $S_1$ are chain models and the elements of $S_2$ are ordinary models. Let $\sim$ be an equivalence relation among the elements of $S_2$.
We say that $(f,g)$ is an [*$\sim$-identitary*]{} Chu transform if $g$ is onto and, moreover, for every chain model $(A_n)_{n<\omega}$ in $S_1$ and ${\mathfrak A}\in S_2$ with ${\mathfrak A}= \bigcup_{n<\omega} A_n$, there is ${\mathfrak B}\in S_2$ such that $$g({\mathfrak B})=\langle A_n:\,n<\omega\rangle\mbox{ and } {\mathfrak B}\sim{\mathfrak A}.$$ We write ${{\cal L}}_1\le_i^\sim {{\cal L}}_2$ if there is an $\sim$-identitary Chu transform $(f,g)$ between ${{\cal L}}_1$ and ${{\cal L}}_2$.
\[chainbigger\] Suppose that $\kappa$ is a singular cardinal of countable cofinality, $\tau$ is a fixed vocabulary containing a binary relation symbol $<$, ${{\cal K}}$ is the class of all models of $\tau$ and ${{\cal K}}^c$ the class of all weak chain models of $\tau$.
[(1)]{} Let $\equiv$ stand for the elementary equivalence in $L_{\kappa,\kappa}$. Then we have: $$(L^{c,\ast}_{\kappa, \kappa}, \models^c, {{\cal K}}^c) \nleq_i^\equiv (L_{\kappa,\kappa}, \models, {{\cal K}}).$$
[(2)]{} Suppose that in addition $\kappa=\beth_\kappa$ and for $\beta<\theta^+<\kappa$ let $\equiv^\beta_\theta$ be the relation defined in Definition \[DG\]. For ${\mathfrak A}, {\mathfrak B}\in{{\cal K}}$ define ${\mathfrak A}\equiv^1_\kappa{\mathfrak B}$ iff for all $\theta<\kappa$ and all $\beta <\theta^+$ we have ${\mathfrak A}\equiv^\beta_\theta{\mathfrak B}$.
Then $$(L^{c,\ast}_{\kappa, \kappa}, \models^c, {{\cal K}}^c) \nleq_i^{\equiv^1_\kappa} (L^1_\kappa, \models_{L^1_\kappa}, {{\cal K}}).$$
Let $\varphi$ be the $L_{\omega_1, \omega_1}$ sentence which says that $<$ is a well-order. In particular, this is an $L^{c,\ast}_{\kappa, \kappa}$-sentence. Similarly to the proof in Example \[dependenceonchain\], let us define an ordinary model ${\mathfrak A}$ of $\neg\varphi$ as the lexicographic sum of $\omega^\ast$ many copies of $\kappa$, i.e. an infinite decreasing sequence of copies of $\kappa$. Then ${\mathfrak A}\models \neg \varphi$. We now define a chain model $(A_n)_{n<\omega}$ such that each $A_n$ consists of the first $n$ copies of $\kappa$ within ${\mathfrak A}$. Therefore $(A_n)_{n<\omega}\models^c\varphi$. Next, we define an $\omega$-sequence $\bar{a}$ by taking exactly one point from each copy of $\kappa$ in ${\mathfrak A}$, so $\bar{a}$ is an infinite decreasing $<$-sequence in ${\mathfrak A}$. Finally, let $B_n=A_n\cup {\mbox{\rm ran}}(\bar{a})$, for each $n$. Then $(B_n)_{n<\omega}\models^c\neg \varphi$ and $\bigcup_{n<\omega}B_n=\bigcup_{n<\omega}A_n={\mathfrak A}$. We shall use these models in both parts of the proof.
[(1)]{} By the definition of an identitary Chu transform, there are $\mathfrak C$ and $\mathfrak D$ in ${{\cal K}}$ such that:
- $g(\mathfrak C)= (A_n)_{n<\omega}$,
- $g(\mathfrak D)= (B_n)_{n<\omega}$ and
- $\mathfrak C\equiv \mathfrak D\equiv {\mathfrak A}$.
By the defining property of elementary equivalence, we have that either $\mathfrak C,\mathfrak D \models f(\varphi)$ or $\mathfrak C,\mathfrak D \models \neg f(\varphi)$. However, the adjointness condition implies that $\mathfrak C \models f(\varphi)$, since $(A_n)_{n<\omega} \models^c \varphi$, while $\mathfrak D\models
\neg f(\varphi)$, since $(B_n)_{n<\omega} \models^c \neg \varphi$. A contradiction.
[(2)]{} Consider $f(\varphi)$. It is an $L^1_\kappa$-sentence, so there are $\beta<\theta^+<\kappa$ such that $f(\varphi)$ is a union of $\equiv^{\beta}_\theta$-equivalence classes. By the definition of an identitary Chu transform with respect to $\equiv^1_\kappa$, there are $\mathfrak C$ and $\mathfrak D$ in $S_2$ such that:
- $g(\mathfrak C)= (A_n)_{n<\omega}$,
- $g(\mathfrak D)= (B_n)_{n<\omega}$,
- $\mathfrak C \equiv^{\beta}_\theta {\mathfrak A}\equiv^{\beta}_\theta
\mathfrak D$.
In particular, $\mathfrak C\equiv^{\beta}_\theta \mathfrak D$, which means by the properties of $f(\varphi)$, that either $\mathfrak C,\mathfrak D \models_{L^1_\kappa} f(\varphi)$ or $\mathfrak C,\mathfrak D \models_{L^1_\kappa} \neg f(\varphi)$. However, the adojintness condition implies that $\mathfrak C
\models_{L^1_\kappa} f(\varphi)$, since $(A_n)_{n<\omega} \models^c \varphi$, while $\mathfrak D
\models_{L^1_\kappa} \neg f(\varphi)$, since $(B_n)_{n<\omega} \models^c \varphi$. A contradiction. ${\bigstar}_{\ref{chainbigger}}$
\[example\] In Theorem \[PCtheorem\] (2) we show that $ (L^{c,\ast}_{\kappa, \kappa}, \models^c, {{\cal K}}^c) \leq_i^\fallingdotseq {{\cal L}}_2$ for a model class ${{\cal L}}_2$ of $L_{\kappa, \kappa}$ and an equivalence relation $\fallingdotseq$ in which all models with the same $\tau$-reduct are equivalent.
Maximality of the Chain Logic {#maximalitysec}
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Chain logic satisfies the undefinability of well order, by a theorem Theorem \[Makkaiudwo\] of Makkai which we recall here. In this section we show that $\udwo_\kappa$ in a sense characterises the chain logic $L^c_{\kappa, \kappa}$, as per Theorem \[maxudwo\].
\[Makkaiudwo\] Suppose that $\tau$ is a vocabulary containing $\{<, U\}$ where $<$ is a binary relation symbol and $U$ is a unary relation symbol and suppose further that $\varphi$ is an $L_{\kappa, \kappa}$-sentence in $\tau$ such that for each $\alpha<\kappa^+$ there is a chain model ${\mathfrak B}_\alpha$ satisfying that $(U^{{\mathfrak B}_\alpha}, <^{{\mathfrak B}_\alpha})$ is a well order of type $\ge\alpha$.
Then there is a strict chain model ${\mathfrak B}$ of $\varphi$ such that $(U^{{\mathfrak B}}, <^{{\mathfrak B}})$ contains a copy of the rationals.
We abstract this property of $L^c_{\kappa,\kappa}$ to arbitrary logics on chain models:
We consider logics $(L,\models_{\mathcal{L},},S)$, where $S $ is a class of weak chain models. We call them [*logics on chain models*]{}. Such a logic is said to satisfy the property $\udwo_\kappa$ if every sentence in $L$ with a distinguished binary predicate $<$ which has for all $\alpha<\kappa^+$ a weak chain model in $S$ in which $<$ is well-ordered in type $\ge\alpha$, has a proper chain model of cardinality $\le\kappa$ in $S$ in which $<$ is non-well-ordered.
A logic is said to *relativize* if for every sentence $\varphi$ of the logic with vocabulary $\tau$ and every unary predicate $P\notin\tau$ there is a sentence $\psi$ in the logic which holds in a model with vocabulary $\tau\cup\{P\}$ if and only if the relativization of the model to $P$ (the “$P$-part" of the model) satisfies $\varphi$. Our chain logics satisfies this property but one has to bear in mind that the relativization of a proper chain model need not be a proper chain model. For details concerning relativization we refer to [@BarwiseFeferman].
\[maxudwo\] Let $\kappa$ be a singular cardinal of countable cofinality, $\tau$ a fixed vocabulary and ${{\cal K}}^c$ the class of all weak chain models of $\tau$. Suppose that ${{\cal L}}^\ast$ is a logic on chain models such that:
- $L^c_{\kappa, \kappa}\le{{\cal L}}^\ast$ and
- $({{\cal L}}^\ast,\models^c, {{\cal K}}^c)$ satisfies $\udwo_\kappa$.
Then for every sentence $\varphi$ of ${{\cal L}}^\ast$ there is $\alpha<\kappa^+$ such that the class of models of $\varphi$ is closed under $\sim^c_{\kappa,\alpha}$. Hence $\varphi$ is definable in $L^c_{\lambda,\kappa}$, where $\lambda=\max(\kappa,(\beth_{\beta+2}(|\tau|)^+)$.
To prove the theorem, we shall need some lemmas. The first lemma is the crucial one and the one where the singularity of $\kappa$ is really essential.
\[Scottwo\] Let $\kappa$ be a singular cardinal of countable cofinality and suppose that $(M_n)_{n<\omega}$ and $(N_n)_{n<\omega}$ are weak chain models of the same vocabulary such that $|M_n|<\kappa$ and $|N_n|<\kappa$ for all $n$. If player II has a winning strategy in $EF^c_{\kappa,\infty}((M_n)_{n<\omega},(N_n)_{n<\omega})$, then $(M_n)_{n<\omega}
\cong^c (N_n)_{n<\omega}$.
We can choose an increasing sequence $\langle \kappa_n:\, n<\omega\rangle$ of cardinals such that $\kappa=\sup_{n\in \omega} \kappa_n$ and so that $|M_n|, |N_n| \le\kappa_n$. We play $EF^c_{\kappa}(M,N)$ so that Player I starts by playing a sequence covering $M_0$, then one covering $N_0$, then $M_1$, then $N_1$, etc. Player II plays according to the winning strategy. In the end we have a chain isomorphism from $(M_n)_{n<\omega}$ to $(N_n)_{n<\omega}$. ${\bigstar}_{\ref{Scottwo}}$
We now prove two lemmas that work for weak as well as proper chain models. Then we formulate the same lemmas appropriately for chain models satisfying extra conditions. The proofs are standard arguments, based on [@MR0209132].
\[dd\] The following conditions are equivalent:
1. Player II has a winning strategy in the game $EF^c_{\kappa,\beta}((M_n)_{n<\omega},(N_n)_{n<\omega})$,
2. There are sets $\langle I_\xi:\,\xi<\beta, n<\omega\rangle $ such that:
1. If $\xi<\zeta<\beta$, then $\emptyset\ne I_\zeta\subseteq I_\xi$.
2. if $f\in I_\xi$, then $f=\langle(a_\zeta,b_\zeta): \zeta<\kappa_n\rangle$ is a partial chain isomorphism from $(M_n)$ to $(N_n)$,
3. if $f\in I_{\xi}$, then for all $\eta<\xi$ and all bounded $A\subseteq \bigcup_{n<\omega} M_n$ of cardinality $<\kappa$ there is $g\in I_\eta$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$.
4. If $f\in I_{\xi}$, then for all $\eta<\xi$ and all bounded $B\subseteq
\bigcup_{n<\omega} N_n$ of cardinality $<\kappa$ there is $g\in I_\eta$ for some $k<\omega$ such that $f\subseteq g$ and $B\subseteq{\mbox{\rm ran}}(g)$.
During the game $EF^c_{\kappa,\beta}$ Player I plays, among other things, some elements of $\beta$ in descending order. Let us call those move *rank moves*. If (1) holds then we let $I_\xi$ be the set of partial chain isomorphisms from $(M_n)$ to $(N_n)$ that arise in the game when II is playing her winning strategy and Player I has played $\xi$ as his rank-move. Conversely, suppose such sets $I_\xi$ exist. The strategy of II is to make sure that the partial functions she plays after Player I plays $\xi$ as his rank move, are in $I_\xi$. More exactly, suppose she has played a partial chain isomorphism $f$ in $I_\xi$ and then Player I plays a bounded set $A$ of cardinality $<\kappa$ as well as $\eta<\xi$ as the rank move. Player II uses (c) above to find $g\in I_\eta$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$. She plays this $g$ in the game. ${\bigstar}_{\ref{d}}$
\[ee\] The following conditions are equivalent:
1. Player II has a winning strategy in the game $EF^c_{\kappa,\infty}((M_n)_{n<\omega},(N_n)_{n<\omega})$.
2. There is a set $J$ such that
1. $J\ne\emptyset$.
2. if $f\in J$, then $f$ is a partial chain isomorphism of cardinality $<\kappa$ from $(M_n)$ to $(N_n)$,
3. if $f\in J$, then for all bounded $A\subseteq \bigcup_{n<\omega} M$ of cardinality $<\kappa$ there is $g\in J$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$.
4. If $f\in J$, then for all bounded $B\subseteq \bigcup_{n<\omega} N_n$ of cardinality $<\kappa$ there is $g\in J$ such that $f\subseteq g$ and $B\subseteq{\mbox{\rm ran}}(g)$.
If (1) holds then we let $J$ be the set of partial chain isomorphisms of cardinality $<\kappa$ from $(M_n)$ to $(N_n)$ that arise in the game when II is playing her winning strategy. Conversely, suppose such sets $J$ exist. The strategy of II is to make sure that the partial functions she plays are in $J$. More exactly, suppose she has played a partial chain isomorphism $f$ of cardinality $<\kappa$ in $J$ and then Player I plays a bounded set $A$ of cardinality $<\kappa$. Player II uses (c) above to find $g\in J$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$. She plays this $g$ in the game. ${\bigstar}_{\ref{e}}$
For singular $\kappa$ we can be more specific:
\[d\]Let $\kappa$ be a singular cardinal of cofinality $\omega$ and $\kappa=\sup_n\kappa_n$, where $\kappa_n<\kappa_{n+1}$. The following conditions are equivalent:
1. Player II has a winning strategy in the game $EF^c_{\kappa,\beta}((M_n)_{n<\omega},(N_n)_{n<\omega})$,
2. There are sets $\langle I^n_\xi:\,\xi<\beta, n<\omega\rangle $ such that:
1. If $\xi<\zeta<\beta$, then $\emptyset\ne I^n_\zeta\subseteq I^n_\xi$.
2. if $f\in I^n_\xi$, then $f=\langle(a_\zeta,b_\zeta): \zeta<\kappa_n\rangle$ is a partial chain isomorphism from $(M_n)$ to $(N_n)$,
3. if $f\in I^n_{\xi}$, then for all $\eta<\xi$ and all bounded $A\subseteq \bigcup_{n<\omega} M_n$ of cardinality $\le\kappa_{m}$, $m>n$, there is $g\in I^{m}_\eta$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$.
4. If $f\in I^n_{\xi}$, then for all $\eta<\xi$ and all bounded $B\subseteq
\bigcup_{n<\omega} N_n$ of cardinality $\le\kappa_{m}$, $m>n$, there is $g\in I^m_\eta$ such that $f\subseteq g$ and $B\subseteq{\mbox{\rm ran}}(g)$.
If (1) holds then we let $I^n_\xi$ be the set of partial chain isomorphisms of cardinality $\le\kappa_n$ from $(M_n)$ to $(N_n)$ which are subsets of cardinality $\le\kappa_n$ of partial functions that arise in the game when II is playing her winning strategy and Player I has played $\xi$ as his rank-move. Conversely, suppose such sets $I^n_\xi$ exist. The strategy of II is to make sure that the partial functions she plays after Player I plays $\xi$ as his rank move, are in $I^n_\xi$ for a suitable $n$. More exactly, suppose she has played a partial chain isomorphism $f$ of cardinality $\le\kappa_n$ in $I^n_\xi$ and then Player I plays a bounded set $A$ of cardinality $\le\kappa_m$, where $m>n$ as well as $\eta<\xi$ as the rank move. Player II uses (c) above to find $g\in I^m_\eta$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$. She plays this $g$ in the game. ${\bigstar}_{\ref{d}}$
\[e\]Let $\kappa$ be a singular cardinal of cofinality $\omega$ and $\kappa=\sup_n\kappa_n$, where $\kappa_n<\kappa_{n+1}$. The following conditions are equivalent:
1. Player II has a winning strategy in the game $EF^c_{\kappa,\infty}((M_n)_{n<\omega},(N_n)_{n<\omega})$.
2. There are sets $J_n$, $n<\omega$, such that
1. $J_n\ne\emptyset$.
2. if $f\in J_n$, then $f$ is a partial chain isomorphism of cardinality $\le\kappa_n$ from $(M_n)$ to $(N_n)$,
3. if $f\in J_n$, then for all bounded $A\subseteq \bigcup_{n<\omega} M$ of cardinality $\le\kappa_m$, $m>n$, there is $g\in J_m$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$.
4. If $f\in J_n$, then for all bounded $B\subseteq \bigcup_{n<\omega} N_n$ of cardinality $\le\kappa_m$, $m>n$, there is $g\in J_m$ such that $f\subseteq g$ and $B\subseteq{\mbox{\rm ran}}(g)$.
If (1) holds then we let $J_n$ be the set of partial chain isomorphisms of cardinality $\le\kappa_n$ from $(M_n)$ to $(N_n)$ which are subsets of cardinality $\le\kappa_n$ of partial functions that arise in the game when II is playing her winning strategy. Conversely, suppose such sets $J_n$ exist. The strategy of II is to make sure that the partial functions she plays are in $J_n$ for a suitable $n$. More exactly, suppose she has played a partial chain isomorphism $f$ of cardinality $\le\kappa_n$ in $J_n$ and then Player I plays a bounded set $A$ of cardinality $\le\kappa_m$, where $m>n$. Player II uses (c) above to find $g\in J_m$ such that $f\subseteq g$ and $A\subseteq{\mbox{\rm dom}}(g)$. She plays this $g$ in the game. ${\bigstar}_{\ref{e}}$
(of Theorem \[maxudwo\]) We argue along the lines of the proof of Theorem 3.2 in [@Barwise]. By Lemma \[TFAEequivalence\], it suffices to show that for all ${{\cal L}}^\ast$-sentences $\varphi$, there is $\alpha<\kappa^+$ such that that all models in the same $\sim^c_{\kappa, \alpha}$-class of ${{\cal K}}^c$ agree on the truth value of $\varphi$.
Suppose that ${{\cal L}}^\ast$ is as in the assumption and let $\varphi$ be an ${{\cal L}}^\ast$-sentence. Suppose for a contradiction that for every $\alpha<\kappa^+$ there are chain models $(M^\alpha_n)_{n<\omega}$ and $(N^\alpha_n)_{n<\omega}$ in ${{\cal K}}^c$ such that $(M^\alpha_n)_{n<\omega}\sim^c_{\kappa, \alpha}(N^\alpha_n)_{n<\omega}$, while $(M^\alpha_n)_{n<\omega}\models^c\varphi$ and $(N^\alpha_n)_{n<\omega}\models^c\neg \varphi$. For each such $\alpha$ we get sets $I^n_\xi, \xi<\alpha,$ from Lemma \[d\].
We now build a new ${{\cal L}}^\ast$ sentence $\varphi^\ast$ in an extended vocabulary in which $U,P,Q$ are new unary relation symbols, $<^\ast$ is a new binary relation symbol and $I_n$ is a new $1+\kappa_n+\kappa_n$-ary relation symbol for each $n<\omega$. The sentence $\varphi^\ast$ is the conjunction of sentences saying:
1. the sentence $\varphi$ holds in the $P$ part of the (chain) model and $\neg\varphi$ holds in the $Q$-part,
2. $\le^\ast$ is a linear order on $U$,
3. If $I_n(i,(a_\xi)_{\xi<\zeta},(b_\xi)_{\xi<\zeta})$, then $i\in U$, $(a_\xi)_{\xi<\zeta}\subseteq P$ and $(b_\xi)_{\xi<\zeta}\subseteq Q$, both bounded,
4. If $i<^*j$ and $I_n(j,\bar{a},\bar{b} )$, then $I_n(i,\bar{a},\bar{b} )$ for all bounded sequences $\bar{a}$ and $\bar{b}$ of length $\kappa_n$,
5. For each $n<\omega$, if $I_n(i,(a_\xi)_{\xi<\kappa_n},(b_\xi)_{\xi<\kappa_n})$ then for all $j<^\ast i$, $m>n$, and all bounded $(a'_\xi)_{\kappa_n\le\xi<\kappa_m}\in P$ ($(b'_\xi)_{\kappa_n\le\xi<\kappa_m}\in Q$) there is a bounded $(b'_\xi)_{\kappa_n\le\xi<\kappa_m}\in Q$ ($(a'_\xi)_{\kappa_n\le\xi<\kappa_m}\in P$) such that $$I_m(j,(a_\xi)_{\xi<\kappa_n}\frown (a'_\xi)_{\kappa_n\le\xi<\kappa_m},(b_\xi)_{\kappa_n\le\xi<\kappa_m}\frown(b'_\xi)_{\kappa_n\le\xi<\kappa_m})).$$
6. If $I_n(i,(a_\xi)_{\xi<\kappa_n},(b_\xi)_{\xi<\kappa_n})$, then for all atomic formulas $\phi((v_\xi)_{\xi<\kappa_n})$ we have $$(M_m)_{m<\omega}\models^c\phi((a_\xi)_{\xi<\kappa_n})\iff (N_m)_{m<\omega}\models^c \phi((b_\xi)_{\xi<\kappa_n}),$$ where $(M_m)_{m<\omega}$ is the $P$-part of $(K_m)_{m<\omega}$ and $(N_m)_{m<\omega}$ is the $Q$-part.
Using the fact that $L^c_{\kappa, \kappa}$ is included in ${{\cal L}}^\ast$, we can express this by an ${{\cal L}}^\ast$-sentence, which we take to be $\varphi^\ast$.
For every $\kappa\le\alpha<\kappa^+$ we build a chain model $(K_n)^\alpha_{n<\omega}$ of $\varphi^\ast$ in which $<^\ast$ has order type $\alpha$, by declaring $K_0^\alpha$ to be the ordinal $\alpha$, interpreting $U$ as $K_0^\alpha$ and $<^\ast$ as the natural order on $\alpha$, letting $K_{n+1}^\alpha=M_n^\alpha\bigcup
N^\alpha_n$ and declaring $\bigcup_{n<\omega} M^\alpha_n$ to be the $P$-part and $\bigcup_{n<\omega} N^\alpha_n$ to be the $Q$-part of the model. The interpretation of the predicates $I_n$ in $(K_m)^\alpha_{m<\omega}$ is as follows: $I_n(\gamma,(a_\xi)_{\xi<\zeta},(b_\xi)_{\xi<\zeta})$, $\zeta<\alpha$, is defined to hold in $(K_m)^\alpha_{m<\omega}$ if and only if the relation $\{(a_\xi,b_\xi):\xi<\zeta\}$ is in the set $I^n_\gamma$.
Now we are in the situation to apply $\udwo_\kappa$ of ${{\cal L}}^\ast$. Doing so, we obtain a strict chain model $(K_n)_{n<\omega}$ of $\varphi^\ast$ in which $<^\ast$ is not well-founded. Let $e_0>^*e_1>^*e_2\ldots$ be a descending chain in $<^*$ in $(K_n)_{n<\omega}$. Let $J_n$ consist of functions $f$ such that $f(a_\xi)=b_\xi$ for some pair $((a_\xi)_{\xi<\zeta},(b_\xi)_{\xi<\zeta})$ for which $(K_n)_{n<\omega}$ satisfies $I_n(e_m,(a_\xi)_{\xi<\zeta},(b_\xi)_{\xi<\zeta})$ for some $m<\omega$. By Lemma \[e\], Player II has a winning strategy in the game $EF^c_{\kappa,\infty}((M_n)_{n<\omega},(N_n)_{n<\omega})$, where $(M_n)_{n<\omega}$ is the $P$-part of $(K_n)_{n<\omega}$ and $(N_n)_{n<\omega}$ is the $Q$-part. By Lemma \[Scottwo\], $(M_n)_{n<\omega} \cong^c (N_n)_{n<\omega}$, contradicting the fact that $(M_n)_{n<\omega} \models^c\varphi$ and $ (N_n)_{n<\omega} \models^c\neg\varphi$. ${\bigstar}_{\ref{maxudwo}}$
The above results in conjunction with Shelah’s characterisation of $L^1_\kappa$ as maximal above $\bigcup_{\lambda<\kappa}L_{\lambda, \omega}$ to satisfy SUDWO, indicate that it is likely that SUDWO is a strict strengthening of UDWO. However, the conclusion cannot be drawn directly since Shelah’s characterisation is obtained in the class of logics with the ordinary meaning of a model, which does not fit the chain logic.
Further properties of chain logic {#sec:further}
=================================
We recall that Theorem 2.6 of Cunnigham [@Cunnigham] shows that $L^{c,\ast}_{\kappa,\kappa}$ has Interpolation. (For a definition of Interpolation, see Definition \[interpolation\]). We now prove some further properties of the chain logic.
The incompactness of $L_{\kappa,\kappa}^c$
------------------------------------------
A property that does not seem to have been studied classically for chain logic is its degree of compactness. Positive compactness results have been obtained for restricted classes of chain models: Green [@green1974] proved a $\Sigma_1$–compactness result for admissible sets, which was given another proof by Makkai [@Makkaichainlogic]. Possibly Karp and her school were aware of the incompactness of full chain logic, but in the absence of any literature on it, we give a proof. In fact we give two proofs, as the proofs for the logics $L^c_{\kappa, \kappa}$ and $L^{c, \ast}_{\kappa, \kappa}$ turn out to be different.
It is not straightforward to determine all cardinals $\lambda$ such that the logic $L_{\theta, \omega}$ is $\lambda$-compact or satisfies some weak version of $\lambda$-compactness. Clearly, if $\theta=\lambda$ is a strongly compact cardinal, this is the case. However, Boos [@Boss] showed that it is even possible to have analogues of weak compactness of $L_{\theta, \omega}$ for $\theta$ below the continuum (though $\theta$ needs to be a large cardinal in $L$).[^10] By the results of Keisler and Tarski [@KeislerTarski], see Theorem 3.1.4 of [@Dickmann], $L_{\kappa, \omega}$ is never $\kappa$-compact for $\kappa$-singular. Moreover, see Corollary 3.1.6 in [@Dickmann], $L_{\lambda, \omega}$ is $\lambda$-compact iff $\lambda$ is a strongly compact cardinal.
\[notcompact\] The logic $L^{c,\ast}_{\kappa,\kappa}$ is not $\kappa$-compact.
Theorem \[lowerbound\] shows that $L^{c,\ast}_{\kappa,\kappa}$ is Chu above $L_{\kappa, \omega}$ and so by Theorem \[kappacompactnesspreservation\], $\kappa$-compactness of $L^{c,\ast}_{\kappa,\kappa}$ would imply the $\kappa$-compactness of $L_{\kappa, \omega}$, a contradiction. ${\bigstar}_{\ref{notcompact}}$
Theorem \[lowerbound\] does not immediately apply to the logic $L^{c}_{\kappa,\kappa}$ so it does not rule out some degree of compactness of this logic. If $L_{\kappa,\omega}$ had some degree of upwards Löwenheim-Skolem theorem, we would be able to replace ${\mathcal M}_{\ge \kappa}$ in Theorem (2) by ${\mathcal M}$ and draw the same conclusion for $L^{c}_{\kappa,\kappa}$ as we did for $L^{c,w}_{\kappa,\kappa}$. However, there is no such upwards theorem.[^11] For example, the following is a well known open question:
\[\] If a complete $L_{\omega_1,\omega}$-sentence has a model of size $\aleph_{\omega_n}$ for every $n$, does it then have a model of size $\aleph_{\omega_\omega}$?
Therefore, we need another argument to deal with the question if the chain logic $L_{\kappa,\kappa}^c$ is $\kappa$-compact.
\[incompactness\] $L_{\kappa,\kappa}^c$ is not $\kappa$-compact.
We construct a set $\Gamma$ of $L_{\kappa,\kappa}^c$-sentences which is $(<\kappa)$-satisfiable but not $\kappa$-satisfiable. We need the following lemma.
\[ordertypekappa\] There is an $L_{\kappa,\omega}$-sentence $\theta$ in the language consisting of one binary predicate $<^\ast$ whose models are exactly the models of size $\kappa$, where $<^\ast$ is a well order of the domain of order type $\kappa$.
By Observation \[moreforless\], it suffices to find an $L_{\kappa^+,\omega}$-sentence $\theta$ with the required properties. Following Dana Scott [@Scottinfinitary], starting from the first order sentence stating that $<^\ast$ is a linear order, one can define by induction on $\alpha<\kappa$ an $L_{\kappa^+,\omega}$ formula $E_\alpha(x)$ so that $$E_\alpha(x)\iff(\forall y)[ y <^\ast x \iff \bigvee_{\beta <\alpha} E_\beta (y)].$$ Then one shows by induction on $\alpha$ that $E_\alpha(x)$ means that the $<^\ast$-predecessors of $x$ form a well-order of order type $\alpha$. The sentence for our proof is $$\bigwedge_{\alpha<\kappa}(\exists x)E_\alpha(x)\wedge(\forall x) \bigvee_{\alpha<\kappa} E_\alpha (x).$$ ${\bigstar}_{\ref{ordertypekappa}}$
Now we formulate $\Gamma$ in the language $\{<^\ast\}\cup \{c_\alpha:\,\alpha<\kappa\}\cup \{d\}$ as $$\Gamma=\{\theta\}\cup \{c_\alpha <^\ast c_\beta:\,\alpha<\beta<\kappa\}\cup \{c_\alpha <^\ast d:\,\alpha<\kappa\}.$$ Then for any subset $\Gamma_0$ of $\Gamma$ of size $<\kappa$, any model $M$ of $\theta$ easily gives rise to an actual model of $\Gamma_0$ by interpreting the relevant $c_\alpha$ as the $\alpha$-th element of $M$ in the well order provided by $<^\ast$ and $d$ as any element of $M$ which is of large order in $M$ than any of these relevant $c_\alpha$s. This model is also a chain model, since all sentences in $\Gamma_0$ are in $L_{\kappa^+,\omega}$. On the other hand, $\Gamma$ does not have any models or chain models. ${\bigstar}_{\ref{incompactness}}$
We now turn to upper bounds for the compactness number of the chain logic.
\[PC\] Let $\tau$ be a vocabulary.
[(1)]{} We define a new vocabulary $\tau'=\tau\cup \{P_n:\,n<\omega\}$ where $P_n$ are new unary predicate symbols.
[(2)]{} For each $n<\omega$ we define an $L_{\kappa,\kappa}$-sentence $\psi_n$ in the vocabulary $\tau'$ as follows $$(\exists (x_\alpha)_{\alpha<\kappa_n}) (\forall y) [P_n(y)\implies \bigvee_{\alpha<\kappa_n} y=x_\alpha].$$ (This sentence expresses that the realization of the predicate $P_n$ has size $\le\kappa_n$.)
[(3)]{} Let $\sigma_0$ be the $L_{\kappa,\kappa}$-sentence $$(\forall x ) \bigvee_{n<\omega}P_n( x) \wedge \bigwedge_{n<\omega}(\forall x )[P_n( x)\implies P_{n+1}( x)].$$ (Notice that if this sentence is true for any $x$ then it is trivially true for finitary ${\bar x}$, but this is not necessarily the case for ${\bar x}$ of infinite length). Let $\sigma_1$ be $\sigma_0\wedge \bigwedge_{n<\omega}\psi_n$.
[(4)]{} For $l<2$, we define ${\mathcal Mod}(\sigma_l)$ to be the class of all $L_{\kappa,\kappa}$-models of $\sigma_l$.
\[PCtheorem\] To avoid problems with extending vocabularies, for this theorem we assume that all statements are relativized to a vocabulary $\tau'$ as in convention \[PC\], in particular, $\tau'$ was obtained from a vocabulary $\tau$ as in that convention. Then:
[(1)]{} (a) $L^{c,\ast}_{\kappa,\kappa} \le (L_{\kappa,\kappa}, \models, {\mathcal Mod}(\sigma_0))$ and (b) $L^{c}_{\kappa,\kappa} \le (L_{\kappa,\kappa}, \models, {\mathcal Mod}(\sigma_1))$.
[(2)]{} For models ${\mathfrak A}$ and ${\mathfrak B}$ in ${\mathcal Mod}(\sigma_0)$ let $\fallingdotseq$ be the equivalence relation saying that the reducts of ${\mathfrak A}$ and ${\mathfrak B}$ to $\tau$ are the same. Then, in addition to (1)(a), $$L^{c,\ast}_{\kappa,\kappa} \le_i^\fallingdotseq (L_{\kappa,\kappa}, \models, {\mathcal Mod}(\sigma_0)).$$
[(1)]{} (a) For the first inequality, we define $f(\psi)$ for sentences $\psi$ in $L^{c,\ast}_{\kappa,\kappa}$ by induction on the complexity of $\psi$. For atomic sentences we have $f(\psi)=\psi$ and this is extended through the logical operations $\neg$ and conjunctions and disjunctions of size $<\kappa$. It remains to define $f(\psi)$ for $\psi$ of the form $\forall {\bar x}\, \varphi ( {\bar x}; {\bar y})$ where ${\bar y}$ is a sequence of parameters and $\bar{x}=
\langle x_\alpha:\alpha<\alpha^\ast\rangle $ for some $\alpha^\ast<\kappa$.
We let $f(\forall \bar{x} \,\varphi ( \bar{x}; {\bar y}))$ be the sentence $\forall \bar{x} ([\bigvee_{n<\omega} \bigwedge_{\alpha<\alpha^\ast}P_n (x_\alpha)]\implies \varphi ( x; {\bar y}))$.
To define $g$, let $M$ be a model of $\sigma_0$ and define $g(M)=(P_0^M, P_1^M, \ldots)_{n<\omega}$, which gives a weak chain model by the choice of $\sigma_0$. The pair $(f,g)$ is well defined, $f$ preserves the logical operations of $L_{\kappa,\kappa}$ and $g$ is onto. It remains to verify the adjointness condition, which will imply that $(f,g)$ is indeed a Chu transform.
Suppose that $\psi$ is in $L^{c,\ast}_{\kappa,\kappa}$ and $M\in S_0'$ is such that $M\models f(\psi)$. By induction on the complexity of $\psi$ we have to check that $(P_0^M, P_1^M, \ldots)_{n<\omega}\models^c \psi$. The nontrivial case of this induction is the quantifier case, so suppose that $\psi$ is of the form $\forall {\bar x}\, \varphi ( {\bar x}; {\bar y})$, where ${\bar y}$ bounded, that is, ${\bar y}$ is such that there is $n$ for which all individual elements of $y$ are in $P_n^M$. Let $\bar{x}$ also be bounded. The definition is made so that $M\models f(\psi)$ implies that $M\models \varphi ( {\bar x}; {\bar y})$, and therefore, since ${\bar y}$ is bounded we have that $g(M)=(P_0^M, P_1^M, \ldots)_{n<\omega}\models^c \forall {\bar x}\, \varphi ( {\bar x}, {\bar y})$.
[(b)]{} The definition of $f$ remains the same for this proof. The definition of $\sigma_1$ is such as to guarantee that for every $M\in S_1'$ and for every $n<\omega$ we have $|P_n|^M\le \kappa_n$. Therefore, leaving the definition of $g$ the same as in the proof of (a), we have that $g(M)=(P_0^M, P_1^M, \ldots)_{n<\omega}$ is a strict chain model. The proof of the adjointness property remains the same. However, $g$ is not necessarily onto, as it is perfectly possible to have chain models $(A_n)_{n<\omega}$ for which there are $n$ that satisfy $|A_n|>\kappa_n$. We shall however show that $g$ satisfies the density condition.
So suppose that $\varphi$ is a sentence of $L^{c}_{\kappa,\kappa}$ which has a proper chain model $(A_n)_{n<\omega}$. We shall define $B_n$ by induction on $n<\omega$. For convenience let us define $A_1=\emptyset$. Let $B_0$ be $A_k$ for the largest $k$ such that $|A_k|\le \kappa_0$. Such a $k$ exists because the sets $A_k$ are increasing and $|\bigcup_{n<\omega} A_n|=\kappa$. Having defined $B_n$ as $A_{k_n}$ with $|A_{k_n}|\le \kappa_n$ and $k_n$ the largest $k$ with that property, let $B_{n+1}=A_k$ for the largest $k$ such that $|A_k|\le \kappa_{n+1}$. In this way we necessarily have $B_n\subseteq B_{n+1}$. The chain model $(B_n)_{n<\omega}=g(M)$ for $M$ in $S'_1$ which is obtained on $\bigcup_{n<\omega} (B_n)$ by interpreting each $P_n$ as the corresponding $B_n$. By the same argument as in Observation \[variouschainmodels\](3), $(B_n)_{n<\omega}$ is a model of $\varphi$.
[(2)]{} We use the same $(f,g)$ as in (1)(a). Suppose that we are given a chain model $(A_n)_{n<\omega}$ and ${\mathfrak A}\in {\mathcal Mod}(\sigma_0)$ with ${\mathfrak A}=\bigcup_{n<\omega} A_n$. Define ${\mathfrak B}$ to be the the model whose reduct to $\tau$ is the same as that one of ${\mathfrak A}$, but $P^{{\mathfrak B}}_n=A_n$ for each $n$. Then it is easily seen that ${\mathfrak B}\fallingdotseq {\mathfrak A}$ and $g({\mathfrak B})=
(A_n)_{n<\omega}$. ${\bigstar}_{\ref{PCtheorem}}$
The following is a classical argument resembling the projective class arguments, for example Proposition II 3.1.9 in [@BarwiseFeferman].
\[Oiko\] Suppose that $\sigma$ is in $L_{\kappa,\kappa}$ and that $\theta$ is a cardinal such that $L_{\kappa,\kappa}$ is $\theta$-compact. Then so is $(L_{\kappa,\kappa}, \models, {\mathcal Mod}(\sigma))$.
Suppose that $\Gamma$ is a set of $\theta$ many sentences of $L_{\kappa,\kappa}$ such that every subset of $\Gamma$ of size $<\theta$ is satisfiable in ${\mathcal Mod}(\sigma)$. In particular, $\Gamma\cup \{\sigma\}$ is a $(<\theta)$-satisfiable set of sentences of $L_{\kappa,\kappa}$ in the vocabulary $\tau'$ obtained by enlarging $\tau$ by the symbols in $\sigma$. Hence, by the assumption, $\Gamma\cup \{\sigma\}$ has a model $M$. Then $M\in {\mathcal Mod}(\sigma)$ by definition. ${\bigstar}_{\ref{Oiko}}$
\[strcompact\] $L^{c,\ast}_{\kappa,\kappa}$ and $L^{c}_{\kappa,\kappa}$ are $\theta$-compact whenever $\theta$ is a strongly compact cardinal $>\kappa$.
By the definition of a strongly compact cardinal, $L_{\theta, \theta}$ is $\theta$-compact. Let $\Gamma$ be a $(<\theta)$-satisfiable set of $L_{\kappa,\kappa}$-sentences. Then $\Gamma$ is in particular a set of $L_{\theta, \theta}$-sentences and hence has a model in the sense of $L_{\theta, \theta}$. By the definition of $\models$ in the two logics, this model is also an $L_{\kappa,\kappa}$ model. So, $L_{\kappa,\kappa}$ is $\theta$-compact. (In fact, all this just says that $L_{\kappa, \kappa}\le L_{\theta, \theta}$). By Theorem \[Oiko\], $(L_{\kappa,\kappa},\models, {\mathcal Mod}(\sigma_l))$ for $l<2$ are $\theta$-compact. Then, by Theorem \[PCtheorem\] and Theorem \[kappacompactnesspreservation\], $L^{c,w}_{\kappa,\kappa}$ and $L^{c}_{\kappa,\kappa}$ are $\theta$-compact. ${\bigstar}_{\ref{strcompact}}$
Let us recall the definition of Craig’s Interpolation:
\[interpolation\] (1) For a logic ${{\cal L}}$ and sentences $\rho_0, \rho_1$ we say $\rho_0\models_{{\cal L}}\rho_1$ if for any model $M$ of ${{\cal L}}$ in a vocabulary including those of $\rho_0$ and $\rho_1$, we have $$M\models_{{\cal L}}\rho_0\implies M\models_{{\cal L}}\rho_1.$$ [(2)]{} A logic ${{\cal L}}$ is said to satisfy [*Interpolation*]{} if for any two vocabularies $\tau_0$ and $\tau_1$ and sentences $\rho_l\in {{\cal L}}(\tau_l) (l <2)$, if $\rho_0\models_{{\cal L}}\rho_1$ then there is a sentence $\rho$ in ${{\cal L}}(\tau_0\cap \tau_1)$ such that $\rho_0\models_{{\cal L}}\rho $ and $\rho\models_{{\cal L}}\rho_1$.
[**Problem 1.4. from [@Sh797]**]{}: Suppose that $\kappa$ is a singular strong limit cardinal of countable cofinality. Is there a logic between $ L_{\kappa^+, \omega}$ and $L_{\kappa^+, \kappa^+}$ which satisfies Interpolation.
\[solves1.4.\] Suppose that $\kappa$ is a singular strong limit cardinal of countable cofinality. Then the chain logic $L^{c,\ast}_{\kappa,\kappa}$ satisfies $L_{\kappa^+,\omega}\le L^{c,\ast}_{\kappa,\kappa} \le (L_{\kappa^+,\kappa^+}, \models, {\mathcal Mod}(\sigma_0))$ and has the Interpolation, so modulo a model class, it gives a solution to Problem 1.4. from [@Sh797].
Theorem 2.6 of Cunnigham [@Cunnigham] shows that $L^{c,\ast}_{\kappa,\kappa}$ has Interpolation. The Chu inequalities are proved in Theorem \[lowerbound\] and Theorem \[PCtheorem\] respectively (note also Observation \[moreforless\]). ${\bigstar}_{\ref{solves1.4.}}$
We recall that $L^1_\kappa$ does not solve Problem 1.4. from [@Sh797], since it is not above $L_{\kappa^+,\omega}$.
Finally, we show that, as in the case of $L^2_{\omega,\omega}$, the second order logics $(L^{c}_{\omega,\omega})^2$ and $(L^{c, \ast}_{\omega,\omega})^2$ are not compact, or even $(\omega,\omega)$-compact. The definition of these logics is the same as in ordinary chain logic, except that we now consider all second order sentences of $L^2_{\omega, \omega}$, only allowing ordinary or set variables which are bounded by an element of the chain.
\[non-compactness\] Neither $(L^{c}_{\omega,\omega})^2$ nor $(L^{c, \ast}_{\omega,\omega})^2$ are $(\omega,\omega)$-compact.
Let $\phi$ be the second order sentence $$\exists X\forall y\, (P(y)\to X(y)).$$
\[claim1\]The chain models of $\phi$ are the chain models in which $P$ is contained in one level of the chain.
Suppose that $M=(M_n)_{n<\omega}$ as above is a chain model of $\phi$. Then there is an interpretation $Q$ of the variable $X$ such that $P^M\subseteq Q$. By the truth definition of chain models the variable $X$ ranges over subsets of $M$ which are included in some $M_n$. Thus $Q\subseteq M_n$ for some $n$. Hence $P^M\subseteq M_n$. For the other direction, suppose that $M=(M_n)_{n<\omega}$ is a chain model in which $P^M$ is contained in some $M_n$. Then we can pick $P^M$ as the interpretation of $X$ and $\phi$ becomes true.
\[claim2\] In models of $\phi$ we have full second order quantification over subsets of $P$.
Suppose $M=(M_n)_{n<\omega}$ is a chain model of $\phi$. By Lemma \[claim1\], there is $n$ such that $P^M\subseteq M_n$. If we relativize second order quantification to $P$ as follows: $$\exists X\psi(X)\mapsto\exists X[(\forall y(X(y)\to P(y))\wedge\psi(X)]$$ $$\forall X\psi(X)\mapsto\forall X[(\forall y(X(y)\to P(y))\to\psi(X)]$$ then we can allow the standard interpretation for second order quantifiers because the relativization forces the bound second order variables to range over subsets of $P$, and hence to be always subsets of $M_n$.
To finish the proof, let $\theta$ be the second order sentence which says that $<$ is a well-order on the predicate $P$. In chain models of $\theta$ we have no guarantee that $<$ is really a well-order because a descending sequence may cross over all the sets $M_n$. In models of $\theta\wedge \phi$ we know by the above that $<$ is really a well-order because any potential descending chain is a subset of $P$ and hence a subset of some $M_n$. We can now form a finitely consistent theory $\{\theta,\phi\}\cup\{c_0>c_1>c_2>\ldots\}\cup \{P(c_n):\, n<\omega\}$, which has no models. ${\bigstar}_{\ref{non-compactness}}$
The Union Lemma
---------------
The Union Lemma features in several proofs about $L^1_\kappa$, see for example Remark 1.17. (b) in [@Sh797][^12] We show that the Union Lemma holds for chain logic.
\[\] For chain models $(A_n)_{n<\omega}$ and $(B_n)_{n<\omega}$ we define
1. $(A_n)_{n<\omega} \subseteq (B_n)_{n<\omega}$ if $\bigcup_n A_n\subseteq\bigcup_n B_n$ and $\forall n\exists m\,(A_n\subseteq B_m)$
2. $(A_n)_{n<\omega} \prec_{L^c_{\kappa, \kappa}} (B_n)_{n<\omega}$ if $(A_n)_{n<\omega} \subseteq (B_n)_{n<\omega}$ and for any bounded sequence $\bar{a}$ in $(A_n)_{n<\omega}$ and formula $\phi(\bar{x})$ in $L^c_{\kappa, \kappa}$ we have $$(A_n)_{n<\omega} \models^c \phi(\bar{a})\Leftrightarrow (B_n)_{n<\omega} \models^c\phi(\bar{a}).$$
\[\] Suppose $(A_n)_{n<\omega} \subseteq (B_n)_{n<\omega}$. Then:
1. If $\bar{a}$ is bounded in $(A_n)_{n<\omega}$, then it is bounded in $(B_n)_{n<\omega}$.
2. If $(A_n)_{n<\omega}\models^c\phi$ and $\phi$ is existential sentence, then $(B_n)_{n<\omega}\models^c\phi$.
\[union\] Suppose that for each $m<\omega $ we have a chain model $(A^m_n)_{n<\omega}$ and that these models satisfy $(A^m_n)\subseteq (A^{m+1}_n)$ for all $m$. Let $A^\omega_n=\bigcup_{m\le n} A^m_n$, for each $n$.
We define the *union* of the sequence $\langle (A_n)_{n<\omega}:\,m<\omega\rangle $, as the chain model $(A^\omega_n)_{n<\omega}$.
\[\] Using the notation of Definition \[union\], we have that $(A^m_n)_{n<\omega}\subseteq (A^{\omega}_n)_{n<\omega}$ for all $m$.
\[unionlemma\] If $(A^m_n)_{n<\omega}\prec_{L^c_{\kappa, \kappa}}(A^{m+1}_n)_{n<\omega}$ for all $m<\omega$, then $(A^m_n)_{n<\omega}\prec_{L^c_{\kappa, \kappa}}(A^{\omega}_n)_{n<\omega}$.
By the usual Tarski-Vaught argument, noticing that every bounded sequence in the union is bounded in one of the models. ${\bigstar}_{\ref{unionlemma}}$
The chain-independent fragment
------------------------------
A point in favour of chain models seems to be the difference between the views that various chain decompositions of the same model can take on the same sentence. The following theorem renders that intuition formal. To avoid trivialities, we formulate it in terms of proper chain models.
\[chainindependent\] An $L_{\kappa, \kappa}$-sentence $\varphi$ is said to be [*chain-independent*]{} if for any proper chain models $(A_n)_{n<\omega}, (B_n)_{n<\omega}$ we have that $$\bigcup_{n<\omega}A_n=\bigcup_{n<\omega}B_n \implies [(A_n)_{n<\omega}\models^c \varphi\iff
(B_n)_{n<\omega}\models^c \varphi].$$ The [*chain-independent fragment of*]{} $L_{\kappa, \kappa}$ is the set of chain-independent sentences of $L_{\kappa, \kappa}$.
It is obvious that $L_{\kappa\omega}$ is included in the chain-independent fragment of $L_{\kappa, \kappa}$.
\[nondependent\] An $L_{\kappa, \kappa}$-sentence $\varphi$ is chain-independent iff for any model ${\mathfrak A}$ of size $\kappa$ $${\mathfrak A}\models \varphi\iff \mbox{ for any proper chain model }(A_n)_{n<\omega}\, [\bigcup_{n<\omega}\, A_n={\mathfrak A}\implies (A_n)_{n<\omega}\models^c \varphi].$$
For the proof we require the following lemma.
\[helpful\] Suppose that ${\mathfrak A}$ is a $\tau$ model and $\varphi$ an $L_{\kappa, \kappa}$-sentence such that $(A_n)_{n<\omega}\models^c \varphi$ for any chain decomposition $\langle A_n:\, n<\omega\rangle$ of ${\mathfrak A}$. Then ${\mathfrak A}\models \varphi$.
[**of Lemma \[helpful\].**]{} By induction on the complexity of $\varphi$. ${\bigstar}_{\ref{helpful}}$
For the proof of the Theorem, first suppose that $\varphi$ is chain-independent, ${\mathfrak A}\models \varphi$ and $(A_n)_{n<\omega}$ is a proper chain decomposition of ${\mathfrak A}$. If $(A_n)_{n<\omega}{\nvDash}^c \varphi$ then by the definition of chain-independence it follows that for any proper chain decomposition $(B_n)_{n<\omega}$ of ${\mathfrak A}$ we have $(B_n)_{n<\omega}\models ^c \neg\varphi$. By Lemma \[helpful\] it follows that ${\mathfrak A}\models \neg\varphi$, a contradiction.
In the other direction, suppose that $\varphi$ satisfies the equivalence, but $\varphi$ is not chain-independent. Then there are proper chain decompositions $(A_n)_{n<\omega}$ and $(B_n)_{n<\omega}$ of ${\mathfrak A}$ such that $(A_n)_{n<\omega}\models^c \varphi$ and $(B_n)_{n<\omega}\models^c \neg\varphi$, a contradiction. ${\bigstar}_{\ref{nondependent}}$
Theorem \[nondependent\] can be used to show that the chain-independent fragment can express a number of classical concepts:
\[Aronszajn\] Suppose that $\lambda<\kappa$ and ${{\rm cf}}(\lambda)>\aleph_0$. The following properties can be expressed by chain-independent sentences of $L_{\kappa, \kappa}$.
- $G$ is a graph which omits a clique of size $\lambda$,
- $T$ is a tree which omits a branch of length $\lambda$,
- $<$ is a linear order with no decreasing sequences of length $\lambda$.
The proof is basically the same for all three examples, so let us show it for the case of graphs. Let $\rho$ be an $L_{\kappa, \kappa}$ sentence which says that $R$ is a binary relation and that it has no clique of size $\lambda$. We claim that $\rho$ is chain-independent.
Let ${\mathfrak A}$ be a model of $\rho$ size $\kappa$ and let $(A_n)_{n<\omega}$ a proper chain model with $\bigcup_{n<\omega} A_n={\mathfrak A}$. If ${\mathfrak A}\models\rho$ then clearly no $A_n$ can have a clique of size $\lambda$. Suppose on the other hand that no $A_n$ has a clique of size $\lambda$, but that ${\mathfrak A}$ has such a clique, say $K$. Since ${{\rm cf}}(\lambda)>\aleph_0$ and ${\mathfrak A}$, there will be $n$ such that $K\cap A_n$ has size $\lambda$. Therefore $(A_n)_{n<\omega}\models^c \neg \rho.$ ${\bigstar}_{\ref{Aronszajn}}$
Concluding remarks {#fin}
==================
We have shown that in the case of a singular cardinal $\kappa$ of countable cofinality which in addition satisfies $\kappa=\beth_\kappa$, the chain logic $L^{c,\ast}_{\kappa, \kappa}$ satisfies all the motivating requirements for the introduction of the logic $L^1_\kappa$, including solving Problem 1.4 from [@Sh797] which was left open by $L^1_\kappa$. A caveat to note, however, is that $L^1_\kappa$ and $L^{c,\ast}_{\kappa, \kappa}$ have a major difference in what they consider a structure: the former has the classical concept of a structure while the latter is based on chain models.
One could look for further desirable properties of a logic at a singular cardinal, most importantly the property of $\kappa$-compactness. Neither $L^1_\kappa$ nor chain logic have it.
[^1]: We thank Stamatis Dimopoulos and Will Boney for discussion involving the comparison of the chain logic and the logic $L_{\kappa,\omega}$. Mirna Džamonja gratefully acknowledges the help of Leverhulme Trust through research Fellowship 2014-2015, l’Institut d’Histoire et de Philosophie des Sciences et des Techniques, CNRS et Universit' e Paris 1, where she is an Associated Member, and the University of Helsinki for their hospitality in October 2014 and June 2016.
Keywords: singular cardinal, strong undefinability of well order, $L^1_\kappa$, chain models, abstract logic.
MSC 2010 Classification: 03C95, 03C85, 03E75.
[^2]: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK, ,
[^3]: Department of Mathematics, University of Helsinki, Helsinki, Finland, and Institute for Logic, Language and Computation, University of Amsterdam, Netherlands,,
[^4]: but see [@JoukoLindstrom] for some examples and further references
[^5]: Some authors denote what we call $\lambda$-compactness by [*strong*]{} $\lambda$-compactness.
[^6]: We note that in the area of cardinal invariants in set theory of the reals and topology, similar reductions were introduced by Vojt[á]{}[š]{} in [@Vojtas].
[^7]: hence, automatically, models of $\tau$, since $\tau$ is relational
[^8]: More historical details and comparisons with the classical $L_{\kappa, \kappa}$ can be found in [@singtrees].
[^9]: These models are typically much bigger than $\theta$ but not necessarily smaller than $\kappa$.
[^10]: Moreover, Stavi [@StaviI] showed that below a measurable cardinal this is true for basically any logic.
[^11]: See Chapter IV of [@Dickmann].
[^12]: It says that some concessions were made in the definition of $L^1_\kappa$ in order to have the Union Lemma hold for $L^1_\kappa$.
|
---
abstract: |
Power density constraints are limiting the performance improvements of modern CPUs. To address this we have seen the introduction of lower-power, multi-core processors, but the future will be even more exciting. In order to stay within the power density limits but still obtain Moore’s Law performance/price gains, it will be necessary to parallelize algorithms to exploit larger numbers of lightweight cores and specialized functions like large vector units. Example technologies today include Intel’s Xeon Phi and GPGPUs.\
Track finding and fitting is one of the most computationally challenging problems for event reconstruction in particle physics. At the High Luminosity LHC, for example, this will be by far the dominant problem. The need for greater parallelism has driven investigations of very different track finding techniques including Cellular Automata or returning to Hough Transform. The most common track finding techniques in use today are however those based on the Kalman Filter [@Fruhwirth]. Significant experience has been accumulated with these techniques on real tracking detector systems, both in the trigger and offline. They are known to provide high physics performance, are robust and are exactly those being used today for the design of the tracking system for HL-LHC.\
Our previous investigations showed that, using optimized data structures, track fitting with Kalman Filter can achieve large speedup both with Intel Xeon and Xeon Phi. We report here our further progress towards an end-to-end track reconstruction algorithm fully exploiting vectorization and parallelization techniques in a realistic simulation setup.
address:
- '$^1$ University of California - San Diego , La Jolla, CA, 92093, USA'
- '$^2$ Department of Physics, Princeton University, Princeton, NJ 08540, USA'
- '$^3$ Center for Advanced Computing, Cornell University, Ithaca NY 14853, USA'
- '$^4$ Laboratory of Elementary Particle Physics, Cornell University, Ithaca NY 14853, USA'
author:
- 'Giuseppe Cerati$^1$, Peter Elmer$^2$, Steven Lantz$^3$, Kevin McDermott$^4$, Dan Riley$^4$, Matevž Tadel$^1$, Peter Wittich$^4$, Frank Würthwein$^1$, Avi Yagil$^1$'
title: Kalman Filter Tracking on Parallel Architectures
---
Introduction
============
Pile-up (PU) represents a challenge for HEP event reconstruction, both in terms of physics performance and in terms of processing time. In fact, for PU values exceeding 100, as expected at the High Luminosity LHC (HL-LHC), the time needed for event reconstruction diverges (Fig. \[reco-time-pu\]); due to power density limitations to Moore’s law, such a large increase is not sufficiently compensated by an increase in CPU clock frequency. For this reason, the reconstruction model currently used is most HEP experiments, based on traditional computers both for offline and online processing, cannot be sustained at HL-LHC without compromises between timing and physics performance that will impact the final sensitivity of the experiments.
As a solution to this problem we investigate a transition to highly parallel computing architectures such as e.g. Intel’s Xeon Phi and NVIDIA GPGPUs. The challenge in this context is that, due to distinct features of these architectures, a simple porting of the current, ’serial’, implementation of the algorithms would be highly suboptimal: code and data-structures need to be redesigned with a strong emphasis on hardware capabilities and limitations.
We initially focus our study on the Xeon Phi architecture. The main reason for this choice is that it shares many features with more conventional architectures, like the Intel Xeon, so that we can optimize and test the algorithm performance on both architectures at the same time; however, there is no real prejudice on the choice of the architectures and in the future we plan to explore also other options, including GPGPUs. Xeon Phi coprocessors are characterized by 60 cores running up to 4 threads each and featuring 512-bit wide vector units; thus, in order to achieve optimal performance all cores must be kept occupied, usage of vector units must be maximized, and code branching must be minimized.
We tackle the problem starting from the the most challenging algorithm, track reconstruction; tracking is by far the most time consuming process in event reconstruction (Fig. \[reco-time-pu\]) so that, if the proposed approach does not work in this case, there is little to gain from the rest of event reconstruction. Other algorithms, such as vertexing, jet clustering and so on will be studied at a later stage. The present report describes the result of early studies and outlines the next steps of the project.
![\[reco-time-pu\] Timing vs luminosity for $t\bar{t}$+PU samples with 25 ns bunch crossing. Samples with different average pile-up are used as reported on the plot. Shown is timing of full CMS reconstruction and of the iterative tracking sequence [@cerati].](RecoTimeLUMI_25_BX_25ns.png){width="20pc"}
Setup and Previous Results
==========================
Below is a brief summary of the test setup and of previous results; a more detailed description can be found in [@acat].
In order to study the problem with maximum flexibility and with gradually increasing complexity, we developed a standalone tracking code based on Kalman Filter and running on a simplified setup consisting of an ideal barrel geometry, a uniform and longitudinal magnetic field, gaussian-smeared hit positions, a particle gun simulation with flat transverse momentum distribution between 0.5 and 10 GeV, no material interaction and no correlation between particles nor decays.
The track reconstruction process can be divided in 3 steps: track seeding, building and fitting. The track fit is the simple application of the Kalman Filter to a pre-determined set of hits, so it was a natural choice as a starting point. It is implemented using Matriplex, a library for vectorized matrix operations we developed. Matriplex is a matrix-major representation, where vector units elements are separately filled by the same element from different matrices; in other terms, $n$ matrices - and thus $n$ tracks - work in sync so that vectorization is used as an additional resource for parallelism. The track fit using Matriplex gives same physics results and, even in serial case, it is faster than an equivalent version based on SMatrix.
We tested the track fit both on Intel Xeon and Xeon Phi (native application) with OpenMP, obtaining similar qualitative results. We observed a large speedup both from vectorization (4x/8x on Xeon/Xeon Phi) and parallelization (10x/80x). The effective utilization efficiency of vector units is about 50%; time scaling is close to ideal in case of 1 thread/core, while there is some overhead with 2 threads/core. Main performance limitations are related to data availability in L1 cache and data re-packing in Matriplex format.
As steps forward, we followed two distinct lines of development. On the one side we consolidated the track fitting results, identifying the performance bottlenecks to vectorization and progressing towards a fully realistic setup. On the other side, we tackled the most time consuming part of the track reconstruction algorithm, redesigning the track building in order to exploit vectorization and parallelization.
Improvements to Track Fit
=========================
A detailed profiling of the code showed that the relative fraction of time used for data input is large with respect to total fit time, where input refers both to data transfer to L1 cache and re-packing into Matriplex format. Figure \[vectorize-data-input\] compares the time spent for data input using different methods to the actual fit time with Xeon Phi on a single-threaded execution using the vector units to the extent possible. We explored different approaches (methods 1-3) and different intrinsics (4-6), either with a direct copy approach (1 and 6) or with a two-stage copy using temporaries (2-5), resulting in substantially different performance. The best method (\#4) relies on two-stage copy and employs the vgather intrinsic.
![\[vectorize-data-input\]\
1. scatter data into Matriplexes, track by track, using simple for-loops\
2. copy 16 tracks into contiguous memory, transpose into Matriplexes via loops\
3. copy 16 tracks into memory, use Intel MKL to transpose each Mplex\
4. copy 16 tracks into memory, vgather into Matriplexes by rows\
5. copy 16 tracks into memory by rows, vscatter into Matriplexes\
6. scatter data into Matriplexes, track by track, using Intel intrinsics ](vectorize-data-input.pdf){width="20pc"}
We also made significant progress towards the larger goal of implementing an end-to-end tracking chain in a realistic simulation, with track fitting as the final step. Our initially simple setup was designed to allow increases in complexity, so we can add more realistic features in an incremental way. In particular, we included a more realistic detector description implementing different barrel-like layouts by using the USolids geometry package [@usolids]. Such changes are transparent to the actual fitting code since the track parameters are propagated to the radius of the next hit in global coordinates. We verified that the timing and physics performance of the track fit are not affected (Fig. \[pTpull-cylindrical\] and \[pTpull-polyhedric\]). Further improvements include the options to account for the effects of multiple scattering in the simulation and, instead of fitting the track hits as obtained from the simulation, to fit the tracks returned by the track building.
![\[pTpull-polyhedric\] Pull of the track transverse momentum using an barrel geometry composed of polyhedral layers.](pTpull-cylindrical.pdf){width="17pc"}
![\[pTpull-polyhedric\] Pull of the track transverse momentum using an barrel geometry composed of polyhedral layers.](pTpull-polyhedric.pdf){width="17pc"}
Track Building
==============
The track building is the part of the track reconstruction process that, starting from a proto-track made of a 2-3 hits (the seed), identifies all other hits belonging to the track. From the computing point of view, it is by far to most expensive task in HEP event reconstruction.
While it entails similar calculations as the track fitting (i.e. track parameter propagation and update, $\chi^2$ of each measurement), it adds two big complications to the problem. First, while during fitting the set of hits to process is already defined, in the building case a large number of hits is present at each layer, typically of the same order of the track multiplicity in the event. Second, when more than one compatible hit is identified on the same layer, the combinatorial nature of the algorithm requires that a new track candidate is created for each compatible hit.
Data locality is the key for reducing the large number of hits problem, so that only a reduced set of hits is considered at a given time and thus needs to be available in lower caches levels. For this purpose, we partition the space both in pseudorapidity ($\eta$) and in azimuthal angle ($\phi$). We exploit the fact that tracks do not bend in $\eta$ to define self-consistent $\eta$ partitions which are redundant in terms of hits, so that there are no boundary effects and track candidates never search outside their $\eta$ bin (Fig. \[eta-segmentation\]); $\eta$ bins constitute a natural splitting for thread definitions. A simple partitioning in $\phi$ is used for fast look-up of hits in the 3$\sigma$ compatibility window, where $\sigma$ is the uncertainty on the propagated track $\phi$ on the considered layer.
![\[eta-segmentation\] Diagrams illustrating one single $\eta$ bin (left) and the full $\eta$ partitioning (right).](eta-segmentation.pdf){width="38pc"}
The two issues described above (i.e. hit multiplicity and branching of candidates) can be addressed separately by factorizing the problem in two stages. In a first version of the algorithm, at each layer all hits within the compatibility window of a given track are probed and only the hit with the lowest $\chi^2$ is chosen, so that there is no multiplication of candidates; this simple and fast version already achieves good hit collection performance: running over 10 events with 20k tracks each, 70% (93%) of tracks are reconstructed with $\geq$90% (60%) of the hits. In this setup we studied the performance in terms of vectorization, obtaining a maximum speedup of $>$2x both on Xeon and Xeon Phi (Fig. \[vector-host\] and \[vector-mic\]). The scaling with vector unit size is monotonic on Xeon while an overhead is observed when vectorization is enabled on Xeon Phi, followed by a gradual speedup; the usage of prefetching and gathering instrinsics gives a further time reduction when the vector units are maximally used. It should be noted that, with respect to the fitting case, worse vectorization performance is expected since the latency due to data input is amplified by the much larger number of hits processed at each layer.
![\[vector-mic\] Track building time (best hit version) vs number of elements in vector unit for Xeon Phi.](./vector-host-bh.pdf){width="17pc"}
![\[vector-mic\] Track building time (best hit version) vs number of elements in vector unit for Xeon Phi.](./vector-mic-bh.pdf){width="17pc"}
The second version is more advanced and targets ultimate physics performance. In this case we allow the branching of candidates, so that at each layer new candidates are created for all hits passing a $\chi^2$ cut of 30; also, in order to account for outlier hits and detector inefficiencies, candidates are allowed to miss the hit on one layer. Up to 10 candidates per seed are considered and, when this number is exceeded, candidates are sorted based on the number of hits and the total $\chi2$ and those in excess are discarded. Running over the same set of events of 20k tracks each, the hit collection performance is significantly improved and 85% (95%) of the tracks are reconstructed with $\geq$90% (60%) of the hits. We tested the performance of this algorithm in terms of parallelization. Threads are evenly distributed threads across 21 eta bins, so that when the number of eta bins (nEtaBins) is a multiple of the number of threads (nThreads), n=nEtaBins/nThreads eta bins are processed in each thread, while for nThreads multiple of nEtaBins, seeds in each eta bin are distributed across n=nThreads/nEtaBins threads. A speedup of 5x on Xeon and $>$10x on Xeon Phi is achieved (Fig. \[parallel-host\] and \[parallel-mic\]); on Xeon Phi, we observe a saturation above nThreads=42 that will require further investigations.
![\[parallel-mic\] Track building time vs number of threads for Xeon Phi.](./parallel-host-fc.pdf){width="17pc"}
![\[parallel-mic\] Track building time vs number of threads for Xeon Phi.](./parallel-mic-fc.pdf){width="17pc"}
Conclusions and Outlook
=======================
In summary, a first implementation of a vectorized and parallelized track building in a simplified setup achieves significant speedup both on Xeon and Xeon Phi: 2x from vectorization on both architectures, 5x on Xeon and 10x on Xeon Phi from parallelization. Track fitting studies have been consolidated by implementing an improved data input method and a more realistic detector geometry description.
The current results are definitely promising but many developments are still needed to deploy a complete tracking on parallel architectures. The comparison with the scaling for ideal performance suggests a large margin for further improvements, so the first task is clearly to identify and address the current bottlenecks in the track building algorithm, both for vectorization and parallelization. In addition, the study of how to exploit vectorization in presence of branching points in the algorithm is far from being trivial and still needs to be addressed. Lastly, the performance gain obtained on our simplified and standalone configuration will have to be propagated to an end-to-end track reconstruction sequence on a fully realistic setup.
Acknowledgments
===============
This work was partially supported by the National Science Foundation, partially under Cooperative Agreement PHY-1120138, and by the U.S. Department of Energy.
References {#references .unnumbered}
==========
[9]{}
G. Cerati \[CMS Collaboration\], “Vertexing and Tracking Algoritms at High Pile-Up,” PoS Vertex [**2014**]{}, 037 (2015).
R. Fruhwirth, “Application of Kalman filtering to track and vertex fitting,” Nucl. Instrum. Meth. A [**262**]{}, 444 (1987).
G. Cerati, P. Elmer, S. Lantz, I. MacNeill, K. McDermott, D. Riley, M. Tadel and P. Wittich [*et al.*]{}, “Traditional Tracking with Kalman Filter on Parallel Architectures,” arXiv:1409.8213 \[physics.ins-det\].
M. Gayer, J. Apostolakis, G. Cosmo, A. Gheata, J. M. Guyader and T. Nikitina, “New software library of geometrical primitives for modeling of solids used in Monte Carlo detector simulations,” J. Phys. Conf. Ser. [**396**]{}, 052035 (2012).
|
---
abstract: 'A study is presented to extend the measurements of photon-photon scattering in ultra-peripheral Pb-Pb collisions at the LHC into the mass region of the pseudoscalar resonances $\eta$ and $\eta''$. The elementary photon-photon scattering cross section is presented. The cross section for photon-photon scattering in Pb-Pb is derived by convoluting the elementary photon-photon cross section with the Pb-Pb photon luminosity. The main background to two-photon final states, arising from double $\pi^{0}$ production with two of the four decay photons escaping detection, is examined, and possible kinematical conditions are discussed to optimize the signal-to-background ratio for such measurements at mid-rapidity.'
address:
- 'Phys. Inst., Heidelberg'
- 'Inst. Nucl. Phys., PAN Krak$\acute{o}$w, Univ. of Rzesz$\acute{o}$w'
author:
- |
Rainer Schicker\
[Mariola Kłusek-Gawenda, Antoni Szczurek]{}
title: 'Photon-photon scattering in the resonance region at midrapidity at the LHC. [^1]'
---
Introduction
============
Classical electrodynamics is epitomised by Maxwell’s equations $$\partial_{\alpha}F^{\alpha\beta} = \frac{4\pi}{c}J^{\beta},\hspace{1.cm}
\partial_{\alpha}{\cal F}^{\alpha\beta} = 0 \, .
\label{Eq:Maxwell}$$
The Maxwell equations in vacuum are linear in the electromagnetic fields [**E**]{} and [**B**]{}. Two electromagnetic waves will pass through each other without scattering. The superposition principle conveniently expresses the non-interaction of electromagnetic fields at the classical level. The electromagnetic field energy carried by the electron, however, poses a conceptual challenge at the classical level. Attempts to circumvent the infinite Coulomb energy of a point charge resulted in Born-Infeld electrodynamics which affirms the linearity of Maxwell’s equations down to some length scale intrinsic to the electron, and introduces non-linear equations for smaller lengths scales [@Born]. The polarisation of the vacuum in view of Dirac’s positron theory led to the Euler-Kockel-Heisenberg Lagrangian which modifies the classical Maxwell’s equation in vacuum by leading non-linear terms [@Heisenberg]. The advent of accelerating heavy-ions at the LHC has opened up the possibility of measuring the photon-photon scattering cross section due to the large associated photon luminosity of the heavy-ion beams. Evidence of such events have been reported by the ATLAS and CMS collaborations at the LHC [@ATLAS; @CMS]. These measurements are, however, restricted to photon-photon invariant masses W$_{\gamma\gamma} >$ 5 and 6 GeV for the CMS and ATLAS data, respectively. The purpose of the analysis presented here is to study the feasibility of measuring photon-photon scattering in the range 0.4$<$W$_{\gamma\gamma}<$5 GeV. As a first step, we analyse the corresponding cross section in this range, and examine the background which is dominated by $\pi^{0}$ pair production with only two of the four decay photons being within the detector acceptance.
The elementary photon-photon scattering cross section
=====================================================
Different mechanisms contribute to the elementary $\gamma\gamma\rightarrow\gamma\gamma$ scattering.
![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_box.pdf "fig:"){width="3.2cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_gluon.pdf "fig:"){width="4.4cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_IPIR.pdf "fig:"){width="4.4cm"}
In Fig. \[Fig:gg\_elem\], the different mechanisms of $\gamma\gamma$ scattering are presented. On the left, the loop diagram for fermions, leptons and quarks, is shown. In the center, a QCD correction is displayed corresponding to a three-loop mechanism. On the right, the analogous process as expressed in the vector dominance approach is shown [@Sz1].
Photon-photon scattering in ultra-peripheral heavy-ion reactions
================================================================
The cross section for photon-photon scattering in ultra-peripheral heavy-ion collisions can be calculated by folding the cross section of the elementary mechanisms shown in Fig. \[Fig:gg\_elem\] with the equivalent photon flux [@Baur], $$N(\omega,b) = \frac{Z^{2}\alpha}{\pi^{2}}
\frac{1}{\beta^{2}b^{2}}
\bigg| \int_0^{\infty} dv\; v^{2} J_{1}(v)\frac{F_{el}(-\frac{u^{2}+v^{2}}{b^{2}})}{u^{2}+v^{2}}\bigg|^{2} \, .
\label{Eq:EPA}$$
The equivalent photon flux is defined in Eq. \[Eq:EPA\]. Here, $\omega$ denotes the energy of the photon, and b represents the transverse distance from the center of the nucleus where the photon density is evaluated. The integral in Eq. \[Eq:EPA\] represents the form factor of the charge distribution of the source.
![Differential cross section d$\sigma$/dM for photon scattering in PbPb-collisions.[]{data-label="Fig:dsigma1"}](dsigmadM_signal.png){width="6.5cm"}
The cross section for $\gamma\gamma\rightarrow\gamma\gamma$ in PbPb-collisions is calculated by convoluting the elementary cross section with the photon flux ,
$$\sigma^{EPA}_{PbPb \rightarrow PbPb\gamma\gamma} = \int\!\!\int dn^{1}_{\gamma}
\; dn^{2}_{\gamma} \; \sigma_{\gamma\gamma \rightarrow \gamma\gamma}(\omega_1,\omega_2) \, .
\label{Eq:dsigmadM_PbPb}$$
The differential cross section for photon-photon scattering in PbPb-collisions at $\sqrt{s}$=5.02 TeV resulting from the convolution of Eq. \[Eq:dsigmadM\_PbPb\] is shown in Fig. \[Fig:dsigma1\]. This cross section is derived with the box diagrams of Fig. \[Fig:gg\_elem\], and with conditions of the two final state photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV.
Resonance signal from $\eta,\eta'$ decays
=========================================
The cross section for photoproduction of $\eta,\eta'$ is taken according to [@Budnev] $$\sigma_{\gamma\gamma\rightarrow R} = 8\pi(2J+1)\frac{\Gamma_{\gamma\gamma}\Gamma_{tot}}{(W^{2}-M^{2}_{R})^{2}+M^{2}_{R}\Gamma^{2}_{tot}} \, .
\label{Eq:sigma_res}$$
The cross section for photoproduction of $\eta,\eta'$ in PbPb-collisions is calculated according to the convolution defined in Eq. \[Eq:dsigmadM\_PbPb\].
![Cross section for photoproduction of $\eta,\eta'.$[]{data-label="Fig:dsigmadM_eta"}](dsigmadM_signaleta.png){width="6.5cm"}
The $\eta,\eta'$ cross section in PbPb-collisions at $\sqrt{s}$= 5.02 TeV multiplied by the branching ratio $\eta,\eta'\rightarrow\gamma\gamma$ is shown in Fig. \[Fig:dsigmadM\_eta\]. The values shown are derived with conditions of the two decay photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV. The finite width of these two resonances results from the detector resolution of the photon measurements which is taken here to be $\sigma_{E_{phot}}/E$ = 0.02.
Photoproduction of $\pi^{0}\pi^{0}$ pairs
=========================================
The main background to the signal of two photons in the final state results from $\pi^{0}\pi^{0}$ production with two of the four decay photons escaping detection. The two measured photons from this $\pi^{0}\pi^{0}$ decay cannot be distinguished from the signal of photon-photon scattering.
![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](pires.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl1.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl2.pdf "fig:"){width="4.cm"}
The contribution of resonance decays to the $\pi^{0}\pi^{0}$ final state is shown in Fig. \[Fig:pion\_pairs\] on the left. Here, the resonances $\sigma$(600), $f_{0}$(980), $f_{0}$(1500), $f_{0}$(1710), $f_{2}$(1270), $f^{`}_{2}$(1525), $f_{2}$(1565), $f_{2}$(1950) and $f_{4}$(2050) are considered [@Sz2]. The cross section $\gamma\gamma\rightarrow\pi^{+}\pi^{-}$ is much larger than for $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$, hence even a small coupling between the charged and neutral pion channel might have an influence on the $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$ cross section. Examples of processes leading to such channel couplings are shown in Fig. \[Fig:pion\_pairs\] in the middle and on the right.
Background from $\pi^{0}\pi^{0}$ decays
=======================================
The $\pi^{0}\pi^{0}$ cross section in PbPb-collisions is calculated the elementary $\pi^{0}\pi^{0}$ cross section with the photon flux according to Eq. \[Eq:EPA\]. The background from $\pi^{0}\pi^{0}$ decays is shown in Fig. \[Fig:bck\_pi0\] by the solid red line. This background cross section is derived by the conditions that exactly one decay photon from each of the two $\pi^{0}$’s is within the pseudorapidity range $|\eta| <$0.9, and that these photons have an energy E$_{phot} >$200 MeV.
![Signal and background from $\pi^{0}\pi^{0}$ decays.[]{data-label="Fig:bck_pi0"}](dsigmadM_signaletabck.png){width="6.5cm"}
Background suppression by asymmetry cuts
----------------------------------------
The two photons of the signal are of equal transverse momentum and are back-to-back in azimuth. These correlations are smeared out due to finite resolution in the measurement of photon energy and azimuthal angle. The two photons from $\pi^{0}\pi^{0}$ decay do not show these correlations. and vector ($A_{V}$) asymmetry can be defined for background suppression, $$A_S=\left|\frac{|\vec{p}_T(1)|-|\vec{p}_T(2)|}
{|\vec{p}_T(1)|+|\vec{p}_T(2)|}\right|,\hspace{1.cm}
A_V=\frac{|\vec{p}_T(1)-\vec{p}_T(2)|}
{|\vec{p}_T(1)+\vec{p}_T(2)|} \, .$$
![Background suppression by asymmetry cut A$_{S}$.[]{data-label="Fig:bck_pi0_asym"}](dsigmadM_sigbck.png){width="6.5cm"}
The background reduction for condition A$_{S}\!<$0.1 and A$_{S}\!<$0.02 is shown in Fig. \[Fig:bck\_pi0\_asym\] by the blue and red line, respectively. The condition A$_{S}<$0.02 reduces the background by about a factor $\sim$10 while keeping 98%
Background correction by sideband subtraction
---------------------------------------------
The background remaining after the cut A$_{S}\!<\!0.02$ can be subtracted by sideband correction. The signal band is given by with sideband 1 and 2 defined by $0.02\!<\!A_{s}\!<\!0.04$ and $0.04\!<\!A_{s}\!<\!0.06$, respectively. An estimator for the background in the signal region can be defined by linear extrapolation of the sidebands 1 and 2 into the signal region.
![Sideband corrected background.[]{data-label="Fig:bck_pi0_asym_corr"}](dsigmadM_sidecorr.png){width="6.5cm"}
The background remaining after sideband correction is shown by the red line. The sideband correction could be further improved by defining more than two sidebands, and by a non-linear extrapolation of the background into the signal region.
ACKNOWLEDGMENTS
===============
This work is supported by the German Federal Ministry of Education and Research under promotional reference 05P15VHCA1.
[99]{} M.Born, Nature 132:3329 (1933) 282. W.Heisenberg, H.Euler, Zeitschrift für Physik 98: 11-12 (1936) 714. ATLAS Collaboration, Nature Phys. 13 (2017) no.9, 852. CMS Collaboration, arXiv:1810.04602. M.Kłusek-Gawenda et al., Phys.Rev. C93 (2016), no.4, 044907. G.Baur et al., Phys.Rept. 364 (2002) 359. V.M.Budnev, $\!$I.F.Ginzburg, $\!$G.V.Meledin, $\!$V.G.Serbo, $\!$Phys.Rept.15, $\!$(1975) $\!$181. M.Kłusek-Gawenda, A.Szczurek, Phys.Rev. C87 (2013), no.5, 054908.
[^1]: Presented at Diffraction and Low-x Conference 2018, Reggio Calabria,
|
---
author:
- 'A. Kalz, A. Honecker, S. Fuchs'
- 'T. Pruschke'
date: 'Received: May 7, 2008 / Revised version: July 18, 2008'
title: Phase diagram of the Ising square lattice with competing interactions
---
Introduction \[intro\]
======================
The search for exotic groundstates in two-dimensional frustrated quantum spin systems is a topic of intense research (see, e.g., Refs. [@P:misguich05; @P:richter04] for recent reviews). In this context, the antiferromagnetic $J_1$-$J_2$ spin-1/2 Heisenberg model on the square lattice has been intensively studied during the past two decades. Nevertheless, the nature of an intermediate non-magnetic phase around $J_2 \approx J_1/2$ has remained under debate until recently (see Refs. [@P:misguich05; @P:mambrini06] and references therein). Among the possible approaches to the antiferromagnetic $J_1$-$J_2$ Heisenberg model, quantum Monte-Carlo (QMC) simulations suffer from a severe sign problem in the region $J_2 \approx J_1/2$. Other approaches include perturbation theory around the Ising limit [@P:oitmaa96; @P:singh03]. A closely related model is given by hard-core bosons on the square lattice with nearest neighbour (NN) and next-nearest neighbour (NNN) hopping and repulsion terms [@P:batrouni00; @P:batrouni01; @P:wessel08; @P:chen08]. In this case, there is no sign problem such that QMC simulations are possible in principle [@P:batrouni00; @P:batrouni01; @P:wessel08; @P:chen08]. However, simple QMC algorithms suffer freezing problems in the intermediate regime at $J_2 = J_1/2$ which can be traced to a groundstate degeneracy of the Ising limit. This motivated us to perform a model study by solving the related freezing problems in Monte-Carlo (MC) simulations of the Ising model.
Investigations of the two-dimensional $J_1$-$J_2$ Ising model on the square lattice have an even longer history than of the corresponding Heisenberg model, including in particular MC simulations (see, e.g., [@P:SK79; @P:landau80; @P:landaubinder80; @P:landaubinder85; @P:bloete87; @P:MKT06; @B:landaubinder]). Nevertheless, certain issues have remained controversial also in the case of the Ising model, in particular the nature of the finite-temperature phase transition for $J_2 > J_1/2$: MC simulations [@P:SK79; @P:landaubinder85; @P:MKT06] and an investigation of the Fisher zeros of the partition function [@P:monroe07] have suggested non-universal critical exponents, whereas a variational approach [@P:lopez93] and a differential operator technique [@P:AVS08] predict a first-order transition for $J_1/2 < J_2 \lesssim J_1$. In this paper we resolve this issue in favour of a weak first-order transition at least for not too large $J_2 > J_1/2$ by providing substantially improved MC results for the phase diagram.
This paper is organized as follows: in section \[sec:model\] we introduce the $J_1$-$J_2$ Ising model on the square lattice and discuss its $T=0$ groundstates. The MC simulation methods are described in section \[sec:methods\] and results are presented in section \[sec:results\]. We conclude with a summary and outlook in section \[sec:discussion\].
Model \[sec:model\]
-------------------
We study the classical Ising model with competing antiferromagnetic interactions $J_1$ on the NN bonds and $J_2$ on the NNN bonds ($J_i > 0$): $$\begin{aligned}
H = J_1 \sum_{\text{NN}} S_i S_j + J_2 \sum_{\text{NNN}} S_i S_j \, ,
\quad S_i =\pm1 \, .
\label{eq:En}\end{aligned}$$ We will study square lattices of linear extent $L$ with periodic boundary conditions.
For $J_2 = 0$ the groundstate of (\[eq:En\]) is the known antiferromagnetic solution which is a Néel ordered lattice (see Fig. \[fig:degeneracy\], left) with energy $E = -2 J_1 N$ ($N = L^2$ is the number of sites). Switching on the repulsive interaction on the diagonal (NNN) bonds of the square lattice yields an increase of the groundstate energy for the Néel-ordered state: $$\begin{aligned}
E_{\text{N\'eel}} = -2 N (J_1-J_2) \, .\end{aligned}$$ For large $J_2$ the system orders in the collinear (or superantiferromagnetic) phase (see Fig. \[fig:degeneracy\], right) where all diagonal bonds are antiferromagnetic, while half of the NN bonds is antiferromagnetic and the other half is ferromagnetic. Thus, the energy in this state depends only on $J_2$: $$\begin{aligned}
E_{\text{Coll}} = -2 N J_2 \, .\end{aligned}$$ The critical point separating these two phases lies at $J_2 = J_1/2$, where the transition temperature is suppressed to $T=0$. At this point the groundstate is highly degenerate. More precisely, there are $2^{L+1}-2$ groundstates which can be obtained as follows: flipping a line of antiparallel spins costs no energy at $J_2 = J_1/2$. Thus, one can generate almost $2^{L+1}$ groundstates from a Néel state by flipping either the $L$ horizontal or the $L$ vertical lines independently (the second Néel state can be reached in both ways). In particular, one can reach a collinear state through a series of intermediate groundstates by $L/2$ line flips, as sketched in Fig. \[fig:degeneracy\]. Accordingly, close to the critical point $J_2 = J_1/2$ the energy landscape is characterized by many local minima which are separated by large energy barriers. Therefore, MC simulations using only single-spin flips have problems to reach the configuration with global minimum energy. To solve this problem and to obtain transition temperatures also in the vicinity of $J_2 = J_1/2$ we implemented improved MC algorithms which we will discuss in the next section.
Methods \[sec:methods\]
=======================
Computational methods \[sec:comp\]
----------------------------------
We first tried to simulate the model with competing interactions using conventional single-spin flip MC simulations [@B:landaubinder]. However, near the critical point this algorithm does not provide proper results and suffers severe freezing problems in the proximity of the critical temperature and below. To overcome these problems we implemented a parallel tempering algorithm [@P:huku96; @P:marinari98; @P:hans97; @P:trebst06]: a number of simulations with the same set of parameters ($J_1$, $J_2$, $L$) but varying temperatures are simulated simultaneously. After a sufficiently large number of sweeps over the complete lattice an additional MC step proposes a configuration swap between neighbouring simulations with an acceptance rate $p(i,i+1)$: $$\begin{aligned}
p(i,i+1) = \min\left\{1, e^{\Delta \beta \Delta E}\right\} \, ,\\
\Delta \beta = \frac{1}{T_{i+1}}-\frac{1}{T_i} \, ,\quad
\Delta E = E_{i+1}-E_i \, .\nonumber\end{aligned}$$ We have chosen the temperatures $T_i$ logarithmically with a density maximum near the estimated transition temperature. A selfadjusting temperature set [@P:trebst06] was not necessary for our purposes. To manage the additional computational workload of the parallel tempering algorithm we implemented a parallelisation of the MC code via OpenMP [@B:omp00; @B:omp07]. To assure the independence of the simulations we used the <span style="font-variant:small-caps;">sprng</span> (version 4.0) [@P:masca00] lagged Fibonacci generator for producing independent random number streams for each simulation. We have checked in some samples that our MC results do not change if we replace the random number generator by the Mersenne Twister algorithm [@P:matsu98], as implemented in the <span style="font-variant:small-caps;">boost</span> libraries version 1.33.1.
Another way to improve the MC simulations is to allow not only single-spin updates but also flipping whole lines of spins in an additional MC step (as sketched in Fig. \[fig:degeneracy\]). This method simulates directly the transition between degenerate groundstates at $J_2 = J_1/2$ and helps to minimize statistical errors for lattice sizes up to $50 \times 50$. For larger lattices the probability to flip a whole line decreases rapidly for ratios $J_2 \neq J_1/2$ while the simulation time to calculate them increases with $L$. We also tried cluster updates [@P:wolff89] but in the case of competing interactions this method does not help: when one approaches the critical temperature, the clusters extend over the whole lattice and therefore do not support the ordering process.
The parallel tempering algorithm gives rise to correlations between different temperature points in a simulation which can affect the independency of the calculated data. Therefore, the meanvalues and errorbars of the shown observables are derived from at least 10 independent MC runs.
Order parameters \[phys\]
-------------------------
To distinguish the two ordered and the disordered phases we use the respective structure factors $$\begin{aligned}
S(\vec q)=\frac{1}{N}\sum_{i,j}e^{i\vec q\cdot(\vec x_i-\vec x_j)}\langle
S_i S_j \rangle\end{aligned}$$ where $\vec q$ is a vector in momentum space and indicates the different magnetic phases: $$\begin{aligned}
\vec q = (0,0) &\rightarrow& \text{ferromagnetic order,} \nonumber \\
\vec q = (0,\pi), (\pi,0) &\rightarrow& \text{collinear order,} \\
\vec q = (\pi,\pi) &\rightarrow& \text{N\'eel order.} \nonumber \end{aligned}$$ We identify $M(\vec q) = \sqrt{S(\vec q)/N}$ as our order parameter. The Néel order parameter can be calculated as staggered magnetization due to a simple sublattice rotation. Using the order of the collinear phase we can also calculate the associated order parameter via the column or line index of the underlying lattice. To find the transition temperature we calculate the fourth order Binder cumulant [@P:binder81a; @P:binder81b; @B:landaubinder] for different system sizes $L$: $$\begin{aligned}
U_4 = 1 -\frac{\langle M^4\rangle}{3\langle M^2\rangle^2}\, .\end{aligned}$$ For large enough $L$ they will meet in a single point at $T_C$ (for an example see Fig. \[fig:cumulants\]).
To analyze the character of the phase transition we calculated the specific heat and recorded time series of the energies to set up histograms [@P:challa86; @P:berg91; @P:borgs92]. Furthermore we calculated the temperature derivative of the Binder cumulant which is related with the critical exponent $\nu$ [@B:landaubinder] to study the critical behaviour of the system: $$\begin{aligned}
a L^{1/\nu} = \left.\frac{\partial U_4}{\partial T}\right|_{T=T_C}\,.
\label{e:derivU4} \end{aligned}$$ To have a closer look on the critical point $J_2 = J_1/2$ we analyzed the peaks of the specific heat via polynomial fitting.
Results {#sec:results}
=======
We calculated the critical temperatures for various ratios of $J_2 /J_1$ especially close to the critical point (Fig. \[fig:phase\]). Our data is in good agreement with MC results from Landau and Binder for $J_2 \geq 0.6J_1$ [@P:landaubinder85] and $J_2 \leq 0.4J_1$ [@P:landau80]. In addition, our improved MC algorithm with the above described parallel tempering mode and line flip updates enabled us to obtain results much closer to $J_2 = J_1/2$ ($|J_2/J_1 - 0.5|= 0.005$) for system sizes up to $N = 250~000$.
To investigate the character of the phase transition we calculated the specific heat and energy histograms for different lattice sizes. Looking at the peaks in the specific heat for different $J_2/J_1$ suggests different character of the phase transition. For $J_2 < J_1/2$ (Fig. \[fig:heat0306\](a)) we have a slowly emerging peak with growing system size. Indeed, for an Ising like transition we expect only a logarithmic divergence in the specific heat[@P:onsager44] and a critical exponent $\nu = 1$ [@P:fisher67]. To verify the Ising like behaviour we had a closer look at the derivative of the Binder cumulant $U_4$ near the critical temperature. We expect a linear correspondence in $L$ for equation (\[e:derivU4\]). An analysis of our results at $J_2 = 0.3J_1$ yields $\nu = 0.99(1)$ which is in good agreement with the expected value $\nu = 1$. On the other hand for $J_2 > J_1/2$ and in particular for the case $J_2 = 0.6 J_1$ shown in Fig. \[fig:heat0306\](b) there are different opinions about the character of the phase transition. Some authors assume a continuous phase transition with non universal exponents [@P:SK79; @P:landaubinder85; @P:MKT06; @P:monroe07] whereas other authors find a first order phase transition [@P:lopez93; @P:AVS08]. Note that our results for the specific heat (Fig. \[fig:heat0306\](b)) differ quantitatively from the approximate results of [@P:lopez93]. In particular, the maximum of the specific heat continues to diverge for growing $L$. We estimated the area under the peak in dependence on the lattice size and find it to converge towards a constant value for large enough $L$. This indicates a $\delta$-peak structure for the specific heat in the thermodynamic limit as expected for a first order phase transition. We should nevertheless mention that the value of the specific heat does not follow the finite size scaling law expected for first order transitions [@P:challa86] very well, which may be due to crossover phenomena.
As further verification of our characterisation of the order of the phase transition we computed energy histograms. First, we extracted a system-size dependent $T_C$ from the maximum of the specific heat. Then we recorded the time series of the energy for each system size and the associated $T_C$. Fig. \[fig:histo\] shows the resulting energy distribution for $J_2 = 0.3J_1$ and $J_2 = 0.6J_1$. For $J_2 = 0.3J_1$ we find a single peak[^1], consistent with a conventional second order phase transition. By contrast, for $J_2 = 0.6 J_1$ a double peak structure emerges for sufficiently large system size, see Fig. \[fig:histo\](b). Such a double peak structure is characteristic for a first order transition [@P:challa86; @P:berg91; @P:borgs92]. The fact that the double peak structure emerges only for large system sizes shows that the transition for $J_2 = 0.6 J_1$ is a weak first order one. Since we have observed similar behavior for nearby ratios of $J_2/J_1$, we believe the transition for $J_2 > J_1/2$ to be of first order, at least for not too large values of $J_2$.
Fig. \[fig:heat05\] shows the specific heat exactly at the critical point $J_2 = J_1/2$ for different system sizes. The purpose of this computation is to verify if a direct phase transition from Néel to collinear order is possible at finite temperature, or if any other finite-temperature phase could exist in this region. Previous publications [@P:landau80; @P:lopez93] have shown curves for the specific heat at $J_2 = J_1/2$ with a rounded peak, but we are not aware of any systematic finite-size analysis. In our results, we observe that the peaks of the specific heat are moving to lower temperatures for increasing system sizes, suggesting that the transition temperature is suppressed to $T_C=0$ in the thermodynamic limit. To further substantiate this conclusion, the inset of Fig. \[fig:heat05\] shows the peak positions ($T_C$) with respect to the inverse lattice length $1/L$. For the given lattice sizes we find a power law behaviour for $T_C$ and $1/L$. This is consistent with $T_C = 0$ in the thermodynamic limit.
Discussion {#sec:discussion}
==========
In this paper, we combined single-spin flip, parallel tempering and line flip MC algorithms to enhance the efficiency of MC simulations of the frustrated square-lattice Ising model. These improvements enabled us to compute critical temperatures in the direct vicinity of the critical point $J_2 = J_1/2$ where the groundstate is highly degenerate. For $J_2 < J_1/2$ there is a finite-temperature phase transition into a Néel-ordered state. This transition belongs to the two-dimensional Ising universality class. We believe that at the critical point $J_2 = J_1/2$ the transition temperature is suppressed to zero. On the right hand side ($J_2 > J_1/2$) there is again a finite-temperature phase transition into a phase with collinear order.
At $J_2 = 0.6 J_1$ we have observed a double peak structure in the histogram of the energy. This identifies the phase transition as a weakly first order one. We believe this first order transition to be generic for moderate $J_2 > J_1/2$. However, for larger $J_2$ it becomes increasingly difficult to identify a double peak structure in the histogram of the energy. Therefore, the transition could in fact become a second order one for $J_2 \gtrsim J_1$. Note that a recent MC investigation [@P:MKT06] has concluded that the transition is second order for $J_2 = J_1$, although in [@P:MKT06] only smaller lattices have been considered than in the present work.
In order to understand the possible nature of the transition at large $J_2$, it is instructive to consider the limit $J_1 = 0$ where one has two decoupled Ising models. The critical theory then consists of two conformal field theories with central charge $c=1/2$ each, *i.e.*, a $c=1$ theory. Thus, the point $J_1 = 0$ lies within the manifold of $c=1$ conformal field theories where continuously varying critical exponents are possible in principle [@P:ginsparg88]. However, the coupling given by $J_1$ turns out to be a relevant perturbation of the critical point such that this coupling should either lead to a first order transition or a fixed point with $c<1$ [@B:cardy96]. According to the classification of minimal conformal field theories with $c<1$ [@P:BPZ84] a possible second order phase transition at $J_2 \gg J_1$ should have universal exponents. Therefore, a scenario with non-universal critical behaviour [@P:SK79; @P:landaubinder85; @P:MKT06; @P:monroe07] does not appear very plausible from a conformal field theory perspective either. Our MC results show that large crossover scales exist in the present model such that very big lattices would be needed for a reliable numerical determination of the nature of the transition for $J_2 \gtrsim J_1$.
A similar phase diagram as in the Ising model is also found for the classical $J_1$-$J_2$ $X$-$Y$ [@P:simon00] and Heisenberg models [@P:weber03]. However, we believe that the nature of the phase transition for $J_2 > J_1/2$ is a different issue in these two models due to the different symmetries in spin space.
After having solved the freezing problems in the Ising model, we are now about to introduce hopping terms and to study finite-temperature properties of hard-core bosons on the square lattice using parallel tempering QMC simulations [@P:melko07].
We acknowledge financial support by the Deutsche Forschungsgemeinschaft under grant No. HO 2325/4-1 and through SFB602.
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[^1]: The histogram shown in Fig. \[fig:histo\](a) is very close to a single Gaussian curve, as is known for second order phase transitions [@P:berg91].
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---
abstract:
- 'We consider online similarity prediction problems over networked data. We begin by relating this task to the more standard class prediction problem, showing that, given an arbitrary algorithm for class prediction, we can construct an algorithm for similarity prediction with “nearly” the same mistake bound, and vice versa. After noticing that this general construction is computationally infeasible, we target our study to [*feasible*]{} similarity prediction algorithms on networked data. We initially assume that the network structure is [*known*]{} to the learner. Here we observe that Matrix Winnow [@w07] has a near-optimal mistake guarantee, at the price of cubic prediction time per round. This motivates our effort for an efficient implementation of a Perceptron algorithm with a weaker mistake guarantee but with only poly-logarithmic prediction time. Our focus then turns to the challenging case of networks whose structure is initially [*unknown*]{} to the learner. In this novel setting, where the network structure is only incrementally revealed, we obtain a mistake-bounded algorithm with a quadratic prediction time per round.'
- 'We consider online similarity prediction problems over networked data, where at each online step the algorithm receives a pair of vertices in a graph and has to decide if they share the same label. We start off by relating this task to the more standard class prediction problem, showing that, given an arbitrary algorithm for class prediction on graph vertices, we can construct an algorithm for similarity prediction with “nearly” the same mistake bound, and vice versa. Because this construction is computationally infeasible, we target our study to feasible similarity prediction on networked data. We initially assume that the network has a structure [*known*]{} to the learner. Here we observe that [Matrix Winnow]{} [@w07] has a near-optimal mistake guarantee at the price of cubic prediction time per round. This motivates our effort for an efficient implementation of a [Perceptron algorithm]{} with a weaker mistake guarantee but with only poly-logarithmic prediction time. Our focus then turns to the challenging case of networks whose structure is initially [*unknown*]{} to the learner. In this novel setting, the network structure is only incrementally revealed. For this case, we obtain a mistake-bounded algorithm with a quadratic prediction time per round.'
author:
- |
Claudio Gentile\
DiSTA, Università dell’Insubria, Italy\
`claudio.gentile@uninsubria.it`
- |
Mark Herbster\
Department of Computer Science, University College London\
Gower Street, London, WC1E 6BT, UK\
`m.herbster@cs.ucl.ac.uk`
- |
Stephen Pasteris\
Department of Computer Science, University College London\
Gower Street, London, WC1E 6BT, UK\
`s.pasteris@cs.ucl.ac.uk`
---
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Proofs
======
This appendix contains all omitted proofs. Notation is as in the main text.
|
---
abstract: 'We classify all tight holomorphic maps between Hermitian symmetric spaces.'
address: |
Department of Mathematics\
Chalmers University of Technology and the University of Gothenburg\
412 96 GÖTEBORG\
SWEDEN
author:
- Oskar Hamlet
title: 'Tight holomorphic maps, a classification'
---
Introduction
============
Let $\Gamma=\pi_1(\Sigma_g)$ be the fundamental group of an oriented compact surface $\Sigma_g$ of genus $g$ and $G=Isom(\mathcal{X})$ the isometry group of a Hermitian symmetric space of noncompact type $\mathcal{X}$. A subject of a lot of recent interest is the study of representations $\rho\colon \Gamma \rightarrow G$. In [@A9] Burger, Wienhard and Iozzi showed that maximal representations, that is representations with maximal Toledo invariant, have some striking geometrical properties. If $H$ is another Hermittian Lie group and $\eta\colon G \rightarrow H$ a positive homomorphism, then $\eta \circ \rho$ is maximal if and only if $\rho$ is maximal and $\eta$ is tight. Tight representations correspond to tight maps between the corresponding Hermitian symmetric spaces. With the exception of the irreducible representations $\rho_n\colon SU(1,1)\rightarrow Sp(2n)$ all known tight representations between Hermitian Lie groups correspond to tight holomorphic maps. In this paper we classify all tight holomorphic maps between Hermitian symmetric spaces.
Description of the problem {#descr}
==========================
Let $\mathcal{X}_1=G_1/K_1, \mathcal{X}_2=G_2/K_2$ be Hermitian symmetric spaces of the non-compact type. Equip $\mathcal{X}_i$ with the unique $G_i$-invariant metric $g_i$ such that the minimal holomorphic sectional curvature is $-1$ on each irreducible factor of $\mathcal{X}_i$. Further let $\omega_i$ be the associated Kähler forms defined by $\omega_i(X,Y):=g_i(J_iX,Y)$, where $J_i$ denotes the complex structure of $\mathcal{X}_i$ . Let $f \colon \mathcal{X}_1\rightarrow \mathcal{X}_2$ be a totally geodesic map, i.e the image of geodesics in $\mathcal{X}_1$ are geodesics in $\mathcal{X}_2$. We have $$\label{tightdef}
sup_{\Delta\subset\mathcal{X}_1}\int_\Delta{f^*\omega_2}\leq sup_{\Delta\subset\mathcal{X}_2}\int_\Delta{\omega_2}$$ where the supremum is taken over all triangles with geodesic sides. The inequality is obvious since $f(\Delta)$ is a geodesic triangle in $\mathcal{X}_2$ for $\Delta$ a geodesic triangle in $\mathcal{X}_1$. If we have equality in (\[tightdef\]) we say that the map is *tight* [@A8].\
Our objective is to classify tight holomorphic maps up to equivalence. We say that two maps are equivalent if they are equal up to an action of an element of $G_2$.
It will be useful to look at the problem from different perspectives. Let $\mathfrak{g}_i$ be the Lie algebra of $G_i$ and $\mathfrak{g}_i=\mathfrak{k}_i+\mathfrak{p}_i$ a Cartan decompostion. Let $0$ be a fixed basepoint of $\mathcal{X}_i$. Identifying $T_{0}\mathcal{X}_i$ with $\mathfrak{p}_i$, totally geodesic maps correspond to Lie algebra homomorphisms respecting the Cartan decomposition. As is well known there exists an element $Z_i$ in the center of $\mathfrak{k}_i$ such that the complex structure on $\mathfrak{p}_i$ is given by $ad(Z_i)$. A map $f$ is holomorphic if and only if the corresponding Lie algebra homomorphism $\rho$ satisfies $$\tag{H1}
\rho \circ ad(Z_1)=ad(Z_2)\circ \rho.$$ A stronger condition, that $$\tag{H2}
\rho(Z_1)=Z_2$$ will also play an important role. See [@A6] and [@A7] for a clarification of the roles of (H1) and (H2) in holomorphic representations. Our goal is reformulated as classifying all Lie algebra homomorphisms satisfying (H1) and corresponding to tight maps. We will later on give conditions in Lie algebraic terms for a homomorphism to correspond to a tight map.\
Yet another point of view is that of the corresponding homomorphisms of the isometry groups and the maps induced by these on continuous bounded cohomology. In a familiar way $\omega_i$ defines a cohomology class $\kappa_{G_i}^b\in H^2_{cb}(G_i)$, defined by $\kappa_{G_i}^b(g_0,g_1,g_2):=\int_{\Delta(g_0 \cdot 0,g_1\cdot 0,g_2\cdot 0)}{\omega_i}$ where $\Delta(g_0\cdot 0,g_1\cdot 0,g_2\cdot 0)$ is the geodesic triangle with vertices $g_0\cdot 0,g_1\cdot 0,g_2\cdot 0$. Continuous bounded cohomology is equipped with a canonical seminorm, namely the supremum norm. For a continuous representation $\rho:G_1\rightarrow G_2$, $\rho^*$ is norm decreasing, i.e $||\rho^*\kappa_{G_2}^b||\leq ||\kappa_{G_2}^b||$. If there is equality we say that $ \rho$ is a tight homomorphism. Each $\rho$ corresponds to a totally geodesic map $f$ and we have, as the name suggests, that $\rho$ is tight if and only if $f$ is tight [@A8]. We can thus view the problem on three different levels. The symmetric space, the isometry group and the corresponding Lie algebra. We will switch frequently between these perspectives as they all have their advantages. While doing so we will abuse the concepts and notation a bit. We will talk about holomorphic representations when what we mean is that the corresponding totally geodesic map is holomorphic. We will also use the same letter to denote a homomorphism and the corresponding totally geodesic map.
We divide the classification into four parts. In the section 3 we reduce the problem to the classification of tight (H2)-representations and tight regular subalgebras. In the section 4 we give the tools needed to determine if a representation or subalgebra is tight. In section 5 we classify the tight regular subalgebras and in section 6 we classify the tight (H2)-representations.
Reduction of the problem {#sectrescur}
========================
The reduction consists of three main lemmas. We start by stating these and explaining their role in the reduction. We then recall some facts about bounded cohomology and prove Lemmas \[lma1\] and \[lma2\]. We then give the definition of regular subalgebras. We will not attempt to prove Lemma \[lma3\] here but refer the reader to [@A6] instead.
\[lma1\] Let $\rho\colon \mathcal{Y}\rightarrow \mathcal{X}_1\times ... \times \mathcal{X}_n$ be a holomorphic embedding. Denote by $\pi_i$ the projection map onto the $\mathcal{X}_i$. Then $\rho$ is tight if and only if all the $\pi_i\circ\rho$ are tight.
Lemma \[lma1\] allows us to reduce our classification to the case when the target space is irreducible.
\[lma2\] Let $\rho\colon \mathcal{X}\rightarrow \mathcal{Y}$ and $\eta\colon \mathcal{Y}\rightarrow \mathcal{Z}$ be totally geodesic holomorphic maps and further assume that $\eta$ is injective. Then $\eta\circ\rho$ is tight if and only if $\rho$ and $\eta$ are tight.
Lemma \[lma2\] quickly reduces our classification to injective maps. Let $\rho\colon\mathcal{X}\rightarrow\mathcal{X}^{'}$ be a non-injective holomorphic totally geodesic map. Let $\rho\colon \mathfrak{g}\rightarrow \mathfrak{g}^{'}$ be the corresponding lie algebra homomorphism. The kernel of $\rho$ is then an ideal $\mathfrak{g}_1$ of $\mathfrak{g}$ corresponding to a subsymmetric space. Since $\mathfrak{g}$ is semisimple we can write it as a direct sum $\mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2$ and factor our homomorphism $\rho$ as $\mathfrak{g}_1\oplus\mathfrak{g}_2\stackrel{\pi_2}{\rightarrow} \mathfrak{g}_2 \stackrel{\tilde{\rho}}{\longrightarrow} \mathfrak{g}^{'}$. As projections are tight, we get that $\rho$ is tight if and only if $\tilde{\rho}$ is tight.
\[lma3\] Let $\rho\colon \mathfrak{g}\rightarrow \mathfrak{g}^{'}$ be a representation of Hermitian Lie algebras respecting the Cartan decomposition and satisfying the condition (H1). Then there exist a regular subalgebra $\mathfrak{g}^{''}\subset\mathfrak{g}^{'}$ such that $\rho(\mathfrak{g})\subset\mathfrak{g}^{''}$ and $\rho\colon \mathfrak{g}\rightarrow \mathfrak{g}^{''}$ satisfies (H2). Further, if $\mathfrak{g}$ is a sum of simple Hermitian Lie algebras $\mathfrak{g}_1\oplus...\oplus\mathfrak{g}_n$ we can choose $\mathfrak{g}^{''}$ as a sum of Hermitian Lie algebras $\mathfrak{g}_1^{''}\oplus ... \oplus \mathfrak{g}_n^{''}$ such that the image of each $\mathfrak{g}_i$ is contained in $\mathfrak{g}^{''}_i$. The restricted maps $\rho|\colon \mathfrak{g}_i\rightarrow \mathfrak{g}_i^{''}$ all satisfy (H2).
The inclusion map of regular subalgebras satisfy (H1) [@A6]. Lemma \[lma3\] is one of the key reductions in classifying holomorphic maps between Hermitian symmetric spaces. It breaks down the classifying into two managable parts, classifying (H2)-representations and regular subalgebras. Together with the Lemma \[lma2\] it allows us to break down our classification into the same parts, classifying tight (H2)-representations $\rho\colon \mathfrak{g}\rightarrow\mathfrak{g}^{''}$ and tight regular subalgebras $\mathfrak{g}^{''}\subset\mathfrak{g}^{'}$. The second part of Lemma \[lma3\] allows us to reduce our classification to the case of $\mathfrak{g}$ simple.
We will now recall some facts about continuous bounded cohomology. We will restrict ourselves to cohomology in the second degree and to Hermitian Lie groups only. For a more thorough review of the theory we recommend [@A10], and for connections to rigidity questions [@A11].
Let $G$ be a Hermitian Lie group, i.e one which is the isometry group of a Hermitian symmetric space. Further let $G=G_1\times...\times G_n$ be a decomposition of $G$ into simple factors and $\mathcal{X}=\mathcal{X}_1\times... \times\mathcal{X}_n$ be the corresponding decomposition of the symmetric space into irreducible symmetric spaces. Let $\pi_i\colon G\rightarrow G_i$ be the projection maps and $\iota_i\colon G_i\rightarrow G$ be inclusion maps. We have $$\label{isom}
H_{cb}^2(G)\cong \prod{H^2_{cb}(G_i)}\cong \prod{\mathbb{R}\kappa_{G_i}^b}$$ where the $\kappa_{G_i}^b$ are the classes we defined in section \[descr\]. The isomorphism is $\kappa\mapsto (\iota_i^{*}\kappa)_{i=1}^n$ with inverse $(\kappa_i)_{i=1}^n\mapsto\sum_{i=1}^n{\pi_i^{*}\kappa_i}$. Under this isomorphism we have $$\kappa_{G}^b=\sum_i{\kappa_{G_i}^b}.$$ The seminorm on $H^2_{cb}(G)$ is in fact a norm [@A11].
For a class $\kappa\in H^2_{cb}(G)$ we can with a slight abuse of notation write $\kappa=\sum_{i}{\lambda_1\kappa_{G_i}^b}$ and we have $$\begin{aligned}
||\kappa||&=&sup_{(g_0,g_1,g_2)\in G}\kappa(g_0,g_1,g_2)=sup_{(g_0,g_1,g_2)\in G}\sum_i{\lambda_i\kappa_{G_i}^b(\pi_ig_0,\pi_ig_1,\pi_ig_2)}\\
&=&\sum_i{sup_{(g_0,g_1,g_2)\in G_i}\lambda_i\kappa_{G_i}^b(g_0,g_1,g_2)}=\sum_i{|\lambda_i| ||\kappa_{G_i}^b||}\end{aligned}$$ This implies that for positive classes $\kappa_1,\kappa_2\in H^2_{cb}(G)$ we have $||\kappa_1 + \kappa_2||=||\kappa_1|| +|| \kappa_2||$.
Fixing a $G$-invariant complex structure on $\mathcal{X}$ determines $G_i$-invariant complex structures on $\mathcal{X}_i$. These in turn determines a choice of Kähler forms $\omega_i$ which in turn determines a choice of basis $\kappa_{G_i}^b$ and hence an orientation on the $H^2_{cb}(G_i)$:s. In the following we assume such a choice of complex structures is made.
We say that a class $\alpha \in H^2_{cb}(G)$ is
1. positive if $\alpha=\sum{\mu_i \kappa_{G_i}^b}$ where $\mu_i\geq 0$ for all $i=1,...,n$, and
2. strictly positive if $\alpha=\sum{\mu_i \kappa_{G_i}^b}$ where $\mu_i> 0$ for all $i=1,...,n$.
We say that a homomorfism $\rho\colon H\rightarrow G$ is (strictly) positive if $\rho^* \kappa_{G}^b$ is (strictly) positive.
\[lma5\] If $\rho\colon G_1\rightarrow G_2$ is holomorphic then it is positive. If it is holomorphic and injective then it is strictly positive.
\[lma5\] Let $\mathcal{X}_i$ denote the corresponding symmetric spaces with Kälher forms $\omega_i$, complex structures $J_i$ and Riemanian metrics $g_i$ for $i=1,2$. We denote the induced map between the symmetric spaces by $\rho$ also. $\rho^*\omega_2(X,Y)=g_2(\rho_*X, J_2\rho_*Y)=g_2(\rho_*X, \rho_*J_1Y)$. Since $\rho$ is an isometry up to scaling we have that $\rho^*\omega_2(X,Y)$ is a positive multiple of $\omega_1(X,Y)$ hence $\rho$ is positive. If we further assume injectivity we can conclude that this multiple is non-zero hence $\rho$ is strictly positive.
\[lma1\] Let $H$ be the isometry group of $\mathcal{Y}$ and $G=G_1\times ... \times G_n$ the isometry group of $\mathcal{X}_1\times ... \times \mathcal{X}_n$. We write $\rho_i$ as $\rho=(\rho_1,...,\rho_n)$. We have $$||\rho^*\kappa^b_G||=||\sum{\rho_i^* \kappa_{G_i}^b}||=\sum{||\rho_i^* \kappa_{G_i}^b||}\leq\sum{||\kappa_{G_i}^b||=||\kappa^b_{G}||}$$ where the second equality follows from positivity. Thus $\rho$ is tight if and only if all the $\rho_i$ are tight.
\[lma2\] Using Lemma \[lma1\] we may assume $\mathcal{Z}$ is irreducible. Let $\mathcal{Y}=\mathcal{Y}_1\times ... \times\mathcal{Y}_n$ be a decomposition of $\mathcal{Y}$ into irreducible parts. Let $ H,G=G_1\times...\times G_n , L$ denote the isometry groups of $\mathcal{X},\mathcal{Y},\mathcal{Z}$. Again we write $\rho$ as $\rho=(\rho_1,...,\rho_n)$. Assume that $\rho$ and $\eta$ are tight.
We have $\eta^*\kappa^b_{L}=\sum{c_i \kappa^b_{G_i}}$ such that $ \sum{||c_i \kappa^b_{G_i}||}=\sum{c_i|| \kappa^b_{G_i}||}=||\kappa^b_{L}||$. We get $||\rho^*\eta^*\kappa^b_{L}|| =||\sum{\rho_i^*c_i \kappa^b_{G_i}}||=\sum{c_i ||\rho_i^* \kappa^b_{G_i}||}=\sum{|c_i| || \kappa^b_{G_i}||}=||\kappa^b_{L}||$ so $\eta\circ\rho$ is tight.
Assume instead that $\eta\circ\rho$ is tight. We have $||\kappa_L^b||=||\rho^{*}\eta^{*}\kappa_L^b||\leq||\eta^{*}\kappa_L^b||\leq||\kappa_L^b||$ which implies that $\eta$ is tight. We have that $\eta^{*}\kappa_L^b=\sum{c_i\kappa^b_{G_i}}$ such that $\sum{c_i ||\kappa^b_{G_i}||}=||\kappa^b_{L}||$. Lemma \[lma5\] assures us that all $c_i>0$. The composition is tight so $||\kappa^b_{L}|| =||\rho^*\eta^*\kappa^b_{L}|| =||\sum{\rho_i^*c_i \kappa^b_{G_i}}||=\sum{c_i ||\rho_i^* \kappa^b_{G_i}||}\leq\sum{c_i || \kappa^b_{G_i}||}=||\kappa^b_{L}||$ , so the inequality in the equation above is an equality. Since we have established that all $c_i>0$ the inequality is an equality if and only if $||\rho_i^{*}\kappa^b_{G_i}||=||\kappa_{G_i}^b||$ for all $i$ which is true if and only $\rho$ is tight.
The norms $||\kappa^b_{G}||$ were computed in [@A4] and equals $r_{\mathcal{X}}\pi$, where $r_{\mathcal{X}}$ is the rank of the symmetric space $\mathcal{X}$ associated to $G$. Another approach using the Maslov index can be found in [@A12].
Regular subalgebras are formed by choosing a subset of the root system. Fulfilling certain conditions this subset works as the set of simple roots for the subalgebra. More precisely, let $\mathfrak{g}$ be a Hermitian Lie algebra with complexification $\mathfrak{g}^{\mathbb{C}}$. Let $\Lambda$ denote the root system of $\mathfrak{g}^{\mathbb{C}}$. A subset $\Delta$ of $\Lambda$ is called a $\Pi$-system if it satisfies\
(i) If $\alpha,\beta \in \Delta$ then $\alpha-\beta\notin\Delta$\
(ii) $\Delta$ is a linearly independent set in $i\mathfrak{h}^*$\
(iii) Each connected component of the Dynkin diagram of $\Delta$ contains at most one non-compact root.\
From this $\Pi$-system we form a new root system $\Lambda(\Delta):=(\sum_{\alpha\in\Delta}{\mathbb{Z}\alpha)\cap\Lambda}$\
and a subalgebra of the complexification by $\mathfrak{g}^{\mathbb{C}}(\Delta):=\sum_{\alpha\in\Delta}{\mathbb{C}H_\alpha}+\sum_{\alpha\in\Lambda(\Delta)}{\mathfrak{g}^\alpha}$.\
Finally we define the regular subalgebra as $\mathfrak{g}(\Delta):=\mathfrak{g}^{\mathbb{C}}(\Delta)\cap\mathfrak{g}$.
Criterions for tightness
========================
\[subalg\] Let $\mathfrak{g}$ be a simple Hermitian Lie algebra with regular subalgebra $\mathfrak{g}_1\oplus...\oplus\mathfrak{g}_k$, each $\mathfrak{g}_i$ being simple, and $\mathcal{X}_1\times...\times\mathcal{X}_k \subset \mathcal{X}$ the corresponding symmetric spaces. Let $\Lambda$ be the root system of $\mathfrak{g}^{\mathbb{C}}$ and $\Lambda_i$ the sub-root systems corresponding to the $\mathfrak{g}_i^{\mathbb{C}}$. Let $\gamma_i$ be the highest root of $\Lambda_i$ with respect to some ordering of $\Lambda_i$ and let $\gamma$ be the highest root of $\Lambda$. Further let $r$ and $r_i$ denote the ranks. If $$\sum_{i}{ \frac{\langle\gamma,\gamma\rangle}{\langle\gamma_i,\gamma_i\rangle}r_i }=r,\label{kvoten}$$ where the brackets denote the Killing form of $\mathfrak{g}^{\mathbb{C}}$, then the inclusion $\mathcal{X}_1\times...\times\mathcal{X}_k\subset \mathcal{X}$ is tight.
Let $g$ and $g_i$ be the invariant metrics on $\mathcal{X}$ respectively $\mathcal{X}_i$, normalized so that the minimal holomorphic sectional curvature is $-1$. Further let $\omega$ and $\omega_i$ be the associated Kähler forms. We have that $g|_{\mathcal{X}_i}=c_i g_i$ for some $c_i$ and since the inclusion is holomorphic we also have $\omega|_{\mathcal{X}_i}=c_i\omega_i$. Here $c_i\geq 1$ since the holomorphic sectional curvature of $g$ is $\geq -1$ on $\mathcal{X}_i$. We get $$\begin{aligned}
sup_{\Delta\subset\mathcal{X}_1\times...\times \mathcal{X}_k } \int_{\Delta}\omega&=&\sum_i{sup_{\Delta\subset\mathcal{X}_i}\int_{\Delta}\omega}\label{kurv1}\\
&=&\sum_i{sup_{\Delta\subset\mathcal{X}_i}\int_{\Delta}c_i \omega_i}=\sum_i{c_i r_i \pi}\nonumber\end{aligned}$$ thus the inclusion is tight if $\sum_i{c_i r_i}=r$. Let us examine the $c_i$ further. The minimal holomorphic sectional curvature on $\mathcal{X}_i$ with respect to $g$, $mincurv(\mathcal{X}_i,g)$, is $-\frac{1}{c_i}$. If we consider the quotient $\frac{mincurv(\mathcal{X},g)}{mincurv(\mathcal{X}_i,g)}=c_i$ we can calculate the $c_i$ without concerns about the normalization of the metric. Identifying $T_0\mathcal{X}$ with $\mathfrak{p}$ with for some reference point $0$, we have $g(X,Y)= b\langle X,Y\rangle $ for some $b>0$ where the brackets denote the Killing form of $\mathfrak{g}^{\mathbb{C}}$. Let $\mathfrak{g}^{\mathbb{C}}$ be the complexification of $\mathfrak{g}$, and $\mathfrak{g}^{\mathbb{C}}=\mathfrak{h}+\sum_\alpha{\mathfrak{g}^\alpha}$ a root space decomposition. Pick $e\in\mathfrak{g}^{\gamma}$ and $\bar{e}\in\mathfrak{g}^{-\gamma}$ such that $\xi=e+\bar{e}\in \mathfrak{p}$. We reach the minimal holomorphic sectional curvature in the complex line spanned by $\xi$. This can be deduced from the construction by Harish-Chandra of the realisation of $\mathcal{X}$ as a bounded symmetric domain, see for example [@A3]. Further let $H_\gamma$ be the element of $\mathfrak{h}$ such that $\langle H,H_\gamma\rangle =\gamma(H)$ for all $H\in\mathfrak{h}$. We get $$\begin{aligned}
mincurv(\mathcal{X},g)&=&\frac{b\langle[\xi,J\xi],[\xi,J\xi]\rangle }{b^2\langle\xi,\xi\rangle \langle J\xi,J\xi\rangle }=\frac{\langle[\xi,J\xi],[\xi,J\xi]\rangle }{b\langle\xi,\xi\rangle ^2}\nonumber\\
&=&\frac{\langle[e+\bar{e},ie-i\bar{e}],[e+\bar{e},ie-i\bar{e}]\rangle }{b\langle e+\bar{e},e+\bar{e}\rangle ^2}=\frac{4\langle[e,\bar{e}],[e,\bar{e}]\rangle }{4b\langle e,\bar{e}\rangle ^2}\\
&=&\frac{\langle\langle e,\bar{e}\rangle H_\gamma,\langle e,\bar{e}\rangle H_\gamma\rangle }{b\langle e,\bar{e}\rangle ^2}=\frac{\langle \gamma,\gamma\rangle }{b}\end{aligned}$$ Now let $\gamma_i$ denote the highest root of $\mathfrak{g}_i^{\mathbb{C}}$ and pick $\bar{e_i},e_i$ like with $\gamma$. We then get $$c_i=\frac{mincurv(\mathcal{X},g)}{mincurv(\mathcal{X}_i,g)}=\frac{\langle \gamma,\gamma\rangle }{\langle \gamma_i,\gamma_i\rangle }.$$
\[cor\] Let $\rho\colon\mathcal{X}_1\times...\times\mathcal{X}_n\rightarrow \mathcal{X}$ be a holomorphic, totally geodesic embedding. Let $r_i$ be the rank of $\mathcal{X}_i$ and $r$ the rank of $\mathcal{X}$. If $\sum{r_i}=r$ then $\rho$ is tight.
Viewing $\mathcal{X}_1\times ... \times\mathcal{X}_n$ as a subsymmetric space of $\mathcal{X}$ we can argue as in the proof of Theorem \[subalg\]. From (\[kurv1\]) we have that the inclusion is tight if $\sum{c_i r_i}=r$. We also have that $c_i\geq 1$, $\sum{c_i r_i}\leq r$ and that $\sum{r_i}=r$ by assumption. Thus all the $c_i=1$ and the inequality is an equality, i.e $\rho$ is tight.
As is well known, we can in each Hermitian symmetric space $\mathcal{X}$ of rank $r$ holomorphically embed the product of $r$ Poincare discs $\mathbb{D}$. This embedding is tight[@A8]. Lemma \[lma1\] also shows that the diagonal embedding $d\colon \mathbb{D}\rightarrow \mathbb{D}\times...\times\mathbb{D}$ is tight. The composition of $d\colon \mathbb{D}\rightarrow \mathbb{D}^{r}$ and $\iota\colon \mathbb{D}^{r}\rightarrow \mathcal{X}$ is then a tight embedding. We call these embeddings *diagonal discs*. They will be needed in the first of the following two Theorems from [@A8].
\[diagd\] Let $\rho\colon\mathcal{X}\rightarrow \mathcal{X}^{'}$ be a holomorphic and totally geodesic embedding. Further let $d$ and $d^{'}$ be embeddings of diagonal discs in $\mathcal{X}$ and $\mathcal{X}^{'}$. Denote by the same letters the corresponding Lie algebra homomorphisms, where we let $\mathfrak{g}$ and $\mathfrak{g}^{'}$ denote the Lie algebras corresponding to $\mathcal{X}$ and $\mathcal{X}^{'}$. Further let $Z_{\mathbb{D}}\in su(1,1)$ and $Z^{'}\in\mathfrak{g}^{'}$ be the complex structures of the disc and of $\mathcal{X}^{'}$. Then $\rho$ is tight if and only if $\langle \rho dZ_{\mathbb{D}},Z^{'}\rangle =\langle d^{'}Z_{\mathbb{D}},Z^{'}\rangle $, where the brackets denote the Killing form of $\mathfrak{g}^{'}$.
\[tube\] Suppose $\mathcal{X}_1$, $\mathcal{X}_2$ are Hermitian symmetric spaces of tube type. Then $\rho\colon \mathcal{X}_1\rightarrow\mathcal{X}_2$ is a tight and holomorphic embedding if and only if the corresponding Lie algebra homomorphism $\rho\colon\mathfrak{g}_1\rightarrow\mathfrak{g}_2$ is an (H2)-homomorphism.
We now have the tools we need to determine if a representation corresponds to a tight map.
regular subalgebras
===================
In this section we examine the regular subalgebras to see which are tight. To do this we will use Theorem \[subalg\] and Corollary \[cor\]. Since the criterion in Theorem \[subalg\] is a quotient of norms we can normalize the Killing form as we please. We can thus pretty much read off the results directly from the Dynkin diagrams. To make calculations easy we normalize so that the length of the shortest roots are 1. We recall some basic facts about root systems that will make the calculations easier. There are at most two lengths of roots for each simple Hermitian Lie algebra. In fact, for the Lie algebras we are concerned with, only $sp(2p)$ and $so(2l+1,2)$ contains roots of different lengths. With the normalization we have chosen the longer roots of these Lie algebras have length $\sqrt{2}$. Also, the highest root of a root system is always a longer root.
We divide this section into subsections, one for each simple Hermitian Lie algebra $\mathfrak{g}^{'}$. In each subsection $\{\alpha_i\}$ denotes the set of simple roots and $\gamma$ the highest root of $\mathfrak{g}^{'}$. We denote the subalgebras by $\mathfrak{g}$ and their decomposition into simple parts by $\mathfrak{g}=\mathfrak{g}_1\oplus...\oplus\mathfrak{g}_n$. For each $\mathfrak{g}_i$ we denote the highest root by $\gamma_i$. We begin each subsection by describing the Dynkin diagram and listing roots that will be needed to describe the subalgebras. We then list the systems of roots corresponding to the maximal regular subalgebras as classified by Ihara [@A6]. We then calculate which of them are tightly embedded in the original Lie algebra. We list only the proper maximal regular subalgebras but we cover in fact all regular subalgebras. We do this by considering chains of proper maximal regular subalgebras. If $\mathfrak{g}$ is a non-maximal regular subalgebra of $\mathfrak{g}^{'}$, we can find a proper maximal regular subalgebra $\mathfrak{g}_1$ containing $\mathfrak{g}$. If $\mathfrak{g}$ is a proper maximal regular subalgebra of $\mathfrak{g}_1$ we are done, otherwise we repeat the process until we have a chain $\mathfrak{g}\subset\mathfrak{g}_k\subset ... \subset\mathfrak{g}_1\subset\mathfrak{g}^{'}$ of proper maximal regular subalgebras. We have that $\mathfrak{g}$ is a tight regular subalgebra of $\mathfrak{g}^{'}$ if all the inclusions in the chain are tight.
Our list looks a little different than Ihara’s as we are only interested in the part of the subalgebras corresponding to a Hermitian symmetric spaces of non-compact type. To avoid overlappings and special cases of the Dynkin diagrams in lower dimensions we restrict param eters as follows:\
$su(p,q)$, $1\leq p\leq q$, $so^*(2p)$, $p\geq 5$, $sp(2p)$, $p\geq 2$, $so(p,2)$ $p\geq 5$\
All cases are still covered due to wellknown isomorphisms, see for example [@A3].\
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=su(p,q)$**]{}\
(50,43) (5,31)(30,0)[7]{} (20,31)(30,0)[[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{q+1}$,$\alpha_{q+2}$,$\alpha_{q+p-1}$,$\alpha_{1}$,$\alpha_{2}$,$\alpha_{q-1}$,$\alpha_{q}$]{}
$$\begin{aligned}
&\gamma=\alpha_1+...+\alpha_{p+q-1}\label{gammat}\end{aligned}$$
$$\begin{aligned}
&su(l,q) ,\,1\leq l <p,\label{su1}\\
& \{\alpha_{p+q-l},\,\alpha_{p+q-l+1},...,\alpha_{p+q-1},\alpha_1,\alpha_2,...,\alpha_q\}\nonumber\\
&su(p,s) ,\,p\leq s <q, \label{su2}\\
&\{\alpha_{q+1},\,\alpha_{q+2},...,\alpha_{p+q-1},\alpha_1,\alpha_2,...,\alpha_s\}\nonumber\\
&su(s,p),\, 1\leq s<p, \, \label{su3}\\
&\{\alpha_{s},\alpha_{s-1},...,\alpha_{2},\alpha_1,\alpha_{p+q-1},\alpha_{p+q-2},...,\alpha_{q+1}\}\nonumber\end{aligned}$$
$$\begin{aligned}
&su(l,s)+su(p-l,q-s),\, 1\leq l\leq s, \, p-l\leq q-s, \, \label{su4}\\
&\begin{array}{l} \{\alpha_{p+q-l},\alpha_{p+q-l+1},...,\alpha_{p+q-2},\alpha_1,\alpha_2,...,\alpha_s\}\cup\\
\{-\alpha_{p+q-l-2},...,-\alpha_{q+1},\gamma,-\alpha_q,...,-\alpha_{s+2}\}
\end{array}\nonumber\\
&su(s,l)+su(p-l,q-s),\, 1\leq s<l <p, \,\label{su5}\\
&\begin{array}{l} \{\alpha_{s},\alpha_{s-1},...,\alpha_1,\alpha_{p+q-1},...,\alpha_{p+q-l}\}\cup\\
\{-\alpha_{p+q-l-2},...,-\alpha_{q+1},\gamma,-\alpha_q,...,-\alpha_{s+2}\}\nonumber
\end{array}\end{aligned}$$
Since all roots in $su(p,q)$ are of the same length, the quotient $\frac{\langle\gamma,\gamma\rangle}{\langle\gamma_i,\gamma_i\rangle}$ in (\[kvoten\]), Theorem \[subalg\] is 1 in all cases. Comparing ranks we see that (\[su2\]) and (\[su4\]) are tight, the rest are not.\
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=so^*(2p)$**]{}\
(50,43) (5,31)(30,0)[5]{} (20,31)(30,0)[[(10,0)[25]{}]{},[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{p-2}$,$\alpha_{p-1}$]{} (35,0) (34,4)[(0,10)[25]{}]{} (20,0)[$\alpha_p$]{}
$$\begin{aligned}
&\gamma=\alpha_1+2(\alpha_2+...+\alpha_{p-2})+\alpha_{p-1}+\alpha_p \\
&\beta=\alpha_2+ \alpha_3+...+\alpha_p\end{aligned}$$
$$\begin{aligned}
&su(l,p-l) ,\, 1\leq l\leq p/2\, ,\label{so1}\\
&\{-\alpha_{p-l+2},...,-\alpha_{p-2},-\alpha_{p-1},\beta,\alpha_1,...,\alpha_{p-l}\}\nonumber\\
&so^*(2l) + so^*(2(p-l)) ,\, [p/2]\leq l \leq p-2 \, , \label{so2}\\
&\Big\{\begin{array}{l}
\alpha_1,...,\alpha_{l-1}\\
\alpha_p
\end{array}\Big\}
\cup
\Big\{\begin{array}{l}
\gamma,-\alpha_{p-2},-\alpha_{p-3}...,-\alpha_{l+1}\\
-\alpha_{p-1}
\end{array}\Big\}\nonumber\\
&so^*(2(p-1)), \,
\Big\{\begin{array}{l}
\alpha_1,...,\alpha_{p-2}\\
\alpha_p
\end{array}\Big\}\label{so3}
$$
Again all roots are of the same length. By comparing ranks we see that (\[so1\]) is tight precisely when $l=[\frac{p}{2}]$.\
For (\[so2\]) we see that the inclusion is tight for all $l$ if $p$ is odd, and for even $l$ if $p$ is even. For (\[so3\]) the inclusion is tight for $p$ odd.\
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=sp(2p)$**]{}\
(50,23) (5,1)(30,0)[5]{} (50,1)(30,0)[[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{}]{} (5,11)(30,0)[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{p-1}$,$\alpha_{p}$]{} (7,2)[(10,0)[25]{}]{} (7,0)[(10,0)[25]{}]{}
$$\begin{aligned}
&\gamma=\alpha_1+2\sum_2^{p}{\alpha_i} \\
&\beta=\sum_2^{p}{\alpha_i}\end{aligned}$$
$$\begin{aligned}
&su(l,p-l) \:1\leq l\leq [p/2],\label{sp1}\\
& \{-\alpha_{p-l+3},...,-\alpha_{p-1},-\alpha_p, \beta,\alpha_1+\alpha_2,\alpha_3,\alpha_4,...,\alpha_{p-l+1}\}\nonumber\\
&sp(2l) + sp(2(p-l)) \:[p/2]\leq l \leq p-1 , \label{sp2}\\
&\{\alpha_{1},...,\alpha_{l}\}\cup\{\gamma,-\alpha_p,\alpha_{p-1},...,-\alpha_{l+2}\}\nonumber\end{aligned}$$
Here there are two possible lengths for roots so we have to make some calculations. For (\[sp1\]) we use the formula (\[gammat\]) for the highest root of $\mathfrak{g}$ of type $su(p,q)$. We get $$\gamma_1=-\sum_{p-l+3}^{p}{\alpha_i}+\sum_2^{p}{\alpha_i}+\sum_1^{p-l+1}{\alpha_i}=\alpha_1+2\sum_2^{p-l+1}{\alpha_i}+\alpha_{p-l+2}$$ as the highest root. We have $$\begin{aligned}
\langle \gamma_1,\gamma_1\rangle = ||\alpha_1||^2+||2\sum_2^{p-l+1}{\alpha_i}||^2+||\alpha_{p-l+2}||^2+2\langle \alpha_1,2\alpha_2\rangle \\ +2\langle 2\alpha_{p-l+1},\alpha_{p-l+2}\rangle =2+4+1-4-2=1\end{aligned}$$ and $\langle \gamma,\gamma\rangle =2$ since it is the highest root. By Theorem \[subalg\] the inclusion is tight if l=p/2. Using Corollary \[cor\] we see that (\[sp2\]) is tight .\
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=so(p,2)$, $p= 2k-2$ even**]{}\
(50,37) (5,31)(30,0)[5]{} (20,31)(30,0)[[(10,0)[25]{}]{},[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{k-2}$,$\alpha_{k-1}$]{} (94,1) (94,4)[(0,10)[25]{}]{} (100,0)[$\alpha_k$]{}
$$\begin{aligned}
&\gamma=\alpha_1+2(\alpha_2+...+\alpha_{k-2})+\alpha_{k-1}+\alpha_k\\
&\beta_1=\alpha_2+2(\alpha_3+...+\alpha_{k-2})+\alpha_{k-1}+\alpha_k\\
&\beta_2=\alpha_{k-2}+\alpha_{k-1}+\alpha_k\end{aligned}$$
$$\begin{aligned}
&su(1,1)+ su(1,1),\, \{\alpha_1\}\cup\{\gamma\}\label{soj1}\\
&su(1,l),\, 2\leq l \leq k-1, \,\{\alpha_1,...,\alpha_l\}\label{soj2}\\
&su(1,k-1), \, \{\alpha_1,...,\alpha_{k-2},\alpha_k\}\label{soj3}\\
&su_(2,2),\, \{\beta_1,\alpha_1,\alpha_2\}\label{soj4}\\
&so(p-2,2),\, \Big\{\begin{array}{l}
\alpha_2,...,\alpha_{k-1}\\
\beta_2
\end{array}\Big\}\label{soj5}\end{aligned}$$
Again we have that all roots have the same length. Comparing ranks, Theorem \[subalg\] then tells us that (\[soj1\]), (\[soj4\]) and (\[soj5\]) are tight inclusions, the rest are not.\
\
(50,57) (5,31)(30,0)[5]{} (20,31)(30,0)[[(10,0)[25]{}]{},[$\cdots$]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{1}$,$\alpha_{2}$,,$\alpha_{k-1}$,$\alpha_{k}$]{} (97,32)[(10,0)[25]{}]{} (97,30)[(10,0)[25]{}]{}
$$\begin{aligned}
&\gamma=\alpha_1+2(\alpha_2+...+\alpha_{k-2})+\alpha_{k-1}+\alpha_k\\
&\beta_1=\alpha_2+2(\alpha_3+...+\alpha_{k-1}+\alpha_k)\\
&\beta_2=\alpha_{k-1}+2\alpha_k\\
&\beta_3=\alpha_{k-1}+\alpha_k\\
&\beta_4=\alpha_1+\alpha_2+...+\alpha_{k-2}+\alpha_{k-1}+\alpha_k\end{aligned}$$
$$\begin{aligned}
&su(1,1)+su(1,1),\, \{\alpha_1\}\cup\{\gamma\}\label{sou1}\\
&su(1,l),\, 2\leq l \leq k-2, \,\{\alpha_1,...,\alpha_l\}\label{sou2}\\
&su(2,2),\, \{\beta_1,\alpha_1,\alpha_2\}\label{sou3}\\
&su(1,1), \, \{\beta_4\}\label{sou4}\\
&so(p-1,2),\, \Big\{\begin{array}{l}
\alpha_2,...,\alpha_{k-1}\\
\beta_2
\end{array}\Big\}\label{sou5}\\
&so(p-2,2),\, \{\alpha_2,...,\alpha_{k-1},\beta_3\}\label{sou6}\end{aligned}$$
Here (\[sou1\]), (\[sou3\]), (\[sou5\]) and (\[sou6\]) are tight by comparing ranks and applying Corollary \[cor\]. Calculating the quotient for (\[sou4\]) gives us $c_1=2$, the inclusion is thus tight by Theorem \[subalg\]. For (\[sou2\]) the quotient is one and the inclusion is not tight.
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=e_{6(-14)}$**]{}\
(50,37) (5,31)(30,0)[5]{} (20,31)(30,0)[[(10,0)[25]{}]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{4}$,$\alpha_{5}$]{} (65,1) (65,4)[(0,10)[25]{}]{} (70,0)[$\alpha_6$]{}
$$\begin{aligned}
&\gamma=\alpha_1+2\alpha_2+3\alpha_3+2\alpha_4+\alpha_5+2\alpha_6\\
&\beta_1=\alpha_2+2\alpha_3+2\alpha_4+\alpha_5+\alpha_6\\
&\beta_2=\alpha_3+\alpha_4+\alpha_5+\alpha_6\end{aligned}$$
$$\begin{aligned}
&su(1,5)+su(1,1), \, \{\alpha_1,...,\alpha_5\}\cup\{\gamma\}\\
&su(1,2)+su(1,2), \, \{\alpha_1,\alpha_2\}\cup\{\gamma,-\alpha_6\}\\
&su(2,4),\, \{\beta_1,\alpha_1,\alpha_2,\alpha_3,\alpha_6\}\\
&so^*(10),\, \Big\{\begin{array}{l}
\alpha_1,\alpha_2,\alpha_3,\alpha_4\\
\beta_2
\end{array}\Big\}\\
&so(8,2),\, \Big\{\begin{array}{l}
\alpha_1,\alpha_2,\alpha_3,\alpha_4\\
\alpha_6
\end{array}\Big\}\end{aligned}$$
We see immeadiately that all the inclusions are tight by comparing ranks and applying Corollary \[cor\].
[**Maximal regular subalgebras $\mathfrak{g}$ of $\mathfrak{g}^{'}=e_{7(-25)}$**]{}\
(50,37) (5,31)(30,0)[6]{} (20,31)(30,0)[[(10,0)[25]{}]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{},[(10,0)[25]{}]{}]{} (5,41)(30,0)[$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{4}$,$\alpha_{5}$,$\alpha_{6}$]{} (65,1) (65,4)[(0,10)[25]{}]{} (50,0)[$\alpha_7$]{}
\
\
$$\begin{aligned}
&\gamma=\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + 2\alpha_7\\
&\beta_1=\alpha_2 + 2\alpha_3 +2 \alpha_5 + \alpha_6 + 2 \alpha_7\\
&\beta_2=\alpha_3 + 2\alpha_4 + 2\alpha_5 + \alpha_6 + \alpha_7\\
&\beta_3=\alpha_4 + \alpha_5 + \alpha_6 + \alpha_7 \end{aligned}$$ $$\begin{aligned}
&su(1,5)+su(1,2), \, \{\alpha_1,...,\alpha_4,\alpha_7\}\cup\{\gamma,-\alpha_6\}\label{e1}\\
&su(1,3)+su(1,3), \, \{\alpha_1,\alpha_2,\alpha_3\}\cup\{\gamma,-\alpha_6\,-\alpha_5\}\label{e2}\\
&su(2,6), \, \{\beta_1,\alpha_1,\alpha_2,...,\alpha_6\}\label{e3}\\
&su(3,3), \, \{-\alpha_7,\beta_1,\alpha_1,\alpha_2,\alpha_3\}\label{e4}\\
&so^*(12),\, \label{e5}\Big\{\begin{array}{l}
\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_7\\
\beta_2
\end{array}\Big\}\\
&so(10,2)+su(1,1), \,
\Big\{\begin{array}{l}
\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5\\
\alpha_7
\end{array}\Big\}
\cup\{\gamma\}\label{e6}\\
&e_{6(-14)},\, \Big\{\begin{array}{l}
\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6\\
\beta_3\}
\end{array}\Big\}\label{e7}\end{aligned}$$
Again all the roots are of length 1. Comparing ranks and applying Theorem \[subalg\] we have that (\[e4\]), (\[e5\]) and (\[e6\]) are tight, the rest are not.
Irreducible representations satisfying (H2)
===========================================
The remaining problem is to classify irreducible tight (H2)-representations. Irreducible (H2)-representations $\rho\colon\mathfrak{g}\rightarrow\mathfrak{g}^{'}$ were classified by Satake and Ihara so we just have to check which of them are tight using Theorem \[diagd\] and Corollary \[tube\]. This will require some explicit calculations. We therefore begin by giving matrix descriptions of $su(p,q), sp(2p)$ and $so^*(2p)$ together with Cartan decompositions and elements defining a complex structure on the corresponding symmetric space. We also define two diagonal discs that will be needed in the calculations.\
Let $V$ be a $(p+q)$-dimensional vector space over $\mathbb{C}$. $su(p,q)$ is defined as the traceless subalgebra of $End(V)$ preserving some sesquilinear form $F$ of signature $(p,q)$, i.e. $X\in su(p,q)$ if $F(Xv,w)+F(v,Xw)=0$ for all $v,w\in V$. Choosing an orthonormal basis $\{e_i\}$ for $F$ with $F(e_i,e_i)=1$ for $i\leq p$ we can represent $F$ with the matrix $\left(\begin{array}{cc}
1_p&0\\
0&-1_q
\end{array}\right) $, by $F(v,w)=v^* \left(\begin{array}{cc}
1_p&0\\
0&-1_q
\end{array}\right) w$. With respect to this basis we can identify $su(p,q)$ as the matrix algebra $$\begin{aligned}
\mathfrak{g}=\{
\left(\begin{array}{cc}
A&B\\
B^*&C
\end{array}\right)
:A\in M_p(\mathbb{C}),B\in M_{p,q}(\mathbb{C}),C\in M_q(\mathbb{C}), A^*=-A, C^*=-C\}\end{aligned}$$ We choose the Cartan decomposition and complex structure $$\begin{aligned}
\mathfrak{k}=\{
\left(\begin{array}{cc}
A&0\\
0&C
\end{array}\right) \}\, ,
\mathfrak{p}=\{
\left(\begin{array}{cc}
0&B\\
B^*&0
\end{array}\right) \}\, ,
Z_{p,q}=\frac{i}{p+q}\left(\begin{array}{cc}
q1_p&0\\
0&-p1_q
\end{array}\right) \\\end{aligned}$$
The algebra $so^*(2p)$ is defined using a symmetric $\mathbb{C}$-bilinear form and a skew-Hermitian form. Fixing a well chosen basis we can present $so^*(2p)$ as the traceless subalgebra of $End(\mathbb{C}^{2p})$ preserving the forms $$\begin{aligned}
F(v,w)&=&v^*
\left(\begin{array}{cc}
i1_p&0\\
0&-i1_p
\end{array}\right) w
\mbox{ and}\label{soett}\\
Q(v,w)&=&v^t \left(\begin{array}{cc}
0&1_p\\
1_p&0
\end{array}\right) w.\label{sotva}\end{aligned}$$ This gives us the following matrix description for $so^*(2p)$: $$\begin{aligned}
\mathfrak{g}=\{
\left(\begin{array}{cc}
A&B\\
B^*&\bar{A}
\end{array}\right)
:A\in M_p(\mathbb{C}),B\in M_{p}(\mathbb{C}), A^*=-A, B^t=-B\}\\
\mathfrak{k}=\{
\left(\begin{array}{cc}
A&0\\
0&\bar{A}
\end{array}\right) \}\, ,
\mathfrak{p}=\{
\left(\begin{array}{cc}
0&B\\
B^*&0
\end{array}\right) \}\, ,
Z=\frac{i}{2}\left(\begin{array}{cc}
1_p&0\\
0&-1_p
\end{array}\right) \end{aligned}$$
In a similar fashion $sp(2p)$ is defined as the algebra preserving a skewsymmetric bilinear form on $\mathbb{R}
^{2p}$. Conjugating this algebra in $End(\mathbb{C}^{2p})$ we arrive at the following description:
$$\begin{aligned}
\mathfrak{g}=\{
\left(\begin{array}{cc}
A&B\\
B^*&\bar{A}
\end{array}\right)
:A\in M_p(\mathbb{C}),B\in M_{p}(\mathbb{C}), A^*=-A, B^t=B\}\\
\mathfrak{k}=\{
\left(\begin{array}{cc}
A&0\\
0&\bar{A}
\end{array}\right) \}\, ,
\mathfrak{p}=\{
\left(\begin{array}{cc}
0&B\\
B^*&0
\end{array}\right) \}\, ,
Z=\frac{i}{2}\left(\begin{array}{cc}
1_p&0\\
0&-1_p
\end{array}\right) \\\end{aligned}$$
We also define the following two diagonal discs
$$\begin{aligned}
&d_{p,q}\colon su(1,1)\rightarrow su(p,q),\, p\geq q\\
&\left(\begin{array}{cc}
ai&z\\
\bar{z}&-ai
\end{array}\right)\mapsto
\left(\begin{array}{ccc}
0&0&0\\
0&ai1_q&z1_q\\
0&\bar{z}1_q&-ai1_q
\end{array}\right)\label{disk}\end{aligned}$$
For $p=2l+1$ we define $$\begin{aligned}
&d_{p}\colon su(1,1)\rightarrow so^*(2p)\\
&\left(\begin{array}{cc}
ai&z\\
\bar{z}&-ai
\end{array}\right)\mapsto
\left(\begin{array}{cccccc}
ai1_l&0&0&0&0&z1_l\\
0&0&0&0&0&0\\
0&0&ia1_l&-z1_l&0&0\\
0&0&-\bar{z}1_l&-ia1_l&0&0\\
0&0&0&0&0&0\\
\bar{z}1_l&0&0&0&0&-ai1_l
\end{array}\right).\end{aligned}$$
From these descriptions we see that we have two inclusions $$\begin{aligned}
\iota_1\colon sp(2p)\rightarrow su(p,p)\\
\iota_2\colon so^*(2p)\rightarrow su(p,p)\end{aligned}$$ By Theorem \[tube\] we have that $\iota_1$ is tight and that $\iota_2$ is tight for $p$ even. For $p=2l+1$ odd we calculate
$$\begin{aligned}
&\iota_2 d_p Z_{1,1}-d_{p,p}Z_{1,1}=\\
&=\frac{i}{2}\left(\begin{array}{cccccc}
1_l&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&1_l&0&0&0\\
0&0&0&-1_l&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&-1_l
\end{array}\right)
-\frac{i}{2}
\left(\begin{array}{cccccc}
1_l&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&1_l&0&0&0\\
0&0&0&-1_l&0&0\\
0&0&0&0&-1&0\\
0&0&0&0&0&-1_l
\end{array}\right)\\
&=\frac{i}{2}
\left(\begin{array}{cccccc}
0&0&0&0&0&0\\
0&-1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&0
\end{array}\right),
Z_{p,p}=\frac{i}{2}\left(\begin{array}{cccccc}
1_l&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&1_l&0&0&0\\
0&0&0&-1_l&0&0\\
0&0&0&0&-1&0\\
0&0&0&0&0&-1_l
\end{array}\right),\end{aligned}$$
$\langle \iota_2 d_p Z_{1,1}-d_{p,p}Z_{1,1}, Z_{p,p}\rangle=4p\mbox{Trace}( (\iota_2 d_p Z_{1,1}-d_{p,p}Z_{1,1}) Z_{p,p})=2p\neq 0$. Applying Theorem \[diagd\] we see that $\iota_2$ is not tight for $p$ odd. Besides these inclusions, identity homomorphisms and the representations corresponding to regular subalgebras we have two classes of irreducible (H2)-representations [@A6],[@A7].\
The first class we have are spin representations of $so(p,2)$ into $su(p^{'},p^{'})$. Here $p^{'}=2^{\frac{p}{2}-1}$ for $p$ even and $p^{'}=2^{\frac{p-1}{2}}$ for $p$ odd. For each even $p$ we have two spin representations, and for $p$ odd we have one. Satake showed that the image of these representations is contained in a copy of $sp(2p^{'})$ if $p\equiv 1,2,3 (8)$ and in a copy of $so^*(2p^{'})$ if $ p \equiv 5,6,7 (8)$.
The spin representations $$\begin{aligned}
&\rho\colon so(p,2)\rightarrow su(p^{'},p^{'}),\\
&\rho\colon so(p,2)\rightarrow sp(2p^{'}),\\
&\rho\colon so(p,2)\rightarrow so^*(2p^{'})\end{aligned}$$ are all tight.
Since all these representations are between tube type domains and satisfy (H2) they are tight by Theorem \[tube\].
The second class of irreducible (H2)-representations are the skewsymmetric tensor representations of $su(p,1)$. Since the symmetric space corresponding to $su(p,1)$ is not of tube type we will have to resort to Theorem \[diagd\] and do some calculations. We begin by describing these representations in detail.
As mentioned $su(p,1)$ acts on $(V,F)$ where $V$ is a vector space over $\mathbb{C}$ of dim $p+1$ and $F$ is a Hermitian form of signature $(p,1)$. We can extend this action to $\bigwedge^m V$, the exterior product of $m$ copies of $V$. We extend $F$ to a Hermitian form $F^m$ on $\bigwedge^m V$ defined as $F^m(x_1\wedge...\wedge x_m,y_1\wedge...\wedge y_m,):=det(F(x_i,y_j))$. If we choose an orthonormal basis $\{e_1,..,e_{p+1}\}$ for $(V,F)$, $\{e_{i_1}\wedge ...\wedge e_{i_m}, i_1<...<i_m\}$ is an orthonormal basis for $(\bigwedge^m V,F^m)$. To shorten notation we write $e_I= e_{i_1}\wedge ...\wedge e_{i_m}$ for an ordered set $I=\{i_1,...,i_m\}$. We get $$F^m(e_I,e_I)=
\begin{cases}
1 \, , \mbox{ if } i_m\leq p\\
-1\, , \mbox{ if } i_m=p+1.
\end{cases}$$ Thus $F^m$ has signature $(\binom{p}{m},\binom{p}{m-1})=:(p^{'},q^{'})$. The extension of the action thus defines a representation $$\rho_m\colon su(p,1)\rightarrow su(p^{'},q^{'}).$$
The skewsymmetric tensor representations $\rho_m \colon su(p,1)\rightarrow su(p^{'},q^{'})$ are not tight except for $m=1,p.$
To see if this is tight we have to calculate $\langle\rho_m d_{p,1}Z_{1,1}-d_{p^{'},q^{'}}Z_{1,1},Z_{p^{'},q^{'}}\rangle $ and apply Theorem \[diagd\]. By \[disk\] we have $$d_{p,1}Z_{1,1}=
\frac{i}{2}\left(\begin{array}{ccc}
0&0&0\\
0&1&0\\
0&0&-1
\end{array}\right)$$ where the zeroes denote zero matrices of appropriate size. Let us see how this matrix acts on the basis elements of $\bigwedge^m V$.\
$$\begin{aligned}
\rho_m d_{p,1} Z_{1,1}(e_I)&=&( d_{p,1} Z_{1,1}(e_{i_1})\wedge...\wedge e_{i_m})+...+(e_{i_1}\wedge...\wedge d_{p,1} Z_{1,1}(e_{i_m}))\\
&=&\begin{cases}
\frac{i}{2}e_I \, , \mbox{ if } i_m=p\\
-\frac{i}{2}e_I\, , \mbox{ if } i_m=p+1 \mbox{ and } i_{m-1}<p\\
0 \mbox{ otherwise}
\end{cases}\end{aligned}$$\
We decompose $\bigwedge^m V$ as $\bigwedge^m V=W^+\oplus W^-=W^+_0\oplus W^+_1\oplus W^-_0\oplus W^-_1$. Here $W^\pm$ denotes the positive respectively the negative part of $\bigwedge^m V$ with respect to $F^m$ and the subscript $0$ and $1$ denotes the kernel of $\rho_m d_{p,1}Z_{1,1}$ and its complement. With respect to this decomposition we can write in matrix form\
$\rho_m d_{p,1}Z_{1,1}=
\frac{i}{2}\left(\begin{array}{cccc}
0&0&0&0\\
0&1_{\binom{p-1}{m-1}}&0&0\\
0&0&0&0 \\
0&0&0&-1_{\binom{p-1}{m-1}}
\end{array}\right)$
and, assuming $p^{'}\geq q^{'}$,\
$d_{p^{'},q^{'}}Z_{1,1}=
\frac{i}{2}\left(\begin{array}{ccc}
0&0&0\\
0&1_{q^{'}}&0\\
0&0&-1_{q^{'}}
\end{array}\right).$
We thus get $\langle \rho_m d_{p,1}Z_{1,1}-d_{p^{'},q^{'}}Z_{1,1},Z_{p^{'},q^{'}}\rangle =-(\binom{p+1}{m}+1)(\binom{p-1}{m-1}-\binom{p}{m-1})$. This is non-zero for all $m$ except $m=1,p$. The case $p^{'}<q^{'}$ has the same result via a slightly different calculation.
If we choose $m=\frac{p+1}{2}$ for $p$ odd, we have $p^{'}=\binom{p}{\frac{p+1}{2}}=\binom{p}{\frac{p-1}{2}}=q^{'}$. Satake showed that in this case the image is contained in a copy of $sp(2p^{'})$ if $p\equiv 1 \, (4)$ and in a copy of $so^*(2p^{'})$ if $p\equiv 3 \, (4)$ [@A7].
For $p\equiv 1 \, (4)$ the skewsymmetric tensor representations $\rho_m \colon su(p,1)\rightarrow sp(2p^{'})$ are not tight. For $p\equiv 3 \, (4)$ the skewsymmetric tensor representations $\rho_m \colon su(p,1)\rightarrow so^*(2p^{'})$ are not tight except for $p=3$.
Since $sp(2p^{'})$ is tightly embedded in $su(p^{'},p^{'})$ the restriction of $\rho_m$ to $\rho_m\colon su(p,1)\rightarrow sp(2p^{'})$ can not be tight since a non-tight embedding can not be factored as two tight ones by Lemma \[lma2\]. The same argument is valid for the representations into $so^*(2p^{'})$ when $p^{'}$ is even.
When $p^{'}$ is odd however, $so^*(2p^{'})$ is not tightly embedded in $su(p^{'},p^{'})$. Thus we have to examine this case further. We begin by showing how the image of $su(p,1)$ lies in $so^*(2p^{'})$. We do this by defining a bilinear form $B$ on $\bigwedge^m V$ by $x\wedge y=B(x,y)e_1\wedge...\wedge e_{p+1}$. For $g\in GL(V)$ we have $B(\rho_m(g)x,\rho_m(g)y)=det(g)B(x,y)$ so clearly $B$ is invariant under $\rho_m(su(p,1))$. We now choose a basis for $\bigwedge^m V$ as follows. Let $k=\binom{p}{m}$, and choose some ordering $I_1,...,I_k$ of the $m$-subsets of $\{1,...,p\}$. Denote by $I_i^c$ the complement of $I_i$ in $\{1,...,p+1\}$. We then choose the ordered basis $e_{I_1},...,e_{I_k},\sigma_1 e_{I_1^c},...,\sigma_k e_{I_k^c}$ for $\bigwedge^m V$. The $\sigma_i$ chosen as $+1$ or $-1$ so that $B(e_{I_i},\sigma_i e_{I_i^c})$=1. With respect to this basis we can represent $F^m$ as the matrix $\left(\begin{array}{cc}
I&0\\
0&-I
\end{array}\right) $ and $B$ as the matrix $\left(\begin{array}{cc}
0&I\\
I&0
\end{array}\right) $. These are the matrices in (\[soett\]), (\[sotva\]) defining our choice of matrix description of $so^*(2p^{'})$ . If we further assume we ordered the $I_i$:s so that the last $\binom{p-1}{m-1}$ indices correspond to subsets such that $i_m=p$, we have $\rho_m d_{p,1}Z_{1,1}=
\frac{i}{2}\left(\begin{array}{cccc}
0&0&0&0\\
0&1_{\binom{p-1}{m-1}}&0&0\\
0&0&0&0 \\
0&0&0&-1_{\binom{p-1}{m-1}}
\end{array}\right)$ and
$
d_{p^{'}}Z_{1,1}=
\frac{i}{2}\left(\begin{array}{cccccc}
1_l&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&1_l&0&0&0\\
0&0&0&-1_l&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&-1_l
\end{array}\right)
$ where $l=[\frac{p^{'}}{2}]$.\
Now $\langle \rho_m d_{p,1}Z_{1,1}-d_{p^{'}}Z_{1,1},J\rangle =(2p^{'}-2)(\binom{p-1}{m-1}-p^{'}+1)\neq 0 $ except for $p=3$. Hence we have a tight representation of $su(3,1)$ into $so^*(6)$ but for no other values of $p$. This is however just one of the special isomorphisms between the simple Lie algebras in lower dimension.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank my advisor Genkai Zhang for valuable discussions in the preparation of this paper.
[9999]{}
\#1\#2\#3[[\#1]{}: [*\#2*]{}, \#3.]{}
|
---
abstract: 'We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge $c_L$ in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of [@Bautista:2019:QuantumGravityTimelike], and its main goal is to completely characterize the cohomology of the timelike theory with $c_L < 1$ which was there defined. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with $c_L > 1$, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models.'
author:
- 'Teresa Bautista[[^1]]{}'
- 'Harold Erbin[[^2]]{}'
- 'Matěj Kudrna[[^3]]{}'
bibliography:
- 'brst\_timelike\_liouville.bib'
- 'brst\_timelike\_liouville.bib'
title: BRST cohomology of timelike Liouville theory
---
------------------------------------------------------------------------
------------------------------------------------------------------------
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge useful discussions with Dileep Jatkar, Sylvain Ribault, Raoul Santachiara and Ashoke Sen. We are particularly grateful to Atish Dabholkar for suggesting us this problem, for ongoing discussions and for collaborations on related topics. The work of H.E. has been conducted under a Carl Friedrich von Siemens Research Fellowship of the Alexander von Humboldt Foundation during part of this project. H.E. is partially supported by the <span style="font-variant:small-caps;">Miur Prin</span> Contract <span style="font-variant:small-caps;">2015Mp2cx4</span> “Non-perturbative Aspects of Gauge Theories and Strings”.
[^1]: [[](mailto:teresa.bautista@aei.mpg.de)]{}
[^2]: [[](mailto:erbin@to.infn.it)]{}
[^3]: [[](mailto:kudrnam@fzu.cz)]{}
|
---
address:
- 'Institute of Astronomy, University of Cambridge, Cambridge, CB3 0HA, UK\'
- 'Cardiff School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK\'
author:
- JONATHAN R GAIR
- GARETH JONES
title: 'Detecting LISA sources using time-frequency techniques'
---
Introduction {#intro}
============
The LISA data stream will contain many gravitational wave (GW) signals from different types of source, overlapping in time and frequency. We expect to detect signals from compact binaries (composed of white dwarfs (WDs) or neutron stars (NSs)), in the nearby Universe. At low frequencies these will form a confusion foreground, but we also hope to individually resolve $\sim10,000$ of these sources[@farmer] at high frequencies. LISA will also detect 1-10 signals per year[@sesana05] from the merger of supermassive black holes (SMBHs) of appropriate mass ($\sim10^5 M_{\odot}$ – $10^7 M_{\odot}$). Thirdly, LISA should detect GWs from extreme mass ratio inspirals (EMRIs) — the inspiral of a compact object (a WD, NS or BH) into a SMBH in the centre of a galaxy. The astrophysical rate is very uncertain, but LISA could resolve as many as several hundred EMRIs [@gair04] and may also see a confusion background from distant events[@bc04].
The development of techniques to analyze LISA data is the subject of much current research. One promising approach is to use Markov Chain Monte Carlo (MCMC) methods. These have proven effective for detecting compact binaries[@cornish05], SMBH mergers[@cornishporter06] and for the detection of a single simplified EMRI signal[@stroeer06emri]. Although MCMC techniques can be used to fit simultaneously for many signals of several types, it is not yet clear whether this will be practical for the EMRI search. This is because of the high computational cost associated with constructing sufficiently accurate EMRI waveform templates, even when using kludge models[@gair04]. It may therefore by impractical to use MCMC for the EMRI search unless some advance estimate has been made of the source parameters. One alternative approach to LISA data analysis is to use time-frequency (t-f) techniques. These could be used to estimate the parameters of the loudest EMRIs in the LISA data stream and for the detection of unexpected GW events. A t-f analysis will consist of two stages — detection of a source in the data and parameter estimation for that source.
Source Detection
================
We consider a simplified model of the LISA data stream in which there is a single source embedded in instrumental noise. We divide the data stream into $M$ segments of length $T$, carry out a Fourier transform on each segment and hence construct a spectrogram of the data, $S^0$, with power $P^0_{i,j}$ in pixel $(i, j)$. We then search this spectrogram for features. The simplest technique is to look for individual pixels that are unusually bright, i.e., with $P^0_{i,j} > \eta$, for some suitably chosen threshold $\eta$. To improve the performance, we generate and search a sequence of binned spectrograms, $S^k$, in which the power in pixel $(i,j)$ is defined to be $$P^k_{i,j} = \frac{1}{n_k \times l_k} \sum_{a=0}^{n_k-1} \sum_{b=0}^{l_k-1} P^0_{i+a,j+b}.$$ Using bins of the form $n_k=2^p$, $l_k=2^q$, for all possible $p$ and $q$, a segment length $T=2^{20}$s, and assuming a $3$ year LISA mission, this simple excess power search has a reach of $\sim2.5$Gpc for a typical EMRI event (we take the reach to be the distance at which the detection rate is only $20\%$ for a search false alarm probability of $10\%$). The range is somewhat higher for EMRIs on nearly circular orbits. This method and these results are described in Wen & Gair 2005[@wengair05] and Gair & Wen 2005[@gairwen05].
A more sophisticated technique is to look for clusters of bright pixels. One algorithm is the Hierarchical Algorithm for Clusters and Ridges (HACR). This involves identifying [*black pixels*]{} with $P_{i,j} > \eta_{up}$, and then counting the number of [*grey pixels*]{} with $P_{i,j} > \eta_{low}$ ($< \eta_{up}$) that are connected to the black pixel. If the number of pixels in the cluster, $N_p$, exceeds a threshold, $N_c$, then the cluster constitutes a detection. The three thresholds can be tuned to make the search sensitive to a particular source or chosen to make the search generally sensitive to a variety of source types. After tuning, HACR has a detection rate $10-15\%$ higher than the simple excess power search at fixed overall false alarm probability for a typical EMRI. This represents a significant improvement in LISA event rate. The HACR search is described in more detail in Gair & Jones 2006[@gairjones06]. HACR can also detect SMBH mergers at redshift up to $\sim 3.5$ and compact binaries at up to $\sim 12$kpc[@gairjones06].
Parameter Extraction
====================
Once a source has been identified in the data, we would like to estimate its parameters to allow a targeted follow up with matched filtering. The time-frequency structure of an event tells us about the type of signal — a WD-WD binary is almost monochromatic (the track is therefore long in time but narrow in frequency), while EMRI and SMBH merger signals “chirp” over time. EMRIs chirp slowly and are likely to be on eccentric orbits, indicated by the presence of several tracks at different frequencies that evolve in a similar fashion. By contrast, SMBH mergers are likely to be circular and evolve much more rapidly. The time, central frequency, frequency derivatives and power profile of an event can all be extracted from a t-f map and provide information on the system, as does the bin size used to generate the spectrogram in which the detection is made. If multiple tracks can be associated with the same event we get this information for each track. The shape of the boundary of a track provides a way to distinguish a single event from two crossing tracks or a noise burst. The shape parameters (curvature, area, perimeter), skeleton and convex hull of a cluster provide further information[@russ02]. This information can be extracted directly from clusters identified by HACR (see discussion in Gair & Jones 2006[@gairjones06] and Gair & Jones 2007 in prep.). The excess power search identifies individual pixels only, so this search must be followed by a second track identification search before information can be extracted[@lisa6proc].
Application to LISA Data Analysis
=================================
Time-frequency searches of the nature described here could play a useful role in the LISA data analysis pipeline. These methods should be able to detect the loudest events in the LISA data stream at much lower computational cost than matched filtering searches. They also provide a method to find unexpected sources in the LISA data, since they do not rely on the observer having a model of the source. The main issue that will limit the sensitivity of time-frequency techniques is source confusion. The analyses described here have considered the detection of single isolated events, which is not the situation we expect for LISA. To deal with confusion, we could apply t-f techniques only to analyze a “cleaned” spectrogram, i.e., with the loudest recognizable events extracted as well as possible by other techniques. This could find events missed at the first stage of the analysis, but the effect of cleaning must be carefully explored. Alternatively, we can use percolation techniques — set a high threshold and gradually reduce it until a track appears. We can then extract this loudest event before lowering the threshold further to find the next event etc. This approach will be examined further in the future. Although our focus has been on LISA, the methods discussed here could also be applied to searches of Advanced LIGO data, e.g., for detection of intermediate mass ratio inspiral sources.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by St. Catharine’s College, Cambridge (JG) and by the School of Physics and Astronomy, Cardiff University (GJ).
[00]{} Farmer A J and Phinney E S, [*Mon. Not. Roy. Astron. Soc.*]{} [**346**]{}, 1197 (2003). Sesana A, Haardt F, Madau P and Volonteri M, [*Astrophys. J.*]{} [**623**]{}, 23 (2005). Gair J R, Barack L, Creighton T, Cutler C, Larson S L, Phinney E S and Vallisneri M, [*Class. Quantum Grav.*]{} [**21**]{}, S1595 (2004). Barack L and Cutler C, [*Phys. Rev.*]{} D [**70**]{}, 122002 (2004). Cornish N J and Crowder J, [*Phys. Rev.*]{} D [**72**]{}, 043005 (2005). Cornish N J and Porter E K, preprint gr-qc/0612091 (2006). Stroeer A, Gair J R and Vecchio A, in [*Proceedings of the Sixth LISA Symposium*]{}, AIP Conference Proceedings [**873**]{} 444 (2006). Wen L and Gair J R, [*Class. Quantum Grav.*]{} [**22**]{}, S445 (2005). Gair J R and Wen L, [*Class. Quantum Grav.*]{} [**22**]{}, S1359 (2005). Gair J R and Jones G, [*Class. Quantum Grav.*]{} [**27**]{}, 1145 (2007). Russ J C, [*Image Processing Handbook*]{} (Boca Raton: CRC Press) (2002). Wen L, Chen, Y and Gair J R, in [*Proceedings of the Sixth LISA Symposium*]{}, AIP Conference Proceedings [**873**]{} 595 (2006).
|
---
abstract: '[Recurrent Neural Networks (RNNs) are powerful models that achieve exceptional performance on a plethora pattern recognition problems. However, the training of RNNs is a computationally difficult task owing to the well-known “vanishing/exploding” gradient problem. Algorithms proposed for training RNNs either exploit no (or limited) curvature information and have cheap per-iteration complexity, or attempt to gain significant curvature information at the cost of increased per-iteration cost. The former set includes diagonally-scaled first-order methods such as [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}, while the latter consists of second-order algorithms like Hessian-Free Newton and K-FAC. In this paper, we present [<span style="font-variant:small-caps;">adaQN</span>]{}, a stochastic quasi-Newton algorithm for training RNNs. Our approach retains a low per-iteration cost while allowing for non-diagonal scaling through a stochastic L-BFGS updating scheme. The method uses a novel L-BFGS scaling initialization scheme and is judicious in storing and retaining L-BFGS curvature pairs. We present numerical experiments on two language modeling tasks and show that [<span style="font-variant:small-caps;">adaQN</span>]{}is competitive with popular RNN training algorithms.]{}'
author:
- 'Nitish Shirish Keskar[^1]'
- 'Albert S. Berahas[^2]'
bibliography:
- 'adaQN\_refs.bib'
title: '[<span style="font-variant:small-caps;">adaQN</span>]{}: An Adaptive Quasi-Newton Algorithm for Training RNNs'
---
Introduction
============
Recurrent Neural Networks (RNNs) have emerged as one of the most powerful tools for modeling sequences [@sutskever2013training; @graves2012supervised]. They are extensively used in a wide variety of applications including language modeling, speech recognition, machine translation and computer vision [@graves2013speech; @robinson1996use; @bengio2003neural; @mikolov2011rnnlm; @graves2009offline]. RNNs are similar to the popular Feed-Forward Networks (FFNs), but unlike FFNs, allow for cyclical connectivity in the nodes. This enables them to have exceptional expressive ability, permitting them to model highly complex sequences. This expressiveness, however, comes at the cost of training difficulty, especially in the presence of long-term dependencies [@pascanu2012difficulty; @bengio1994learning]. This difficulty, commonly termed as the “vanishing/exploding” gradient problem, arises due to the recursive nature of the network. Depending on the eigenvalues of the hidden-to-hidden node connection matrix during the Back Propagation Through Time (BPTT) algorithm, the errors either get recursively amplified or diminished making the training problem highly ill-conditioned. Consequently, this issue precludes the use of methods which are unaware of the curvature of the problem, such as Stochastic Gradient Descent (SGD), for RNN training tasks.
Many attempts have been made to address the problem of training RNNs. Some propose the use of alternate architectures; for e.g. Gated Recurrent Units (GRUs) [@cho2014learning] and Long Short-Term Memory (LSTM) [@hochreiter1997long] models. These network architectures do not suffer as severely from gradient-related problems, and hence, it is possible to use simple and well-studied methods like SGD for training, thus obviating the need for more sophisticated methods. Other efforts for alleviating the problem of training RNNs have been centered around designing training algorithms which incorporate curvature information in some form; see for e.g. Hessian-Free Newton [@martens2011learning; @martens2010deep] and Nesterov Accelerated Gradient [@nesterov1983method].
First-order methods such as [<span style="font-variant:small-caps;">Adagrad</span>]{}[@duchi2011adaptive] and [<span style="font-variant:small-caps;">Adam</span>]{}[@kingma2014adam], employ diagonal scaling of the gradients and consequently achieve invariance to diagonal re-scaling of the gradients. These methods have low per-iteration cost and have demonstrated good performance on a large number of deep learning tasks. Second-order methods like Hessian-Free Newton [@martens2010deep] and K-FAC [@martens2015optimizing], allow for non-diagonal-scaling of the gradients using highly expressive Hessian information, but tend to either have higher per-iteration costs or require non-trivial information about the structure of the graph. We defer the discussion of these algorithms to the following section.
In this paper, we present [<span style="font-variant:small-caps;">adaQN</span>]{}, a novel (stochastic) quasi-Newton algorithm for training RNNs. The algorithm attempts to reap the merits of both first- and second-order methods by judiciously incorporating curvature information while retaining a low per-iteration cost. Our algorithmic framework is inspired by that of Stochastic Quasi-Newton (SQN) [@byrd2014stochastic], which is designed for stochastic convex problems. The proposed algorithm is designed to ensure practical viability for solving RNN training problems.
The paper is organized as follows. We end the introduction by establishing notation that will be used throughout the paper. In Section \[sec:rel\_work\], we discuss popular algorithms for training RNNs and also discuss stochastic quasi-Newton methods. In Section \[sec:adaQN\], we describe our proposed algorithm in detail and emphasize its distinguishing features. We present numerical results on language modeling tasks in Section \[sec:num\_res\]. Finally, we discuss possible extensions of this work and present concluding remarks in Sections \[sec:disc\] and \[sec:conc\] respectively.
Notation
--------
The problem of training RNNs can be stated as the following optimization problem, $$\begin{aligned}
\label{eqn:prob}
\min_{w\in\mathbb{R}^n}f(w)=\frac{1}{m}\sum_{i=1}^{m} f_{i}(w).\end{aligned}$$ Here, $f_i$ is the RNN training error corresponding to a data point denoted by index $i$. We assume there are $m$ data points. During each iteration, the algorithm samples data points $\mathcal{B}_k \subseteq \{1,2,\cdots,m\}$. The iterate at the $k^{th}$ iteration is denoted by $w_k$ and the (stochastic) gradient computed on this mini-batch is denoted by $\hat{\nabla}_{\mathcal{B}_k} f(w_k)$. In particular, the notation $\hat{\nabla}_{\mathcal{B}_j} f(w_k)$ can be verbally stated as the gradient computed at $w_k$ using the mini-batch used for gradient computation during iteration $j$. For ease of notation, we may eliminate the subscript $\mathcal{B}_k$ whenever the batch and the point of gradient evaluation correspond to the iterate index; in other words, we use $\hat{\nabla} f(w_k)$ to mean $\hat{\nabla}_{\mathcal{B}_k} f(w_k)$. Unless otherwise specified, $H_k$ denotes any positive-definite matrix and the step-length is denoted by $\alpha_k$. The positive-definiteness of a matrix $H$ is expressed using the notation $H \succ 0$. Lastly, we denote the $i^{th}$ component of a vector $v\in\mathbb{R}^n$ by $[v]_i$ and use $v^2$ to represent element-wise square.
Related Work {#sec:rel_work}
============
In this section, we discuss several methods that have been proposed for training RNNs. In its most general form, the update equation for these methods can be expressed as $$\label{genupd}
w_{k+1}=w_{k} - \alpha_{k}H_{k}(\hat{\nabla} f(w_k) + v_kp_k)$$ where $\hat{\nabla} f(w_k)$ is a stochastic gradient computed using batch $\mathcal{B}_k$; $H_k$ is a positive-definite matrix representing an approximation to the [inverse]{}-Hessian matrix; $p_k$ is a search direction (usually $w_{k}-w_{k-1}$) associated with a momentum term; and $v_k\geq0$ is the relative scaling of the direction $p_k$.
Stochastic First-Order Methods
------------------------------
Inarguably, the simplest stochastic first-order method is SGD whose updates can be represented in the form of by setting $H_k=I$ and $v_k,p_k=0$. Momentum-based variants of SGD (such as Nesterov Accelerated Gradient [@nesterov1983method]) use $p_k=(w_{k}-w_{k-1})$ with a tuned value of $v_k$. While SGD has demonstrated superior performance on a multitude of neural network training problems [@bengio2015deep], in the specific case of RNN training, SGD has failed to stay competitive owing to the “vanishing/exploding” gradients problem [@pascanu2012difficulty; @bengio1994learning].
There are diagonally-scaled first-order algorithms that perform well on the RNN training task. These algorithms can be interpreted as attempts to devise second-order methods via inexpensive diagonal Hessian approximations. [<span style="font-variant:small-caps;">Adagrad</span>]{}[@duchi2011adaptive] allows for the independent scaling of each variable, thus partly addressing the issues arising from ill-conditioning. [<span style="font-variant:small-caps;">Adagrad</span>]{}can be written in the general updating form by setting $v_k,p_k=0$ and by updating $H_k$ (which is a diagonal matrix) as $$\begin{aligned}
[H_k]_{ii} &= \frac{1}{\sqrt{\sum_{j=0}^k[\hat{\nabla} f(w_j)]_i^2 + \epsilon}},
\end{aligned}$$ where $\epsilon>0$ is used to prevent numerical instability arising from dividing by small quantities.
Another first-order stochastic method that is known to perform well empirically in RNN training is [<span style="font-variant:small-caps;">Adam</span>]{}[@kingma2014adam]. The update, which is a combination of [<span style="font-variant:small-caps;">RMSProp</span>]{}[@tieleman2012lecture] and momentum, can be represented as follows in the form of , $$\begin{aligned}
v_k &= \beta_1, &
p_k &= \sum_{j=0}^{k-1} \beta_1^{(k-j-1)}(1-\beta_1) \hat{\nabla} f(w_j) - \hat{\nabla} f(w_k),\\
r_k &= \sum_{j=0}^k \beta_2^{(k-j)}(1-\beta_2) \hat{\nabla} f(w_j)^2, &
[H_k]_{ii} &=
\frac{1}{\sqrt{[r_k]_i+\epsilon}}. \end{aligned}$$
The diagonal scaling of the gradient elements in [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}allows for infrequently occurring features (with low gradient components) to have larger step-sizes in order to be effectively learned, at a rate comparable to that of frequently occurring features. This causes the iterate updates to be more stable by controlling the effect of large (in magnitude) gradient components, to some extent reducing the problem of “vanishing/exploding” gradients. However, these methods are not completely immune to curvature problems. This is especially true when the eigenvectors of $\nabla^2 f(w_k)$ do not align with the co-ordinate axes. In this case, the zig-zagging (or bouncing) behavior commonly observed for SGD may occur even for methods like [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}.
Stochastic Second-Order Methods
-------------------------------
Let us first consider the Hessian-Free Newton methods (HF) proposed in [@martens2010deep; @martens2011learning]. These methods can be represented in the form of by setting $H_k$ to be an approximation to the inverse of the Hessian matrix ($\nabla^2 f(w_k)$), as described below, with the circumstantial use of momentum to improve convergence. HF is a second-order optimization method that has two major ingredients: (i) it implicitly creates and solves quadratic models using matrix-vector products with the Gauss-Newton matrix obtained using the “Pearlmutter trick” and (ii) it uses the Conjugate Gradient method (CG) for solving the sub-problems inexactly. Recently, [@martens2015optimizing] proposed K-FAC, a method that computes a second-order step by constructing an invertible approximation of a neural networks’ Fisher information matrix in an online fashion. The authors claim that the increased quality of the step offsets the increase in the per-iteration cost of the algorithm.
Our algorithm [<span style="font-variant:small-caps;">adaQN</span>]{}belongs to the class of stochastic quasi-Newton methods which use a non-diagonal scaling of the gradient, while retaining low per-iteration cost. We begin by briefly surveying past work in this class of methods.
Stochastic Quasi-Newton Methods
-------------------------------
Recently, several stochastic quasi-Newton algorithms have been developed for large-scale machine learning problems: oLBFGS [@schraudolph2007stochastic; @mokhtari2014global], RES [@mokhtari2014res], SDBFGS [@wang2014stochastic], SFO [@sohl2013fast] and SQN [@byrd2014stochastic]. These methods can be represented in the form of by setting $v_k,p_k=0$ and using a quasi-Newton approximation for the matrix $H_k$. The methods enumerated above differ in three major aspects: (i) the update rule for the curvature pairs used in the computation of the quasi-Newton matrix, (ii) the frequency of updating, and (iii) the applicability to non-convex problems. With the exception of SDBFGS, all aforementioned methods have been designed to solve convex optimization problems. In all these methods, careful attention must be taken to monitor the quality of the curvature information that is used. In deterministic optimization, the (L-)BFGS curvature information is constructed by first computing the iterate and gradient differences, $$\begin{aligned}
s_k &= w_{k+1} - w_{k}\\
y_k &= \nabla f(w_{k+1}) - \nabla f(w_k)
\end{aligned}$$ and then using them to determine the Hessian-approximation through the (L-)BFGS update rule. $$\begin{aligned}
\label{eqn:bfgs_update}
H_k^{-1} = B_{k+1} &= B_k + \frac{y_k^Ty_k}{y_k^Ts_k} - \frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k}
\end{aligned}$$ The positive-definiteness of the approximation is either guaranteed through skipping or through a Wolfe line-search.
The RES and SDBFGS algorithms control the quality of the steps by modifying the BFGS update rule [@nocedal2006numerical]. Specifically, the update equations take on the following form, $$\begin{aligned}
s_k &= w_{k+1} - w_{k},\\
y_k &= \hat{\nabla}_{\mathcal{B}_k} f(w_{k+1}) - \hat{\nabla}_{\mathcal{B}_k} f(w_k) - \delta s_k \label{eqn:res_y},\\
H_{k+1}^{-1} = B_{k+1} &= B_k + \frac{y_k^Ty_k}{y_k^Ts_k} - \frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k} + \delta I.\end{aligned}$$ This ensures that the Hessian approximations are uniformly bounded away from singularity, thus preventing the steps from becoming arbitrarily large. Further, in these methods, the line-search is replaced by a decaying step-size rule. Note that at the $k^{th}$ iteration, the gradients used during updates are both evaluated on $\mathcal{B}_k$. oLBFGS, is similar to the above methods except no $\delta$-modification is used. In the equations above, $B_k$ and $H_k$ denote approximations to the Hessian and inverse-Hessian matrices respectively.
Finally, in [@byrd2014stochastic], the authors propose a novel quasi-Newton framework, SQN, in which they recommend the decoupling of the stochastic gradient calculation from the curvature estimate. The BFGS matrix is updated once every $L$ iterations as opposed to every iteration, which is in contrast to other methods described above. The authors prescribe the following curvature pair updates, $$\begin{aligned}
s_t &= \bar{w}_{t} - \bar{w}_{t-1}, & \text{where } \bar{w}_t = \frac{1}{L} \sum_{i=(t-1)L}^{tL} w_i,\label{eq:curvSQN_s}\\
y_t &= \hat{\nabla}^2_{\mathcal{H}_t}F(\bar{w}_t)s_t, & \label{eq:curvSQN_y}\end{aligned}$$ where $t$ is the curvature pair update counter, $L$ is the update frequency (also called the aggregation length) and $\mathcal{H}_t$ is a mini-batch used for computing the sub-sampled Hessian matrix. The iterate difference, $s$, is based on the average of the iterates over the last $2L$ iterations, intuitively allowing for more stable approximations. On the other hand, the gradient differences, $y$, are not computed using gradients at all, rather they are computed using a Hessian-vector product representing the approximate curvature along the direction $s$. The structure of the curvature pair updates proposed in SQN has several appealing features. Firstly, updating curvature information, and thus the Hessian approximation, every $L$ iterations (where $L$ is typically between $2$ and $20$) considerably reduces the computational cost. Additionally, more computational effort can be expended for the curvature computation since this cost is amortized over $L$ iterations. Further, as explained in [@byrd2014stochastic], the use of the Hessian-vector product in lieu of gradient differences allows for a more robust estimation of the curvature, especially in cases when $\|s\|$ is small and the gradients are noisy.
The SQN algorithm was designed specifically for convex optimization problems arising in machine learning, and its extension to RNN training is not trivial. In the following section, we describe [<span style="font-variant:small-caps;">adaQN</span>]{}, our proposed algorithm, which uses the algorithmic framework of SQN as a foundation. More specifically, it retains the ability to decouple the iterate and update cycles along with the associated benefit of investing more effort in gaining curvature information.
[<span style="font-variant:small-caps;">adaQN</span>]{} {#sec:adaQN}
=======================================================
In this section, we describe the proposed algorithm in detail. Specifically, we address key ingredients of the algorithm, including (i) the initial L-BFGS scaling, (ii) step quality control, (iii) choice of Hessian matrix for curvature pair computation, and (iv) the suggested choice of hyper-parameters. The pseudo-code for [<span style="font-variant:small-caps;">adaQN</span>]{}is given in Algorithm \[alg:adaQN\].
[**Inputs:**]{} $w_0$, $L$, $\alpha$, sequence of batches $\mathcal{B}_k$ with $|\mathcal{B}_k|=b$ for all $k$, $m_L=10$, $m_F=100$, $\epsilon=10^{-4}$, $\gamma=1.01$
Set $t \leftarrow 0$ and $\bar{w}_{o}, w_s=0$ Initialize accumulated Fisher Information matrix FIFO container $\tilde{F}$ of maximum size $m_F$ and L-BFGS curvature pair containers $S,Y$ of maximum size $m_L$. Randomly choose a mini-batch as monitoring set $\mathcal{M}$
$w_{k+1} = w_{k} - \alpha H_k \hat{ \nabla}f(w_k)$ Store $\hat{ \nabla} f(w_k) \hat{ \nabla} f(w_k)^T$ in $\tilde{F}$
$w_s = w_s + w_{k+1}$
$\bar{w}_n = \frac{w_s}{L}$ $w_s = 0$
Clear L-BFGS memory and the accumulated Fisher Information container $\tilde{F}$. $w_k = \bar{w}_{o}$ **continue**
$s = \bar{w}_n - \bar{w}_{o}$ $y = \frac{1}{|\tilde{F}|}(\sum_{i=1}^{|\tilde{F}|} \tilde{F}_i \cdot s)$
Store curvature pairs $s_t$ and $y_t$ in containers $S$ and $Y$ respectively $\bar{w}_{o} = \bar{w}_n$
$\bar{w}_{o} = \bar{w}_n$
$t \gets t + 1$
We emphasize that the storage of $(\hat{ \nabla} f(w_k)\hat{ \nabla} f(w_k)^T)$ in Step 6 is for ease of notation; in practice it is sufficient to store $\hat{ \nabla} f(w_k)$ and compute $y$ in Step 18 without explicitly constructing the matrix. Also, the search direction $p_k=-H_k\hat{\nabla}f_k$ is computed via the two-loop recursion using the available curvature pairs $(S,Y)$, and thus the matrix $H_k$ (the approximation to the inverse-Hessian matrix) is never constructed; refer to Section \[scn:two-loop\]. Further, in Algorithm \[alg:adaQN\], we specify a fixed monitoring set $\mathcal{M}$, a feature of the algorithm that was set for ease of exposition. In practice, this set can be changed to allow for lower bias in the step acceptance criterion.
Choice of $H^{(0)}_k$ for L-BFGS {#scn:hk0}
--------------------------------
Firstly, we discuss the most important ingredient of the proposed algorithm: the initial scaling of the L-BFGS matrix. For L-BFGS, in both the deterministic and stochastic settings, a matrix $H^{(0)}_k$ must be provided, which is an estimate of the scale of the problem. This choice is crucial since the relative scale of the step (in each direction) is directly related to it. In deterministic optimization, $$\begin{aligned}
H^{(0)}_k &= \frac{s_k^T y_k}{y_k^Ty_k} I \label{eqn:bb_scaling} \end{aligned}$$ is found to work well on a wide variety of applications, and is often prescribed [@nocedal2006numerical]. Stochastic variants of L-BFGS, including oBFGS, RES and SQN, prescribe the use of this initialization (). However, this is dissatisfying in the context of RNN training for two reasons. Firstly, as mentioned in the previous sections, the issue of “vanishing/exploding” gradients makes the problems highly ill-conditioned; using a scalar initialization of the L-BFGS matrix does not address this issue. Secondly, since $s$ and $y$ are noisy estimates of the true iterate and gradient differences, the scaling suggested in could introduce adversarial scale to the problem, causing performance deterioration.
To counter these problems, we suggest an initialization of the inverse-Hessian matrix based on accumulated gradient information. Specifically, we set $$\label{adaQN_scale}
[H_k^{(0)}]_{ii} = \frac{1}{\sqrt{\sum_{j=0}^k[\hat{\nabla} f(w_j)]_i^2 +\epsilon}}, \forall i={1,...,n}.$$ We direct the reader to Section \[scn:two-loop\] for details on how the above initialization is used as part of the L-BFGS two-loop recursion. We emphasize that this initialization is: (i) a diagonal matrix with non-constant diagonal entries, (ii) has a cost comparable to , and (iii) is identical to the scaling matrix used by [<span style="font-variant:small-caps;">Adagrad</span>]{}at each iteration. This choice is motivated by our observation of [<span style="font-variant:small-caps;">Adagrad</span>]{}’s stable performance on many RNN learning tasks. By initializing L-BFGS with an [<span style="font-variant:small-caps;">Adagrad</span>]{}-like scaling matrix, we impart a better scale in the L-BFGS matrix, and also allow for implicit safeguarding of the proposed method. Indeed, in iterations where no curvature pairs are stored, the [<span style="font-variant:small-caps;">adaQN</span>]{}and [<span style="font-variant:small-caps;">Adagrad</span>]{}steps are identical in form.
Step Acceptance and Control
---------------------------
While curvature information can be used to improve convergence rates, noisy or stale curvature information may in fact deteriorate performance [@byrd2014stochastic]. SQN attempts to prevent this problem by using large-batch Hessian-vector products in . Other methods attempt to control the quality of the steps by modifying the L-BFGS update rule to ensure that $H_k \succ 0$ for all $k$. However, we have found that these do not work well in practice. Instead, we control the quality of the steps by judiciously choosing the curvature pairs used by L-BFGS. We attempt to store curvature pairs during each cycle but skip the updating if the calculated curvature is small; see [@nocedal2006numerical] for details regarding skipping in quasi-Newton methods. Further, we flush the memory when the step quality deteriorates, allowing for more reliable steps till the memory builds up again.
The proposed criterion (Line 12 of Algorithm \[alg:adaQN\]) is an inexpensive heuristic wherein the functions are evaluated on a monitoring set, and $\gamma$ approximates the effect of noise on the function evaluations. A step is rejected if the function value of the new aggregated point is significantly worse (measured by $\gamma$) than the previous. In this case, we reset the memory of L-BFGS which allows the algorithm to preclude the deteriorating effect of any stored curvature pairs. The algorithm resumes to take [<span style="font-variant:small-caps;">Adagrad</span>]{}steps and build up the curvature estimate again. We report that, as an alternative to the proposed criterion, a more sophisticated criterion such as *relative improvement*, $$\begin{aligned}
\frac{f_\mathcal{M}(\bar{w}_n)-f_\mathcal{M}(\bar{w}_o)}{f_\mathcal{M}(\bar{w}_o)} &> \tilde{\gamma} \in (0,1)\end{aligned}$$ delivered similar performance on our test problems.
In the case when the sufficient curvature condition (Line 19 of Algorithm \[alg:adaQN\]) is not satisfied, the storage of the curvature pair is skipped. In deterministic optimization, this problem is avoided by conducting a Wolfe line-search. If the curvature information for a given step is inadequate, the line-search attempts to look for points further along the search path. We extend this idea to the RNN setting by not updating $\bar{w}_{o}$ when this happens. This allows us to move further, and possibly glean curvature information in subsequent update attempts. We have experimentally found this safeguarding to be crucial for the robust performance of [<span style="font-variant:small-caps;">adaQN</span>]{}. That being said, such rejection happens infrequently and the average L-BFGS memory per epoch remains high for all of our reported experiments (see Section \[sec:num\_res\]).
Choice of Curvature Information Matrix {#scn:curv}
--------------------------------------
As in SQN, the iterate difference $s$ in our algorithm is computed using aggregated iterates and the gradient difference $y$ is computed through a matrix-vector product; refer to equations and . The choice of curvature matrix for the computation of $y$ must address the trade-off between obtaining informative curvature information and the computational expense of its acquisition. Recent work suggests that the Fisher Information matrix (FIM) yields a better estimate of the curvature of the problem as compared to the true Hessian matrix (which is a natural choice); see for e.g. [@martens2014new].
Given a function $f$ parametrized by a random variable $\mathcal{X}$, the (true) FIM at a point $w$ is given by $$\begin{aligned}
F(w) &= \mathbb{E}_{\mathcal{X}} [\nabla f_{\mathcal{X}}(w) \nabla f_{\mathcal{X}}(w)^T].\end{aligned}$$ Since the distribution for $\mathcal{X}$ is almost never known, the *empirical* Fisher Information matrix (eFIM) is often used in practice. The eFIM can be expressed as follows $$\begin{aligned}
\label{eq:emp_fish}
\hat{F}(w) &= \frac{1}{|\mathcal{H}|} \sum_{i\in\mathcal{H}} \nabla_i f(w) \nabla_i f(w)^T,\end{aligned}$$ where $\mathcal{H} \subseteq \{1,2,\cdots,m\}$.
Notice from equation that the eFIM is guaranteed to be positive semi-definite, a property that does not hold for the true Hessian matrix. The use of the FIM (or eFIM) in second-order methods allows for attractive theoretical and practical properties. We exclude these results for brevity and refer the reader to [@martens2014new] for a detailed survey regarding this topic.
Given these observations and results, the use of the eFIM may seem like a reasonable choice for the Hessian matrix approximation used in the computation of $y_t$ (see equation ). However, the use of this matrix, even infrequently, increases the amortized per-iteration cost as compared to state-of-the-art first-order stochastic methods. Further, unlike second-order methods which rely on relatively accurate curvature information to generate good steps, quasi-Newton methods are able to generate high-quality steps even with crude curvature information [@nocedal2006numerical]. In this direction, we propose the use of a modified version of the empirical Fisher Information matrix that uses historical values of stochastic gradients, which were already computed as part of the step, thus reducing the computational cost considerably. This reduction, comes at the expense of storage and potentially noisy estimates due to stale gradient approximations. We call this approximation of the eFIM the *accumulated* Fisher Information matrix (aFIM) and denote it by $\bar{F}$. Given a memory budget of $m_F$, the aFIM at the $k^{th}$ iteration is given by $$\begin{aligned}
\label{eq:acc_fish}
\bar{F}(w_k) &= \frac{1}{\sum_{j=k-m_F+1}^k \left| \mathcal{B}_j \right|} \sum_{j=k-m_F+1}^k \nabla_{\mathcal{B}_j} f(w_{j})\nabla_{\mathcal{B}_j} f(w_{j})^T.\end{aligned}$$
For the purpose of our implementation, we maintain a finite-length FIFO container $\tilde{F}$ for storing the stochastic gradients as they are computed. Whenever the algorithm enters lines 12–16, we reject the step, and the contents of $\tilde{F}$ along with the L-BFGS memory are cleared. By clearing $\tilde{F}$, we also allow for additional safeguarding of future iterates against noisy gradients in the $\tilde{F}$ container that may have contributed in the generation of the poor step.
Choice of Hyper-Parameters
--------------------------
[<span style="font-variant:small-caps;">adaQN</span>]{}has a set of hyper-parameters that require tuning for competitive performance. Other than the step-size and batch-size, which needs to be tuned for all aforementioned methods, the only hyper-parameter exposed to the user is $L$. We prescribe $L$ to be chosen from $\{2,5,10,20\}$. We experimentally observed that the performance was not highly sensitive to the choice of $\alpha$ and $L$. Often, $L=5$ and the same step-length as used for [<span style="font-variant:small-caps;">Adagrad</span>]{}gave desirable performance. The other hyper-parameters have intuitive default values which we have found to work well for a variety of applications. Additional details about the offline tuning costs of [<span style="font-variant:small-caps;">adaQN</span>]{}as compared to [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}can be found in Section \[sec:num\_res\].
Cost {#scn:cost}
----
Given the nature of the proposed algorithm, a reasonable question is about the per-iteration cost. Let us begin by first considering the per-iteration cost of other popular methods. For simplicity, we assume that the cost of the gradient computation is $\mathcal{O}(n)$, which is a reasonable assumption in the context of deep learning. SGD has one of the cheapest per-iteration costs, with the only significant expense being the computation of the mini-batch stochastic gradient. Thus, SGD has a per-iteration complexity of $\mathcal{O}(n)$. [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}also have the same per-iteration complexity since the auxiliary operations only involve dot-products and elementary vector operations. Further, these algorithms have $\mathcal{O}(1)$ space complexity. On the other hand, second-order methods have higher per-iteration complexity since each iteration requires an inexact solution of a linear system, and possibly, storage of the pre-conditioning matrices.
The per-iteration time complexity of our algorithm consists of three components: (i) the cost of gradient computation, (ii) the cost of the L-BFGS two-loop recursion, and (iii) the amortized cost of computing the curvature pair. Thus, the overall cost can be written as $$\begin{aligned}
\label{eqn:complexity}
\underbrace{\mathcal{O}(n)}_{\text{gradient computation}} + \underbrace{4m_Ln}_{\text{two-loop recursion}} + \underbrace{m_FnL^{-1}}_{\text{cost of computing curvature pair}}.\end{aligned}$$ Given the prescription of $L\approx 5$, $m_F=100$ and $m_L=10$, the cost per-iteration remains at $\mathcal{O}(n)$. The memory requirement of our algorithm is also $\mathcal{O}(n)$ since we require the storage of up to $m_F + 2m_L$ vectors of size $n$.
This result is similar to the one presented in [@byrd2014stochastic]. The difference in the complexity arises in the third term of due to our choice of the accumulated Fisher Information matrix as opposed to using a sub-sampled Hessian approximation. It is not imperative for our algorithm to use aFIM for the computation of $y_t$ . We can instead use the eFIM , which would allow for a lower memory requirement (from $(m_F+m_L)n$ to $m_Ln$) at the expense of added computation during curvature pair estimation. However, the time complexity would remain linear in $n$ for either choice. As we mention in Section \[scn:curv\], by using the accumulated Fisher Information matrix, we avoid the need for additional computation at the expense of memory; a choice we have found to work well in practice.
Numerical Results {#sec:num_res}
=================
In this section, we present numerical evidence demonstrating the viability of the proposed algorithm for training RNNs. We also present meta-data regarding the experiments which suggests that the performance difference between [<span style="font-variant:small-caps;">adaQN</span>]{}and its competitors (and [<span style="font-variant:small-caps;">Adagrad</span>]{}in particular) can be attributed primarily to the incorporation of curvature.
Language Modeling {#scn:numerical_lm}
-----------------
For benchmarking, we compared the performance of [<span style="font-variant:small-caps;">adaQN</span>]{}against [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}on two language modeling (LM) tasks: character-level LM and word-level LM. For the character-level LM task [@karpathy2015visualizing], we report results on two data sets: The Tale of Two Cities (Dickens) and The Complete Works of Friedrich Nietzsche (Nietzsche) . The former has 792k characters while the latter has 600k. We used the Penn-Tree data set for the word-level LM task [@zaremba2014recurrent]. This data set consists of 929k training words with 10k words in its vocabulary.
For all tasks, we used an RNN with 5 recurrent layers. The input and output layer sizes were determined by the vocabulary of the data set. The character-level and word-level LMs were constructed with 100 and 400 nodes per layer respectively. The weights were randomly initialized from $\mathcal{N}(0,0.01)$. Unless otherwise specified, the activation function used was $\tanh$. The sequence length was chosen to be $50$ for both cases. For readability, we exclude other popular methods that did not consistently perform competitively. In particular, SGD (with or without momentum) was not found to be competitive despite significant tuning. For [<span style="font-variant:small-caps;">adaQN</span>]{}, [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}, all hyper-parameters were set using a grid-search. In particular, step-sizes were tuned for all three methods. [<span style="font-variant:small-caps;">Adam</span>]{}needed coarse-tuning for $(\beta_1,\beta_2)$ in the vicinity of the suggested values. For [<span style="font-variant:small-caps;">adaQN</span>]{}, the value of $L$ was chosen from $\{2,5,10,20\}$. The rest of the hyper-parameters ($m_F,m_L,\epsilon, \gamma$) were set at their recommended values for all experiments (refer to Algorithm \[alg:adaQN\]). It can thus be seen that the offline tuning costs of [<span style="font-variant:small-caps;">adaQN</span>]{}are comparable to those of [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}. We ran all experiments for 100 epochs and present the results (testing error) in Figure \[fig:performance\].
![Numerical Results on LM Tasks[]{data-label="fig:performance"}](fig_performance.eps){width="75.00000%"}
![Average L-BFGS Memory Per Epoch[]{data-label="fig:memory"}](memory.eps){width="75.00000%"}
It is clear from Figure \[fig:performance\] that [<span style="font-variant:small-caps;">adaQN</span>]{}presents a non-trivial improvement over both [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}on all tasks with $\tanh$ activation function. Specifically, we emphasize the performance gain over [<span style="font-variant:small-caps;">Adagrad</span>]{}, the method which [<span style="font-variant:small-caps;">adaQN</span>]{}is safeguarded by. On the character-level task with ReLU activation, [<span style="font-variant:small-caps;">adaQN</span>]{}performed better than [<span style="font-variant:small-caps;">Adam</span>]{}but worse than [<span style="font-variant:small-caps;">Adagrad</span>]{}. We point out that experiments with other (including larger) data sets yielded results of similar nature.
Average L-BFGS Memory per Epoch {#scn:avg_memory}
-------------------------------
Given the safeguarded nature of our algorithm, a natural question regarding the numerical results presented pertains to the effect of the safeguarding on the performance of the algorithm. To answer this question, we report the average L-BFGS memory per epoch in Figure \[fig:memory\]. This is computed by a running sum initialized at $0$ at the start of each new epoch. A value greater than $1$ indicates that at least one curvature pair was present in the memory (in expectation) during a given epoch. Higher average values of L-BFGS memory suggest that more directions of curvature were successfully explored; thus, the safeguarding was less necessary. Lower values, on the other hand, suggest that the curvature information was either not informative (leading to skipping) or led to deterioration of performance (leading to step rejection).
The word-level LM task with the ReLU activation function has interesting outcomes. It can be seen from Figure \[fig:performance\] that the performance of [<span style="font-variant:small-caps;">Adagrad</span>]{}is similar to that of [<span style="font-variant:small-caps;">adaQN</span>]{}for the first 50 epochs but then [<span style="font-variant:small-caps;">Adagrad</span>]{}continues to make progress while the performance of [<span style="font-variant:small-caps;">adaQN</span>]{}stagnates. During the same time, the average L-BFGS memory drops significantly suggesting that safeguarding was necessary and that, the curvature information was not informative enough and even caused deterioration in performance (evidenced by occasional increase in the function value).
MNIST Classification from Pixel Sequence {#scn:mnist}
----------------------------------------
A challenging toy problem for RNNs is that of image classification given pixel sequences [@le2015simple]. For this problem, the image pixels are presented sequentially to the network one-at-a-time and the network must predict the corresponding category. This long range dependency makes the RNN difficult to train. We report results for the popular MNIST data set. For this experiment, we used a setup similar to that of [@le2015simple] with two modifications: we used $\tanh$ activation function instead of ReLU and initialized all weights from $\mathcal{N}(0,0.01)$ instead of using their initialization trick. The results are reported in Figure \[fig:irnn\].
![Numerical Results on MNIST with Sequence of Pixels[]{data-label="fig:irnn"}](IRNN_fig.eps){width="75.00000%"}
As can be seen from the Figure \[fig:irnn\], [<span style="font-variant:small-caps;">Adam</span>]{}and [<span style="font-variant:small-caps;">Adagrad</span>]{}struggle to make progress and stagnate at an error value close to that of the initial point. On the other hand, [<span style="font-variant:small-caps;">adaQN</span>]{}is able to significantly improve the error values, and also achieves superior classification accuracy rates. Experiments on other toy problems with long range dependencies, such as the addition problem [@hochreiter1997long], yielded similar results.
LSTMs
-----
In order to ascertain the viability of [<span style="font-variant:small-caps;">adaQN</span>]{}on other architectures, we conducted additional experiments using the LSTM models. The experimental setup is similar to the one discussed in Section \[scn:numerical\_lm\] with the modification that 2 recurrent (LSTM) layers were used instead of 5. The results are reported in Figure \[fig:LSTM\].
![Numerical Results on LSTMs[]{data-label="fig:LSTM"}](LSTM.eps){width="75.00000%"}
The results in Figure \[fig:LSTM\] suggest mixed results. For the character-level LM tasks, the performance of [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">adaQN</span>]{}was comparable while the performance of [<span style="font-variant:small-caps;">Adam</span>]{}was better. For the word-level LM task, the performance of [<span style="font-variant:small-caps;">adaQN</span>]{}was superior to that of both [<span style="font-variant:small-caps;">Adagrad</span>]{}and [<span style="font-variant:small-caps;">Adam</span>]{}.
Discussion {#sec:disc}
==========
The results presented in the previous section suggest that [<span style="font-variant:small-caps;">adaQN</span>]{}is competitive with popular algorithms for training RNNs. However, [<span style="font-variant:small-caps;">adaQN</span>]{}is not restricted to this class of problems. Indeed, preliminary results on other architectures (such as Feed-Forward Networks) delivered promising performance. It may be possible to further improve the performance of the algorithm by modifying the update rule and frequency. In this direction, we discuss the practicality of using momentum in such an algorithm and possible heuristics to allow the algorithm to adapt the cycle length $L$ as opposed to tuning it to a constant value.
Recent work by [@sutskever2013importance] suggests superior performance of momentum-based methods on a wide variety of learning tasks. These methods, with the right initialization, have been shown to outperform sophisticated methods such as the Hessian-Free Newton method. However, recent efforts suggest the use of second-order methods *in conjunction* with momentum [@martens2015optimizing; @martens2014new]. In this case, one interpretation of momentum is that of providing a pre-conditioner to the CG sub-solver. Significant performance gains through the inclusion of momentum have been reported when the gradients are reliable [@martens2014new]. We hypothesize that performance gains can be obtained through careful inclusion of momentum for methods like [<span style="font-variant:small-caps;">adaQN</span>]{}as well. However, the design of such an algorithm, and efficacy of using momentum-like ideas is an open question for future research.
Lastly, we discuss the role of the aggregation cycle length $L$ on the performance of the algorithm. If $L$ is chosen to be too large, the aggregation points will be too far-apart possibly leading to incorrect curvature estimation. If $L$ is too small, then the iterates change insufficiently before an update attempt is made leading to skipping of update pairs. Besides the issue of curvature quality, the choice of $L$ also has ramifications on the cost of the algorithm as discussed in Section \[scn:cost\]. Thus, a natural extension of [<span style="font-variant:small-caps;">adaQN</span>]{}is an algorithm where $L$ can be allowed to adapt during the course of the algorithm. $L$ could be increased or decreased depending on the quality of the estimated curvature, while being bounded to ensure that the cost of updating is kept at a reasonable level. The removal of this hyper-parameter will not only obviate the need for tuning, but will also allow for a more robust performance.
Conclusions {#sec:conc}
===========
In this paper, we present a novel quasi-Newton method, [<span style="font-variant:small-caps;">adaQN</span>]{}, for training RNNs. The algorithm judiciously incorporates curvature information while retaining a low per-iteration cost. The algorithm builds upon the framework proposed in [@byrd2014stochastic], which was designed for convex optimization problems. We discuss the key ingredients of our algorithm, such as, the scaling of the L-BFGS matrices using historical gradients, curvature pair updating and step acceptance criterion, and, suggest the use of an accumulated Fisher Information matrix during the computation of a curvature pair. We examine the per-iteration time and space complexity of [<span style="font-variant:small-caps;">adaQN</span>]{}and show that it is of the same order of magnitude as popular first-order methods. Finally, we present numerical results for two language modeling tasks and demonstrate competitive performance of [<span style="font-variant:small-caps;">adaQN</span>]{}as compared to popular algorithms used for training RNNs.
Appendix
========
L-BFGS Two-Loop Recursion {#scn:two-loop}
-------------------------
We describe the two-loop recursion used to compute the step $p_k$ in Algorithm \[alg:two\_loop\]. We refer the reader to [@nocedal2006numerical] for additional details.
[**Inputs:**]{} Curvature pair containers $S$ and $Y$, $\hat{\nabla} f(w_k)$
$\tau = \text{length}(S)$
$q \leftarrow \hat{ \nabla} f(w_k)$
$\alpha_i\leftarrow \rho_i s_i^Tq$
$q \leftarrow q - \alpha_iy_i$
$r \leftarrow H_k^{(0)} q$
$\beta \leftarrow \rho_i y_i ^Tr$
$r \leftarrow r + s_i(\alpha_i - \beta)$
$H_k \hat{\nabla} f(w_k) = r$
[**Output:**]{} $p_k = -H_k \hat{\nabla} f(w_k)$
L-BFGS is an overwriting process as opposed to an updating process; each curvature pair modifies $H_k^{(0)}$ via a rank-2 update. Thus, it is possible to reduce the effect of an incorrect estimation of $H^{(0)}_k$ through sufficient curvature pair updates. However, the initial estimate of scaling $H^{(0)}_k$ remains an important ingredient of L-BFGS updating, especially when the L-BFGS memory is low. If this value is incorrectly estimated, not only does it affect the scale of step, it also affects its quality. If the L-BFGS memory $\tau$ is low, it presents limited avenue for the algorithm to overwrite the poor scale imparted by $H^{(0)}_k$. Indeed, if $\tau$ is $0$, i.e. no curvature pairs are stored, the step that is returned by the two-loop recursion is $p_k = -H_k^{(0)} \hat{\nabla} f(w_k)$. This necessitates the choice of a good scaling matrix for $H_k^{(0)}$ in the case of stochastic quasi-Newton methods. As we mention in Section \[scn:hk0\], [<span style="font-variant:small-caps;">adaQN</span>]{}uses an [<span style="font-variant:small-caps;">Adagrad</span>]{}-like scaling matrix for $H_k^{(0)}$.
[^1]: Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA.
[^2]: Department of Engineering Sciences and Applied Mathematics, Northwestern University Evanston, IL, USA.
|
---
abstract: 'results have revealed that the electric dipole emission from polycyclic aromatic hydrocarbons (PAHs) is the most reliable explanation for anomalous microwave emission that interferes with cosmic microwave background (CMB) radiation experiments. The emerging question is to what extent this emission component contaminates to the polarized CMB radiation. We present constraints on polarized dust emission for the model of grain size distribution and grain alignment that best fits to observed extinction and polarization curves. Two stars with a prominent polarization feature at $\lambda=2175$ Å, HD 197770 and HD 147933-4, are chosen for our study. For HD 197770, we find that the model with aligned silicate grains plus weakly aligned PAHs can successfully reproduce the 2175 Å polarization feature; whereas, for HD 147933-4, we find that the alignment of only silicate grains can account for that feature. The alignment function of PAHs for the best-fit model to the HD 197770 data is employed to constrain polarized spinning dust emission. We find that the degree of polarization of spinning dust emission is about $1.6\%$ at frequency $\nu \approx 3\GHz$ and declines to below $0.9\%$ for $\nu>20\GHz$. We also predict the degree of polarization of thermal dust emission at $353\GHz$ to be $P_{\rm em}\approx 11\%$ and $14\%$ for the lines of sight to the HD 197770 and HD 147933-4 stars, respectively.'
author:
- 'Thiem Hoang, A. Lazarian, and P. G. Martin'
bibliography:
- 'ms.bib'
title: Constraint on the Polarization of Electric Dipole Emission from Spinning Dust
---
\[sec61\]Introduction
=====================
Cosmic Microwave Background (CMB) experiments (see @Bouchet:1999p4616; @Tegmark:2000p5597; @Efstathiou:2003p4793; @Bennett:2003p4582) are of great importance for studying the early universe and its subsequent expansion. Precision cosmology with [*Wilkinson Microwave Anisotropy Probe*]{} ([*WMAP*]{}) and requires a good model of the microwave foreground emission to allow the reliable subtraction of foreground contamination from the CMB radiation.
In the 10–60 GHz frequency range, electric dipole emission (@1998ApJ...508..157D) from rapidly spinning tiny dust grains (mostly polycyclic aromatic hydrocarbons, hereafter PAHs; ; ) is an important component of Galactic foregrounds that dominates the CMB signal (see @Collaboration:2013vx). In the last several years, significant progress has been made in understanding spinning dust in terms of both theory (@2009MNRAS.395.1055A;; @2010ApJ...715.1462H; @2011ApJ...741...87H; @Silsbee:2011p5567; @2013AdAst2013E...2A; see @Hoang:2012bb for a review) and observation (@Dickinson:2009p607; @2011ApJ...734....4K; @2012ApJ...754...94T). In particular, [*Planck*]{} results have confirmed spinning dust emission as the most reliable source of anomalous microwave emission (AME) (@PlanckCollaboration:2011p515). [*Planck*]{} is poised to release interesting results on the CMB polarization; however, the question to what extent the spinning dust emission contaminates to the polarized CMB signal remains unclear.
The degree of polarization of spinning dust emission depends on the alignment efficiency of PAHs in the interstellar magnetic field. [@2000ApJ...536L..15L] (hereafter LD00) suggested that PAHs can be aligned via resonance paramagnetic relaxation–a mechanism which extends the classical Davis-Greenstein (@1951ApJ...114..206D) mechanism for very fast rotating grains for which the Barnett magnetization arising from fast rotation of grains cannot be neglected. They predicted the polarization of spinning dust typically $\leq 1\%$ for frequency $\nu>20$ GHz.
Observational studies (@2006ApJ...645L.141B; @2009ApJ...697.1187M; @2011MNRAS.418L..35D; @2011MNRAS.418..888M) showed that the upper limit for the AME polarization is between $1-5\%$. In addition, an upper limit of $1\%$ for the AME polarization is reported in various media (see @LopezCaraballo:2011p508; @RubinoMartin:2012ji). Since and other CMB experiments provide extremely precise measurements of polarization, a reliable prediction for the spinning dust polarization is useful for separation of polarized Galactic foreground components from the CMB.
The problem of grain alignment, especially for PAHs[^1] is complicated in general (see @2007JQSRT.106..225L for a review), but one can derive constraints on grain alignment observationally (see @2007EAS....23..165M). An important attempt to obtain observational constraints on grain alignment was carried out by [@1995ApJ...444..293K]. The authors applied maximum entropy method to infer the mass distribution of aligned grains through fitting theoretical polarization curves to observational data. They discovered that interstellar silicate grains of size $a \ge 0.05{\,\mu{\rm m}}$ are aligned, whereas smaller grains, including PAHs, are weakly aligned. [@Draine:2009p3780] performed simultaneous fitting to the typical extinction and polarization curves of the diffuse interstellar medium (ISM) and came to the similar conclusion as in [@1995ApJ...444..293K] that small ($a<0.05{\,\mu{\rm m}}$) grains are weakly aligned and large ($a>0.1{\,\mu{\rm m}}$) grains are efficiently aligned.
Since the tiny dust grains that radiate spinning dust emission are likely the same as those that produce the ultraviolet (UV) extinction bump at $\lambda=2175$Å,[^2] a good way to search for the alignment of PAHs is through its imprint on the UV polarization. While the UV extinction bump is ubiquitous in the ISM, the UV polarization bump is rarely seen, except for two stars HD 197770 and HD147933-4, which show a prominent $2175$Å polarization feature (@1992ApJ...385L..53C; @1993ApJ...403..722W; @1997ApJ...478..395W). In addition to the UV polarization bump, HD 197770 exhibits an excess UV polarization that cannot be accounted for with the typical ISM polarization curve–Serkowski law (@Serkowski:1973p6351) as do some stars with the peak wavelength of polarization curve $\lambda_{\max}<0.55{\,\mu{\rm m}}$ (@1995ApJ...445..947C; @1999ApJ...510..905M). Such an excess UV polarization may be due to the enhanced alignment of small silicate grains by paramagnetic relaxation or radiative torques (Hoang et al. 2013, submitted).
A number of studies have suggested that the UV polarization bump can arise from aligned, small graphite grains (@1988ApJ...333..848D; @1993ApJ...403..722W; @1997ApJ...478..395W); however, a detailed study quantifying the alignment efficiency of small graphite grains is not yet available.
The main goal of this paper is to find the degrees of alignment of interstellar grains including PAHs that reproduces the 2175Å polarization bump seen in HD 197770 and HD 147933-4 and employ the inferred degrees of alignment to predict the degree of polarization of spinning dust emission. The paper is structured as follows.
In §\[sec:optical\] we calculate extinction cross section and polarization cross section for silicate and carbonaceous oblate spheroidal grains. In §\[sec:fitUV\], we describe a procedure to derive grain size distributions and degree of grain alignment by fitting theoretical predictions to observed extinction and polarization curves and present the obtained results. In §\[sec:polspdust\] polarized spinning dust emission is calculated using the degree of grain alignment obtained for the best-fit model. Discussion and summary are presented in §\[sec:discus\] and §\[sec:summ\], respectively.
Optical properties of dust grains {#sec:optical}
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Extinction and Polarization Cross Section
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Interstellar dust grains are widely known to induce the extinction and polarization of starlight due the absorption and scattering of light out of the line of sight by dust grains.
Let us consider an oblate spheroidal grain with the symmetry axis $\ba_{1}$ having an effective size $a$, which is the size of an equivalent sphere of the same volume as the grain. A perfectly polarized electromagnetic wave with the electric field vector $\bE$ is assumed to propagate along the line of sight. Let $C_{\ext}(\bE\perp \ba_{1})$ and $C_{\ext}(\bE\| \ba_{1})$ be the extinction of radiation by the grain for the cases in which $\bE$ is parallel and perpendicular to $\ba_{1}$, respectively. For the sake of simplification, we denote these extinction cross sections by $C_{\perp}$ and $C_{\|}$.
For the general case in which $\bE$ makes an angle $\theta$ with $\ba_{1}$, the extinction cross section becomes C\_=\^[2]{}C\_+\^[2]{}C\_.\[eq:Cext0\]
Since the original starlight is unpolarized, one can compute the total extinction cross section for a randomly oriented grain by integrating Equation (\[eq:Cext0\]) over the isotropic distribution of $\theta$, i.e., $f_{\rm iso} d\theta \sim \sin\theta d\theta$. As a result, C\_=(2C\_+C\_).\[eq:Cext\]
The polarization coefficient for oblate spheroidal grains is equal to C\_=.
The dust extinction and polarization cross section can be represented through the extinction and polarization efficiency, which are respectively defined by: Q\_(a,)=,\
Q\_(a,)=.
Silicate grains
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For silicate grains, we employ the publicly available DDSCAT code (@2012arXiv1202.3424D) to compute the extinction cross section $C_{\ext}^{\sil}(a,\lambda)$ using the dielectric functions of amorphous silicate from [@2003ApJ...598.1026D].
Figure \[fig:Qext\_waveinv\] shows $\lambda Q_{\ext}/a$ and $\lambda Q_{\pol}/a$ as functions of $\lambda^{-1}$ for a silicate spheroidal grain with axial ratio $r=2$ and for the different grain sizes.
Carbonaceous grains
-------------------
Carbonaceous grains consist of a fraction $\zeta_{\PAH}$ of PAHs and $1-\zeta_{\PAH}$ of graphite grains with \_(a)=(1-q\_), where $a_{\zeta}=50$Å, $q_{\gra}=0.01$ are assumed as in [@2007ApJ...657..810D].
For graphite grains, which are an anisotropic material, we adopt the dielectric functions from [@2003ApJ...598.1026D] and compute $C_{\ext}^{\gra}(a,\lambda)$ using DDSCAT. We consider two cases in which $\bE$ are parallel and perpendicular to the grain optical axis (c-axis) with the corresponding the dielectric function $\epsilon_{\|}$ and $\epsilon_{\perp}$. Using the $\frac{1}{3}-\frac{2}{3}$ approximation (i.e., 1/3 of graphite grains have $\epsilon=\epsilon_{\|}$ and 2/3 of them have $\epsilon=\epsilon_{\perp}$), one can obtain the extinction cross section $C_{\ext}^{\gra}=\left(C_{\ext}(\bE\| c)+2C_{\ext}(\bE\perp c)\right)/3$ for a given grain orientation. Using Equation (\[eq:Cext\]) we obtain $C_{\ext}$ for randomly oriented grains.
For PAHs, the extinction $C_{\ext}^{\PAH}(\lambda)$ is computed as in Equations (5)-(10) in [@2001ApJ...554..778L]. The polarization cross section of PAHs is taken to be the same as graphite grains of the same size. It has been shown in that the absorption properties (and then polarization properties) of PAHs are similar to those of bulky graphite grains when the dielectric constant of bulky graphite is adopted for PAHs.
![Extinction efficiency $\lambda Q_{\ext}/a$ (upper) and polarization efficiency $\lambda Q_{\pol}/a$ (lower) of silicate grains as functions of $1/\lambda$ for the different grain sizes.[]{data-label="fig:Qext_waveinv"}](f1a.eps "fig:"){width="45.00000%"} ![Extinction efficiency $\lambda Q_{\ext}/a$ (upper) and polarization efficiency $\lambda Q_{\pol}/a$ (lower) of silicate grains as functions of $1/\lambda$ for the different grain sizes.[]{data-label="fig:Qext_waveinv"}](f1b.eps "fig:"){width="45.00000%"}
Extinction and Polarization Curves {#sec:theory}
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The extinction induced by randomly oriented grains in units of magnitude is defined by A()&=&2.5[log]{}\_[10]{}()=1.086\_,\
&=&1.086\_[j=,]{}\_[i=0]{}\^[N\_[a]{}-1]{} C\_\^[j]{}(a\_[i]{}) n\^[j]{}(a\_[i]{})dz,\[eq:Aext\] where $F_{\lambda}^{\star}$ is the intrinsic flux from the star, $F_{\lambda}^{\obs}=F_{\lambda}^{\star}e^{-\tau{_\lambda}}$ is the observed flux, $\tau_{\lambda}$ is the optical depth, and the integration is performed along the line of sight.
To find the polarization by aligned grains, let us define an observer’s reference system in which the line of sight is directed along the $z-$axis, and $x-$ and $y-$ axes constitute the sky plane. The polarization of starlight arising from the dichroic extinction by aligned grains in a cell of $dz$ is computed as dp()==\_[i=0]{}\^[N\_[a]{}-1]{}(C\_[x]{}-C\_[y]{}) n(a\_i)dz, \[eq:dplam\] where $N_{a}$ is the number of size bin, $n(a_{i})\equiv (dn/da)da$ is the number of grains of size $a_{i}$, $C_{x}$ and $C_{y}$ are the grain cross section along the $x-$ and $y-$ axes, respectively.
For the case of perfect internal alignment of grain symmetry axis $\ba_{1}$ with angular momentum $\bJ$, by transforming the grain coordinate system to the observer’s reference system and taking corresponding weights, we obtain C\_[x]{}&=&C\_-\^[2]{},\
C\_[y]{}&=&C\_-(2\^[2]{}\^[2]{}+\^[2]{}\^[2]{}), where $\gamma$ is the angle between the magnetic field $\Bv$ and the sky plane and $\beta$ is the angle between $\bJ$ and $\Bv$.
The polarization efficiency then becomes C\_[x]{}-C\_[y]{}=C\_\^[2]{}.\[eq:Cpol\] Taking the average of $C_{x}-C_{y}$ over the distribution of the alignment angle $\beta$, it yields C\_[x]{}-C\_[y]{}=C\_ Q\_[J]{}\^[2]{},\[eq:Cx-Cy\] where $Q_{J}=\langle G_{J}\rangle$ is the ensemble average of $G_{J}=\left(3\cos^{2}\beta-1\right)/2$ that describes the alignment of $\bJ$ and $\Bv$.
When the internal alignment is not perfect, following the similar procedure, we obtain C\_[x]{}-C\_[y]{}=C\_G\_[J]{}G\_[X]{} \^[2]{}C\_R\^[2]{},\[eq:Cpol\] where $G_{X}=\left(3\cos^{2}\theta-1\right)/2$ with $\theta$ being the angle between $\ba_{1}$ and $\bJ$, and $R=\langle G_{J}G_{X}\rangle$ is the Rayleigh reduction factor (see also @1999MNRAS.305..615R). The degree of internal alignment is described by $Q_{X}=\langle G_{X}\rangle$.
For a perpendicular magnetic field, i.e., $\Bv$ lies on the sky plane, Equation (\[eq:Cpol\]) simply becomes $C_{x}-C_{y}=C_{\pol}R$. For an arbitrary magnetic field geometry, let $f=R\cos^{2}\gamma$ be the effective degree of grain alignment, which is a function of grain size $a$. Thus, in the following, $f(a)$ is referred as the alignment function.
Plugging this above equation into Equation (\[eq:dplam\]) and integrating along the line of sight, we obtain p()=\_[j=,]{}\_[i=0]{}\^[N\_[a]{}-1]{} C\_\^[j]{}(a\_[i]{})f\^[j]{}(a\_[i]{}) n\^[j]{}(a\_i)dz,\[eq:Plam\] where $f^{j}(a_i)$ is the effective degree of grain alignment for the grain specie $j$ of size $a_i$.
It is more convenient to represent the extinction (polarization) through the extinction (polarization) cross section. Hence, the above equations can be rewritten as A()&=&1.086 \_()N\_,\[eq:sigext\]\
p()&=&\_()N\_,\[eq:sigpol\] where $N_{\H}(\cm^{-2})$ is the column density, and $\sigma_{\ext}$ and $\sigma_{\pol}$ in units of $\cm^{2} \H^{-1}$ are the dust extinction cross section and polarization cross section, respectively.
Grain size distribution and alignment functions constrained by observations {#sec:fitUV}
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Nonlinear Chi-square Fitting
----------------------------
In this section, we find grain size distributions and alignment functions by fitting theoretical predictions to the observational data for the HD 197770 and HD 147933-4 stars. The parameters for these stars, including the optical depth $\tau_{V}$, ratio of visual to selective extinction $R_{V}=A_{V}/E_{B-V}$, peak polarization $p_{\max}$, peak wavelength $\lambda_{\max}$, and polarization efficiency $p_{\max}/A(\lambda_{\max})$, are listed in Table \[tab:HD\] (see also @1997ApJ...478..395W).
[l l l l l l]{}\
Star HD & $R_{V}$ & $\tau_{V}$ & $\lambda_{\max}(\mu\rm m)$ & $p_{\max}(\%)$ & $p_{\max}/A(\lambda_{\max})$\
197770 & 3.1 & 1.66 & $0.511\pm 0.001$ & $3.90 \pm 0.01$ & 1.95 $\%$/mag147933-4 & 4.3 & 1.86 & $0.683\pm 0.003$ & $2.72 \pm 0.01$ & $1.67$ $\%$/mag
We adopt the mixture dust model consisting of amorphous silicate grains, graphite grains and PAHs (see @2001ApJ...548..296W ; @2007ApJ...657..810D). Since there is no observational evidence for a particular shape of interstellar grains, here we assume that both silicate and carbonaceous grains are oblate spheroidal as in previous studies (e.g., [@1995ApJ...444..293K]; @2006ApJ...652.1318D; @Draine:2009p3780). These studies also show that a wide range of the grain axial ratio $r$ can successfully reproduce the observational data. Moreover, our principal interest is focused on PAHs, a reasonable assumption for the axial ratio of PAHs is necessary to infer a reliable constraint for their degree of alignment. Indeed, a disklike shape with radius $R$ and height $L$ is usually assumed for PAHs with $a \le 6$Å (see e.g. @1998ApJ...508..157D). The grain volume is $V= 4\pi a^3/3= 4\pi r^{-1} b^3/3 = \pi R^2 L$ with $L=3.35$Å for planar PAHs. For the smallest PAH with $a =3.56$Å that is the most abundant, we obtain $R^2=4a^3/3L \approx 25$ or $R \approx 5$Å. If we approximate the disklike shape as an oblate spheroid such that the length of the major semiaxis is equal to the disk radius (i.e., $b=R$), then the axial ratio $r=(4/3) b^{3}/(L R^2) \approx 4R/3L \approx 2$. As a result, we conservatively assume the axial ratio $r=2$ for all grains throughout the paper.
In the present paper, we assume that graphite grains are randomly oriented, whereas silicate grains and PAH can be aligned. Such an assumption allows us to obtain an upper limit on the alignment efficiency of PAHs since aligned graphite grains would contribute some polarization to the $2175$Å feature. As in [@Draine:2009p3780], we simultaneously fit theoretical predictions to the observed extinction and polarization curves.
Following [@1995ApJ...444..293K], we find the grain size distribution and alignment function by minimizing a objective function $\chi^{2}$, which is given by \^[2]{}=\_\^[2]{}+\_\^[2]{}+\_[con]{}\^[2]{},\[eq:chisq\] where \_\^[2]{}=w\_\_[i=0]{}\^[N\_-1]{}\^[2]{},\
\_\^[2]{}=w\_\_[i=0]{}\^[N\_-1]{}\^[2]{}, with $w_{\ext}$ and $w_{\pol}$ being the fitting weights for the extinction and polarization, respectively, and the last term $\chi_{\rm con}^{2}$ contains the additional constraints of the fitting model. Here, the summation is performed over $N_{\lambda}$ wavelength bins. We consider $N_{a}=64$ size bins from $a=3.56$Å to $1{\,\mu{\rm m}}$ and $N_{\lambda}=64$ from $\lambda=0.15$ to $2.5{\,\mu{\rm m}}$. Both grain size and wavelength grids have their logarithms equally spaced.
The “observed" extinction $A_{\obs}(\lambda)$ is calculated using the extinction law (@1989ApJ...345..245C; @1994ApJ...422..158O) for the measured values $R_{V}$ of the two stars (see Table \[tab:HD\]), and the “observed" polarization $p_{\obs}(\lambda)$ is obtained by interpolating the observational data from [@1997ApJ...478..395W] for $N_{\lambda}$ wavelength bins. The extinction $A_{\mod}(\lambda)$ and polarization $p_{\mod}(\lambda)$ are given by Equations (\[eq:Aext\]) and (\[eq:Plam\]).
Following [@2006ApJ...652.1318D], we introduce a number of constraints for grain size distribution $dn/da$ and alignment function $f(a)$. The most important constraint for $f(a)$ is given by the polarization efficiency $p_{\max}/A(\lambda_{\max})$. For the case of the maximum $p_{\max}/A(\lambda_{\max})=3\% {\rm mag}^{-1}$ (@Serkowski:1975p6681), the conditions for grain alignment are expected to be optimal such that the alignment of big grains can be perfect, and the magnetic field is regular and perpendicular to the line of sight. Thus, $f(a=a_{\max})=1$ has been taken for this optimal case (@Draine:2009p3780; Hoang et al. 2013). For HD 197770 and HD 147933-4 having lower $p_{\max}/A(\lambda_{\max})$, the constraint for $f(a)$ should be adjusted such that $f(a=a_{\max})=(1/3)p_{\max}/A(\lambda_{\max})$. Moreover, for relatively large ($a>0.05{\,\mu{\rm m}}$) grains that are aligned by radiative torques (see @2007JQSRT.106..225L), we expect the degree of alignment increasing with $a$. In particular, our detailed calculations in Hoang et al. (2013) show that $f(a)$ of small grains that are aligned by Davis-Greenstein paramagnetic relaxation tends to increase monotonically with $a$ whereas $f(a)$ of very small ($a<50$Å) grains aligned by resonance relaxation does not. Thus, a constraint for the monotonic increase of $f(a)$ versus $a$ is introduced for grains larger than $50$Å. Other constraints include the non-smoothness of $dn/da$ and $f(a)$ (see @2006ApJ...652.1318D).
The fitting procedure is started with an initial size distribution $n(a)$ that best reproduces the observational data for the diffuse ISM, which corresponds to model 3 in [@Draine:2009p3780]. By doing so, we implicitly assume that dust properties are similar throughout the ISM and the difference in the polarization of starlight is mainly due to the efficiency of grain alignment, which depends on environment conditions along the line of sight, e.g., radiation field, magnetic fields and gas density. For silicate grains, we take the alignment function for the ISM from [@Draine:2009p3780] as an initial alignment function. For PAHs, we adopt a physically motivated initial alignment function that is directly computed in Hoang et al. 2013. This initial alignment function peaks at $a\sim 10$Å.
The nonlinear least square fitting is carried out using the Monte Carlo method. Basically, for each size bin, we generate $N_{\rm rand}$ random samples in the range $[-\zeta,\zeta]$ from a uniform distribution for $f(a)$ and $n(a)$, $\alpha_{f}$ and $\alpha_{n}$, respectively. The new values of $f$ and $n$ are given by $\tilde{f}=(\alpha_{f}+1)f(a)$ and $\tilde{n}=(\alpha_{n}+1)n(a)$. Then we calculate $A_{\mod}$ and $p_{\mod}$ for the new values $\tilde{f}$ and $\tilde{n}$ using Equations (\[eq:Aext\]) and (\[eq:Plam\]). The values of $\chi^{2}$ obtained from Equation (\[eq:chisq\]) are used to find the minimum $\chi^{2}$. The range $[-\zeta,\zeta]$ of the uniform distribution is adjusted after each iterative step. Initially $\zeta=0.5$ is assumed, which allows more room for the random sampling, and when the convergence is close (i.e., the variation of $\chi^{2}$ is small) $\zeta$ is decreased to $\zeta=0.1$.
The above process is repeated until convergence criterion is satisfied. Here, we employ the convergence criterion which is based on the decrease of $\chi^{2}$ after one step: $\epsilon=(\chi^{2}(n,f)-\chi^{2}(\tilde{n},\tilde{f}))/\chi^{2}(n,f)$. If $\epsilon\le \epsilon_{0}$ with $\epsilon_{0}$ sufficiently small, then the convergence is said to be achieved. For $\epsilon_{0}=10^{-3}$, we found that the convergence is achieved after $32$ steps for HD 197770. The convergence is hard to achieve for HD 147933-4 due to the rather low UV polarization relative to the peak polarization, so we stop the iterative process after 43 steps.
Results
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Figure \[fig:extcurve\] shows the extinction curves from our best-fit model compared to the “observed" data for the HD 197770 and HD 147933-4 stars. As shown the model can successfully reproduce the observed extinction curves for both stars.
Figure \[fig:polfit\] shows the polarization curves from our best-fit model versus the “observed" data from [@1997ApJ...478..395W]. Here the error bars of $3\sigma$ are shown. For HD 197770, it can be seen that our model provides an excellent fit to the observed polarization, successfully reproducing both the $2175$ Å polarization feature and excess UV polarization.
The $2175$ Å polarization feature in HD 147933-4 can also be reproduced, but the model appears to overestimate the UV polarization for $\lambda^{-1}>5{\,\mu{\rm m}}^{-1}$ (see the lower panel in Figure \[fig:polfit\]). Such a poor fit to the low UV polarization present in the line of sight with large $\lambda_{\max}$ arises from the fact that the UV polarization signal is dominated by relatively large grains, which produce the optical and infrared polarization. Thus, any improvement in the fit to the UV polarization comes with the expense of a poorer fit to the optical and infrared polarization (see also @1995ApJ...444..293K).
To see quantitatively the goodness of our best-fit model, we calculate the chi-square test statistics: \_\^[2]{}=\_[i=0]{}\^[N\_-1]{},\
X\_\^[2]{}=\_[i=0]{}\^[N\_-1]{}, using the best-fit parameters. We obtain $X^{2}_{\ext}=0.16$ and $X^{2}_{\pol}=0.05$ for HD 197770. The similar values $X^{2}_{\ext}=0.27$ and $X^{2}_{\pol}=0.59$ for HD 147933-4. Thus, the best-fit model appears to be good for both stars.
![Observed extinction curve (filled ellipses) versus model for HD 197770 and HD 147933-4. An excellent fit achieved for both stars.[]{data-label="fig:extcurve"}](f2a.eps "fig:"){width="45.00000%"} ![Observed extinction curve (filled ellipses) versus model for HD 197770 and HD 147933-4. An excellent fit achieved for both stars.[]{data-label="fig:extcurve"}](f2b.eps "fig:"){width="45.00000%"}
![Polarization curves for our model (solid lines) versus the observed polarization data (filled circles) for HD 197770 (upper) and HD 147933-4 (lower). Dotted lines show the observational data interpolated to our wavelength bins used for model fitting. The 3$\sigma$ error bars are shown. An excellent fit achieved for HD 1977770.[]{data-label="fig:polfit"}](f3a.eps "fig:"){width="45.00000%"} ![Polarization curves for our model (solid lines) versus the observed polarization data (filled circles) for HD 197770 (upper) and HD 147933-4 (lower). Dotted lines show the observational data interpolated to our wavelength bins used for model fitting. The 3$\sigma$ error bars are shown. An excellent fit achieved for HD 1977770.[]{data-label="fig:polfit"}](f3b.eps "fig:"){width="45.00000%"}
Figure \[fig:mafit\] shows the mass distributions of silicate and carbonaceous grains for the best-fit model for HD 197770 (upper) and HD 1479333-4 (lower). The distribution functions (DL07 and DF09) from [@2007ApJ...657..810D] and [@Draine:2009p3780] are shown for comparison.
For HD 197770, it can be seen that the mass of PAHs in our model is similar to DL07 and DF09. The mass distribution of graphite grains is nearly the same as DF09 and distinct from DL07. For silicate grains, the mass distribution peaks at $a\sim 0.009{\,\mu{\rm m}}, 0.06{\,\mu{\rm m}}$ and $0.2{\,\mu{\rm m}}$. Compared to DF09, our model requires a slight increase of silicate mass in the range $a=[0.012{\,\mu{\rm m}}-0.05{\,\mu{\rm m}}]$ to reproduce the excess UV continuum polarization seen in HD 197770.
For HD 147933-4, the mass distribution of silicate grains peaks at a larger size ($a\approx 0.25{\,\mu{\rm m}}$) than for HD 197770, whereas the mass of small grains ($a<0.2{\,\mu{\rm m}}$) is subtantially reduced. The peak in the mass of ($\sim 0.1{\,\mu{\rm m}}$) graphite grains is increased compared to DF09. Such a decrease in the mass of small grains and increase in the mass of large grains is necessary to reproduce the higher value $R_{V}$ along the sightline to HD 147933-4 ($R_{V}=4.3$) compared to the typical value ($R_{V}=3.1$) along most of the sightlines of the diffuse ISM.
![Mass distribution of silicate grains and carbonaceous grains for the best-fit model (solid lines) for the HD 197770 (upper) and HD 147933-4 (lower) stars. Dotted (DF09) and dashed (DL07) lines show the grain mass distributions from [@Draine:2009p3780] and [@2007ApJ...657..810D].[]{data-label="fig:mafit"}](f4a.eps "fig:"){width="45.00000%"} ![Mass distribution of silicate grains and carbonaceous grains for the best-fit model (solid lines) for the HD 197770 (upper) and HD 147933-4 (lower) stars. Dotted (DF09) and dashed (DL07) lines show the grain mass distributions from [@Draine:2009p3780] and [@2007ApJ...657..810D].[]{data-label="fig:mafit"}](f4b.eps "fig:"){width="45.00000%"}
Figure \[fig:Rfit\] (upper) shows the alignment function $f(a)$ that reproduces the polarization curve in Figure \[fig:polfit\] (upper). We can see that the alignment of silicate grains increases monotonically with the increasing $a$ and becomes nearly constant with $f\sim 0.6$ for $a>0.2{\,\mu{\rm m}}$. The degree of alignment declines rapidly for $a<0.01{\,\mu{\rm m}}$. The alignment of $a<0.003{\,\mu{\rm m}}$ (i.e. 30Å) silicate grains is not shown because the polarization by tiny silicate grains containing minor dust mass is negligible. In particular, one can see from the figure that the best-fit model corresponds to a peaky alignment function of PAHs with $f_{\rm peak}\approx 0.004$ at $a\approx 10$Å.[^3] It indicates that a rather low alignment degree of PAHs could be sufficient to reproduce the UV polarization feature.
For HD 147933-4 (see Figure \[fig:Rfit\], lower), we find that the best-fit model requires no alignment of PAHs (i.e., the alignment of only silicate grains can reproduce the observed polarization). The degree of alignment $f\sim 0.55$ for $a>0.15{\,\mu{\rm m}}$ silicate grains. Moreover, the alignment of silicate grains decline steeply for $a<0.1{\,\mu{\rm m}}$ and becomes negligible ($f<0.003$) for $a<0.05{\,\mu{\rm m}}$ (small grains). It turns out that small silicate grains have negligibly small contribution to the UV polarization, which is different from HD 197770. It is noted that the sharp decline in alignment of small ($a<0.1{\,\mu{\rm m}}$) silicate grains (i.e., the polarization is dominated by larger silicate grains) is required to account for the large value of $\lambda_{\max}$ seen in HD 147933-4.
![[Upper panel:]{} effective degree of alignment as a function of grain size for silicate grains (blue line) and carbonaceous grains (red line) obtained from the fitting for HD 197770. Dotted line shows $a=3\times 10^{-7}\cm$ from which the polarization by aligned small silicate grains becomes negligible. [Lower panel:]{} similar to upper panel but for HD 147933-4. The alignment of silicate grains is shown only since the presence of PAH alignment is unnecessary (see the text).[]{data-label="fig:Rfit"}](f5a.eps "fig:"){width="45.00000%"} ![[Upper panel:]{} effective degree of alignment as a function of grain size for silicate grains (blue line) and carbonaceous grains (red line) obtained from the fitting for HD 197770. Dotted line shows $a=3\times 10^{-7}\cm$ from which the polarization by aligned small silicate grains becomes negligible. [Lower panel:]{} similar to upper panel but for HD 147933-4. The alignment of silicate grains is shown only since the presence of PAH alignment is unnecessary (see the text).[]{data-label="fig:Rfit"}](f5b.eps "fig:"){width="45.00000%"}
Polarization of spinning dust emission {#sec:polspdust}
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Degree of grain alignment for spinning dust emission
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To calculate polarized spinning dust emissivity, it is important to understand the relation between the degree of alignment of PAHs that is responsible for the polarization in microwave emission and the degree of alignment of PAHs that results in starlight polarization by extinction (i.e., Rayleigh reduction factor).
As discussed in [@2007JQSRT.106..225L], these two measures are expected to be different. Indeed, even spherical PAHs aligned by paramagnetic mechanism can produce polarized microwave emission. Naturally, the polarization by UV absorption or polarized infrared emission is not expected from the spherical PAHs. Although we do not really believe that most of PAHs are spherical, this illustrates the problem that we face comparing different observational consequences of alignment. For instance, fast wobbling and flipping grains are also expected to deliver substantially reduced degrees of polarization in terms of UV absorption. However, this just decreases the polarization of microwave emission by a factor of unity[^4].
In the following, to find the upper limit of polarized spinning dust, we employ the degree of alignment of angular momentum with the magnetic field, $Q_{J}$. Thus, from the inferred alignment function $R(a)$, we can derive $Q_{J}$ using the following relationship (see Section \[sec:theory\]) R(a)G\_[X]{}G\_[J]{}=Q\_[J]{}Q\_[X]{}(1+f\_[corr]{}),\[eq:Ra\] where $Q_{X}$ describes the degree of alignment of grain axes with the angular momentum and $f_{\rm corr}$ describes the correlation between the internal and external alignment. The case $f_{\rm corr}=0$ indicates that the internal alignment is independent from the external alignment. Our calculations show that $f_{\rm corr}\sim 0.6$ for $a=10$Å grains.
Figure \[fig:QX\] shows the different measures of grain alignment as functions of grain size. The effective Rayleigh reduction factor $R\cos^{2}\gamma$ is obtained from the best-fit model (Figure \[fig:Rfit\], upper). $Q_{X}$ is obtained from detailed calculations of grain alignment by paramagnetic relaxation in Hoang et al. (2013) for the cold neutral medium (CNM: $n_{\H}=30\cm^{-3}, T_{\gas}=100\K$) and a constant dust temperature $T_{\d}=60\K$, assuming a typical interstellar magnetic field $B=10\mu$G. $Q_{J}$ is calculated using Equation (\[eq:Ra\]) with $f_{\rm corr}$ from Hoang et al. (2013). It can be seen that $Q_{J}$ is larger than $R$ by a factor of $3$.
![Alignment measures as functions of grain size $a$: effective degree of alignment $f=R\cos^{2}\gamma$, degree of alignment of grain axes with the angular momentum $Q_{X}$, and effective degree of alignment of angular momentum with the magnetic field $Q_{J}\cos^{2}\gamma$. $Q_{X}$ is calculated for paramagnetic alignment of PAHs with $T_{\d}=60\K$ in the CNM with the typical magnetic field $B=10\mu$G.[]{data-label="fig:QX"}](f6.eps){width="50.00000%"}
Polarized spinning dust emissivity
----------------------------------
The polarized emissivity and unpolarized emissivity of spinning dust emission can be given by q\_&=&\_[a\_[l]{}]{}\^[a\_[u]{}]{} Q\_[J]{}(a)\^[2]{}j\_(a) da ,\
j\_&=&\_[a\_[l]{}]{}\^[a\_[u]{}]{} j\_(a) da, where $j_{\nu}(a)$ is the spinning dust emissivity at frequency $\nu$ from a grain of size $a$, and $a_{l}=3.56$Å and $a_{u}=100$Å. The frequency dependence of polarization of spinning dust emission is $p(\nu)=q_{\nu}/j_{\nu}$. We adopt the spinning dust model from [@2011ApJ...741...87H] for oblate spheroidal grains with ratio axis $r=2$ and a typical electric dipole moment $\beta=0.4$ Debye.
![Upper panel: Unpolarized (solid line) and polarized (dotted line) spinning dust emissivity from the diffuse ISM along the line of sight to HD 197770. Lower panel: Frequency dependence of degree of polarization of spinning dust emission.[]{data-label="fig:polspec"}](f7a.eps "fig:"){width="50.00000%"} ![Upper panel: Unpolarized (solid line) and polarized (dotted line) spinning dust emissivity from the diffuse ISM along the line of sight to HD 197770. Lower panel: Frequency dependence of degree of polarization of spinning dust emission.[]{data-label="fig:polspec"}](f7b.eps "fig:"){width="50.00000%"}
Figure \[fig:polspec\] (upper panel) shows the total (unpolarized) and polarized spinning dust emissivity for the best-fit model. The degree of polarization is shown in Figure \[fig:polspec\] (lower panel). It can be seen that the maximum degree of polarization is about $1.6\%$ and peaks at $\nu\approx 3\GHz$. The polarization is $\le 0.9\%$ for $\nu\ge 20 \GHz$.
Discussion {#sec:discus}
==========
In the era, electric dipole emission from rapidly rotating PAHs has become an accepted component of diffuse galactic foreground emissions that contaminate to the CMB radiation. In light of more exciting results on CMB polarization, a pressing issue remains is to quantify the level of contamination by polarized spinning dust emission to the polarized CMB signal. The present paper seeks constraints on the polarization of spinning dust emission based on the models that best fit to observed polarization data of the stars having potential evidence of PAHs alignment.
$2175$Å polarization bump and alignment of PAHs
-----------------------------------------------
The $2175$Å polarization bump observed for two stars, HD197770 and HD147933-4, was discovered a long time ago (@1992ApJ...385L..53C; @1993ApJ...403..722W), and it was suggested that the polarization bump originates from aligned small graphite (see e.g. @1989IAUS..135..313D). Employing the observational data for these stars, we obtain the grain size distributions and alignment functions that best reproduce both the observed extinction and polarization curves.
For HD 197770, we found that a model with aligned silicate grains plus weakly aligned PAHs can successfully reproduce the $2175$Å polarization bump as well as the excess UV polarization. The effective degree of PAH alignment inferred for HD 197770 varies with the grain size and has peak $R\cos^{2}\gamma\approx 0.004$ at $a \approx 10$Å. Although the degree of PAH alignment is rather small, due to the dominance of PAHs for the dust mass for $a<20$Å, it is sufficient to reproduce the $2175$Å polarization bump. In fact, the low degree of alignment for PAHs is not unexpected from theoretical predictions based on resonance paramagnetic alignment, which was proposed by [@2000ApJ...536L..15L] and numerically studied in Hoang et al. 2013.
For HD 147933-4, on the other hand, we found no indication of alignment of PAHs. The model with only aligned silicate grains can account for the $2175$Å polarization bump; although the model overestimates the observed polarization for $\lambda^{-1}>5{\,\mu{\rm m}}^{-1}$ (see Figure \[fig:Rfit\]).[^5]
Before understanding why there is such a difference in two stars, let us recall the significant difference in the optical and polarization properties of HD 197770 and HD 147933-4 (see Table \[tab:HD\]). Indeed, the latter has a much higher ratio of visual-to-selective extinction $R_{V}$ and a much larger peak wavelength $\lambda_{\max}$. In addition, the polarization ratio $p$(2175Å)/$p_{\max}$ is much lower in the later case. Qualitatively, $R_{V}$ reflects the average size of dust grains, whereas $\lambda_{\max}$ reflects the average size of [*aligned*]{} grains. Thus, compared to HD 197770, HD 147933-4 essentially has the larger average grain size and the larger average size of [*aligned*]{} grains. As a result, the UV polarization is dominated by the alignment of large grains. Moreover, since the large silicate grains have optical properties with oscillating features in the UV due to interference effects,[^6] they can induce the broad polarization feature seen beyond $\lambda^{-1}\approx 4.6{\,\mu{\rm m}}^{-1}$.
Why is the 2175Å polarization bump not seen for most stars?
-----------------------------------------------------------
The question now is that if the PAHs are potentially aligned, why we do not see the UV polarization feature for most of stars. To answer this question, let us consider the lines of sight of similar $R_{V}$ and $\lambda_{\max}$. Then the presence of $2175$Å polarization bump depends on the degree of alignment of PAHs in the ambient magnetic field. Although the alignment mechanism for PAHs is still not clear, the most promising mechanism is based on the resonance paramagnetic relaxation (LD00). Since the paramagnetic alignment is sensitive to the magnetic field and gas randomization, the variation of the magnetic field (both strength and direction) and gas density along different lines of sight can result in the absence of the UV polarization feature.
In addition, the resonance paramagnetic alignment depends on spin-spin and spin-lattice relaxation within PAHs. If the relaxation is reduced, the expected degree of grain alignment is also reduced. The point of the relaxation in PAHs, as it was claimed in LD00 can be settled via laboratory studies.
The possibility to identify the $2175$Å polarization bump depends on both the alignment of PAHs and small silicate grains because the latter is responsible for the UV continuum polarization. If the alignment of small silicates is inefficient, then the bump can be detected due to high contrast. If the alignment of small silicate grains is considerable, the UV polarization produced by such grains tends to smooth out the bumpy polarization by PAHs, which makes the detection of 2175Å bump more difficult.
One interesting point with HD 197770 is that it has an excess UV polarization much lower than other stars with the same $\lambda_{\max}=0.51{\,\mu{\rm m}}$ (see @1995ApJ...445..947C). Moreover, HD 197770 has an excess emission at $60{\,\mu{\rm m}}$, indicating that the radiation field intensity along this line of sight is higher than the averaged interstellar radiation field and the dust is hotter than the typical ISM. Since hotter, small grains tend to have the lower degree of alignment due to stronger thermal fluctuations within the grain, the UV continuum polarization is reduced accordingly, favoring the detection of the 2175Å polarization bump along this line of sight.
Finally, the alignment degree of PAHs in the magnetic field in general is rather weak ($R\cos^{2}\gamma\le 0.004$) for which the resulting UV polarization excess would be small. As pointed out in [@1997ApJ...478..395W], this feature might not be detected by low signal-to-noise observations.
Constraint on the polarization of spinning dust emission
--------------------------------------------------------
We calculated the degree of polarization for spinning dust emission using the alignment function for the best-fit model to the HD 197770 data. We found that the upper limit for the polarization of spinning dust emission is $\sim 1.6\%$ at $\nu\approx 3\GHz$ and the degree of polarization declines rapidly to below $0.9\%$ for $\nu>20\GHz$.
It is noted that our calculations assume that the UV polarization bump is produced by PAHs only (i.e., the contribution of large graphite grains is disregarded). In fact, if graphite grains can be aligned, then their contribution to the UV polarization bump would reduce our estimate for spinning dust polarization. Therefore, this constraint is indeed the upper limit for the spinning dust polarization.
Our study assumed oblate spheroidal grains with axial ratio $r=2$ for both PAHs and silicate grains. It is obvious that for some grain shape with a smaller axial ratio, which corresponds to lower polarization efficiency $C_{\pol}$, the degree of alignment required to reproduce the $2175$Å polarization bump is higher. As a result, the polarization of spinning dust is higher. However, ultrasmall grains or PAHs are expected to be planar, and the axial ratio appears to be $r \ge 2$ if the grains are approximated as oblate spheroid. Therefore, our constraint for spinning dust polarization obtained for $r=2$ plays as an upper limit.
Observationally, a number of studies indicate that anomalous microwave emission (AME) is weakly polarized (@2006ApJ...645L.141B; @2009ApJ...697.1187M; @LopezCaraballo:2011p508). Using WMAP data, [@2011MNRAS.418..888M] found an upper limit for the polarization of AME between $1.4-2\%$ for $\nu=20-40\GHz$. [@RubinoMartin:2012ji] reviewed in great detail on observational constraints for the AME polarization, and an upper limit of $1\%$ for $\nu= 20-30\GHz$ is placed for the various environments. It appears that our upper limit on spinning dust polarization is consistent with the current observational data for the polarization of AME.
It is worth noting that in addition to spinning dust emission, magnetic dipole emission from dust was suggested to contribute some emission to the AME (@1999ApJ...512..740D). The degree of polarization of magnetic dipole emission may be considerable for the AME. However, an improved study in [@2013ApJ...765..159D] shows that the magnetic dipole emission is important for the AME at frequencies $\nu=20-40\GHz$ if a large fraction of the Fe is in metallic iron nanoparticles with extreme elongation.
Implication for polarized far-infrared thermal dust emission
------------------------------------------------------------
[@Collaboration:2013ww] showed that the degree of polarization of thermal dust emission can reach a high level $P_{\rm em}>15\%$ at $\nu=353\GHz$. Let us estimate the degree of dust polarization at this frequency along the lines of sight toward two stars HD 197770 and HD 147933-4.
For simplicity, we assume that all grains have the same temperature. Thus, using the grain size distribution and alignment function from the model, we can obtain the degree of polarization of thermal dust emission as follows: P\_[em]{}=, where $I_{\lambda,\pol}$ is the polarized emission, $I_{\lambda}$ is the total emission, and $\sigma_{\ext}$ and $\sigma_{\pol}$ are present in Equations (\[eq:sigext\]) and (\[eq:sigpol\]).
At $\nu=353\GHz$ (i.e., $\lambda\approx 850{\,\mu{\rm m}}$), we found the degree of polarization $P_{\rm em}\approx 11\%$ and $14\%$ for HD 197770 and 147933-4, respectively. The higher degree of polarization for the HD 147933-4 star is obtained because this star has a higher fraction of large, aligned silicate grains which dominate far-infrared dust emission (see Figure \[fig:Rfit\]). It can be seen that a high level ($\ge 15\%$) of dust polarization seen by can be expected from our model.
To study a potential correlation between the degree of polarization of far-infrared dust emission to that by dust extinction, we introduce a correlation parameter $r_{\pol}$, which is equal to the ratio of $P_{\rm em}$ at $\nu=353\GHz$ to $P_{\ext}/\tau$ at $\lambda=\lambda_{\max}$. We obtain $r_{\pol} \approx 5.2$ for HD 197770 and $r_{\pol}\approx 7.7$ for HD 147933-4.
Summary {#sec:summ}
=======
In the present paper, we obtain the constraint on the degree of polarization of spinning dust emission based on the observed extinction and polarization curves of starlight. Our principal results are summarized as follows:
- A model of grain size distribution and alignment function for interstellar grains that best fits to the observed extinction and polarization curve is obtained for two stars HD 197770 and HD147933-4 with the prominent polarization features at $2175$Å.
- We find that a small degree of alignment ($R\cos^{2}\gamma\approx 0.004$) of PAHs included to the aligned silicate grains is sufficient to reproduce the $2175$Åpolarization feature for HD 197770. For HD 147933-4, we find no indication of alignment for PAHs and the alignment of silicate grains can account for the feature.
- We calculate the polarized spinning dust emissivity using the alignment function from the best-fit model. We show that the degree of polarization for spinning dust emission has a peak of $\approx 1.6\%$ at $\nu\approx 3\GHz$ and rapidly declines to below $0.9\%$ for $\nu>20 \GHz$.
- The degree of polarization for thermal dust emission at $353\GHz$ is estimated to be $P_{\rm em}\approx 11\%$ for the line of sight to the HD 197770 star and $P_{\rm em}\approx 14\%$ for the line of sight to the HD 147933-4 star.
We thank the anonymous referee for valuable comments and suggestions that significantly improved our paper. T.H. thanks Brandon Hensley for useful discussions. A.L. acknowledges the support of the NASA Grant NNX11AD32G and the NSF grant AST-1109469, as well as the Vilas Associate Award and the support of the NSF Center for Magnetic Self-Organization. A part of the work was performed during A.L. stay in the stimulating atmosphere of the International Institute of Physics (Brazil).
[^1]: The accepted theory of grain alignment for large grains is that based on radiative torques (; @1996ApJ...470..551D; @1997ApJ...480..633D; @2007MNRAS.378..910L; @2008ApJ...676L..25L; @2008MNRAS.388..117H; @2009ApJ...695.1457H; @2009ApJ...697.1316H) but the radiative torques are negligibly small for PAHs. This encourages us to explore the possibilities provided by other alignment mechanisms.
[^2]: Such a UV bump is believed to arise from the electronic transition $\pi-\pi^{*}$ in the $sp^{2}$-bonded carbon sheets of small carbonaceous grains (see @1989IAUS..135..313D; @2007ApJ...657..810D).
[^3]: This value is a factor of 2 larger than the rough estimate $\Delta p/\Delta \tau \sim 0.0017$ in [@1997ApJ...478..395W].
[^4]: One can say that for the microwave polarization from PAHs the alignment in terms of the PAH angular momenta is important, while for the UV absorption polarization, it is the alignment in terms of grain axes that is essential. The two alignments are different (see @1997ApJ...484..230L;@2008MNRAS.388..117H; @2011ApJ...741...87H).
[^5]: More discussions on why it is challenging to obtain a satisfactory fit to the observed polarization data for the stars with large $\lambda_{\max}$ can be found in [@1995ApJ...444..293K].
[^6]: The oscillating feature is prominent for dust with the dielectric function having small imaginary part (weakly absorbing material).
|
---
abstract: |
We verify the conjecture formulated in [@NN] for any normal surface singularity which admits a good $\C^*$-action. The main result connects the Seiberg-Witten invariant of the link (associated with a certain “canonical” $spin^c$ structure) with the geometric genus of the singularity.
As a by-product, we compute the Seiberg-Witten monopoles of the link (associated with the canonical $spin^c$ structure and the natural Thurston metric) in terms of its Seifert invariants. Additionally, we also determine in terms of the Seifert invariants all the Reidemeister-Turaev sign-refined torsion of the link (associated with any $spin^c$ structure).
address:
- |
Department of Mathematics\
Ohio State University\
Columbus, OH 43210
- |
University of Notre Dame\
Notre Dame, IN 46556
author:
- András Némethi
- 'Liviu I. Nicolaescu'
title: |
Seiberg-Witten invariants and surface singularities II\
(singularities with good $\C^*$-action)
---
[^1]
Introduction
============
The present article is a natural continuation of [@NN], where the authors formulated a very general conjecture which relates the topological and the analytical invariants of a complex normal surface singularity whose link is a rational homology sphere.
Let $(X,0)$ be a normal two-dimensional analytic singularity. It is well-known that from a topological point of view, it is completely characterized by its link $M$, which is an oriented 3-manifold. Moreover, by a result of Neumann [@NP], any decorated resolution graph of $(X,0)$ carries the same information as $M$. A property of $(X,0)$ will be called *topological* if it can be determined from $M$, or equivalently, from any resolution graph of $(X,0)$. For example, for a given resolution, if we take the canonical divisor $K$, and the number $\# {\mathcal V}$ of irreducible components of the exceptional divisor of the resolution, then $K^2+\#{\mathcal V}$ is independent of the choice of the resolution, it is an invariant of the link $M$ (cf. \[ss: 2.5\]).
Our interest is to investigate the possibility to express the geometric genus (which is, by its very definition, an analytic invariant of the singularity) in terms of topological invariants of the link.
Let us start with a brief historical survey. M. Artin proved in [@Artin; @Artin2] that the rational singularities (i.e. $p_g=0$) can be characterized completely from the graph. In [@Lauferme], H. Laufer extended Artin’s results to minimally elliptic singularities, showing that Gorenstein singularities with $p_g=1$ can be characterized topologically. Additionally, he noticed that the program breaks for more complicated singularities (see also the comments in [@NN] and [@Neminv]). On the other hand, the first author noticed in [@Neminv] that Laufer’s counterexamples do not signal the end of the program. He conjectured that if we restrict ourselves to the case of those Gorenstein singularities whose links are rational homology spheres then $p_g$ is topological. This was carried out explicitly for elliptic singularities in [@Neminv] (partially based on some results of S. S.-T. Yau, cf. e.g. with [@Yau]).
For Gorenstein singularities, in the presence of a smoothing with Milnor fiber $F$, the above question can be reformulated in terms of the signature $\si(F)$ and/or the topological Euler characteristic $\chi_{top}(F)$ of $F$. Indeed, via some results of Laufer, Durfee, Wahl and Steenbrink, for Gorenstein singularities, any of $p_g$, $\si(F)$ and $\chi_{top}(F)$ determines the remaining two modulo $K^2+\#{\mathcal V}$ (see e.g. [@LW]).
This fact creates the bridge connecting the above problem with the following list of results about the signature $\si(F)$. Fintushel and Stern proved in [@FS] that for a hypersurface Brieskorn singularity whose link is an [*integral*]{} homology sphere, the Casson invariant $\lambda(M)$ of the link $M$ equals $-\si(F)/8$. This fact was generalized by Neumann and Wahl in [@NW]. They proved the same statement for all Brieskorn-Hamm complete intersections and suspensions of plane curve singularities (with the same assumption about the link). Moreover, they conjectured the validity of the formula for any isolated complete intersection singularity (with the same restriction about the link). For some other conjectures about these singularities, the reader can also consult [@NW2].
The goal of [@NN] was to generalize this conjecture for smoothing of Gorenstein singularities with rational homology sphere link. In fact, more generally (i.e. even if the singularity is not smoothable) [@NN] conjecturally introduced an [*“optimal” topological upper bound*]{} for $p_g$ in the following sense: it is a topological upper bound for $p_g$ for [*any*]{} normal surface singularity, but additionally, for Gorenstein singularities it yields exactly $p_g$. (Such an “optimal” topological upper bound for [*elliptic*]{} singularities is the length of the elliptic sequence, introduced and studied by S. S.-T. Yau, see e.g. [@Yau], and Laufer.)
The conjecture in [@NN] replaces the Casson invariant $\lambda(M)$ by a certain Seiberg-Witten invariant of the link, i.e. by the sum of the Casson-Walker invariant and a certain Reidemeister-Turaev sign-refined torsion invariant.
We recall (for details, see the first part [@NN], and the references listed there) that the set of Seiberg-Witten invariants associates with any $spin^c$ structure $\si$ of $M$ a rational number ${{\bf sw}}^0_M(\si)$. In [@NN] we introduced a “canonical” $spin^c$ structure $\si_{can}$ of $M$. This can be done as follows. The (almost) complex structure on $X\setminus\{0\}$ induces a natural $spin^c$ structure on $X\setminus \{0\}$. Then $\si_{can}$, by definition, is its restriction to $M$. (An equivalent definition can be done using a canonical quadratic function constructed by Looijenga and Wahl in [@LW].) The point is that this structure depends only on the topology of $M$ alone.
We are now ready to recall the conjecture from [@NN].
Conjecture. {#MC}
-----------
[@NN]
*Assume that $(X,0)$ is a normal surface singularity whose link $M$ is a rational homology sphere. Let $\si_{can}$ be the canonical $spin^c$ structure on $M$. Then, conjecturally, the following facts hold.*
\(1) For any $(X,0)$, there is a topological upper bound for $p_g$ given by: $${{\bf sw}}^0_M(\si_{can})-\frac{K^2+\#{\mathcal V}}{8}\geq p_g.$$ (2) If $(X,0)$ is rational or Gorenstein, then in (1) one has equality.\
(3) In particular, if $({\mathcal X},0)$ is a smoothing of a Gorenstein singularity $(X,0)$ with Milnor fiber $F$, then $${{\bf sw}}^0_M(\si_{can})=-\frac{\si(F)}{8}.$$
\[Notice that if $(X,0)$ is numerically Gorenstein and $M$ is a $\Z_2$–homology sphere then $\si_{can}$ is the unique spin structure of $M$; if $M$ is an integral homology sphere then in the above formulae ${{\bf sw}}^0_M(\si_{can})=\lambda(M)$, the Casson invariant of $M$.\]
In [@NN], the conjecture was verified for cyclic quotient and du Val singularities, Brieskorn-Hamm complete intersections, and some rational and minimally elliptic singularities.
In [@NN] Remark 4.6(2), we noticed that it is very likely that parts (2) and (3) of the conjecture are true even for a larger class of singularities, e.g. for $\Q$-Gorenstein singularities, or some families of singularities with some kind of additional analytical rigidity.
In fact, this is exactly the case for normal surface singularities with a (good) $\C^*$-action. It is well-known that a complex affine algebraic variety $X$ admits a $\C^*$-action if and only if the affine coordinate ring $A$ admits a grading $A=\oplus_kA_k$. Following Orlik-Wagreich, we say that the action is [*good*]{} if $A_k=0$ for $k<0$ and $A_0=\C$. This means that the point $0$ corresponding to the maximal ideal $\oplus_{k>0}A_k$ is the only fixed point of the action. Additionally, we will assume that $(X,0)$ is normal.
For these singularities, in this article we prove:
Theorem. {#Th1}
--------
$${{\bf sw}}^0_M(\si_{can})-\frac{K^2+\#{\mathcal V}}{8}=p_g.$$
The proof is based in part on Pinkham’s formula [@Pinkh] of $p_g$ in terms of the Seifert invariants of the link (cf. also with Dolgachev’s work about weighted homogeneous singularities; see e.g. [@Dolg]). On the other hand, in the proof we use the formulae for $K^2+\#{\mathcal V}$ and the Reidemeister-Turaev torsion determined in [@NN], and a formula for the Casson-Walker invariant proved in [@Lescop]. The Reidemeister-Turaev torsion formally shows big similarities with the Poincaré series associated with the graded coordinate ring of the universal abelian cover of $(X,0)$. In the proof we also borrowed some technique of Neumann applied by him for this Poincaré series [@Neu].
Notice that in the above theorem we do not require for $(X,0)$ to be Gorenstein. (This is replaced by the existence of the $\C^*$-action.) On the other hand, the theorem has the following corollary (which can also be applied for singularities without $\C^*$-action).
Corollary. {#C1}
----------
[*Assume that the link of a normal surface singularity $(X,0)$ is a rational homology sphere Seifert 3-manifold. If $(X,0)$ is rational, or minimally elliptic, or Gorenstein elliptic, then the statements of the above conjecture are true for $(X,0)$. (I.e., $(X,0)$ satisfies (1) with equality; and also (3), provided that the additional assumptions of (3) are satisfied.)*]{}
Indeed, in the case of these singularities, all the numerical invariants involved in the conjecture are characterized by the link. Moreover, each family contains a special representative which admits a good $\C^*$-action. In fact, the above corollary can automatically be extended to any family of singularities with these two properties.
The paper is organized as follows. In section 2 we review the needed definitions and results. For a more complete picture and list of references the reader is invited to consult [@NN]. Section 3 starts with a theorem (cf. \[m1\]) which connects four topological numerical invariants of the link. These invariants are: the Reidemeister-Turaev torsion, the Casson-Walker invariant, the Dolgachev-Pinkham invariant $DP_M$ (which is the topological candidate for $p_g$), and finally $K^2+\#{\mathcal V}$ (which can be identified with the Gompf invariant, cf. \[mlb\]). This result implies the above theorem via Pinkham’s result [@Pinkh] (cf. \[ss: inv\](10)).
Finally, we have included at the end of section 3 a complete and explicit description the Reidemeister-Turaev torsion ${\EuScript{T}}_{M,\si}(1) $ for [*any*]{} $spin^c$ structure $\si$ in terms of the Seifert invariants of $M$ (the computation follows Neumann’s method mentioned above).
Remark. {#C2}
-------
Theorem \[m1\] has a consequence interesting from the point of gauge theory as well. Recall that the modified Seiberg-Witten invariant is the sum ${{\bf sw}}^0_M(\si)={{\bf sw}}_M(\si,u)+KS_M(\si,u)/8$ of the Seiberg-Witten monopoles and ($1/8$ of) the Kreck-Stolz invariant (cf. \[ss: 3.2\]). The $(\si_{can},u)$-monopoles were described in [@MOY; @Nico2], and the Kreck-Stolz invariant was described in [@Nico2]. Nevertheless, their explicit computation in terms of the Seifert invariants in the general case meets some difficulties. Our theorem \[m1\] implies the following.
Assume that $M$ is a rational homology sphere Seifert 3-manifold with $e<0$. Then, for any *good* parameter $u$, the signed number of Seiberg-Witten $u$-monopoles can be computed as follows: $${{\bf sw}}_M(\si_{can},u)=-\frac{KS_M(\si_{can},u)}{8}+\frac{K^2+\#
{\mathcal V}}{8}+DP_M.$$ Recall that the Seifert manifold $M$ admits a natural metric $g_0$, the so called Thurston metric. If $u=(g_0,0)$, the invariants appearing on the right hand side have very explicit expression in terms of the Seifert invariants of $M$. Indeed, $KS_M(\si_{can},g_0,0)$ is determined in [@Nico2] (see also [@NN]), $K^2+\#{\mathcal V}$ is determined in [@NN] (see \[ss: inv\](6) here), and $DP_M$ was introduced in [@Pinkh] (it is \[ss: inv\](9) here); in fact the last two are $u$-independent. In particular, when the parameter $(g_0,0)$ is good, the above identity determines ${{\bf sw}}_M(\si_{can},g_0,0)$ explicitly in terms of the Seifert invariants.
For results in this direction, see [@MOY; @Nico2].
Preliminaries
=============
Definitions. {#ss: 4.1}
------------
Let $(X,0)$ be a normal surface singularity. Consider the holomorphic line bundle $\Omega^2_{X\setminus\{0\}}$ of holomorphic 2-forms on $X\setminus
\{0\}$. If this line bundle is holomorphically trivial then we say that $(X,0)$ is *Gorenstein*. Let $\pi:\tilde{X}\to X$ be a resolution over a sufficiently small Stein representative $X$ of the germ $(X,0)$. Then $p_g(X,0):=\dim \, H^1(\tilde{X}, {\mathcal
O}_{\tilde{X}})$ is finite and independent of the choice of $\pi$. It is called the [*geometric genus* ]{} of $(X,0)$.
The link and its canonical $spin^c$ structure. {#ss: 2.1}
----------------------------------------------
Let $(X,0)$ be a normal surface singularity embedded in $(\C^N,0)$. Then for $\epsilon$ sufficiently small the intersection $M:=X\cap S_{\epsilon}^{2N-1}$ of a representative $X$ of the germ with the sphere $S_{\epsilon}^{2N-1}$ (of radius ${\epsilon}$) is a compact oriented 3-manifold, whose oriented $C^{\infty}$ type does not depend on the choice of the embedding and $\epsilon$. It is called the link of $(X,0)$. In this article we will assume that $M$ is a rational homology sphere, and we write $H:=H_1(M,\Z)$.
The almost complex structure on $X\setminus\{0\}$ determines a $spin^c$ structure on $X\setminus \{0\}$, whose restriction to $M$ will be denoted by $\si_{can}\in Spin^c(M)$. It turns out that $\si_{can}$ depends only on the oriented $C^{\infty}$ type of $M$ (for details see [@NN], cf. also with [@LW]).
The Seiberg-Witten invariants of $M$. {#ss: 3.2}
-------------------------------------
To describe the Seiberg-Witten invariants one has to consider an additional geometric data belonging to the space of parameters $$\p=\{ u=(g,\eta);\quad g=\mbox{Riemann metric},\;\;\eta=\mbox{closed
two-form} \}.$$ Then for each $spin^c$ structure $\si$ on $M$ one defines the $(\si,g,\eta)$-[*Seiberg-Witten monopoles*]{}. For a generic parameter $u$, the Seiberg-Witten invariant ${{\bf sw}}_M(\si,u)$ is the signed monopole count. This integer depends on the choice of the parameter $u$ and thus it is not a topological invariant. To obtain an invariant of $M$, one needs to alter this monopole count. The needed additional contribution is the [*Kreck-Stolz*]{} invariant $KS_M(\si,u)$ (associated with the data $(\si,u)$), cf. [@Lim] (or see [@KrSt] for the original “spin version”). Then, by [@Chen1; @Lim; @MW], the rational number $$\frac{1}{8}KS_M(\si, u)+{{\bf sw}}_M(\si, u)$$ is independent of $u$ and thus it is a topological invariant of the pair $(M,\si)$. We denote this [*modified Seiberg-Witten invariant*]{} by ${{\bf sw}}_M^0(\si)$.
The Reidemeister-Turaev torsion and the Casson-Walker invariant. {#ss: 3.4}
----------------------------------------------------------------
For any $spin^c$ structure $\si$ on $M$, we denote by $${\EuScript{T}}_{M,\si}=\sum_{h\in H}{\EuScript{T}}_{M,\si}(h)\, h\in {\bQ}[H]$$ the sign refined [*Reidemeister-Turaev torsion*]{} associated with $\si$ (for its detailed description, see [@Tu5]). It is convenient to think of ${\EuScript{T}}_{M,\si}$ as a function $H\ra {\bQ}$ given by $h\mapsto {\EuScript{T}}_{M,\si}(h)$. The [*augmentation map*]{} $\aug: {\bQ}[H]\ra {\bQ}$ is defined by $\sum a_h\, h\mapsto \sum a_h$. It is known that $\aug({\EuScript{T}}_{M, \si})=0$.
Denote by $\lambda(M)$ the Casson-Walker invariant of $M$ normalized as in [@Lescop §4.7]. Then by a result of the second author [@Nico5], one has: $${{\bf sw}}^0_M(\si)=\frac{1}{|H|}\lambda(M)+{\EuScript{T}}_{M,\si}(1).
\tag{1}
\label{3.7.2}$$ Below we will present a formula for ${\EuScript{T}}_{M,\si}$ in terms of Fourier transform. For this, consider the Pontryagin dual $\hat{H}:=\operatorname{{\rm Hom}}(H, U(1))$ of $H$. Then a function $f: H \ra {\bC}$ and its Fourier transform $\hat{f}:\hat{H}\ra {\bC}$ satisfy: $$\quad \hat{f}(\chi)=\sum_{h\in H} f(h)\bar{\chi}(h);\ \
f(h)=\frac{1}{|H|}\sum_{\chi\in \hat{H}}\hat{f}(\chi)\chi(h).$$ Notice that $\hat{f}(1)=\aug(f)$, in particular $\hat{{\EuScript{T}}}_{M,\si}(1)=\aug({\EuScript{T}}_{M, \si})=0$.
$M$ as a plumbing manifold. {#ss: 2.5}
---------------------------
Fix a sufficiently small (Stein) representative $X$ of $(X,0)$ and let $\pi:\tilde{X}\to X$ be a resolution of the singular point $0\in X$. In particular, $\tilde{X}$ is smooth, and $\pi$ is a biholomorphic isomorphism above $X\setminus \{0\}$. We will assume that the exceptional divisor $E:=\pi^{-1}(0)$ is a normal crossing divisor with irreducible components $\{E_v\}_{v\in {\mathcal V}}$. Let $\Gamma(\pi)$ be the dual resolution graph associated with $\pi$ decorated with the self intersection numbers $\{E_v\cdot E_v\}_v$. $\Gamma(\pi)$ can be identified with a plumbing graph, and $M$ with a plumbing 3-manifold constructed from $\Gamma(\pi)$ as its plumbing graph. Since $M$ is a rational homology sphere, all the irreducible components $E_v$ of $E$ are rational, and $\Gamma(\pi)$ is a tree.
Let $D_v$ be a small transversal disc to $E_v$. In fact $\d D_v$ can be considered as the generic fiber of the $S^1$-bundle over $E_v$ used in the plumbing construction of $M$. Consider the elements $g_v:=[\d D_v]$ ($v\in {\mathcal V}$) in $H$. It is not difficult to see that they, in fact, generate $H$.
For the degree of any vertex $v$ (i.e. for $\#\{w: E_w\cdot E_v=1\}$) we will use the notation $\delta_v$.
Next, we define the *canonical cycle* $Z_K$ of $(X,0)$ associated with the resolution $\pi$. This is a [*rational*]{} cycle $Z_K=\sum_{v\in {\mathcal V}}r_vE_v$, $r_v\in\Q$, supported by the exceptional divisor $E$, and defined by (the adjunction formula): $$Z_K\cdot E_v=E_v\cdot E_v+2 \ \ \ \mbox{for any $v\in {\mathcal V}$}.
\label{2.5*}$$ Since the intersection matrix $\{E_v\cdot E_w\}_{v,w}$ is nondegenerate, the above equation has a unique solution. $(X,0)$ is called [*numerically Gorenstein*]{} if $r_v\in \Z$ for each $v\in {\mathcal V}$.
The rational number $Z_K\cdot Z_K$ will be denoted by $K^2$. Let $\#{\mathcal V}$ denote the number of irreducible components of $E=\pi^{-1}(0)$. Then $K^2+\#{\mathcal V}$ does not depend on the choice of the resolution $\pi$, it is an invariant of $M$.
The main object of this paper is a normal surface singularity $(X,0)$ with a [good]{} $\C^*$–action. It is well–known that the link of such a singularity is a Seifert 3-manifold, and the minimal resolution graph is a star-shaped graphs. In these case it is convenient to express all the topological invariants of $M$ in terms of their Seifert invariants. In the next subsections we list briefly the definitions, notations and some of the needed properties.
The Seifert invariants. {#ss: s1}
-----------------------
[@JN; @Neu; @NeR]Consider a Seifert fibration $\pi:M\to \Sigma$. In our situation, since $M$ is a rational homology sphere, the base space $\Sigma$ has genus zero.
Consider a set of points $\{x_i\}_{i=1}^{\nu}$ in such a way that the set of fibers $\{\pi^{-1}(x_i)\}_i$ contains the set of singular fibers. Set $O_i:= \pi^{-1}(x_i)$. Let $D_i$ be a small disc in $X$ containing $x_i$, $\Sigma':=\Sigma\setminus \cup_iD_i$ and $M':=\pi^{-1}(\Sigma')$. Now, $\pi:M'\to \Sigma'$ admits sections, let $s:\Sigma'\to M'$ be one of them. Let $Q_i:=s(\d D_i)$ and let $H_i$ be a circle fiber in $\pi^{-1}(\d D_i)$. Then in $H_1(\pi^{-1}(D_i),\Z)$ one has $H_i\sim \alpha_iO_i$ and $Q_i\sim -\beta_iO_i$ for some integers $\alpha_i>0 $ and $
\beta_i$ with $(\alpha_i,\beta_i)=1
$. The set $((\alpha_i,\beta_i)_{i=1}^\nu)$ constitute the set of *(unnormalized) Seifert invariants*. The number $$e:=- \sum\frac{\beta_i}{\alpha_i}$$ is called the *(orbifold) Euler number* of $M$. $M$ is a link of singularity if and only if $e<0$.
Replacing the section by another one, a different choice changes each $\beta_i$ within its residue class modulo $\alpha_i$ in such a way that the sum $e=-\sum_i(\beta_i/\alpha_i)$ is constant.
The set of [*normalized*]{} Seifert invariants $((\alpha_i,\omega_i)_{i=1}^\nu)$ are defined as follows. Write $$e=b+\sum \omega_i/\alpha_i
\tag{2}$$ for some integer $b$, and $0\leq \omega_i<\alpha_i$ with $\omega_i\equiv -\beta_i \ (\mmod \ \alpha_i)$. Clearly, these properties define $\{\omega_i\}_i$ uniquely. Notice that $b\leq
e<0$. For the uniformity of the notations, in the sequel we assume $\nu\geq 3$. (Recall that for cyclic quotient singularities Conjecture was verified in [@NN].)
For each $i$, consider the continued fraction $\alpha_i/ \omega_i=
b_{i1}-1/(b_{i2}-1/(\cdots
-1/b_{i\nu_i})\cdots)$. Then (a possible) plumbing graph of $M$ is a star-shaped graph with $\nu$ arms. The central vertex has decoration $b$ and the arm corresponding to the index $i$ has $\nu_i$ vertices, and they are decorated by $b_{i1},\ldots, b_{i\nu_i}$ (the vertex decorated by $b_{i1}$ is connected by the central vertex).
We will distinguish those vertices $v\in {\mathcal V}$ of the graph which have $\delta_v\not= 2$. We will denote by $\bar{v}_0$ the central vertex (with $\delta=\nu$), and by $\bar{v}_i$ the end-vertex of the $i^{th}$ arm (with $\delta=1$) for all $1\leq
i\leq \nu$. In this notation, $g_{\bar{v}_0}$ is exactly the class of the generic fiber. The group $H$ has the following representation: $$H=\mbox{ab}\langle g_{\bar{v}_0}, g_{\bar{v}_1},\ldots g_{\bar{v}_\nu}\ |\
g_{\bar{v}_0}^{-b}=\prod_{i=1}^\nu g_{\bar{v}_i}^{\omega_i},
\ \ g_{\bar{v}_0}=g_{\bar{v}_i}^{\alpha_i}
\ \mbox{for all $i$} \rangle.
\tag{3}$$
Let $\alpha:=\mbox{lcm}(\alpha_1,\ldots,\alpha_n)$. The order of the group $H$ and the order ${o}$ of the subgroup $\langle g_{\bar{v}_0}
\rangle$ can be determined by (cf. [@Neu]): $$|H|=\alpha_1\cdots \alpha_\nu |e|, \ \ \ \ {o}:=|\langle
g_{\bar{v}_0} \rangle|=\alpha|e|.
\tag{4}$$
Invariants computed from the plumbing graph {#ss: inv}
-------------------------------------------
In the sequel we will also use the Dedekind sums. They are defined as follows [@Ra; @RG]. Let $\lfloor x\rfloor$ be the integer part, and $\{x\}:= x-\lfloor x\rfloor$ the fractional part of $x$. Then $${\mbox{\boldmath$s$}}(h,k)=\sum_{\mu=0}^{k-1}\Big(\Big( \frac{\mu}{k}\Big)\Big)
\Big(\Big( \frac{h\mu}{k} \Big)\Big),$$ where $((x))$ denotes the Dedekind symbol $$((x))=\left\{
\begin{array}{ccc}
\{x\} -1/2 & {\rm if} & x\in {\bR}\setminus {\bZ}\\
0 & {\rm if} & x\in {\bZ}.
\end{array}
\right.$$ Assume that $M$ is a Seifert manifold with $e<0$. Then one has the following formulae for its invariants.
$\bullet$ [**The Casson-Walker invariant.**]{} [@Lescop] (6.1.1): $$\frac{24}{|H|}\lambda(M)=
\frac{1}{e}\Big(2-\nu +\sum_{i=1}^\nu\frac{1}{\alpha_i^2}\Big)+
e+3+12\sum_{i=1}^\nu \, {\mbox{\boldmath$s$}}(\beta_i,\alpha_i).
\tag{5}$$
$\bullet$ [**$K^2+\#{\mathcal V}$.**]{} [@NN] (5.4): $$K^2+\#{\mathcal V}=
\frac{1}{e}\Big(2-\nu +\sum_{i=1}^\nu \frac{1}{\alpha_i}\Big)^2 +e +
5+12\sum_{i=1}^\nu \, {\mbox{\boldmath$s$}}(\beta_i,\alpha_i).
\tag{6}$$
$\bullet$ [**The Reidemeister-Turaev sign-refined torsion.**]{} For any $\chi\in \hat{H}$ (and free variable $t\in \C$) set $$\hat{P}_{\chi}(t):=
\frac{\bigl(t^{\alpha}\chi(g_{\bar{v}_0})-1\bigr)^{\nu-2}}
{\prod_{i=1}^\nu\ \bigl(t^{\alpha/\alpha_i}\,
\chi(g_{\bar{v}_i})-1\bigr)}.
\tag{7}$$ Then, by [@NN] (5.8), the Fourier transform $\hat{{\EuScript{T}}}_{M,\si_{can}}$ of ${\EuScript{T}}_{M,\si_{can}}$ is given by $$\hat{{\EuScript{T}}}_{M,\si_{can}}(\bar{\chi})=
\lim_{t\to 1}\, \hat{P}_{\chi}(t)\ \ \
\mbox{for any $\chi\in \hat{H}\setminus \{1\}$}.
\tag{8}$$
$\bullet$ [**The geometric genus of $(X,0)$.**]{} Let $M$ be a Seifert manifold with $e<0$ and Seifert invariants as above. Define the Dolgachev-Pinkham (topological) invariant of $M$ by $$DP_M:=\sum_{l\geq 0}\, \max\big( 0\, , \, -1+lb-\sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big).
\tag{9}$$ Assume that $(X,0)$ is a normal surface singularity with a good $\C^*$–action (see e.g. [@Pinkh]) such that its link $M$ is a rational homology sphere. Then, by [@Pinkh], (5.7): $$p_g(X,0)=DP_M.
\tag{10}$$
The main result
===============
The key identity of this article is presented in the following theorem.
Theorem. {#m1}
--------
[*Let $M$ be a Seifert 3–manifold with $e<0$. Then the invariants ${\EuScript{T}}_{M,\si_{can}}(1)$, $\lambda(M)$, $K^2+\#{\mathcal V}$ and $DP_M$ are connected by the following identity:*]{} $${\EuScript{T}}_{M,\si_{can}}(1)+\frac{\lambda(M)}{|H|}=\frac{K^2+\#{\mathcal V}}{8}
+DP_M.$$
Remark. {#mlb}
-------
Using \[ss: 3.4\](1), for the (modified) Seiberg-Witten invariant one obtains: $${{\bf sw}}^0_M(\si_{can})=\frac{K^2+\#{\mathcal V}}{8}+DP_M.$$ If $M$ is a singularity link, one can define on $M$ a canonical [*contact structure*]{} $\xi_{can}$ (induced by the natural almost complex structure on $TM\oplus \R_M$; for details, see [@GS p. 420] or [@NN (4.8)] ) with $c_1(\xi_{can})$ torsion. On the other hand, in [@Gompf], Gompf associates with such a contact structure $\xi$ an invariant $\theta_M(\xi)$. It turns out that in our case (see [@NN 4.8]) $\theta_M(\xi_{can})=K^2+\#{\mathcal V}-2$. Therefore, for any link of singularity which is a Seifert 3-manifold one has: $${{\bf sw}}^0_M(\si_{can})=\frac{\theta_M(\xi_{can})+2}{8}+DP_M.$$
The proof of \[m1\] is carried out in several steps.
Proposition. {#m2}
------------
$$\frac{1}{|H|}\sum_{\chi\in \hat{H}}\, \hat{P}_{\chi}(t)=\sum_{l\geq 0}\, \max
\big( \, 0\, ,\, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big)\ t^{{o}l}.$$ [*In particular,* ]{} $${\EuScript{T}}_{M,\si_{can}}(1)=
\lim_{t\to 1}\,\Big(
\sum_{l\geq 0}\, \max \big(\, 0 \,, \, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big)\ t^{{o}l}
-\frac{1}{|H|}\cdot \hat{P}_1(t)\, \Big).$$ Before we start the proof, we draw to the reader’s attention the “mysterious” similarity between our formula (7) for the (Fourier transform) of the Reidemeister-Turaev torsion, and the formula [@Neu 4.2] of W. Neumann of the Poincaré series of the graded affine ring associated with the universal abelian cover of $(X,0)$.
In fact, the next proof is based completely on Neumann’s computation about this graded ring on \[loc. cit.\], page 241; (he attributed the idea to D. Zagier).
Using the identity $g_{\bar{v}_i}^{\alpha_i}=g_{\bar{v}_0}$ in $H$ (cf. \[ss: s1\](3)), first write $\hat{P}_{\chi}(t)$ as $$(1-t^\alpha\chi(g_{\bar{v}_0}))^{-2}\prod_i\frac{
1-\big(\, t^{\alpha/\alpha_i}\chi(g_{\bar{v}_i})\big)^{\alpha_i}}{
1-t^{\alpha/\alpha_i}\,
\chi(g_{\bar{v}_i})}=
\Big(\, \sum_{s_0=0}^{\infty}\, (1+s_0)\chi(g_{\bar{v}_0})^{s_0}\, t^{\alpha
s_0}\Big)\cdot \prod_i\, \sum_{s_i=0}^{\alpha_i-1} t^{s_i\alpha/\alpha_i}\big(
\chi(g_{\bar{v}_i})\big)^{s_i}$$ $$=\sum\, (1+s_0)\, t^{\alpha s_0+\sum_i \alpha s_i/\alpha_i}\
\chi\big(g_{\bar{v}_0}^{s_0}g_{\bar{v}_1}^{s_1}\cdots g_{\bar{v}_\nu}^{s_\nu}
\big),$$ where the (unmarked) sum is over $s_0\geq 0$ and $0\leq s_i< \alpha_i$ for each $i$. But $\sum_{\chi\in \hat{H}}(h)$ is non-zero only if $h=1$, and in that case it is $|H|$. Using the group structure \[ss: s1\](3) one gets that all the relations in $H$ have the form $$g_{\bar{v}_0}^{l_1+\cdots +l_\nu-lb}\prod_i
g_{\bar{v}_i}^{-\omega_il-\alpha_il_i}=1,$$ where $l_1,\ldots, l_\nu$ and $l$ are integers. Therefore, $
g_{\bar{v}_0}^{s_0}g_{\bar{v}_1}^{s_1}\cdots g_{\bar{v}_\nu}^{s_\nu}=1$ if and only if $s_0=l_1+\cdots +l_\nu-lb$ and $s_i=-\omega_il-\alpha_il_i$ ($1\leq i\leq \nu$) for some integers $l_1,\ldots, l_\nu, l$. Since $0\leq s_i<\alpha_i$ one obtains that $$l_i=
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\, \Big\rfloor.$$ In particular, $$1+s_0=1-lb +\sum_i
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\, \Big\rfloor,$$ and only those integers $l$ are allowed for which this number $1+s_0$ is $\geq 1$. It is easy to see that this cannot happen for $l<0$. Indeed, for $l<0$ $$-lb +\sum_i
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\, \Big\rfloor\leq
-lb +\sum_i \frac{-l\omega_i}{\alpha_i}=-le<0.$$ Finally notice that the exponent $\alpha (s_0+\sum_i s_i/\alpha_i)$ of $t$ is $-l\alpha e=l{o}$ by \[ss: s1\](4). This ends the proof of the first formula. For the second part, recall that $\hat{{\EuScript{T}}}_{M,\si}(1)=0$ (cf. \[ss: 3.4\]), hence by \[ss: inv\](8), $${\EuScript{T}}_{M,\si_{can}}(1)=\lim_{t\to 1}\frac{1}{|H|}
\sum_{\chi\in\hat{H}\setminus \{1\}}\hat{P}_{\chi}(t).$$
Remark. {#remark}
-------
Let $(X,0)$ be a normal singularity with a good $\C^*$-action and affine graded coordinate ring $A=\oplus_kA_k$. Then its Poincaré series is defined by $p_{(X,0)}(t)=\sum_k \dim(A_k)t^k$. For such an $(X,0)$, the expression from \[m2\] provides exactly $p_{(X,0)}(t^o)$ (cf. [@Neu], page 241). More precisely: $$p_{(X,0)}(t)=\sum_{l\geq 0}\, \max
\big( \, 0\, ,\, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big)\ t^{l}.$$ Moreover, if $(X_{ab},0)$ denotes the universal abelian cover of $(X,0)$, then $p_{(X_{ab},0)}(t)=\hat{P}_1(t)$ (cf. [@Neu], page 240). Therefore, \[m2\] reads as follows: $${\EuScript{T}}_{M,\si_{can}}(1)=\lim_{t\to 1}\,\big(
p_{(X,0)}(t^o)-p_{(X_{ab},0)}(t)/|H| \, \big).$$ Notice that for many special families, the Poincaré series $p_{(X,0)}(t)$ is computed very explicitly, see e.g. [@Wag].
Corollary. {#m3}
----------
$${\EuScript{T}}_{M,\si_{can}}(1)-DP_M=
\lim_{t\to 1}\,\Big(
\sum_{l\geq 0}\, \big( \, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big)\ t^{{o}l}
-\frac{1}{|H|}\cdot \hat{P}_1(t)\, \Big).$$
Use \[ss: inv\](9), \[m2\] and the identity $\max(0,x)-\max(0,-x)=x$.
On the right hand side we have a difference of two series, both having poles of order two at $t=1$. The next results provide their Laurent series at $t=1$. In fact, we prefer to expand the series in terms of the powers of $t^{o}-1$ (instead of $t-1$).
Proposition. {#m4}
------------
[*Define $\chi_M:=2-\sum_{i=1}^\nu\,
(\alpha_i-1)/\alpha_i$ (cf. e.g. with [@Neu]). Then $$\sum_{l\geq 0}\, \big( \, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i}{\alpha_i}\Big\rfloor\, \big)\ t^{{o}l}=
\frac{-e}{(t^{o}-1)^2}+\frac{-e-\chi_M/2}{t^{o}-1}+
\frac{2-\chi_M}{4}+ \sum_{i=1}^\nu{\mbox{\boldmath$s$}}(\beta_i,\alpha_i)+
R(t),$$ with $\lim_{t\to 1}R(t)=0$.* ]{}
We follow again [@Neu], page 241. The left hand side of \[m4\], via \[ss: s1\](2) transforms into $$\sum_{l\geq 0}\, \Big( -le+\frac{\chi_M}{2}\Big)\,t^{{o}l}
+ \sum_{i=1}^\nu \Big(
-\Big\{\, \frac{-l\omega_i}{\alpha_i}\Big\}+\frac{\alpha_i-1}{2
\alpha_i}\, \Big)\, t^{{o}l}.$$ Evidently $$\sum_{l\geq 0}\, \Big( \, -le+\frac{\chi_M}{2}\Big) \ t^{{o}l}=
\frac{-et^{{o}}}{(1-t^{o})^2}+\frac{\chi_M/2}{1-t^{o}},$$ which gives the non-holomorphic part. The second contribution is a sum over $1\leq i\leq\nu$. For each fixed $i$, write $l=\alpha_i m+q$ with $m\geq 0$ and $0\leq q<\alpha_i$. Using the notation $\sum_q:=\sum_{q=0}^{\alpha_i-1}$ and $\sum_q':=
\sum_{q=1}^{\alpha_i-1}$, the $i^{th}$ summand is $$\sum_q\Big(
-\Big\{\, \frac{-q\omega_i}{\alpha_i}\Big\}+\frac{\alpha_i-1}{2\alpha_i}\,
\Big)\ \sum_{m\geq 0}t^{{o}\alpha_i m+{o}q}=
\frac{\sum_q\big(
-\big\{\, \frac{-q\omega_i}{\alpha_i}\big\}+\frac{\alpha_i-1}{2\alpha_i}\,
\big)\ t^{{o}q}}{1-t^{{o}\alpha_i}}.$$ Separating the two cases $q=0$ and $q>0$, and using the definition of the Dedekind symbol and the identity $\{-x\}=1-\{x\}$ for $x\not\in \Z$, this is transformed into $$A(t):=\frac{
\frac{\alpha_i-1}{2\alpha_i}+\sum_q'\big(\big( \frac{q\omega_i}
{\alpha_i}\big)\big) t^{{o}q}-\frac{1}{2\alpha_i}(t^{{o}}+t^{2{o}}+\cdots
+t^{(\alpha_i-1){o}})}
{1-t^{{o}\alpha_i}}.$$ By L’Hospital theorem (and by some simplifications): $$\lim_{t\to 1}A(t)=-{\sum_q}'\Big(\Big(\frac{q\omega_i}{\alpha_i}\Big)\Big)
\frac{q}{\alpha_i}+\frac{\alpha_i-1}{4\alpha_i}.$$ Since $\sum_q'((q\omega_i/\alpha_i))=0$ and $\omega_i\equiv -\beta_i$ ($\mmod\ \alpha_i$), the result follows from the definition of the Dedekind symbol and the Dedekind sums.
Remarks. {#m5}
--------
Cf. [@Neu], page 242. In fact, $$\sum_{l\geq 0}\, \big( \, 1-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{\omega_i}{\alpha_i}(-l)\Big\rfloor\, \big)\ t^{{o}l}=
\frac{-e}{(t^{o}-1)^2}+\frac{-e-\chi_M/2}{t^{o}-1}+
\sum_{i=1}^\nu \frac{1}{\alpha_i} {\sum_{\Z_{\alpha_i}}}'
\frac{1}{(1-\xi)(1-\xi^{\omega_i}t^{o})},$$ where the last sum is over $\xi^{\alpha_i}=1$, $\xi\not=1$. Since we do not need this statement now, we will skip its proof. The interested reader can prove it easily using the expression $A(t)$ above and the property (16c) of the Dedekind symbol from [@RG], page 14.
Paradoxically, the Laurent expansion of the elementary rational fraction $\hat{P}_1(t)$ is more complicated.
Proposition. {#m6}
------------
[*$$\frac{\hat{P}_1(t)}{|H|}=
\frac{-e}{(t^{o}-1)^2}+\frac{-e-\chi_M/2}{t^{o}-1}+ E+ Q(t),$$ where $\lim_{t\to 1}Q(t)=0$ and* ]{} $$E:=-\frac{(e+1)(e+5)}{12e}+\frac{1}{4}\sum_i\big(1-\frac{1}{\alpha_i}\big)
+\frac{1}{12e}\sum_i\big(1-\frac{1}{\alpha_i}\big)
\big(4+\frac{1}{\alpha_i}\big)-
\frac{1}{4e}\sum_{i<j}\big(1-\frac{1}{\alpha_i}\big)
\big(1-\frac{1}{\alpha_j}\big).$$
First notice that one has the following Taylor expansion; $$\frac{t^\gamma-1}{t^\tau-1}=\frac{\gamma}{\tau}+\frac{\gamma}
{2{o}\tau}( \gamma-\tau)\cdot (t^{o}-1)+
\frac{\gamma}{{o}\tau}( \gamma-\tau)\Big(
\frac{2\gamma-\tau}{12{o}}-\frac{1}{4}\Big) \cdot (t^{o}-1)^2+\cdots .$$ Now, use this formula $\nu+2$ times in the expression $$\hat{P}_1(t)=\frac{1}{(t^{o}-1)^2}\cdot \Big(
\frac{t^{o}-1}{t^\alpha-1}\Big)^2\cdot
\prod_i\frac{t^\alpha-1}{t^{\alpha/\alpha_i}-1}.$$ A long (but elementary) computation, together with \[ss: s1\](4), gives the result.
[*Proof of \[m1\].*]{} Apply \[ss: inv\](5) and (6), respectively \[m3\], \[m4\] and \[m6\]. The verification is elementary.
Remark. The Reidemeister-Turaev torsion revisited. {#rt}
--------------------------------------------------
In the above results we needed to determine only ${\EuScript{T}}_{M,\si_{can}}(1)$, i.e. only that sign refined torsion which is associated with the canonical $spin^c$ structure. But, via similar computations, we can obtain the complete set of invariants $\{{\EuScript{T}}_{M,\si}(1)\}_{\si\in Spin^c(M)}$.
Indeed, we recall that $Spin^c(M)$ is an $H$-torsor. Let $\si$ be an arbitrary element of $Spin^c(M)$, and take (the unique) $h_{\si}\in H$ so that $h_{\si}\cdot \si_{can}=\si$. Then, by [@NN 5.7], one has: $$\hat{{\EuScript{T}}}_{M,\si}(\bar{\chi})=\bar{\chi}(h_{\si})\cdot \lim_{t\to 1}
\hat{P}_{\chi}(t)\ \ \mbox{for any $\chi\in \hat{H}\setminus \{1\}$}.$$ Using the representation \[ss: s1\](3) of $H$, $h_\si$ evidently can be written as $$h_\si=g_{\bar{v}_0}^{a_0}g_{\bar{v}_1}^{a_1}\cdots g_{\bar{v}_\nu}^{a_\nu}
\ \ \ \mbox{for some integers} \ a_0,a_1,\ldots,a_\nu.$$ Then repeating the arguments from the proof of \[m2\], one gets: $$\bar{\chi}(h_\si)\hat{P}_{\chi}(t)
=\sum\, (1+s_0)\, t^{\alpha s_0+\sum_i \alpha s_i/\alpha_i}\
\chi\big(h_{\si}^{-1}
g_{\bar{v}_0}^{s_0}g_{\bar{v}_1}^{s_1}\cdots g_{\bar{v}_\nu}^{s_\nu}\big),$$ with $s_0\geq 0$ and $0\leq s_i< \alpha_i$ for each $i$. But $h_\si
g_{\bar{v}_0}^{s_0}g_{\bar{v}_1}^{s_1}\cdots g_{\bar{v}_\nu}^{s_\nu}=1$ if and only if $s_0=a_0+l_1+\cdots +l_\nu-lb$ and $s_i=a_i-\omega_il-\alpha_il_i$ ($1\leq i\leq \nu$) for some integers $l_1,\ldots, l_\nu, l$. Since $0\leq s_i<\alpha_i$ one obtains that $$l_i=
\Big\lfloor\, \frac{-l\omega_i+a_i}{\alpha_i}\, \Big\rfloor.$$ In particular, $$1+s_0=1+a_0-lb +\sum_i
\Big\lfloor\, \frac{-l\omega_i+a_i}{\alpha_i}\, \Big\rfloor,$$ and only those integers $l$ are allowed for which $1+s_0\geq 1$. Therefore, one gets:
Proposition. {#tors}
------------
[*Assume that $M$ is a Seifert 3-manifold with $e<0$. Fix an arbitrary $spin^c$ structure $\si\in Spin^c(M)$ characterized by $h_{\si}\cdot \si_{can}
=\si$. Write $h_\si$ as $g_{\bar{v}_0}^{a_0}g_{\bar{v}_1}^{a_1}\cdots g_{\bar{v}_\nu}^{a_\nu}$ for some integers $a_0,a_1,\ldots, a_\nu$. Finally, define $\tilde{a}:=\alpha\cdot ( a_0+\sum_ia_i/\alpha_i)$. Then $${\EuScript{T}}_{M,\si}(1)=
\lim_{t\to 1}\,\Big(
\sum_{l \in \Z}\, \max \big(\, 0 \,, \, 1+a_0-lb+ \sum_{i=1}^\nu
\Big\lfloor\, \frac{-l\omega_i+a_i}{\alpha_i}\Big\rfloor\, \big)\ t^{{o}l+
\tilde{a}}
-\frac{1}{|H|}\cdot \hat{P}_1(t)\, \Big).$$ (Above, there exists an integer $m$, such that for $l<m$ the contribution in the above sum is zero.)* ]{}
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[^1]: The first author is partially supported by NSF grant DMS-0088950; the second author is partially supported by NSF grant DMS-0071820.
|
---
author:
- |
*Ferruccio Colombini*\
–\
*Daniele Del Santo*\
–\
*Francesco Fanelli*\
–\
*Guy Métivier*\
–
---
### Abstract {#abstract .unnumbered}
[In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension $N\geq1$. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.]{}
### Mathematical Subject classification (2010) {#mathematical-subject-classification-2010 .unnumbered}
[Primary: 35L15; Secondary: 35B65, 35S50, 35B45]{}
### Keywords {#keywords .unnumbered}
[Hyperbolic operators, non-Lipschitz coefficient, log-Zygmund regularity, energy estimates, well-posedness]{}
Introduction
============
This paper is devoted to the study of the Cauchy problem for a second order strictly hyperbolic operator defined in a strip $[0,T]\times{\mathbb{R}}^N$, for some $T>0$ and $N\geq1$. Consider a second order operator of the form $$\label{def:op}
Lu\;:=\;\d^2_tu\,-\,\sum_{j,k=1}^N\d_j\left(a_{jk}(t,x)\,\d_ku\right)$$ (with $a_{jk}=a_{kj}$ for all $j$, $k$) and assume that $L$ is strictly hyperbolic with bounded coefficients, i.e. there exist two constants $0<\lambda_0\leq\Lambda_0$ such that $$\lambda_0\,|\xi|^2\;\leq\;\sum_{j,k=1}^N a_{jk}(t,x)\,\xi_j\,\xi_k\;\leq\;\Lambda_0\,|\xi|^2$$ for all $(t,x)\in[0,T]\times{\mathbb{R}}^N$ and all $\xi\in{\mathbb}{R}^N$.
It is well-known (see [@H-S]; see also e.g. [@Horm Ch. IX] or [@Miz Ch. 6] for analogous results) that, if the coefficients $a_{jk}$ are Lipschitz continuous with respect to $t$ and only measurable in $x$, then the Cauchy problem for $L$ is well-posed in $H^1\times L^2$. If the $a_{jk}$’s are Lipschitz continuous with respect to $t$ and ${\mathcal}{C}^\infty _b$ (i.e. ${\mathcal}{C}^\infty$ and bounded with all their derivatives) with respect to the space variables, one can recover the well-posedness in $H^{s+1}\times H^s$ for all $s\in{\mathbb{R}}$. Moreover, in the latter case, one gets, for all $s\in{\mathbb{R}}$ and for a constant $C_s$ depending only on it, the following energy estimate: $$\begin{aligned}
&&\sup_{0\le t \le T} \biggl(\|u(t, \cdot)\|_{H^{s+1}}\, +
\|\partial_t u(t,\cdot)\|_{H^s}\biggr) \label{est:no-loss} \\
&&\qquad\qquad\qquad\qquad\qquad\qquad
\leq\,C_s\,\,\left(\|u(0, \cdot)\|_{H^{s+1}}+
\|\partial_t u(0,\cdot)\|_{H^s} + \int_0^{T} \| L u(t,\cdot)\|_{H^s}\, dt\right) \nonumber\end{aligned}$$ for all $u\in{\mathcal}{C}([0,T];H^{s+1}({\mathbb{R}}^N))\,\cap\,{\mathcal}{C}^1([0,T];H^s({\mathbb{R}}^N))$ such that $Lu\in L^1([0,T];H^s({\mathbb{R}}^N))$. Let us explicitly remark that previous inequality involves no loss of regularity for the function $u$: estimate holds for every $u\in{\mathcal}{C}^2([0,T];H^\infty({\mathbb{R}}^N))$ and the Cauchy problem for $L$ is well-posed in $H^\infty$ *with no loss of derivatives*.
If the Lipschitz continuity (in time) hypothesis is not fulfilled, then is no more true. Nevertheless, one can still try to recover $H^\infty$-well-posedness, possibly with a [*loss of derivatives*]{} in the energy estimate. The first case to consider is the case of the coefficients $a_{jk}$ depending only on $t$: $$Lu\;=\;\d^2_tu\,-\,\sum_{j,k=1}^N a_{jk}(t)\,\d_j\d_ku\,.$$ In [@C-DG-S], Colombini, De Giorgi and Spagnolo assumed the coefficients to satisfy an integral log-Lipschitz condition: $$\label{hyp:int-LL}
\int^{T-{\varepsilon}}_0\left|a_{jk}(t+{\varepsilon})\,-\,a_{jk}(t)\right|dt\;\leq\;C\,{\varepsilon}\,\log\left(1\,+\,\frac{1}{{\varepsilon}}\right)\,,$$ for some constant $C>0$ and all ${\varepsilon}\in\,]0,T]$. To get the energy estimate, they first smoothed out the coefficients using a mollifier kernel $\left(\rho_{\varepsilon}\right)_{\varepsilon}$. Then, by Fourier transform, they defined an approximated energy $E_{\varepsilon}(\xi,t)$ in phase space, where the problem becomes a family of ordinary differential equations. At that point, the key idea was to perform a different approximation of the coefficients in different zones of the phase space: in particular, they set ${\varepsilon}\,=\,|\xi|^{-1}$. Finally, they obtained an energy estimate *with a fixed loss of derivatives*: there exists a constant $\delta>0$ such that, for all $s\in{\mathbb{R}}$, the inequality $$\begin{aligned}
&&\sup_{0\le t \le T} \biggl(\|u(t, \cdot)\|_{H^{s+1-\delta}}\, +
\|\partial_t u(t,\cdot)\|_{H^{s-\delta}}\biggr) \label{est:c-loss} \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\leq\,C_s\,\left(\|u(0, \cdot)\|_{H^{s+1}}+
\|\partial_t u(0,\cdot)\|_{H^s} + \int_0^{T} \| L u(t,\cdot)\|_{H^s}\, dt\right) \nonumber\end{aligned}$$ holds true for all $u\in{\mathcal}{C}^2([0,T];H^\infty({\mathbb{R}}^N))$, for some constant $C_s$ depending only on $s$. Let us remark that if the coefficients $a_{jk}$ are not Lipschitz continuous then a loss of regularity cannot be avoided, as shown by Cicognani and Colombini in [@Cic-C]. Besides, in that paper the authors proved that, if the regularity of the coefficients $a_{jk}$ is measured by a modulus of continuity, any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz ones entails necessarily a loss of regularity, which, however, can be made arbitrarily small.
Recently Tarama (see paper [@Tar]) analysed the problem when coefficients satisfy an integral log-Zygmund condition: there exists a constant $C>0$ such that, for all $j$, $k$ and all ${\varepsilon}\in\,]0,T/2[$, one has $$\label{hyp:int-LZ}
\int^{T-{\varepsilon}}_{\varepsilon}\left|a_{jk}(t+{\varepsilon})\,+\,a_{jk}(t-{\varepsilon})\,-\,2\, a_{jk}(t)\right|dt\;\leq\;
C\,{\varepsilon}\,\log\left(1\,+\,\frac{1}{{\varepsilon}}\right)\,.$$ On the one hand, this condition is somehow related, for $a$ of class ${\mathcal}{C}^2([0,T])$, to the pointwise condition $|a(t)|+|t\,a'(t)|+|t^2\,a''(t)|\,\leq\,C$, considered by Yamazaki in [@Yama]. On the other hand, it’s obvious that, if the $a_{jk}$’s satisfy , then they satisfy also : so, a more general class of functions is considered. Again, Fourier transform, smoothing out the cofficients and linking the approximation parameter with the dual variable were fundamental tools in the analysis of Tarama. The improvement with respect to paper [@C-DG-S], however, was obtained defining a new energy, which involved (by differentiation in time) second derivatives of the approximated coefficients. Finally, he got an estimate analogous to , which implies, in particular, well-posedness in the space $H^\infty$.
In paper [@C-L], Colombini and Lerner considered instead the case in which coefficients $a_{jk}$ depend both in time and in space variables. In particular, they assumed an isotropic punctual log-Lipschitz condition, i.e. there exists a constant $C>0$ such that, for all $\zeta=(\tau,\xi)\in{\mathbb{R}}\times{\mathbb{R}}^N$, $\zeta\neq0$, one has $$\sup_{z=(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^N}\;\left|a_{jk}(z+\zeta)\,-\,a_{jk}(z)\right|\;\leq\;C\,|\zeta|\,
\log\left(1\,+\,\frac{1}{|\zeta|}\right)\,.$$ Again, smoothing coefficients with respect to the time variable is required; on the contrary, one cannot use the Fourier transform, due to the dependence of $a_{jk}$ on $x$. The authors bypassed this problem taking advantage of Littlewood-Paley decomposition and paradifferential calculus. Hence, they considered the energy concerning each localized part $\Delta_\nu u$ of the solution $u$, and then they performed a weighted summation to put all these pieces together. Also in this case, they had to consider a different approximation of the coefficients in different zones of the phase space, which was obtained setting ${\varepsilon}=2^{-\nu}$ (recall that $2^\nu$ is the “size” of the frequencies in the $\nu$-th ring, see subsection \[ss:L-P\] below). In the end, they got the following statement: for all $s\in\,]0,1/4]$, there exist positive constants $\beta$ and $C_s$ and a time $T^*\in\,]0,T]$ such that $$\begin{aligned}
&&\sup_{0\le t \le T^*} \biggl(\|u(t, \cdot)\|_{H^{-s+1-\beta t}}\, +
\|\partial_t u(t,\cdot)\|_{H^{-s-\beta t}}\biggr) \label{est:t-loss} \\
&&\qquad\qquad\qquad\qquad\qquad
\leq\,C_s\,\left(\|u(0, \cdot)\|_{H^{-s+1}}+
\|\partial_t u(0,\cdot)\|_{H^{-s}} + \int_0^{T^*} \| L u(t,\cdot)\|_{H^{-s-\beta t}}\, dt\right) \nonumber\end{aligned}$$ for all $u\in{\mathcal}{C}^2([0,T];H^\infty({\mathbb{R}}^N))$. Let us point out that the bound on $s$ was due to this reason: the product by a log-Lipschitz function is well-defined in $H^s$ if and only if $|s|<1$. Note also that this fact gives us a bound on the lifespan of the solution: the regularity index $-s+1-\beta T^*$ has to be strictly positive, so one can expect only local in time existence of a solution. Moreover in the case the coefficients $a_{jk}$ are ${\mathcal}{C}^\infty_b$ in space, the authors proved inequality for all $s$: so, they still got well-posedness in $H^\infty$, but *with a loss of derivatives increasing in time*.
The case of a complete strictly hyperbolic second order operator, $$Lu\,=\,\sum_{j,k=0}^N\d_{y_j}\left(a_{jk}\,\d_{y_k}u\right)\,+\,
\sum_{j=0}^N\left(b_j\,\d_{y_j}u\,+\,\d_{y_j}\!\left(c_j\,u\right)\right)\,+\,d\,u$$ (here we set $y=(t,x)\in{\mathbb{R}}_t\times{\mathbb{R}}^N_x$), was considered by Colombini and Métivier in [@C-M]. They assumed the same isotropic log-Lipschitz condition of [@C-L] on the coefficients of the second order part of $L$, while $b_j$ and $c_j$ were supposed to be $\alpha$-Hölder continuous (for some $\alpha\in\,]1/2,1[\,$) and $d$ to be only bounded. The authors headed towards questions such as local existence and uniqueness, and also finite propagation speed for local solutions.
Recently, Colombini and Del Santo, in [@C-DS] (for a first approach to the problem see also [@DS], where smoothness in space was required), came back to the Cauchy problem for the operator , mixing up a Tarama-like hypothesis (concerning the dependence on the time variable) with the one of Colombini and Lerner (with respect to $x$). More precisely, they assumed a pointwise log-Zygmund condition in time and a pointwise log-Lipschitz condition in space, uniformly with respect to the other variable (see relations and below). However, they had to restrict themselves to the case of space dimension $N=1$: as a matter of fact, a Tarama-kind energy was somehow necessary to compensate the bad behaviour of the coefficients with respect to $t$, but it was not clear how to define it in higher space dimensions. Again, localizing energy by Littlewood-Paley decomposition and linking approximation parameter and dual variable lead to an estimate analogous to . The aim of the present paper is to extend the result of Colombini and Del Santo to any dimension $N\geq1$. As just pointed out, the main difficulty was to define a suitable energy related to the solution. So, the first step is to pass from functions $a(t,x)$ with low regularity modulus of continuity, to more general symbols $\sigma_a(t,x,\xi)$ (obviously related to the initial function $a$) satisfying the same hypothesis in $t$ and $x$, and then to consider paradifferential operators associated to these symbols. Nevertheless, positivity hypothesis on $a$ (required for defining a strictly hyperbolic problem) does not translate, in general, to positivity of the corresponding operator, which is fundamental in obtaining energy estimates. At this point, paradifferential calculus depending on a parameter ${\gamma}\geq1$, defined and developed by Métivier in [@M-1986] (see also [@M-Z]), comes into play and allows us to recover positivity of the (new) paradifferential operator associated to $a$. Defining a localized energy and an approximation of the coefficients depending on the dual variable are, once again, basic ingredients in closing estimates. Hence, in the end we will get an inequality similar to , for any $s\in\,]0,1[$.
The paper is organized as follows.
First of all, we will introduce the work hypothesis for our strictly hyperbolic problem, and we will state our main results.
Then, we will present the tools we need, all from Fourier Analysis. In particular, we will recall Littlewood-Paley decomposition and some results about (classical) paradifferential calculus, as introduced first by J.-M. Bony in the famous paper [@Bony]. We will need also to define a different class of Sobolev spaces, of logarithmic type, as done in [@C-M]: they will come into play in our computations. Moreover, we will present also paradifferential calculus depending on a parameter (which is basic in our analysis, as already pointed out), as introduced in [@M-1986] and [@M-Z]. A complete treatement about functions with low regularity modulus of continuity will end this section. In particular, we will focus on log-Zygmund and log-Lipschitz conditions: taking advantage of paradifferential calculus, we will establish some properties of functions satisfying such hypothesis. Hence, we will pass to consider more general symbols and the associated paradifferential operators, for which we will develop also a symbolic calculus and we will state a fundamental positivity estimate.
This having been done, we will be then ready to tackle the proof of our main result, for which we will go back to the main ideas of paper [@C-DS]. First of all, by use of a convolution kernel, we will smooth out the coefficients, but with respect to the time variable only. As a matter of facts, low regularity in $x$ will be compensated by considering paradifferential operators associated to our coefficients. Then, we will decompose the solution $u$ to the Cauchy problem for into dyadic blocks $\Delta_\nu u$, for which we will define an approximate localized energy $e_\nu$: the dependence on the approximation parameter ${\varepsilon}$ will be linked to the phase space localization, setting ${\varepsilon}=2^{-\nu}$. The piece of energy $e_\nu$ will be of Tarama type, but this time multiplication by functions will be replaced by action of paradifferential operators associated to them. A weighted summation of these pieces will define the total energy $E(t)$ associated to $u$. The rest of the proof is classical: we will differentiate $E$ with respect to time and, using Gronwall Lemma, we will get a control for it in terms of the initial energy $E(0)$ and the external force $Lu$ only.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The work was mostly prepared during the Ph.D. thesis of the third author at SISSA and Université Paris-Est: he is grateful to both these institutions. During the final part, the third author was partially supported by Grant MTM2011-29306-C02-00, MICINN, Spain, ERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research Networking Programme OPTPDE and Grant PI2010-04 of the Basque Government.
Basic definitions and main result {#s:results}
=================================
This section is devoted to the presentation of our work setting and of our main results.
Let us consider the operator over $[0,T_0]\times{\mathbb{R}}^N$ (for some $T_0>0$ and $N\geq1$) $$\label{eq:op}
Lu\,=\,\d^2_tu\,-\,\sum_{i,j=1}^N\d_i\left(a_{ij}(t,x)\,\d_ju\right)\,,$$ and let us suppose $L$ to be strictly hyperbolic with bounded coefficients, i.e. there exist two positive constants $0<\lambda_0\leq\Lambda_0$ such that, for all $(t,x)\in{\mathbb}{R}_t\times{\mathbb}{R}^N_x$ and all $\xi\in{\mathbb}{R}^N$, one has $$\label{h:hyp}
\lambda_0\,|\xi|^2\,\leq\,\sum_{i,j=1}^N a_{ij}(t,x)\,\xi_i\,\xi_j\,\leq\,\Lambda_0\,|\xi|^2\,.$$
Moreover, let us suppose the coefficients to be log-Zygmund continuous in the time variable $t$, uniformly with respect to $x$, and log-Lipschitz continuous in the space variables, uniformly with respect to $t$. This hypothesis reads as follow: there exists a constant $K_0$ such that, for all $\tau>0$ and all $y\in{\mathbb}{R}^N\setminus\{0\}$, one has $$\begin{aligned}
\sup_{(t,x)}\left|a_{ij}(t+\tau,x)+a_{ij}(t-\tau,x)-2a_{ij}(t,x)\right| & \leq &
K_0\,\tau\,\log\left(1\,+\,\frac{1}{\tau}\right) \label{h:LZ_t} \\
\sup_{(t,x)}\left|a_{ij}(t,x+y)-a_{ij}(t,x)\right| & \leq & K_0\,|y|\,\log\left(1\,+\,\frac{1}{|y|}\right). \label{h:LL_x}\end{aligned}$$
Now, let us state our main result, i.e. an energy estimate for the operator .
\[t:en-est\] Let us consider the operator $L$ defined in , and let us suppose $L$ to be strictly hyperbolic, i.e. relation holds true. Moreover, let us suppose that coefficients $a_{ij}$ satisfy both conditions and .
Then, for all fixed $\theta\in\,]0,1[\,$, there exist a $\beta^*>0$, a time $T>0$ and a constant $C>0$ such that the following estimate, $$\begin{aligned}
&&\sup_{0\le t \le T} \biggl(\|u(t, \cdot)\|_{H^{-\theta+1-\beta^* t}}\, +
\|\partial_t u(t,\cdot)\|_{H^{-\theta-\beta^* t}}\biggr) \label{est:thesis} \\
&&\qquad\qquad\qquad\qquad\qquad
\leq\,C\,\left(\|u(0, \cdot)\|_{H^{-\theta+1}}+
\|\partial_t u(0,\cdot)\|_{H^{-\theta}} + \int_0^{T} \| L u(t,\cdot)\|_{H^{-\theta-\beta^* t}}\, dt\right)\,, \nonumber\end{aligned}$$ holds true for all $u\in{\mathcal}{C}^2([0,T];H^\infty({\mathbb{R}}^N))$.
So, it’s possible to control the Sobolev norms of solutions to in terms of those of initial data and of the external force only: the price to pay is a loss of derivatives, increasing (linearly) in time.
Tools
=====
In this section we will introduce the main tools, basically from Fourier Analysis, we will need to prove our statement.
First of all, we will recall classical Littlewood-Paley decomposition and some basic results on dyadic analysis. By use of it, we will also define a different class of Sobolev spaces, of logarithmic type.\
Then, we will need to introduce a paradifferential calculus depending on some parameter ${\gamma}\geq1$: the main ideas are the same of the classical version, but the presence of the parameter allows us to perform a more refined analysis. This will play a crucial role to get our result.\
After this, we will consider functions with low regularity modulus of continuity. In particular, we will focus on log-Zygmund and log-Lipschitz functions: dyadic decomposition allows us to get some of their properties. Moreover, we will analyse the convolution of a log-Zygmund function by a smoothing kernel.\
Finally, taking advantage of paradifferential calculus with parameters, we will consider general symbols having low regularity in time and space variables. Under suitable hypothesis on such a symbol, we will also get positivity estimates for the associated operator.
Littlewood-Paley decomposition {#ss:L-P}
------------------------------
Let us first define the so called “Littlewood-Paley decomposition”, based on a non-homogeneous dyadic partition of unity with respect to the Fourier space variable. We refer to [@B-C-D], [@Bony] and [@M-2008] for the details.
So, fix a smooth radial function $\chi$ supported in the ball $B(0,2),$ equal to $1$ in a neighborhood of $B(0,1)$ and such that $r\mapsto\chi(r\,e)$ is nonincreasing over ${\mathbb{R}}_+$ for all unitary vectors $e\in{\mathbb{R}}^N$. Set also $\varphi\left(\xi\right)=\chi\left(\xi\right)-\chi\left(2\xi\right)$.
For convenience, we immediately introduce the following notation: $$\chi_j(\xi)\,:=\,\chi(2^{-j}\xi)\qquad\mbox{ and }\qquad {\varphi}_j(\xi)\,:=\,{\varphi}(2^{-j}\xi)\,.$$ We will indifferently use it or the previous one.
The dyadic blocks $(\Delta_j)_{j\in{\mathbb{Z}}}$ are defined by[^1] $$\Delta_j:=0\ \hbox{ if }\ j\leq-1,\quad\Delta_{0}:=\chi(D)\quad\hbox{and}\quad
\Delta_j:=\varphi(2^{-j}D)\ \text{ if }\ j\geq1.$$ We also introduce the following low frequency cut-off operator: $$S_j\,:=\,\chi(2^{-j}D)\,=\,\sum_{k\leq j}\Delta_{k}\quad\text{for}\quad j\geq0.$$
The following classical properties will be freely used throughout the paper:
- for any $u\in{\mathcal}{S}',$ the equality $u=\sum_{j}\Delta_ju$ holds true in ${\mathcal}{S}'$;
- for all $u$ and $v$ in ${\mathcal}{S}'$, the sequence $\left(S_{j-3}u\,\,\Delta_jv\right)_{j\in{\mathbb{N}}}$ is spectrally supported in dyadic annuli.
Let us also mention a fundamental result, which explains, by the so-called *Bernstein’s inequalities*, the way derivatives act on spectrally localized functions.
\[l:bern\] Let $0<r<R$. A constant $C$ exists so that, for any multi-index $\alpha\in{\mathbb{N}}^N$, any couple $(p,q)$ in $[1,+\infty]^2$ with $p\leq q$ and any function $u\in L^p$, we have, for all $\lambda>0$, $$\displaylines{
{\rm supp}\, \widehat u \subset B(0,\lambda R)\quad
\Longrightarrow\quad
\|\d^\alpha u\|_{L^q}\, \leq\,
C^{|\alpha|+1}\,\lambda^{|\alpha|+N\left(\frac{1}{p}-\frac{1}{q}\right)}\,\|u\|_{L^p}\;;\cr
{\rm supp}\, \widehat u \subset \{\xi\in{\mathbb{R}}^N\,|\, r\lambda\leq|\xi|\leq R\lambda\}
\quad\Longrightarrow\quad C^{-|\alpha|-1}\lambda^{|\alpha|}\|u\|_{L^p}\,
\leq\,\|\d^\alpha u\|_{L^p}\,\leq\,
C^{|\alpha|+1} \lambda^{|\alpha|}\|u\|_{L^p}\,.
}$$
Let us recall the characterization of (classical) Sobolev spaces via dyadic decomposition: for all $s\in{\mathbb}{R}$ there exists a constant $C_s>0$ such that $$\label{est:dyad-Sob}
\frac{1}{C_s}\,\,\sum^{+\infty}_{\nu=0}2^{2\,\nu\, s}\,\|u_\nu\|^2_{L^2}\;\leq\;\|u\|^2_{H^s}\;\leq\;
C_s\,\,\sum^{+\infty}_{\nu=0}2^{2\,\nu\, s}\,\|u_\nu\|^2_{L^2}\,,$$ where we have set $u_\nu:=\Delta_\nu u$.
So, the $H^s$ norm of a tempered distribution is the same as the $\ell^2$ norm of the sequence $\left(2^{s\nu}\,\left\|\Delta_\nu u\right\|_{L^2}\right)_{\nu\in{\mathbb{N}}}$. Now, one may ask what we get if, in the sequence, we put weights different to the exponential term $2^{s\nu}$. Before answering this question, we introduce some definitions. For the details of the presentation, we refer also to [@C-M].
Let us set $\Pi(D)\,:=\,\log(2+|D|)$, i.e. its symbol is $\pi(\xi)\,:=\,\log(2+|\xi|)$.
\[d:log-H\^s\] For all $\alpha\in{\mathbb{R}}$, we define the space $H^{s+\alpha\log}$ as the space $\Pi^{-\alpha}H^s$, i.e. $$f\,\in\,H^{s+\alpha\log}\quad{\Leftrightarrow}\quad\Pi^\alpha f\,\in\,H^s\quad{\Leftrightarrow}\quad
\pi^\alpha(\xi)\left(1+|\xi|^2\right)^{s/2}{\widehat}{f}(\xi)\,\in\,L^2\,.$$
From the definition, it’s obvious that the following inclusions hold true for any $s_1>s_2$ and any $\alpha_1>\alpha_2>0$: $$H^{s_1+\alpha_1\log}\;{\hookrightarrow}\;H^{s_1+\alpha_2\log}\;{\hookrightarrow}\;H^{s_1}\;{\hookrightarrow}\;
H^{s_1-\alpha_2\log}\;{\hookrightarrow}\;H^{s_1-\alpha_1\log}\;{\hookrightarrow}\;H^{s_2}\,.$$
We have the following dyadic characterization of these spaces (see [@M-2008 Prop. 4.1.11]).
\[p:log-H\] Let $s$, $\alpha\,\in{\mathbb{R}}$. A tempered distribution $u$ belongs to the space $H^{s+\alpha\log}$ if and only if:
- for all $k\in{\mathbb{N}}$, $\Delta_ku\in L^2({\mathbb{R}}^N)$;
- set $\,\delta_k\,:=\,2^{ks}\,(1+k)^\alpha\,\|\Delta_ku\|_{L^2}$ for all $k\in{\mathbb{N}}$, the sequence $\left(\delta_k\right)_k$ belongs to $\ell^2({\mathbb{N}})$.
Moreover, $\|u\|_{H^{s+\alpha\log}}\,\sim\,\left\|\left(\delta_k\right)_k\right\|_{\ell^2}$.
Hence, this proposition generalizes property .
This new class of Sobolev spaces, which are in a certain sense of logarithmic type, will come into play in our analysis. As a matter of fact, operators associated to log-Zygmund or log-Lipschitz symbols give a logarithmic loss of derivatives. We will clarify in a while what we have just said; first of all, we need to introduce a new version of paradifferential calculus, depending on a parameter ${\gamma}\geq1$.
Paradifferential calculus with parameters
-----------------------------------------
Let us present here the paradifferential calculus depending on some parameter ${\gamma}$. One can find a complete and detailed treatement in Appendix B of [@M-Z] (see also [@M-1986]).
Fix $\gamma\geq1$ and take a cut-off function $\psi\in{\mathcal}{C}^\infty({\mathbb{R}}^N\times{\mathbb{R}}^N)$ which verifies the following properties:
- there exist $0<{\varepsilon}_1<{\varepsilon}_2<1$ such that $$\psi(\eta,\xi)\,=\,\left\{\begin{array}{lcl}
1 & \mbox{for} & |\eta|\leq{\varepsilon}_1\left(\gamma+|\xi|\right) \\ [1ex]
0 & \mbox{for} & |\eta|\geq{\varepsilon}_2\left(\gamma+|\xi|\right)\,;
\end{array}
\right.$$
- for all $(\beta,\alpha)\in{\mathbb{N}}^N\times{\mathbb{N}}^N$, there exists a constant $C_{\beta,\alpha}$ such that $$\left|\d^\beta_\eta\d^\alpha_\xi\psi(\eta,\xi)\right|\,\leq\,C_{\beta,\alpha}\left(\gamma+|\xi|\right)^{-|\alpha|-|\beta|}\,.$$
\[r:indep\_gamma\] We remark that ${\varepsilon}_1$, ${\varepsilon}_2$ and the different $C_{\beta,\alpha}$ occurring in the previous estimates must not depend on ${\gamma}$.
For instance, if $\gamma=1$, one can take $$\psi(\eta,\xi)\,\equiv\,\psi_{-3}(\eta,\xi)\,:=\,\sum_{k=0}^{+\infty}\chi_{k-3}(\eta)\,{\varphi}_k(\xi)\,,$$ where $\chi$ and ${\varphi}$ are the localization (in phase space) functions associated to a Littlewood-Paley decomposition, see [@M-2008 Ex. 5.1.5]. Similarly, if $\gamma>1$ it is possible to find a suitable integer $\mu\geq0$ such that $$\label{pd_eq:pp_symb}
\psi(\eta,\xi)\,\equiv\,\psi_\mu(\eta,\xi)\,:=\,\chi_{\mu}(\eta)\,\chi_{\mu+2}(\xi)\,+\,\sum_{k=\mu+3}^{+\infty}\chi_{k-3}(\eta)\,{\varphi}_k(\xi)$$ is a function with the just described properties.
Define now $$G^\psi(x,\xi)\,:=\,\left({\mathcal}{F}^{-1}_\eta\psi\right)(x,\xi)\,,$$ the inverse Fourier transform of $\psi$ with respect to the variable $\eta$.
\[l:G\] For all $(\beta,\alpha)\in{\mathbb{N}}^N\times{\mathbb{N}}^N$, there exist constants $C_{\beta,\alpha}$, independent of ${\gamma}$, such that: $$\begin{aligned}
\left\|\d^\beta_x\d^\alpha_\xi G^\psi(\cdot,\xi)\right\|_{L^1({\mathbb{R}}^N_x)} & \leq &
C_{\beta,\alpha}\left(\gamma+|\xi|\right)^{-|\alpha|+|\beta|}\,, \label{pd_est:G_1} \\
\left\||\cdot|\,\log\left(2+\frac{1}{|\cdot|}\right)\,\d^\beta_x\d^\alpha_\xi G^\psi(\cdot,\xi)\right\|_{L^1({\mathbb{R}}^N_x)} & \leq &
C_{\beta,\alpha}\left(\gamma+|\xi|\right)^{-|\alpha|+|\beta|-1}\,\log(1+\gamma+|\xi|)\,. \label{pd_est:G_2}\end{aligned}$$
See [@M-2008 Lemma 5.1.7].
Thanks to $G^\psi$, we can smooth out a symbol $a$ in the $x$ variable and then define the paradifferential operator associated to $a$ as the pseudodifferential operator related to this smooth function. We set the classical symbol associated to $a$ to be $$\sigma_a(x,\xi)\,:=\,\left(\,\psi(D_x,\xi)\,a\,\right)(x,\xi)\,=\,\left(G^\psi(\cdot,\xi)\,*_x\,a(\cdot,\xi)\right)(x)\,,$$ and then the paradifferential operator associated to $a$: $$T_a\,:=\,\sigma_a(x,D_x)\,,$$ where we have omitted $\psi$ because the definition is independent of it, up to lower order terms.
\[r:p-prod\] Let us note that if $a=a(x)\in L^\infty$ and if we take the cut-off function $\psi_{-3}$, then $T_a$ is actually the usual paraproduct operator. If we take $\psi_\mu$ as defined in , instead, we get a paraproduct operator which starts from high enough frequencies, which will be indicated with $T^\mu_a$(see [@C-M Par. 3.3]).
Let us point out that we can also define a $\gamma$-dyadic decomposition. First of all, we set $$\Lambda(\xi,{\gamma})\,:=\,\left({\gamma}^2\,+\,|\xi|^2\right)^{1/2}\,.$$ Then, taken the usual smooth function $\chi$ associated to a Littlewood-Paley decomposition, we can define $$\chi_\nu(\xi,{\gamma})\,:=\,\chi\left(2^{-\nu}\Lambda(\xi,{\gamma})\right)\,,\quad
S^{\gamma}_\nu\,:=\,\chi_\nu(D_x,{\gamma})\,,\quad
\Delta^{\gamma}_\nu\,:=\,S^{\gamma}_{\nu+1}-S^{\gamma}_\nu\,.$$ The usual properties of the support of the localization functions still hold, and for all fixed ${\gamma}\geq1$ and all tempered distributions $u$, we have $$u\,=\,\sum_{\nu=0}^{+\infty}\,\Delta^{\gamma}_\nu\,u\qquad\mbox{in }\;\;{\mathcal}{S}'\,.$$ Moreover, with natural modifications in definitions, we can introduce the space $H^{s+\alpha\log}_\gamma$ as the set of tempered distributions for which $$\left\|u\right\|^2_{H^{s+\alpha\log}_{\gamma}}\,:=\,\int_{{\mathbb{R}}^N_\xi}\Lambda^{2s}(\xi,{\gamma})\,\log^{2\alpha}(1+{\gamma}+|\xi|)\,
\left|{\widehat}{u}(\xi)\right|^2\,d\xi\;\;<\;+\infty\,.$$ For the details see [@M-Z Appendix B]. What is important to retain is that, once we fix ${\gamma}\geq1$ (for example, to obtain positivity of paradifferential operators involved in our computations), the whole previous construction is equivalent to the classical one; in particular, the space $H^{s+\alpha\log}_\gamma$ coincides with $H^{s+\alpha\log}$, the respective norms are equivalent and the characterization given by Proposition \[p:log-H\] still holds true.
On log-Lipschitz and log-Zygmund functions
------------------------------------------
Let us now give the rigorous definitions of the moduli of continuity we are dealing with, and state some of their properties.
\[d:LL\] A function $f\in L^\infty({\mathbb{R}}^N)$ is said to be log-Lipschitz, and we write $f\in LL({\mathbb{R}}^N)$, if the quantity $$|f|_{LL}\,:=\,\sup_{x,y\in{\mathbb{R}}^N,\,|y|<1}
\left(\frac{\left|f(x+y)\,-\,f(x)\right|}{|y|\,\log\left(1\,+\,\frac{1}{|y|}\right)}\right)\,<\,+\infty\,.$$ We define $\|f\|_{LL}\,:=\,\|f\|_{L^\infty}\,+\,|f|_{LL}$.
Let us define also the space of log-Zygmund functions. We will give the general definition in ${\mathbb{R}}^N$, even if the one dimensional case will be the only relevant one for our purposes.
\[d:LZ\] A function $g\in L^\infty({\mathbb{R}}^N)$ is said to be log-Zygmund, and we write $g\in LZ({\mathbb{R}}^N)$, if the quantity $$|g|_{LZ}\,:=\,\sup_{x,y\in{\mathbb{R}}^N,\,|y|<1}
\left(\frac{\left|g(x+y)\,+\,g(x-y)\,-\,2\,g(x)\right|}{|y|\,\log\left(1\,+\,\frac{1}{|y|}\right)}\right)\,<\,+\infty\,.$$ We define $\|g\|_{LZ}\,:=\,\|g\|_{L^\infty}\,+\,|g|_{LZ}$.
\[r:log-g\] Let us immediately point out that, by monotonicity of logarithmic function, we can replace the factor $\log\left(1+1/|y|\right)$ in previous definitions with $\log\left(1+{\gamma}+1/|y|\right)$, for all parameters ${\gamma}\geq1$. As paradifferential calculus with parameters will play a fundamental role in our computations, it’s convenient to perform such a change, and to do so also in hypothesis and of section \[s:results\].
Let us give a characterization of the space $LZ$. Recall that the space of Zygmund functions is actually $B^1_{\infty,\infty}$ (see e.g. [@Ch1995]): arguing in the same way, one can prove the next proposition.
\[p:LZ\] The space $LZ({\mathbb{R}}^N)$ coincides with the logarithmic Besov space $B^{1-\log}_{\infty,\infty}$, i.e. the space of tempered distributions $u$ such that $$\label{est:LZ}
\sup_{\nu\geq0}\biggl(2^{\nu}\,(\nu+1)^{-1}\,\left\|\Delta_\nu u\right\|_{L^\infty}\biggr)\,<\,+\infty\,.$$
- Let us first consider a $u\in B^{1-\log}_{\infty,\infty}$ and take $x$ and $y\,\in{\mathbb{R}}^N$, with $|y|<1$. For all fixed $n\in{\mathbb{N}}$ we can write: $$\begin{aligned}
u(x+y)+u(x-y)-2u(x) & = & \sum_{k<n}\left(\Delta_ku(x+y)+\Delta_ku(x-y)-2\Delta_ku(x)\right) \\
& & \qquad\qquad+\,\sum_{k\geq n}\left(\Delta_ku(x+y)+\Delta_ku(x-y)-2\Delta_ku(x)\right)\,.\end{aligned}$$ First, we take advantage of the Taylor’s formula up to second order to handle the former terms; then, we use property . Hence we get $$\begin{aligned}
\left|u(x+y)+u(x-y)-2u(x)\right| & \leq & C\,|y|^2\sum_{k<n}\left\|\nabla^2\Delta_ku\right\|_{L^\infty}\,+\,
4\,\sum_{k\geq n}\left\|\Delta_ku\right\|_{L^\infty} \\
& \leq & C\left(|y|^2\sum_{k<n}2^k\,(k+1)\,+\,\sum_{k\geq n}2^{-k}(k+1)\right) \\
& \leq & C\,(n+1)\left(|y|^2\,2^n\,+\,2^{-n}\right)\,.\end{aligned}$$ Now, as $|y|<1$, the choice $n=1+\left[\log_2\left(1/|y|\right)\right]$ (where with $[\sigma]$ we mean the greatest positive integer less than or equal to $\sigma$) completes the proof of the first part.
- Now, given a log-Zygmund function $u$, we want to estimate the $L^\infty$ norm of its localized part $\Delta_ku$.
Let us recall that applying the operator $\Delta_k$ is the same of the convolution with the inverse Fourier transform of the function ${\varphi}(2^{-k}\cdot)$, which we call $h_k(x)=2^{kN}h(2^k\cdot)$, where we set $h={\mathcal}{F}^{-1}_\xi({\varphi})$. As ${\varphi}$ is an even function, so does $h$; moreover we have $$\int h(z)\,dz\,=\,\int{\mathcal}{F}^{-1}_\xi({\varphi})(z)\,dz\,=\,{\varphi}(\xi)_{|\xi=0}\,=\,0\,.$$ Therefore, we can write: $$\Delta_ku(x)\,=\,2^{kN-1}\int h(2^ky)\left(u(x+y)+u(x-y)-2u(x)\right)\,dy\,,$$ and noting that $\sigma\,\mapsto\,\sigma\,\log\left(1+{\gamma}+1/\sigma\right)$ is increasing completes the proof of the second part.
From definitions \[d:LL\] and \[d:LZ\], it’s obvious that $LL({\mathbb{R}}^N)\,{\hookrightarrow}\,LZ({\mathbb{R}}^N)$: Proposition 3.3 of [@C-L] explains this property in terms of dyadic decomposition.
\[p:dyadic-LL\] There exists a constant $C$ such that, for all $a\in LL({\mathbb{R}}^N)$ and all integers $k\geq0$, we have $$\left\|\Delta_k a\right\|_{L^\infty}\,\leq\,C\,(k+1)\,2^{-k}\,\|a\|_{LL}\,. \label{est:a_k}$$ Moreover, for all $k\in{\mathbb{N}}$ we have $$\begin{aligned}
\left\|a\,-\,S_k a\right\|_{L^\infty} & \leq & C\,(k+1)\,2^{-k}\,\|a\|_{LL} \label{est:a-S_ka} \\
\left\|S_k a\right\|_{{\mathcal}{C}^{0,1}} & \leq & C\,(k+1)\,\|a\|_{LL}\,. \label{est:S_ka}\end{aligned}$$
\[r:LL\] Note that, again from Proposition 3.3 of [@C-L], property is a characterization of the space $LL$.
Using dyadic characterization of the space $LZ$ and arguing as for proving Proposition \[p:LZ\], we can establish the following property. For our purposes, it’s enough to consider a log-Zygmund function $a$ depending only on the time variable $t$, but the same reasoning holds true also in higher dimensions.
\[l:LZ\] For all $a\in LZ({\mathbb{R}})$, there exists a constant $C$, depending only on the $LZ$ norm of $a$, such that, for all $\gamma\geq1$ and all $0<|\tau|<1$ one has $$\label{est:lip-LZ}
\sup_{t\in{\mathbb{R}}}\left|a(t+\tau)-a(t)\right|\,\leq\,C\,|\tau|\,\log^2\!\left(1+\gamma+\frac{1}{|\tau|}\right).$$
As done in proving Proposition \[p:LZ\], for all $n\in{\mathbb{N}}$ we can write $$a(t+\tau)-a(t)\,=\,\sum_{k<n}\left(\Delta_ka(t+\tau)-\Delta_ka(t)\right)\,+\,
\sum_{k\geq n}\left(\Delta_ka(t+\tau)-\Delta_ka(t)\right)\,,$$ where, obviously, the localization in frequencies is done with respect to the time variable. For the former terms we use the mean value theorem, while for the latter ones we use characterization ; hence, we get $$\begin{aligned}
\left|a(t+\tau)-a(t)\right| & \leq & \sum_{k<n}\left\|\frac{d}{dt}\Delta_ka\right\|_{L^\infty}|\tau|\,+\,
2\sum_{k\geq n}\left\|\Delta_ka\right\|_{L^\infty} \\
& \leq & C\left(n^2\,|\tau|\,+\,\sum_{k\geq n}2^{-k}k\right).\end{aligned}$$ The series in the right-hand side of the previous inequality can be bounded, up to a multiplicative constant, by $2^{-n}n$; therefore $$\left|a(t+\tau)-a(t)\right|\,\leq\,C\,n\left(n\,|\tau|\,+\,2^{-n}\right)\,,$$ and the choice $n=1+[\log_2(1/|\tau|)]$ completes the proof.
Now, given a log-Zygmund function $a(t)$, we can regularize it by convolution. So, take an even function $\rho\in{\mathcal}{C}^\infty_0({\mathbb}{R}_t)$, $0\leq\rho\leq1$, whose support is contained in the interval $[-1,1]$ and such that $\int\rho(t)dt=1$, and define the mollifier kernel $$\rho_{\varepsilon}(t)\,:=\,\frac{1}{{\varepsilon}}\,\,\rho\!\left(\frac{t}{{\varepsilon}}\right)\qquad\qquad\forall\,{\varepsilon}\in\,]0,1]\,.$$ We smooth out the function $a$ setting, for all ${\varepsilon}\in\,]0,1]$, $$\label{eq:a^e}
a_{\varepsilon}(t)\,:=\,\left(\rho_{\varepsilon}\,*\,a\right)(t)\,=\,\int_{{\mathbb}{R}_s}\rho_{{\varepsilon}}(t-s)\,a(s)\,ds\,.$$ The following proposition holds true.
\[p:LZ-reg\] Let $a$ be a log-Zygmund function. There exist constants $C$ such that, for all ${\gamma}\geq1$, one has $$\begin{aligned}
\left|a_{\varepsilon}(t)-a(t)\right| & \leq & C\,\|a\|_{LZ}\,\,{\varepsilon}\,\log\left(1+\gamma+\frac{1}{{\varepsilon}}\right) \label{est:a^e-a} \\
\left|\d_ta_{\varepsilon}(t)\right| & \leq & C\,\|a\|_{LZ}\,\log^2\!\left(1+\gamma+\frac{1}{{\varepsilon}}\right) \label{est:d_t-a^e} \\
\left|\d^2_ta_{\varepsilon}(t)\right| & \leq & C\,\|a\|_{LZ}\,\,\frac{1}{{\varepsilon}}\,
\log\!\left(1+\gamma+\frac{1}{{\varepsilon}}\right) \label{est:d_tt-a^e}\,.\end{aligned}$$
For first and third inequalities, the proof is the same as in [@C-DS]. We have to pay attention only to . As $\rho'$ has null integral, the relation $$\d_ta_{\varepsilon}(t)\,=\,\frac{1}{{\varepsilon}^2}\int_{|s|\leq{\varepsilon}}\rho'\left(\frac{s}{{\varepsilon}}\right)\,
\left(a(t-s)-a(t)\right)ds$$ holds, and hence, taking advantage of , it implies $$\left|\d_ta_{\varepsilon}(t)\right|\,\leq\,\frac{C}{{\varepsilon}^2}\,\int_{|s|\leq{\varepsilon}}
\left|\rho'\left(\frac{s}{{\varepsilon}}\right)\right|\,|s|\,\log^2\!\left(1+\gamma+\frac{1}{|s|}\right)ds\,.$$ Observing that the function $\sigma\mapsto\sigma\log^2(1+\gamma+1/\sigma)$ is increasing in the interval $[0,1]$, and so does in $[0,{\varepsilon}]$, allows us to complete the proof.
Low regularity symbols and calculus
-----------------------------------
For the analysis of our strictly hyperbolic problem, it’s important to pass from $LZ_t-LL_x$ functions to more general symbols in variables $(t,x,\xi)$ which have this same regularity in $t$ and $x$.
We want to investigate properties of these symbols and of the associated operators. For reasons which will appear clear in the sequel, we will have to take advantage not of the classical paradifferential calculus, but of the calculus with parameters. Therefore, we will allow the symbols to depend also on a parameter ${\gamma}\geq1$.
We point out that in our calculus the time can be treated as an additional parameter, while $\xi$ represents, as usual, the dual variable to $x$.
Let us start with a definition (see also [@M-Z Def. B.4]).
\[d:symbols\] A *symbol of order $m+\delta\log$* (for $m,\delta\,\in\,{\mathbb{R}}$) is a function $a(t,x,\xi,{\gamma})$ which is locally bounded on $[0,T_0]\times{\mathbb{R}}^N\times{\mathbb{R}}^N\times[1,+\infty[\,$, of class ${\mathcal}{C}^\infty$ with respect to $\xi$ and which satisfy the following property: for all $\alpha\in{\mathbb{N}}^N$, there exists a constant $C_\alpha>0$ such that $$\label{est:def_symb}
\left|\d^\alpha_\xi a(t,x,\xi,{\gamma})\right|\,\leq\,C_\alpha\,\left({\gamma}+|\xi|\right)^{m-|\alpha|}\,
\log^\delta\left(1+{\gamma}+|\xi|\right)$$ for all $(t,x,\xi,{\gamma})$.
So, take a symbol $a(t,x,\xi,{\gamma})$ of order $m\geq0$, which is log-Zygmund in $t$ and log-Lipschitz in $x$, uniformly with respect to the other variables. Now we smooth $a$ out with respect to time, as done in . Next lemma provides us some estimates on classical symbols associated to $a_{\varepsilon}$ and its time derivatives.
\[l:lzll-symb\] The classical symbols associated to $a_{\varepsilon}$ and its time derivatives satisfy: $$\begin{aligned}
\left|\d^\alpha_\xi\sigma_{a_{\varepsilon}}\right| & \leq & C_\alpha\left(\gamma+|\xi|\right)^{m-|\alpha|} \\
\left|\d^\beta_x\d^\alpha_\xi\sigma_{a_{\varepsilon}}\right| & \leq & C_{\beta,\alpha}\left(\gamma+|\xi|\right)^{m-|\alpha|+|\beta|-1}\,
\log\left(1+\gamma+|\xi|\right) \\
\left|\d^\alpha_\xi\sigma_{\d_ta_{\varepsilon}}\right| & \leq & C_\alpha\left(\gamma+|\xi|\right)^{m-|\alpha|}\,
\log^2\!\!\left(1+\gamma+\frac{1}{{\varepsilon}}\right) \\
\left|\d^\beta_x\d^\alpha_\xi\sigma_{\d_ta_{\varepsilon}}\right| & \leq & C_{\beta,\alpha}\,\,\left(\gamma+|\xi|\right)^{m-|\alpha|+|\beta|-1}\,
\log\left(1+\gamma+|\xi|\right)\,\frac{1}{{\varepsilon}} \\
\left|\d^\alpha_\xi\sigma_{\d^2_ta_{\varepsilon}}\right| & \leq & C_\alpha\left(\gamma+|\xi|\right)^{m-|\alpha|}\,
\log\!\!\left(1+\gamma+\frac{1}{{\varepsilon}}\right)\,\frac{1}{{\varepsilon}} \\
\left|\d^\beta_x\d^\alpha_\xi\sigma_{\d^2_ta_{\varepsilon}}\right| & \leq & C_{\beta,\alpha}\,\left(\gamma+|\xi|\right)^{m-|\alpha|+|\beta|-1}\,
\log\left(1+\gamma+|\xi|\right)\,\frac{1}{{\varepsilon}^2}\,\,,\end{aligned}$$ where all the constants which occur here don’t depend on the parameter ${\gamma}$.
The first inequality is a quite easy computation.
For the second one, we have to observe that $$\int\d_iG(x-y,\xi)dx\,=\,\int\d_iG(z,\xi)dz\,=\,\int{\mathcal}{F}^{-1}_\eta\left(\eta_i\,\psi(\eta,\xi)\right)dz\,=\,
\left(\eta_i\,\psi(\eta,\xi)\right)_{|\eta=0}\,=\,0\,.$$ So, we have $$\d_i\sigma_{a_{\varepsilon}}\,=\,\int\d_iG(y,\xi)\left(a_{\varepsilon}(t,x-y,\xi,{\gamma})-a_{\varepsilon}(t,x,\xi,{\gamma})\right)dy\,,$$ and from this, remembering lemma \[l:G\], we get the final control.
The third estimate immediately follows from the hypothesis on $a$ and from .
Moreover, in the case of space derivatives, we can take advantage once again of the fact that $\d_iG$ has null integral: $$\begin{aligned}
\d_i\sigma_{\d_ta_{\varepsilon}} & = & \int\d_iG(x-y,\xi)\,\d_ta_{\varepsilon}(t,y,\xi,{\gamma})\,dy \\
& = & \int_{{\mathbb{R}}_s}\frac{1}{{\varepsilon}^2}\,\,\rho'\left(\frac{t-s}{{\varepsilon}}\right)
\left(\int_{{\mathbb{R}}^N_y}\d_iG(y,\xi)\left(a(s,x-y,\xi,{\gamma})-a(s,x,\xi,{\gamma})\right)dy\right)ds\,.\end{aligned}$$ Hence, the estimate follows from the log-Lipschitz continuity hypothesis and from inequality .
About the $\d^2_ta_{\varepsilon}$ term, the first estimate comes from , while for the second one we argue as before: $$\begin{aligned}
\d_i\sigma_{\d^2_ta_{\varepsilon}} & = & \int\d_iG(x-y,\xi)\,\d^2_ta_{\varepsilon}(t,y,\xi,{\gamma})\,dy \\
& = & \int_{{\mathbb{R}}^N_y}\d_iG(x-y,\xi)\,\frac{1}{{\varepsilon}^3}
\left(\int_{{\mathbb{R}}_s}\rho''\left(\frac{t-s}{{\varepsilon}}\right)\left(a(s,y,\xi,{\gamma})-a(s,x,\xi,{\gamma})\right)ds\right)dy \\
& = & \frac{1}{{\varepsilon}^3}\int_{{\mathbb{R}}_s}\rho''\left(\frac{t-s}{{\varepsilon}}\right)
\left(\int_{{\mathbb{R}}^N_y}\d_iG(y,\xi)\left(a(s,x-y,\xi,{\gamma})-a(s,x,\xi,{\gamma})\right)dy\right)ds\,,\end{aligned}$$ and the thesis follows again from log-Lipschitz continuity and from .
\[r:symb\_calc\] Note that first and second inequalities are satisfied also by the symbol $a$ (not smoothed in time).
Now let us quote some basic facts on symbolic calculus, which follow from previous lemma. Before doing this, we recall a definition (see also [@M-Z Def. B.8]).
\[d:op\_order\] We say that an operator $P$ is of order $\,m+\delta\log\,$ if, for every $(s,\alpha)\in{\mathbb{R}}^2$ and every ${\gamma}\geq1$, $P$ maps $H^{s+\alpha\log}_{\gamma}$ into $H^{(s-m)+(\alpha-\delta)\log}_{\gamma}$ continuously.
\[p:symb-calc\]
- Let $a$ be a symbol of order $m$ which is $L^\infty$ in the $x$ variable. Then $T_a$ is of order $m$, i.e. it maps $H^{s+\alpha\log}_{\gamma}$ into $H^{s-m+\alpha\log}_{\gamma}$.
- Let us take two symbols $a$, $b$ of order $m$ and $m'$ respectively. Suppose that $a$, $b$ are $LL$ in the $x$ variable. The composition of the associated operators can be approximated by the symbol associated to the product of these symbols, up to a remainder term: $$T_{a}\,\circ\,T_b\,=\,T_{ab}\,+\,R\,.$$ $R$ has order $m+m'-1-\log$: it maps $H^{s+\alpha\log}_{\gamma}$ into $H^{s-m-m'+1+(\alpha+1)\log}_{\gamma}$.
- Let $a$ be a symbol of order $m$ which is $LL$ in the $x$ variable. The adjoint (over $L^2$) operator of $T_a$ is, up to a remainder operator, $T_{{\overline}{a}}$. The remainder operator is of order $m-1+\log$, and it maps $H^{s+\alpha\log}_{\gamma}$ into $H^{s-m+1+(\alpha+1)\log}_{\gamma}$.
Let us end this subsection stating a basic positivity estimate. In this situation, paradifferential calculus with parameters comes into play.
\[p:pos\] Let $a(t,x,\xi,{\gamma})$ be a symbol of order $m$, which is $LL$ in the $x$ variable and such that $${{\rm Re}\,}\,a(t,x,\xi,{\gamma})\,\geq\,\lambda_0\,\left(\gamma+|\xi|\right)^m\,.$$
Then, there exists a constant $\lambda_1$, depending only on $|a|_{LL}$ and on $\lambda_0$ (so not on ${\gamma}$), such that, for $\gamma$ large enough, one has $${{\rm Re}\,}\!\left(T_au,u\right)_{L^2({\mathbb{R}}^N_x)}\,\geq\,\lambda_1\,\|u\|^2_{H^{m/2}_\gamma}\,.$$
The result is an immediate consequence of Theorem B.18 of [@M-Z], together with Proposition \[p:symb-calc\] about the remainder for composition and adjoint operators.
\[r:pos\] Let us note the following fact, which comes again from Theorem B.18 of [@M-Z]. If the positive symbol $a$ has low regularity in time and we smooth it out by convolution with respect to this variable, we obtain a family $\left(a_{\varepsilon}\right)_{\varepsilon}$ of positive symbols, with same constant $\lambda_0$. Now, all the paradifferential operators associated to these symbols will be positive operators, uniformly in ${\varepsilon}$: i.e. the constant $\lambda_1$ of the previous inequality can be chosen independently of ${\varepsilon}$.
Let us observe that the previous proposition generalizes Corollary 3.12 of [@C-M] (stated for the paraproduct by a positive $LL$ function) to the more general case of a paradifferential operator related to a strictly positive symbol of order $m$.
Finally, thanks to Theorem B.17 of [@M-Z] about the remainder operator for the adjoint, we have the following corollary, which turns out to be fundamental in our energy estimates.
\[c:pos\] Let $a$ be a positive symbol of order $1$ and suppose that $a$ is $LL$ in the $x$ variable.
Then there exists $\gamma\geq1$, depending only on the symbol $a$, such that $$\left\|T_au\right\|_{L^2}\,\sim\,\left\|u\right\|_{H^1_{\gamma}}$$ for all $u\in H^1_{\gamma}({\mathbb{R}}^N)$.
Proof of the energy estimate for $L$
====================================
Finally, we are able to tackle the proof of Theorem \[t:en-est\]. We argue in a standard way: we prove energy estimates, for some suitable energy associated to a solution of equation .
The key idea to the proof is to split the total energy into localized components $e_\nu$, each one of them associated to the dyadic block $\Delta_\nu u$, and then to put all these pieces together (see also [@C-L] and [@C-DS]). Let us see the proof into details.
Approximate and total energy
----------------------------
Let us first regularize coefficients $a_{ij}$ in the time variable by convolution, as done in , and let us define the $0$-th order symbol $$\alpha_{\varepsilon}(t,x,\xi)\,:=\,\left(\gamma^2+|\xi|^2\right)^{-1/2}\,
\left(\gamma^2\,+\,\sum_{i,j}a_{ij, {\varepsilon}}(t,x)\,\xi_i\,\xi_j\right)^{1/2}\,.$$ We take ${\varepsilon}\,=\,2^{-\nu}$ (see also [@C-L] and [@C-DS]), and (for notation convenience) we will miss out the ${\varepsilon}$.
Before going on, let us fix a real number $\gamma\geq1$, which will depend only on $\lambda_0$ and on the $\sup_{i,j}\left|a_{ij}\right|_{LL_x}$, such that (see Corollary \[c:pos\]) $$\label{est:param}
\left\|T_{\alpha^{-1/2}}\,w\right\|_{L^2}\,\geq\,\frac{\lambda_0}{2}\,\left\|w\right\|_{L^2}\qquad\mbox{ and }\qquad
\left\|T_{\alpha^{1/2}({\gamma}^2+|\xi|^2)^{1/2}}\,w\right\|_{L^2}\,\geq\,
\frac{\lambda_0}{2}\,\left\|w\right\|_{H^1}$$ for all $w\in H^{\infty}$. Let us remark that the choice of $\gamma$ is equivalent to the choice of the parameter $\mu$ in (\[pd\_eq:pp\_symb\]) (see Remark \[r:p-prod\]): hence, from now on we will consider paraproducts starting from this ${\mu}$, according to definition , even if we will omit it in the notations.
Consider in a function $u\in{\mathcal}{C}^2([0,T_0];H^{\infty})$. We want to get energy estimates for $u$. We rewrite the equation using paraproduct operators by the coefficients $a_{ij}$: $$\d^2_tu =\sum_{i,j}\d_i\left(a_{ij}\,\d_ju\right)\,+\, Lu\\
= \sum_{i,j}\d_i\left(T_{a_{ij}} \d_ju\right) \,+\, {\widetilde}Lu,$$ where ${\widetilde}Lu= Lu + \sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)$. Let us apply the operator $\Delta_\nu$: we get $$\label{eq:loc}
\d^2_tu_\nu\,=\sum_{i,j}\d_i\left(T_{a_{ij}}\,\d_ju_\nu\right)\,+\,
\sum_{i,j}\d_i\left(\left[\Delta_\nu,T_{a_{ij}}\right]\,\d_ju\right)+\,({\widetilde}Lu)_\nu,$$ where $u_\nu=\Delta_\nu u$, $({\widetilde}Lu)_\nu=\Delta_\nu ({\widetilde}Lu)$ and $\left[\Delta_\nu,T_{a_{ij}}\right]$ is the commutator between $\Delta_\nu$ and the paramultiplication by $a_{ij}$.
Now, we set $$\begin{aligned}
v_\nu(t,x) & := & T_{\alpha^{-1/2}}\,\d_tu_\nu\,-\,T_{\d_t(\alpha^{-1/2})}\,u_\nu \\
w_\nu(t,x) & := & T_{\alpha^{1/2}(\gamma^2+|\xi|^2)^{1/2}}\,u_\nu \\
z_\nu(t,x) & := & u_\nu\end{aligned}$$ and we define the approximate energy associated to the $\nu$-th component of $u$ (as done in [@C-DS]): $$\label{eq:appr_en}
e_{\nu}(t)\,:=\,\left\|v_\nu(t)\right\|^2_{L^2}\,+\,\left\|w_\nu(t)\right\|^2_{L^2}\,+\,\left\|z_\nu(t)\right\|^2_{L^2}\,.$$
\[r:w\_nu\] Note that $\left\|w_\nu(t)\right\|^2_{L^2}\,\sim\,\left\|\nabla u_\nu\right\|^2_{L^2}\,\sim\,
2^{2\nu}\,\|u_\nu\|^2_{L^2}$, thanks to hypothesis and the choice of the frequency $\mu$ from which the paraproduct starts, recall also .
Now, we fix a $\theta\in\,]0,1[\,$, as required in the hypothesis, and we take a $\beta>0$ to be chosen later; we can define the total energy associated to the solution $u$ to be the quantity $$\label{eq:tot_E}
E(t)\,:=\,\sum_{\nu\geq0}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t)\,.$$
It’s not difficult to prove (see also inequality below) that there exist constants $C_\theta$ and $C'_\theta$, depending only on the fixed $\theta$, for which one has: $$\begin{aligned}
\left(E(0)\right)^{1/2} & \leq & C_\theta\left(\|\d_tu(0)\|_{H^{-\theta}}\,+\,\|u(0)\|_{H^{-\theta+1}}\right) \label{est:E(0)} \\
\left(E(t)\right)^{1/2} & \geq & C'_\theta\left(\|\d_tu(t)\|_{H^{-\theta-\beta^*t}}\,+\,
\|u(t)\|_{H^{-\theta+1-\beta^*t}}\right)\,, \label{est:E(t)}\end{aligned}$$ where we have set $\beta^*=\beta\left(\log2\right)^{-1}$.
Time derivative of the approximate energy
-----------------------------------------
Let’s find an estimate on the time derivative of the energy. We start analysing each term of .
### $z_\nu$ term
For the third term we have: $$\label{eq:d_t-z}
\frac{d}{dt}\|z_\nu(t)\|^2_{L^2}\,=\,2\,{{\rm Re}\,}\left(u_\nu\,,\,\d_tu_\nu\right)_{L^2}\,.$$ Now, we have to control the term $\d_tu_\nu$: using positivity of operator $T_{\alpha^{-1/2}}$, we get $$\label{est:d_t-u_nu}
\left\|\d_tu_\nu\right\|_{L^2}\;\leq\;C\,\left\|T_{\alpha^{-1/2}}\d_tu_\nu\right\|_{L^2}
\;\;\leq\;\;C\left(\|v_\nu\|_{L^2}\,+\,\left\|T_{\d_t\left(\alpha^{-1/2}\right)}u_\nu\right\|_{L^2}\right)
\;\leq\;C\left(e_{\nu}\right)^{1/2}\,.$$ So, we find the estimate: $$\label{est:d_t-z}
\frac{d}{dt}\|z_\nu(t)\|^2_{L^2}\,\leq\,C\,e_{\nu}(t)\,.$$
### $v_\nu$ term
Straightforward computations show that $$\d_tv_\nu(t,x)\,=\,T_{\alpha^{-1/2}}\d^2_tu_\nu\,-\,T_{\d^2_t(\alpha^{-1/2})}u_\nu\,.$$
Therefore, keeping in mind relation , we get: $$\begin{aligned}
\label{eq:d_t-v}
\frac{d}{dt}\|v_\nu(t)\|^2_{L^2} & = & 2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\left({\widetilde}Lu\right)_{\!\!\nu}\,\right)_{L^2}\,-\,2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\d^2_t(\alpha^{-1/2})}u_\nu\right)_{L^2} \\
& & \qquad\qquad+\,
2\sum_{i,j}{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\d_i\left(T_{a_{ij}}\,\d_ju_\nu\right)\right)_{L^2} \nonumber \\
& & \qquad\qquad\qquad\qquad+\,2\sum_{i,j}{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\d_i\left[\Delta_\nu,T_{a_{ij}}\right]\d_ju\right)_{L^2}\,. \nonumber\end{aligned}$$
Obviously, we have $$\label{est:Lu_nu}
\left|2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\left({\widetilde}Lu\right)_{\!\!\nu}\,\right)_{L^2}\right|\,\leq\,
C\,\left(e_{\nu}\right)^{1/2}\,\left\|\left({\widetilde}Lu\right)_{\!\!\nu}\,\right\|_{L^2}\,,$$ while from Lemma \[l:lzll-symb\] one immediately recovers $$\begin{aligned}
\left|2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\d^2_t(\alpha^{-1/2})}u_\nu\right)_{L^2}\right| & \leq & C\,
\|v_\nu\|_{L^2}\,\log\!\left(1+\gamma+\frac{1}{{\varepsilon}}\right)\frac{1}{{\varepsilon}}\,\|u_\nu\|_{L^2} \label{est:zeta} \\
&\leq & C\,(\nu+1)\,e_{\nu}\,, \nonumber\end{aligned}$$ where we have used the fact that $\varepsilon=2^{-\nu}$. The other two terms of will be treated later.
### $w_\nu$ term
We now differentiate $w_\nu$ with respect to the time variable: thanks to a broad use of symbolic calculus, we get the following sequence of equalities: $$\begin{aligned}
\frac{d}{dt}\left\|w_\nu\right\|^2_{L^2} & = & 2\,{{\rm Re}\,}\left(T_{\d_t(\alpha^{1/2})(\gamma^2+|\xi|^2)^{1/2}}u_\nu\,,\,w_\nu\right)_{L^2}
\,+\,2\,{{\rm Re}\,}\left(T_{\alpha^{1/2}(\gamma^2+|\xi|^2)^{1/2}}\d_tu_\nu\,,\,w_\nu\right)_{L^2} \label{eq:d_t-w} \\
& = & 2\,{{\rm Re}\,}\left(T_{\alpha(\gamma^2+|\xi|^2)^{1/2}}T_{-\d_t(\alpha^{-1/2})}u_\nu\,,\,w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(R_1u_\nu\,,\,w_\nu\right)_{L^2} \nonumber \\
& & +\,2\,{{\rm Re}\,}\left(T_{\alpha(\gamma^2+|\xi|^2)^{1/2}}T_{\alpha^{-1/2}}\d_tu_\nu\,,\,w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(R_2\d_tu_\nu\,,\,w_\nu\right)_{L^2} \nonumber \\
& = & 2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha(\gamma^2+|\xi|^2)^{1/2}}w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(v_\nu\,,\,R_3w_\nu\right)_{L^2} \nonumber\\
& & +\,2\,{{\rm Re}\,}\left(R_1u_\nu\,,\,w_\nu\right)_{L^2}\,+\,2\,{{\rm Re}\,}\left(R_2\d_tu_\nu\,,\,w_\nu\right)_{L^2} \nonumber \\
& = & 2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}T_{\alpha^{3/2}(\gamma^2+|\xi|^2)^{1/2}}w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(v_\nu\,,\,R_4w_\nu\right)_{L^2} \nonumber \\
& & +\,2\,{{\rm Re}\,}\left(v_\nu\,,\,R_3w_\nu\right)_{L^2}\,+\,2\,{{\rm Re}\,}\left(R_1u_\nu\,,\,w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(R_2\d_tu_\nu\,,\,w_\nu\right)_{L^2} \nonumber \\
& = & 2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}T_{\alpha^2(\gamma^2+|\xi|^2)}u_\nu\right)_{L^2} \nonumber \\
& & +\,2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}R_5u_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(v_\nu\,,\,R_4w_\nu\right)_{L^2} \nonumber \\
& & +\,2\,{{\rm Re}\,}\left(v_\nu\,,\,R_3w_\nu\right)_{L^2}\,+\,2\,{{\rm Re}\,}\left(R_1u_\nu\,,\,w_\nu\right)_{L^2}\,+\,
2\,{{\rm Re}\,}\left(R_2\d_tu_\nu\,,\,w_\nu\right)_{L^2}\,. \nonumber\end{aligned}$$
The important issue is that remainders can be controlled in terms of the approximate energy. As a matter of facts, taking advantage of Proposition \[p:symb-calc\] and Lemma \[l:lzll-symb\], we get the following estimates.
- $R_1$ has principal symbol equal to $\d_\xi\left(\alpha(\gamma^2+|\xi|^2)^{1/2}\right)\d_x\d_t(\alpha^{-1/2})$, so $$\begin{aligned}
\left|2\,{{\rm Re}\,}\left(R_1u_\nu\,,\,w_\nu\right)_{L^2}\right| & \leq
& C\,(\nu+1)\,e_{\nu}\,. \label{est:R_1}\end{aligned}$$
- The principal symbol of $R_2$ is instead $\d_\xi\left(\alpha(\gamma^2+|\xi|^2)^{1/2}\right)\d_x(\alpha^{-1/2})$, so, remembering also the control on $\|\d_tu_\nu\|_{L^2}$, we have: $$\label{est:R_2}
\left|2\,{{\rm Re}\,}\left(R_2\d_tu_\nu\,,\,w_\nu\right)_{L^2}\right|\,\leq\,C\,\nu\,\left(e_{\nu}\right)^{1/2}\,\|w_\nu\|_{L^2}\,\leq\,
C\,(\nu+1)\,e_{\nu}\,.$$
- Symbolic calculus tells us that the principal part of $R_3$ is given by $\d_\xi\d_x\left(\alpha(\gamma^2+|\xi|^2)^{1/2}\right)$, therefore $$\label{est:R_3}
\left|2\,{{\rm Re}\,}\left(v_\nu\,,\,R_3w_\nu\right)_{L^2}\right|\,\leq\,C\,\|v_\nu\|_{L^2}\,\nu\,\|w_\nu\|_{L^2}\,\leq\,
C\,(\nu+1)\,e_{\nu}\,.$$
- Now, $R_4$ has $\d_\xi\left(\alpha^{-1/2}\right)\d_x\left(\alpha^{3/2}(\gamma^2+|\xi|^2)^{1/2}\right)$ as principal symbol, so $$\label{est:R_4}
\left|2\,{{\rm Re}\,}\left(v_\nu\,,\,R_4w_\nu\right)_{L^2}\right|\,\leq\,C\,\|v_\nu\|_{L^2}\,\nu\,\|w_\nu\|_{L^2}\,\leq\,
C\,(\nu+1)\,e_{\nu}\,.$$
- $R_5$ is given, at the highest order, by the product of the symbols $\d_\xi\left(\alpha^{3/2}(\gamma^2+|\xi|^2)^{1/2}\right)$ and $\d_x\left(\alpha^{1/2}(\gamma^2+|\xi|^2)^{1/2}\right)$, and then we get $$\label{est:R_5}
\left|2\,{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}R_5u_\nu\right)_{L^2}\right|\,\leq\,
C\,\|v_\nu\|_{L^2}\,2^\nu\,\nu\,\|u_\nu\|_{L^2}\,\leq\,
C\,(\nu+1)\,e_{\nu}\,.$$
### Principal part of the operator $L$
Now, thanks to previous computations, it’s natural to pair up the second term of with the first one of the last equality of . As $\alpha$ is a symbol of order $0$, we have $$\left|2\,{{\rm Re}\,}\!\left(v_\nu\,,\,T_{\alpha^{-1/2}}\sum_{i,j}\d_i\left(T_{a_{ij}}\d_ju_\nu\right)\right)_{\!\!\!L^2}+
2\,{{\rm Re}\,}\!\left(v_\nu\,,\,T_{\alpha^{-1/2}}T_{\alpha^2(\gamma^2+|\xi|^2)}u_\nu\right)_{L^2}\right|\,\leq\,
C\,\|v_\nu\|_{L^2}\,\|\zeta_\nu\|_{L^2}\,,$$ where we have set $$\label{eq:zeta_nu}
\zeta_\nu\,:=\,T_{\alpha^2(\gamma^2+|\xi|^2)}u_\nu\,+\,\sum_{i,j}\d_i\left(T_{a_{ij}}\,\d_ju_\nu\right)\,=\,
\sum_{ij}\,T_{a_{ij, {\varepsilon}}\xi_i\xi_j+\gamma^2}u_\nu\,+\,\d_i\left(T_{a_{ij}}\,\d_ju_\nu\right)\,.$$ We remark that $$\d_i\left(T_{a_{ij}}\,\d_ju_\nu\right) = T_{\d_ia_{ij}}\d_ju_\nu\,-\,T_{a_{ij}\xi_i\xi_j}u_\nu,$$ where, with a little abuse of notation, we have written $\d_ia_{ij}$ meaning that we are taking the derivative of the classical symbol associated to $a_{ij}$.
First of all, using also spectral localization properties, we have $$\begin{aligned}
\left\|T_{\d_ia_{ij}}\d_ju_\nu\right\|_{L^2} & \leq &
\|S_{\mu}\,\d_ia_{ij}\|_{L^\infty}\,\|S_{\mu+2}\d_ju_\nu\|_{L^2}\,+\,
\sum_{k\geq{\mu+3}} \left\|\nabla S_{k-3}a_{ij}\right\|_{L^\infty}\,
\left\|\Delta_k\nabla u_\nu\right\|_{L^2} \label{est:T_1} \\
& \leq & C\left(\mu+1\right)\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|\nabla u_\nu\|_{L^2} \nonumber \\
& & \qquad\qquad\qquad\quad+\,
\sum_{k\geq\mu+3\,,\,k\sim\nu}\left(k+1\right)\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|\nabla\Delta_k u_\nu\|_{L^2} \nonumber \\
& \leq & C_\mu\,(\nu+1)\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\left(e_{\nu}\right)^{1/2}\,, \nonumber\end{aligned}$$ where $\mu$ is the parameter fixed in and we have used also .
Next, we have to control the term $$T_{a_{ij,{\varepsilon}}\xi_i\xi_j+\gamma^2}u_\nu\,-\,T_{a_{ij}\xi_i\xi_j}u_\nu\,=\,T_{\left(a_{ij,{\varepsilon}}-a_{ij}\right)\xi_i\xi_j}u_\nu\,+\,
T_{\gamma^2}u_\nu\,.$$ It’s easy to see that $$\left\|T_{\left(a_{ij,{\varepsilon}}-a_{ij}\right)\xi_i\xi_j}u_\nu\right\|_{L^2}\,\leq\,
C\,{\varepsilon}\,\log\left(1+\frac{1}{{\varepsilon}}\right)2^\nu\,\|\nabla u_\nu\|_{L^2}\,,$$ and so, keeping in mind that ${\varepsilon}=2^{-\nu}$, $$\label{est:delta-T}
\left\|T_{\left(a_{ij,{\varepsilon}}-a_{ij}\right)\xi_i\xi_j+\gamma^2}u_\nu\right\|_{L^2}\,\leq\,
C_{\gamma}\,(\nu+1)\left(e_{\nu}\right)^{1/2}\,.$$
Therefore, from and we finally get $$\label{est:sec-ord}
\left|2\,{{\rm Re}\,}\left(v_\nu,T_{\alpha^{-1/2}}\sum_{i,j}\d_i\left(T_{a_{ij}}\d_ju_\nu\right)\right)_{L^2}+
2\,{{\rm Re}\,}\left(v_\nu,T_{\alpha^{-1/2}}T_{\alpha^2({\gamma}^2+|\xi|^2)}u_\nu\right)_{L^2}\right|\,\leq\,C\,(\nu+1)\,e_{\nu}\,,$$ where the constant $C$ depends on the log-Lipschitz norm (with respect to space) of the coefficients $a_{ij}$ and on the fixed parameters $\mu$ and ${\gamma}$.
To sum up, from inequalities , , and and from estimates of remainder terms -, we can conclude that $$\begin{aligned}
\frac{d}{dt}e_{\nu}(t) & \leq & C_1\,(\nu+1)\,e_{\nu}(t)\,+\,
C_2\,\left(e_{\nu}(t)\right)^{1/2}\,\left\|\left({\widetilde}Lu\right)_{\!\!\nu}(t)\right\|_{L^2} \label{est:d_t-e} \\
& & \qquad\qquad\qquad\qquad\qquad
+\left|2\sum_{i,j}{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\d_i\left[\Delta_\nu,T_{a_{ij}}\right]\d_ju\right)_{L^2}\right|\,. \nonumber\end{aligned}$$
Commutator term {#ss:comm}
---------------
We want to estimate the quantity $$\left|\sum_{i,j}{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\d_i\left[\Delta_\nu,T_{a_{ij}}\right]\d_ju\right)_{L^2}\right|\,.$$
We start remarking that $$[\Delta_\nu, T_{a_{ij}}]w= [\Delta_\nu, S_\mu a_{ij}]S_{\mu+2}w+\sum_{k=\mu+3}^{+\infty}[\Delta_\nu, S_{k-3}a_{ij}]\,\Delta_k w,$$ where $\mu$ is fixed, as usual (see Remark \[r:p-prod\]). In fact $\Delta_\nu$ and $\Delta_k$ commute, so that $$\Delta_\nu(S_\mu{a_{ij}}S_{\mu+2}w)-
S_\mu{a_{ij}}(S_{\mu+2}\Delta_\nu w)= \Delta_\nu(S_\mu{a_{ij}}S_{\mu+2}w)-
S_\mu{a_{ij}}\Delta_\nu (S_{\mu+2}w),$$ and similarly $$\Delta_\nu(S_{k-3}{a_{ij}}\Delta_kw)-
S_{k-3}{a_{ij}}\Delta_k(\Delta_\nu w)= \Delta_\nu(S_{k-3}{a_{ij}}\Delta_kw)-
S_{k-3}{a_{ij}}\Delta_\nu (\Delta_kw).$$ Consequently, taking into account also that $S_{k+2}$ and $\Delta_k$ commute with $\partial_j$, we have $$\partial_i\left([\Delta_\nu, T_{a_{ij}}]\,\partial_ju\right)\,=\,\partial_i\left(
[\Delta_\nu, S_\mu a_{ij}]\,\partial_j (S_{\mu+2}u)\right)\,+\,
\partial_i\left(\sum_{k=\mu+3}^{+\infty}[\Delta_\nu, S_{k-3}{a_{ij}}]\, \partial_j (\Delta_ku)\right).$$
Let’s consider first the term $$\partial_i\left(
[\Delta_\nu, S_\mu a_{ij}]\,\partial_j (S_{\mu+2}u)\right).$$ Looking at the support of the Fourier transform of $[\Delta_\nu, S_\mu a_{ij}]\,\partial_j (S_{\mu+2}u)$, we have that it is contained in $\{|\xi|\leq 2^{\mu+4}\}$ and moreover $[\Delta_\nu, S_\mu a_{ij}]\,\partial_j (S_{\mu+2}u)$ is identically $0$ if $\nu\geq \mu+5$. Then, from Bernstein’s inequalities and [@Coi-Mey Th. 35] we have that $$\left\|\partial_i\left([\Delta_\nu, S_\mu a_{ij}]\,\partial_j (S_{\mu+2}u)\right)\right\|_{L^2}\,\leq\,
C_\mu\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|S_{\mu+2}u\|_{L^2}\,,$$ hence, putting all these facts together, we have $$\begin{aligned}
& & \left|\sum_{\nu=0}^{+\infty}\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,\sum_{ij}
2\,{{\rm Re}\,}\!\biggl(v_\nu\,,\,T_{\alpha^{-1/2}}\,\partial_i\left(
[\Delta_\nu, S_\mu a_{ij}]\partial_j (S_{\mu+2}u)\,\right)\biggr)_{L^2}\right| \label{est:comm0} \\
&& \qquad\qquad\qquad \leq\, C_\mu\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,
\sum_{\nu=0}^{\mu+4}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\|v_\nu\|_{L^2}\left(\sum_{h=0}^{\mu+2}\|u_h\|_{L^2}\right) \nonumber\\
&& \qquad\qquad\qquad\leq\, C_\mu\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\, e^{\beta(\mu +5)T}\,2^{(\mu+4)\theta} \nonumber \\
&& \qquad\qquad\qquad\qquad
\times\left(\sum_{\nu=0}^{\mu+4}\,e^{-\beta(\nu+1)t}\,2^{-\nu\theta}\|v_\nu\|_{L^2}\right)
\left(\sum_{h=0}^{\mu+4}\,e^{-\beta(h+1)t}\,2^{-h\theta}\|u_h\|_{L^2}\right) \nonumber \\
&& \qquad\qquad\qquad \leq\, C_\mu\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,e^{\beta(\mu +5)T}\,2^{(\mu+4)\theta}
\sum_{\nu=0}^{\mu+4} e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}e_\nu(t)\,. \nonumber\end{aligned}$$
Next, let’s consider $$\partial_i\left(\sum_{k=\mu+3}^{+\infty}[\Delta_\nu, S_{k-3}{a_{ij}}]\,\partial_j(\Delta_k u)\right).$$ Looking at the support of the Fourier transform, it is possible to see that $$[\Delta_\nu, S_{k-3}a_{ij}]\,\partial_j(\Delta_k u)$$ is identically 0 if $|k-\nu|\geq 3$. Consequently the sum over $k$ is reduced to at most 5 terms: $\partial_i([\Delta_\nu, S_{\nu-5}a_{ij}]\,\partial_j(\Delta_{\nu-2} u))+
\dots +\partial_i([\Delta_\nu, S_{\nu-1}a_{ij}]\,\partial_j(\Delta_{\nu+2} u))$, each of them having the support of the Fourier transform contained in $\{|\xi|\leq 2^{\nu+1}\}$. Let’s consider one of these terms, e.g. $\partial_i([\Delta_\nu, S_{\nu-3}a_{ij}]\,\partial_j(\Delta_{\nu} u))$, the computation for the other ones being similar. We have, from Bernstein’s inequalities, $$\left\|\partial_i\left([\Delta_\nu, S_{\nu-3}a_{ij}]\,\partial_j(\Delta_{\nu} u)\right)\right\|_{L^2}\,
\leq\, C\,2^\nu\, \|[\Delta_\nu, S_{\nu-3}a_{ij}]\,\partial_j(\Delta_{\nu} u)\|_{L^2}\,.$$ On the other hand, using [@Coi-Mey Th. 35] again, we have: $$\|[\Delta_\nu, S_{\nu-3}a_{ij}]\partial_j(\Delta_{\nu} u)\|_{L^2}\,\leq\, C\,
\|\nabla S_{\nu-3}a_{ij}\|_{L^\infty}\,\|\Delta_\nu u\|_{L^2}\,,$$ where $C$ does not depend on $\nu$. Consequently, using also (\[est:S\_ka\]), we deduce $$\left\|\partial_i\left([\Delta_\nu, S_{\nu-3}a_{ij}]\,\partial_j(\Delta_{\nu} u)\right)\right\|_{L^2}
\,\leq\, C\, 2^\nu\, (\nu+1)\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|\Delta_{\nu} u\|_{L^2}\,.$$ From this last inequality and similar ones for the other terms, we infer $$\left|\sum_{i,j}{{\rm Re}\,}\left(v_\nu\,,\,T_{\alpha^{-1/2}}\partial_i
\left(\sum_{k=\mu+3}^{+\infty}[\Delta_\nu, S_{k-3}{a_{ij}}]\,\partial_j(\Delta_k u)\right)\right)_{L^2}\right|
\,\leq\, C\, \left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\, (\nu+1) \,e_{\nu}(t)$$ and then $$\begin{aligned}
& & \left|\sum_{\nu=0}^{+\infty}\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,\sum_{ij}
2\,{{\rm Re}\,}\!\left(v_\nu\,,\,T_{\alpha^{-1/2}}\partial_i\left(\sum_{k=\mu+3}^{+\infty}[\Delta_\nu, S_{k-3}{a_{ij}}]\,
\partial_j(\Delta_k u)\right)\right)_{L^2}\right| \label{est:comm1} \\
& & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\leq\,
C \,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\sum_{\nu=0}^{+\infty}\,(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t)\,. \nonumber\end{aligned}$$ Collecting the informations from and , we finally obtain $$\begin{aligned}
& & \left|\sum_{\nu=0}^{+\infty}\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,\sum_{ij}
2\,{{\rm Re}\,}\!\left(v_\nu\,,\,T_{\alpha^{-1/2}}\d_i\left[\Delta_\nu,T_{a_{ij}}\right]\d_ju\right)_{L^2}\right| \label{est:comm} \\
& & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\leq\,
C_3\,\sum_{\nu=0}^{+\infty}\,(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t)\,, \nonumber\end{aligned}$$ where $C_3$ depends on $\mu$, $\sup_{i,j}\|a_{ij}\|_{LL_x}$, on $\theta$ and on the product $\beta\,T$.
Final estimate {#ss:f-est}
--------------
From and we get $$\begin{aligned}
\frac{d}{dt}E(t) & \leq & \left(C_1\,+\,C_3\,-\,2\beta\right)
\sum_{\nu=0}^{+\infty}\,(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t) \\
& & \qquad\qquad+\,C_2\,\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,
\left\|\left({\widetilde}Lu(t)\right)_{\!\!\nu}\right\|_{L^2} \\
& \leq & \left(C_1\,+\,C_3\,-\,2\beta\right)
\sum_{\nu=0}^{+\infty}\,(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t) \\
& & +\,C_2\,\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu} \right\|_{L^2} \\
& & \qquad+\,C_2\,\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,
\left\|( Lu(t))_\nu \right\|_{L^2}\,.\end{aligned}$$ Now, applying Hölder inequality for series implies $$\begin{aligned}
&&\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu} \right\|_{L^2} \\
&& \qquad\qquad\qquad\leq\,
\left(\sum_{\nu=0}^{+\infty}(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t)\right)^{1/2} \\
&& \qquad\qquad\qquad\qquad\times \left(\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,(\nu+1)^{-1}
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu}\right\|^2_{L^2}\right)^{1/2}\,,\end{aligned}$$ and, by definition, one has $$\begin{aligned}
&&\hspace{-1cm}
\left(\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,(\nu+1)^{-1}
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu}\right\|^2_{L^2}\right)^{1/2} \\
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
=\,\left\|\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right\|_{H^{-\theta-\beta^*t-{\frac{1}{2}}\log}}\,.\end{aligned}$$ From [@C-M Prop. 3.4] we have that $$\left\| \sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right\|_{H^{-s-{\frac{1}{2}}\log}}\,\leq\, C\,
\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|u\|_{H^{1-s+{\frac{1}{2}}\log}}\,, \label{a-T_a_est}$$ with $C$ uniformly bounded for $s$ in a compact set of $]0,1[$. Consequently, $$\begin{aligned}
&&\left(\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,(\nu+1)^{-1}
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu}\right\|^2_{L^2}\right)^{1/2} \\
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\leq\, C\,\left(\sup_{i,j}\|a_{ij}\|_{LL_x}\right)\,\|u\|_{H^{1-\theta-\beta^*t+{\frac{1}{2}}\log}} \\
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\leq\,C\left(\sum_{\nu=0}^{+\infty}(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,
e_{\nu}(t)\right)^{1/2}\,,\end{aligned}$$ and finally $$\begin{aligned}
&& \sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,
\left\|\left(\sum_{i,j}\d_i\left((a_{ij}-T_{a_{ij}})\d_ju\right)\right)_{\!\!\!\!\nu} \right\|_{L^2} \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\leq\, C_4\, \sum_{\nu=0}^{+\infty}(\nu+1)e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t)\,,\end{aligned}$$ with $C_4$ uniformly bounded for $\beta^* t +\theta$ in a compact set of $]0,1[\,$. So, if we take $\beta>0$ (recall that $\beta^*=\beta(\log2)^{-1}$) and $T\in\,]0,T_0]$ such that $$\label{eq:T}
\beta^*\,T\;=\;\delta\;<\;1-\theta\,,$$ we have $0<\theta\leq\theta+\beta^*t\leq\theta+\delta<1$. Therefore we obtain $$\begin{aligned}
\frac{d}{dt}E(t) & \leq & \left(C_1\,+C_4\,C_2\,+\,C_3-\,2\beta\right)
\sum_{\nu=0}^{+\infty}\,(\nu+1)\,e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\,e_{\nu}(t) \\
& & \qquad\qquad\qquad\qquad
+\,C_2\,\sum_{\nu=0}^{+\infty}e^{-2\beta(\nu+1)t}\,2^{-2\nu\theta}\left(e_{\nu}(t)\right)^{1/2}\,\left\|( Lu(t))_\nu\right\|_{L^2}\,.\end{aligned}$$ Now we fix $\beta$ large enough, such that $C_1\,+\,C_4\,C_2\,+\,C_3-\,2\beta\,\leq\,0$: this corresponds to take $T>0$ small enough in . Therefore we finally arrive to the estimate $$\frac{d}{dt}E(t)\,\leq\,C_2\,\left(E(t)\right)^{1/2}\,\left\| Lu(t)\right\|_{H^{-\theta-\beta^*t}}\,;$$ applying Gronwall’s Lemma and keeping in mind and give us estimate .
\[r:T\] Let us point out that relation gives us a condition on the lifespan $T$ of a solution to the Cauchy problem for . It depends on $\theta\in\,]0,1[$ and on $\beta^*>0$, hence on constants $C_1\ldots C_4$. Going after the guideline of the proof, one can see that, in the end, the time $T$ depends only on the index $\theta$, on the parameter $\mu$ defined by conditions , on constants $\lambda_0$ and $\Lambda_0$ defined by and on the quantities $\sup_{i,j}\left|a_{ij}\right|_{LZ_t}$ and $\sup_{i,j}\left|a_{ij}\right|_{LL_x}$.
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[^1]: Throughout we agree that $f(D)$ stands for the pseudo-differential operator $u\mapsto{\mathcal}{F}^{-1}(f\,{\mathcal}{F}u)$.
|
---
abstract: 'Here we present the analysis of 3D spectroscopic data of three Blue Compact Galaxies (Mrk 324, Mrk 370, and III Zw 102). Each of the more than 22500 spectra obtained for each galaxy has been fitted by a single gaussian from which we have inferred the velocity dispersion ($\sigma$), the peak intensity (I$_{peak}$), and the central wavelength ($\lambda_c$). The analysis shows that the $\sigma$ [*vs*]{} I$_{peak}$ diagrams look remarkably similar to those obtained for giant extragalactic H [II]{} regions. They all present a supersonic narrow horizontal band that extends across all the range of intensities and that result from the massive nuclear star-forming regions of every galaxy. The $\sigma$ [*vs*]{} I$_{peak}$ diagrams present also several inclined bands of lower intensity and an even larger $\sigma$, arising from the large galactic volumes that surround the main central emitting knots. Here we also show that the $\sigma$ [*vs*]{} $\lambda_c$ and $\lambda_c$ [*vs*]{} I$_{peak}$ diagrams, are powerful tools able to unveil the presence of high and low mass stellar clusters, and thus allow for the possibility of inferring the star formation activity of distant galaxies, even if these are not spatially resolved.'
author:
- 'Ismael Martínez-Delgado , Guillermo Tenorio-Tagle, Casiana Muñoz-Tuñón, Alexei V. Moiseev'
- 'Luz M. Cairós'
title: '3D Spectroscopy of Blue Compact Galaxies. Diagnostic Diagrams'
---
Introduction
============
High resolution panoramic spectroscopy with good spatial and spectral resolution is known to be a powerful tool for studying the kinematics of ionized nebulae as it leads to a simultaneous mapping, at seeing limited resolution, of a particular emission line over the whole nebula. This however can easily lead to several tens of thousands of spectra, making it difficult to issue a detailed and/or a global interpretation of the data. An analysis procedure that has proven to be simple and powerful results from fitting a single gaussian to each of the resultant emission line profiles [see @Cas94; @Cas95]. A single gaussian regardless of the actual degree of asymmetry or splitting in the line profiles. The fit is to conserve the flux of the line profile, and thus lower intensity but broader lines would result from the most asymmetric or largely splitted line profiles. From the resultant fits one can derive the velocity dispersion ($\sigma$), the peak intensity (I$_{peak}$) of the fitted lines as well as their central wavelength ($\lambda_c$). Such a method when applied to giant extragalactic H [II]{} region (GHIIR) data, leads to two distinct regions in the $\sigma$ [*vs*]{} I$_{peak}$ diagram [see @Cas96]: A supersonic ($\sigma$ $>$ c$_{\mathrm H\, \mbox{\tiny II}}$), relatively narrow horizontal band with all possible peak intensity values, and a second region populated by lower intensity points presenting even larger supersonic $\sigma$ values, that crowd along multiple inclined bands. For the case of GHIIRs it has been shown [see @Cas94; @Cas96; @Nancy95; @Oriol00; @Telles01 the last reference for the case of H [II]{} Galaxies data] that the horizontal band is conformed by data with truly gaussian profiles arising mainly from the brightest portions of the nebula, from the small regions that enclose groups or clusters of stars. The inclined lower intensity and highly supersonic bands on the other hand, emanate from multiple photoionized shells that surround the star-forming centers and are likely to result from the stellar mechanical energy impact into the ISM, and thus from single gaussian fits to highly asymmetric or strongly splitted lines. The analysis of the $\sigma$ [*vs*]{} I$_{peak}$ diagram for GHIIRs in nearby galaxies (closer than 1 Mpc) was then proposed as an excellent tool to determine their degree of evolution on kinematic bases [@Cas96]. Here we extend the analysis to Blue Compact Galaxies (BCGs) at even larger distances and show that they experience a similar dynamical evolution (ie. their $\sigma$ [*vs*]{} I$_{peak}$ diagrams look very much like those of GHIIRs). We further extend our analysis by looking also at the $\sigma$ [*vs*]{} $\lambda_c$ and $\lambda_c$ [*vs*]{} I$_{peak}$ diagrams, and show them as ideal tools to distinctly unveil the presence of high and low mass stellar clusters.
Here section 2 provides a short description of the observations. Section 3 contains a detailed analysis of the data by means of the diagnostic diagrams here proposed and section 4 gives a summary and a discussion of our findings.
The observations
================
We use the 6m Big Telescope Alta-azimuthal (BTA) of the Special Astrophysical Observatory (SAO) in Nizhnij Arkhyz (Russia), equipped with the multi-mode focal reducer SCORPIO in the scanning Interferometer Fabry-Perot (IFP) observational mode [see @Afanasiev05], for the observations of three BCGs: Mrk 370, III Zw 102, and Mrk 324. SCORPIO is provided with a EEV 42-40 CCD array of 2048 $\times$ 2048 pixels with an instrumental scale of 0.18 $''$ pixel$^{-1}$. A set of interference filters centered at the H$\alpha$ line, covering an interval of systemic velocities that range from $-$200 to 10000 km s$^{-1}$, is used at the IFP observational mode. The IFP offers two configuration depending on the interference order, in our case at the H$\alpha$ line, the IFP260 (at the interference order 235) and the IFP500 (at the interference order 501). Our observation runs were performed using the IFP500 mode, that offers a better spectral resolution in the Ha emission line.
The output from the IFP results on a 3D data cube with [*x*]{} and [*y*]{} being the spatial plane and [*z*]{} the wavelength sampling or ethalon step, that in the case of IFP500 mode, it translates into 32-40 channels. The original data cube dimensions were (522,522,36) although for our analysis we reduced the size, matching the objects size, to (150,150,36). The Free Spectral Range (FSR) was 13 Å, enough to sample the H$\alpha$ line, that was scanned with a spectral sampling of 0.7 Å $\approx$ 32 km s$^{-1}$. A binning of 4$\times$4 pixels was applied in the spatial directions in order to reduce the time exposure and improve the signal to noise of the final data cube, resulting in a final spatial scale of 0.7 $''$ pixel$^{-1}$. The instrumental width taken from a fitting of the calibration lamp data cube was $\sigma_{inst}=20\pm3$ km s$^{-1}$. Table \[logs\] lists a log of the observations together with some data about the galaxies: Column 1 gives the name of the object; columns 2 and 3 show the coordinates (from NED); columns 4 shows the linear scale for each galaxy; column 5 lists the total time exposure of the observations; column 6 gives the seeing conditions; column 7 shows the systemic velocity of each galaxy and column 8 gives the distance in Mpc. Each of our SCORPIO trimmed data cubes contains a total of 22500 spectra from an area of $\sim$3 arcmin$^2$. The bulk of the data thus requires of automatic techniques of analysis to extract the full power of 3D spectroscopy. For each of the three galaxies, their H${\alpha}$ emission lines in every pixel were fitted by single gaussians, to then obtain their I$_{peak}$, $\sigma$, and $\lambda$c, as described in [@Cas95; @Cas96]. The velocity dispersion is given in km s$^{-1}$ and is corrected for the instrumental and thermal broadening ($\sigma^2=\sigma^2_{obs}-\sigma^2_{inst}-\sigma^2_{th}$). We then used [*MoisSoft*]{}, the software package designed for the manipulation of SCORPIO-IFP spectral line data cubes. For a detailed explanation about the data reduction procedure, see [@Alexei02].
Three Dimensional Spectroscopy of BCDs
======================================
Mrk 324
-------
Mrk 324 is a galaxy classified as a Blue Compact Dwarf [BCD, see @Thuan81] and is included as a Nucleated BCD in the morphological classification of Cairos et al. (2001), as it is basically powered by a luminous central knot. The star formation history of the galaxy fits well with an instantaneous star formation law with a Salpeter IMF ($\alpha=2.35$) and a range of masses between 1$\leq{M/M_{\odot}}\leq$100, in which a central knot of 4.4 Myr and 1.4$\times10^5$ M$_\odot$ in stars dominates 95% of the emission flux of the galaxy [see @Ismael06].
Figure \[Mrk324diag\] (upper row of panels) shows the results from the single gaussian fit to the full H$\alpha$ SCORPIO data set in the $\sigma$ [*vs*]{} I$_{peak}$, $\sigma$ [*vs*]{} $\lambda_c$, and $\lambda_c$ [*vs*]{} I$_{peak}$ diagrams for Mrk 324. The last frame, upper row, shows the H$\alpha$ image of the nucleated galaxy [see @Cai01b] obtained at the NOT 2.52m telescope of the Roque de Los Muchachos Observatory (ORM) in La Palma (Spain). The resemblance of the $\sigma$ [*vs*]{} I$_{peak}$ diagram with those of GHIIRs is remarkable. It presents a horizontal band that limits the values of $\sigma$ to supersonic values ($\sigma$ $\geq$ 20 km s$^{-1}$) and a bunch of low intensity points at even higher $\sigma$ values, that delineate a triangular structure above the horizontal band. If one selects from the $\sigma$ [*vs*]{} I$_{peak}$ diagram the highest intensity data points, those belonging to the horizontal band (see second row of panels in Figure \[Mrk324diag\]), these in the $\sigma$ [*vs*]{} $\lambda_c$ diagram appear clumped at a given $\lambda_c$ ($\sim$ 6598.2 A) what implies that they all arise from a very similar location in the galaxy, that they all belong in principle to a single entity. Note however, that this is not exactly coincident with the rest wavelength of the galaxy (marked in the $\lambda_c$ plots with a red line, see Figure \[Mrk324diag\]). The selected points in the $\lambda_c$ [*vs*]{} I$_{peak}$ diagram defined also a pointed structure towards high intensity values. These points in the emission-line peak intensity map (last frame) display their location in the galaxy, which indeed coincides with the major burst of star formation in Mrk 324. A major burst with an estimated mass in stars $\sim 4 \times 10^5$ M$_\odot$ and an age of about 4.4 Myr [see @Ismael05; @Ismael06] that causes an H [II]{} region with a radius of $\sim$ 250 pc. On the other hand, the data belonging to the inclined bands with a higher velocity dispersion and a lower intensity in the $\sigma$ [*vs*]{} I$_{peak}$ diagram (see third row of panels in Figure \[Mrk324diag\]) appear well scattered in $\lambda_c$ and arise from a large-scale (600 pc in thickness) expanding rim that surrounds the central emitting knot, as shown in the last panel of the third row. The inserts in the last column of Figure \[Mrk324diag\] display the H$\alpha$ line and its single gaussian fit at various locations in the galaxy. As in GHIIRs, the data corresponding to the horizontal band in the $\sigma$ [*vs*]{} I$_{peak}$ diagram leads to the highest intensity true gaussian profiles, while the profiles fitted to lower intensity and larger $\sigma$ data points, arising from the outer expanding rim, are fits to asymmetric and/or double peak line profiles. Figure \[Mrk324diag\] shows the quality of typical line profiles from each of the regions, displaying their intensity, central wavelength, and $\sigma$ values.
Mrk 370
-------
This BCD galaxy belongs to the Chained morphological type [see @Cai01b], in which the star formation takes place along a privileged direction. It follows the same star formation law as Mrk 324, with 25 young and massive exciting clusters hosted over an underlying stellar component with ages larger than 5 Gyr, ([see @Cai02]). The clusters have an average age of 6.6 Myr with a dispersion of 0.9 Myr, suggesting a coeval and global star formation event. For the ones located in the inner nuclear region, the stellar mass content reaches up to several 10$^5$ M$_\odot$, [see @Ismael06].
Figure \[Mrk370diag\] displays similar diagnostic diagrams for Mrk 370. The last frame in the upper row of panels shows the H$\alpha$ image of the chained galaxy [see @Cai01b] obtained at the CAHA 2.2m telescope of the Centro Astronómico Hispano Alemán (CAHA), from Calar Alto Observatory (Spain). The ionized gas in Mrk 370 (see the H$\alpha$ image in Figure \[Mrk370diag\]) presents much more structure than Mrk 324. However, its $\sigma$ [*vs*]{} I$_{peak}$ diagram looks almost identical to that shown in Figure \[Mrk324diag\] for Mrk 324. The only exception is perhaps the low intensity data points with values of $\sigma$ below the supersonic lower limit imposed by the horizontal band ($\leq $ 21 km s$^{-1}$, see first panel in Figure \[Mrk370diag\]). The structure of the photoionized gas in Mrk 370 begins to be disentangled when one selects the data from the horizontal band in the $\sigma$ [*vs*]{} I$_{peak}$ diagram (see Figure \[Mrk370diag\], second row of panels). Such a data set appears clumped at three different $\lambda_c$ values in the $\sigma$ vs $\lambda_c$ diagram. The brightest one sits at the rest wavelength of the galaxy (6580.1Å). There is a second one at $\sim$6579.4Å and a third one between these two at $\sim$6579.8Å. Two of them are much more apparent in the $\lambda_c$ [*vs*]{} I$_{peak}$ diagram where they clearly end up as pointed structures with different intensity (factors of 2) and different central wavelengths. The emission-line peak intensity map tracing the selected points (last frame in the second row) shows two major bursts of stellar formation and two adjacent much less intense, associated to the massive central stellar clusters with masses that range between (8$-$50)$\times 10^4$ M$_\odot$ [see @Ismael05; @Ismael06], also resolved by [@Cai02] using the INTEGRAL spectrograph with a single gaussian fit to the emission lines. All of these, although near the center of the galaxy, are kinematically resolved as well detached structures. This multiplicity of structures would have been missed if one had only use the $\sigma$ [*vs*]{} I$_{peak}$ diagram as a diagnostic for this galaxy (see first panel in Figure \[Mrk370diag\]), as the brightest points present almost the same supersonic gaussian profiles. These massive regions also match the area of the galaxy that shows the highest ionization values (see the H$\alpha$ emission-line and the \[O[III]{}\]/H$\beta$ maps in [@Cai02]).
The third row of panels in Figure \[Mrk370diag\] displays the data from the inclined bands with low intensity and large $\sigma$ values in the $\sigma$ [*vs*]{} I$_{peak}$ diagram. This data set, as the corresponding one in Mrk 324, presents a large range of values of $\lambda_c$, around the systemic velocity of the galaxy. The spread of the data is also noticeable in the $\lambda_c$ [*vs*]{} I$_{peak}$ plane. The bulk of the data arises from a large galactic volume shown in the emission-line peak intensity map as an expanding rim of gas around the main centers of star formation all over the galaxy. This is most intense around the nuclear starburst zone and although less intense, it is still noticeable around smaller bursts of stellar formation some 2 kpc away from the nuclear region.
The final row of panels show the low intensity and low velocity dispersion data from the $\sigma$ [*vs*]{} I$_{peak}$ diagram (see Figure \[Mrk370diag\]), which appears as vertical well detached low $\sigma$ structures in the $\sigma$ [*vs*]{} $\lambda_c$ plane. These structures, given their low values of $\sigma$, detach well from the bulk of the data, most of which presents supersonic values, and due to their largely different $\lambda_c$ values they also appear as well detached entities along the $\lambda$ scale. Note however that given their low intensity values, they are much less apparent in the $\lambda_c$ [*vs*]{} I$_{peak}$ diagram. These data points correspond in the emission-line peak intensity map (last panel in the 4$^{th}$ row of Figure \[Mrk370diag\]) to multiple knots of emitting gas, or H [II]{} regions excited by low mass stellar groups [with masses $\le$ 10$^4$ M$_\odot$, see @Ismael05; @Ismael06] at large distances from the centre of the galaxy. The line profiles arising from these regions are also gaussian profiles although with a very low intensity when compared to those arising from the central starburst regions (see inserts in the last frame of Figure \[Mrk370diag\]).
III Zw 102
----------
This galaxy is believed to be the result of an interaction event [see @Vor59; @Vor77] and has been considered by some authors to belongs to the polar-ring galaxy type [@Whitmore90]. It is morphologycally considered as an Extended Blue Compact Galaxy [@Cai01b], in which 65 star-forming regions were identified spreaded over the whole main body of the galaxy, including the arms. The age of the knots and their low age dispersion (6.1 $\pm$ 0.6 Myr) suggest a coeval starburst event [see @Ismael06].
Figure \[IIIZw102diag\] displays the results for III Zw 102. The last frame in the upper row of panels, shows the H$\alpha$ image of the extended galaxy [see @Cai01b] obtained at the NOT 2.52m telescope of the Roque de Los Muchachos Observatory (ORM) in La Palma (Spain).
The data for III Zw 102 confirms the power of all of the diagnostic diagrams here used. The galaxy has a plethora of small H [II]{} regions spread along two arms as well as several massive centers of stellar formation in the densest nuclear region with masses between $\sim$ $10^5-10^6$ M$_\odot$, [see @Ismael05; @Ismael06]. All of these, as well as the large galactic volumes undergoing a supersonic expansion leave their signature in the diagnostic diagrams. In this way, the horizontal band with supersonic $\sigma$ values at all possible intensities noticeable in the $\sigma$ [*vs*]{} I$_{peak}$ diagram (see Figure 3, 2$^{nd}$ row of panels) is spread over a large range of $\lambda_c$ values in the $\sigma$ [*vs*]{} $\lambda_c$ diagram and into at least four pointed structures (at $\lambda_c $ = 6596.7Å, 6597.25Å, 6598Å, and 6600Å) of different intensities in the $\lambda_c$ [*vs*]{} I$_{peak}$ diagram. All of these in the emission-line peak intesity map are well resolved as giant structures within the galactic nuclear region. On the other hand, the larger $\sigma$ points at all peak intensities in the $\sigma$ [*vs*]{} I$_{peak}$ diagram arise from gas expanding around the nuclear zone (see third row of panels in Figure 3). Among this data set is the inclined structure between I$_{peak}$=1100-1400 which is produced by gas between the four major centers of stellar formation in the nuclear region. The broad line structure appears at the center of the galaxy with a $\lambda_c$ very similar to the systemic velocity of III Zw 102.
Finally, the lowest $\sigma$ and lowest intensity data points in the $\sigma$ [*vs*]{} I$_{peak}$ diagram (last row of panels in Figure 3) appear, as in the case of Mrk370, as a set of well detached vertical stalactites in the $\sigma$ [*vs*]{} $\lambda_c$ diagram, hanging from the bulk of the data at very distinct $\lambda_c$ values. The latter as shown in the last panel are produced by the numerous small H [II]{} regions located along the galactic arms. As shown by the inserts in the last column of panels, the quality of the gaussian fits is totally different in the three regions. The nuclear zone shows supersonic perfect gaussians while the surrounding gas presents much broader and asymmetric lines. Single gaussians with small $\sigma$ values fit well the data from small centers of star formation.
Summary
=======
Tridimensional spectroscopy with good spatial and spectral resolution, sampling a particular emission line over the whole area covered by an emitting nebula, is known to be the most suitable observational technique for analyzing the global kinematics of GHIIRs. In particular if a single gaussian fit is carried out over the emission lines in every pixel and from those, one infers the velocity dispersion, the central wavelength, and the peak intensity of the lines. The resultant $\sigma$ [*vs*]{} I$_{peak}$ diagram [@Cas96] has been shown to be an excellent diagnostic diagram to separate the main broadening mechanisms affecting the emission lines, e.g., those that lead to shells and loops generated by the violent action from massive stars (see Dyson 1979, Roy et al. 1986) which finally lead to cloud dispersal, and the one(s) affecting the regions of massive star formation (either gravity; see Terlevich & Melnick 1981, Tenorio-Tagle et al. 1993, or turbulence; see Chu & Kennicutt 1994). The sequences shown by the various GHIIRs so far analyzed, led the authors also to propose the $\sigma$-I$_{peak}$ plot as a tool able to trace the evolutionary status of a GHIIR, from its formation to the total dispersal of the ionized gas.
Here we have confirmed that the data from the inclined supersonic bands in the $\sigma$ [*vs*]{} I$_{peak}$ diagrams are caused by the mechanical energy deposited by massive stellar sources. These, depending on the age of the stellar clusters and on the density of the surrounding gas may affect small or large galactic volumes. However in all cases, the interaction leads to very asymmetric line profiles, evident in the $\sigma$ [*vs*]{} I$_{peak}$ diagram for their highly supersonic $\sigma$ values.
Here we have extended the analysis to Blue Compact Galaxies. We have selected sources with a single nuclear starburst (as Mrk 324) as well as galaxies with multiple stellar bursts both within their nuclei as well as at large galactocentric distances. In all cases the $\sigma$ [*vs*]{} I$_{peak}$ diagram resembles those from GHIIRs in which the data define a supersonic $\sigma$ (larger than $c_{\mathrm H\, \mbox{\tiny II}}\approx$ 10 km s$^{-1}$) horizontal band with all possible intensity values. Note that the value of the limiting $\sigma$ is different in every case. For III Zw 102 the horizontal band peaks at $\sim33$ km s$^{-1}$, while for Mrk 324 and Mrk 370 it occurs at 24 and 25 km s$^{-1}$, respectively.
The $\sigma$ horizontal band, for the case of multiple massive nuclear bursts of stellar formation, has been shown to split into different bands in the $\lambda_c$ [*vs*]{} I$_{peak}$ diagram. In this new diagram, the largest nuclear regions of massive star formation become evident due to the good spectral resolution. And thus even with a poor spatial resolution, or when dealing with further away sources, the sampling of the velocity field, with our 3D spectrographs, would reveal the exciting sources. Similarly, the $\sigma$ [*vs*]{} $\lambda_c$ new diagram picks up the small bursts of stellar formation by tracing their lower intensity and slower expansion of their immediate surroundings.
The stalactites detected in the $\sigma$ [*vs*]{} $\lambda_c$ diagram in the case of III Zw 102 and Mrk 370 correspond to the ionized knots located in the outskirts of the galaxies and their spread in $\sigma$ result from the H [II]{} regions expansion into a low density medium [see @Franco90].
On the whole, the new diagnostic diagrams here presented provide the possibility of inferring from 3D spectroscopy, the magnitude of the star formation activity of distant galaxies, even if these are not spatially resolved.
IMD acknowledges the FPI grant (FP-2001-2506) of the Spanish Government through the collaboration of the project AYA2004-08260-C03-01 (ESTALLIDOS,\
http://www.iac.es/project/GEFE/estallidos). AVM acknowledges the the Russian Foundation for Basic Research (project 05-02-16454). This work is partly based on observations carried out at the 6m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences, operated under the financial support of the Science Department of Russia (registration number 01-43), and it has been partly supported by the ESTALLIDOS Project (AYA2004-08260-C03-01) and the grant AYA 2004-02703 from the Spanish Ministerio de Educación y Ciencia.
Afanasiev, V. L., & Moiseev, A. V. 2005, Astronomy Letters, 31, 194 Cairós, L. M., Caon, N., Vílchez, J. M., Gónzalez-Pérez, J. N., & Muñoz-Tuñón, C. 2001, , 136, 2 Cair[ó]{}s, L. M., Caon, N., Garc[í]{}a-Lorenzo, B., V[í]{}lchez, J. M., & Mu[ñ]{}oz-Tu[ñ]{}[ó]{}n, C. 2002, , 577, 164 Chu, Y.-H., & Kennicutt, R. C. 1994, , 425, 720 Dyson, J. E. 1979, , 73, 132 Franco, J., Tenorio-Tagle, G., & Bodenheimer, P. 1990, , 349, 126
Fuentes-Masip, O., Mu[ñ]{}oz-Tu[ñ]{}[ó]{}n, C., Casta[ñ]{}eda, H. O., & Tenorio-Tagle, G. 2000, , 120, 752 Martínez-Delgado, I., 2005, DEA: Catalogues and Parameters of Starburst Knots in a Sample of Blue Compact Dwarf Galaxies. [*Spain: University of La Laguna/Instituto de Astrofí[i]{}ca de Canarias*]{}. Martínez-Delgado, I., Muñoz-Tuñón, C.,, Cairós, L. M., Tenorio-Tagle, G. 2006, [*submitted to*]{} . Moiseev, A. V. 2002, Bull. Special Astrophys. Obs., 54, 74 Muñoz-Tuñón, C., 1994, Violent Star Formation: From 30 Doradus to QSOs. [*In: Tenorio-Tagle G. (eds.). Cambridge University Press, p.25*]{}. Muñoz-Tuñón, C., Gavryusev, V., & Casta[ñ]{}eda, H. O. 1995, , 110, 1630 Muñoz-Tuñón, C., Tenorio-Tagle, G., Casta[ñ]{}eda, H. O., & Terlevich, R. 1996, , 112, 1636 Roy, J.-R., Arsenault, R., & Joncas, G. 1986, , 300, 624 Sabalisck, N. S. P., Tenorio-Tagle, G., Casta[ñ]{}eda, H. O., & Mu[ñ]{}oz-Tu[ñ]{}[ó]{}n, C. 1995, , 444, 200 Telles, E., Mu[ñ]{}oz-Tu[ñ]{}[ó]{}n, C., & Tenorio-Tagle, G. 2001, , 548, 671 Tenorio-Tagle, G., Mu[ñ]{}oz-Tu[ñ]{}[ó]{}n, C., & Cox, D. P. 1993, , 418, 767 Terlevich, R., & Melnick, J. 1981, , 195, 839 Thuan, T. X., & Martin, G. E. 1981, , 247, 823 Vorontsov-Velyaminov, B. A. 1959, VV, 0 Vorontsov-Vel’Yaminov, B. A. 1977, , 28, 1 Whitmore, B. C., Lucas, R. A., McElroy, D. B., Steiman-Cameron, T. Y., Sackett, P. D., & Olling, R. P. 1990, , 100, 1489
![The figure shows in the first row of panels the velocity dispersion $\sigma$ [*vs*]{} peak intensity (the $\sigma$ was corrected for the instrumental and thermal broadening), for all the gaussian fits to the individual spectra of Mrk 324. The second and third panels display the corresponding $\sigma$ [*vs*]{} central lambda ($\lambda_c$, indicating also the velocity range in km s$^{-1}$) and $\lambda_c$ [*vs*]{} peak intensity, respectively. The solid line in these two panels indicates the rest velocity of the galaxy. The last panel shows the H$\alpha$ image of the galaxy and indicates its scale in kpc. The second and following rows of panels highlight in colour the various sets of data points selected in the $\sigma$ [*vs*]{} peak intensity diagram and their location in the $\sigma$ [*vs*]{} $\lambda_c$ and $\lambda_c$ [*vs*]{} peak intensity planes. The last panels in every row, with a peak intensity scale, display the locations in the galaxy that produce the selected data points. Inserts in these panels correspond to typical gaussian fits (solid red lines) to the data arising from different regions (axis correspond to intensity, in counts, vs the wavelength in Å). The fitted $\sigma$ values are indicated in every frame. \[Mrk324diag\]](f1.eps){width="22cm"}
![Mrk 370. The same as Figure \[Mrk324diag\] for Mrk 370. \[Mrk370diag\]](f2.eps){width="19cm"}
![III Zw 102. The same as Figure \[Mrk324diag\] for III Zw 102.\[IIIZw102diag\]](f3.eps){width="20cm"}
[lccccccc]{}
Mrk370 & 0240290 & +191750& 52 & 4680 & 1.7 & 790$^\dag$ & 10.85\
IIIZw102 & 2320301 & +171332& 110 & 6480 & 1.8 & 1626$^\dag$ & 22.71\
Mrk324 & 2326328 & +181559& 108 & 4320 & 1.6 & 1600$^\dag$ & 22.43\
|
---
abstract: |
Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules.
Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.
address:
- 'Goethe-Universität Frankfurt, FB Informatik und Mathematik, 60054 Frankfurt am Main, Germany'
- 'Universitat Jaume I, Campus de Riu Sec, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, 12071 Castellón de la Plana, Spain'
- 'Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany'
author:
- Lukas Katthän
- 'Julio José Moyano-Fernández'
- Jan Uliczka
bibliography:
- 'Girlanden\_v18.bib'
title: 'Which series are Hilbert series of graded modules over polynomial rings?'
---
[^1]
Introduction
============
Let ${\mathbb{K}}$ be a field. We consider the polynomial ring $R={\mathbb{K}}[X_1, \ldots , X_m]$ in $m$ indeterminates equipped with a ${\mathbb{Z}}^n$-grading, such that the degree of each variable is one of the unit vectors ${\mathbf{e}}_i$ of ${\mathbb{Z}}^n$. This setup includes the standard ${\mathbb{N}}$-grading, as well as the *fine grading*, where $m=n$ and $\deg{X_i}={\mathbf{e}}_i$ for $i \in [n]$.
Let $M=\bigoplus_{{\mathbf{a}}\in {\mathbb{Z}}^n} M_{{\mathbf{a}}}$ be a finitely generated ${\mathbb{Z}}^n$-graded $R$-module. The of $M$ is the formal Laurent series $$H_M :=\sum_{{\mathbf{a}}\in {\mathbb{Z}}^n} (\dim_{{\mathbb{K}}} M_{{\mathbf{a}}}) \cdot {\bm{t}}^{{\mathbf{a}}} \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!),$$ where we set ${\bm{t}}^{{\mathbf{a}}} := t_1^{a_1}\dotsm t_n^{a_n}$ for ${\mathbf{a}}= (a_1, \dotsc, a_n)$. This is well-defined, because the graded components $M_{a}$ of $M$ are finite-dimensional ${\mathbb{K}}$-vector spaces, and, since $R$ is positively graded, there exists a ${\mathbf{b}}\in {\mathbb{Z}}^n$ such that $M_{{\mathbf{a}}}=0$ if ${\mathbf{a}}\ngeq {\mathbf{b}}$ (componentwise). The Hilbert series is known to carry important information about $M$, for example its dimension or its multiplicity. In the present work, we consider the following question:
\[Q\] Which formal Laurent series arise as Hilbert series of $R$-modules (in a certain class)?
An obvious property of Hilbert series is that all their coefficients are nonnegative. If we allow non–finitely generated modules then this is already all that can be said: Any nonnegative series $H=\sum_{{\mathbf{a}}} c_{\mathbf{a}}{\bm{t}}^{{\mathbf{a}}}$ is the Hilbert series of the $R$–module $\bigoplus_{\mathbf{a}}\left(R/{\mathfrak{m}}\right)(-{\mathbf{a}})^{c_{\mathbf{a}}}$, for ${\mathfrak{m}}=(X_1, \ldots , X_m)$.
So we restrict our attention to finitely generated modules. This condition yields a second necessary condition for a series $H$ to be a Hilbert series: It has to be a rational function with denominator $\prod_i (1-{\bm{t}}^{\deg X_i})$. In the standard-${\mathbb{N}}$-graded situation, these two conditions already characterize Hilbert series, as it was shown by the third author in [@U Corollary 2.3]. However, they are not sufficient in the multigraded situation, cf. .
One of the main results of the present paper is a complete answer to Question \[Q\] in the ${\mathbb{Z}}^n$-graded situation, see . This generalization is nontrivial, as there are new phenomena in this setting. While this result is somewhat technical, we obtain very satisfying specializations in the fine-graded and the bigraded situations, respectively; see and . Also, we clarify the relation between Hilbert series and merely nonnegative series in .
It seems natural to further generalize these ideas to arbitrary multigradings. However, in this generality, arithmetical issues arise. For instance, there exists a formal Laurent series with integral coefficients which is not a Hilbert series, but after multiplication with $2$ it is, cf. ; the example already shows that one cannot hope for a characterization using linear inequalities in this setting. In the present paper, we do not further pursue this direction.
One of the difficulties of Question \[Q\] is that if $H$ is a Hilbert series of some module, then any nonnegative series which coincides with $H$ in all but finitely many coefficients is also a Hilbert series (to see this, if $H = H_M$ for some module $M$, then one might replace finitely many components of $M$ in the lowest degrees by copies of ${\mathbb{K}}$). Thus it seems natural to rule this out, i.e. to consider modules which do not contain a copy of the residue field as a submodule. Algebraically, this amounts to requiring that the depth of $M$ should be positive. Generalizing this idea leads to the notion of , introduced by the third author in [@U]. Recall that the Hilbert depth[^2] of a formal Laurent series $H$ is defined as $$\operatorname{Hdep}(H) := \sup {\left\{\, \operatorname{depth}(N) ~|~ N \text{ f.~g.~gr.~$R$--module with } H_N=H \,\right\}}.$$ We consider the Hilbert depth only for those formal Laurent series which actually arise as Hilbert series of some module. Again, in the standard-${\mathbb{N}}$-graded setting, Hilbert series of a given Hilbert depth have been classified in [@U]. Our next main result is a quite general class of linear inequalities which are satisfied by every Hilbert series with a given Hilbert depth. We formulate our result in terms of the projective dimension, but at least over the polynomial ring, this is equivalent to the depth via the Auslander-Buchsbaum formula.
[theorem]{}[TorUngl]{}\[thm:tor\] Let $R = \bigoplus_{\mathbf{a}}R_{\mathbf{a}}$ be a (commutative Noetherian) ${\mathbb{Z}}^n$-graded ${\mathbb{K}}$-algebra, such that $\dim_{\mathbb{K}}R_0 < \infty$. Let further $p \in {\mathbb{N}}$ and $M, N$ be finitely generated $R$-modules. If $\operatorname{pdim}M \leq p$ and $N$ is a $p$-th syzygy module, then the following inequality holds: $$\label{eq:allgungl}
\frac{H_M H_N}{H_R} \geq 0.$$
In general, a classification of Hilbert series with a given Hilbert depth seems to be very difficult. Therefore, we contempt ourselves with an important special case. Our third main result is a complete characterization of the Hilbert series with positive Hilbert depth in the bigraded (i.e. ${\mathbb{Z}}^2$-graded) setup, see . The condition that we obtain is similar to the general inequalities of , but quite different in nature to the condition of .
In the previous paper [@MU], the second and the third author characterized Hilbert series of positive Hilbert depth over a bivariate polynomial ring ${\mathbb{K}}[X,Y]$ endowed with a non-standard ${\mathbb{Z}}$-grading. In we show that this result can be restated in a way analogous to .
#### **Related work**
The classical result of Macaulay [@Mac] answers Question \[Q\] for *cyclic* modules in the standard-${\mathbb{N}}$-graded situation. This work has recently been extended by Boij and Smith in [@BS]. These authors study Hilbert series in the standard-${\mathbb{N}}$-graded setup, with the additional assumption that only modules generated in degree $0$ are considered. The main difference to the present work, however, is that in [@BS] *closure* of the set of Hilbert series is considered (with respect to a suitable topology on the space of formal Laurent series).
Notation and preliminaries
==========================
Let us fix some notation before continuing. We will use boldface letters ${\mathbf{a}}, {\mathbf{u}}, {\mathbf{g}}, \dotsc$ to denote elements of ${\mathbb{Z}}^n$ or ${\mathbb{N}}^n$. For such a vector ${\mathbf{a}}$, we write $a_i$ for its $i$-th component.
For $i \in [n]$ we denote by ${\mathbf{e}}_i$ the $i$-th unit vector. Moreover, for ${\mathbf{d}}= (d_1, \dotsc, d_n) \in {\mathbb{N}}^n$ and ${\bm{t}}= (t_1, \dotsc, t_n)$, we define $$\begin{aligned}
{\bm{t}}^{\mathbf{d}}&:= \prod_{i=1}^n t_i^{d_i} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{and}\\
(1-{\bm{t}})^{\mathbf{d}}&:= \prod_{i=1}^n (1-t_i)^{d_i}.
\end{aligned}$$
For a formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ we write ${\mathfrak{c}({H}, {{\mathbf{a}}})}$ for the coefficient of ${\bm{t}}^{\mathbf{a}}$ in $H$. We call $H$ if every coefficient of $H$ is nonnegative and we denote this by $H \geq 0$.
We consider the partial order on ${\mathbb{Z}}^n$ which is given by coordinatewise comparison. In other words, for ${\mathbf{u}}, {\mathbf{v}}\in {\mathbb{Z}}^n$ we write ${\mathbf{u}}\leq {\mathbf{v}}$ if and only if $u_i \leq v_i$ for all $i \in [n]$. Moreover, we denote the coordinate minimum resp. maximum of ${\mathbf{u}}$ and ${\mathbf{v}}$ by ${\mathbf{u}}\wedge{\mathbf{v}}$ resp. ${\mathbf{u}}\vee{\mathbf{v}}$.
Multivariate Hilbert Polynomials
--------------------------------
Let $H = \sum_i h_i t^i$ be a univariate formal Laurent series, such that $H\cdot (1-t)^d$ is a Laurent polynomial. Then it is well-known that there exist a polynomial $p \in {\mathbb{Q}}[Z]$ and $i_0 \in {\mathbb{Z}}$, such that $h_i = p(i)$ for all $i \geq i_0$. If $H$ is a Hilbert series, then $p$ is called the .
Here we give a multivariate generalization of this fact.
\[lem:HPoly\] Let $Q \in {\mathbb{Z}}[t_1^{\pm 1}, \dotsc, t_n^{\pm 1}]$ be a Laurent polynomial, ${\mathbf{d}}\in {\mathbb{N}}^n$ and $$H = \frac{Q}{(1-{\bm{t}})^{{\mathbf{d}}}}.$$ Then there exist a ${\mathbf{u}}\in {\mathbb{Z}}^n$ and a polynomial $p \in {\mathbb{Z}}[Z_1, \dotsc, Z_n]$ such that ${\mathfrak{c}({H}, {{\mathbf{a}}})} = p({\mathbf{a}})$ for ${\mathbf{a}}\geq {\mathbf{u}}$. Moreover, $p$ is uniquely determined (but ${\mathbf{u}}$ is not), and $\deg_{Z_i} p \leq d_i - 1$ for $1 \leq i \leq n$.
Write $Q = \sum_{{\mathbf{u}}\in \Omega} c_{\mathbf{u}}{\bm{t}}^{\mathbf{u}}$ for a suitable finite set $\Omega \subseteq {\mathbb{Z}}^n$. Recall the series expansion $$\frac{t^u}{(1-t)^d} = \sum_{a \geq u} \binom{a-u+d-1}{d-1} t^i.$$ From this, it is clear that $${\mathfrak{c}({H}, {{\mathbf{a}}})} = \sum_{{\mathbf{u}}\in \Omega} c_{\mathbf{u}}\binom{a_1 - u_1 + d_1 -1 }{d_1-1}\dotsm \binom{a_n - u_n + d_n -1 }{d_n-1}$$ for all ${\mathbf{a}}\geq \bigvee_{{\mathbf{u}}\in \Omega} {\mathbf{u}}$. Hence ${\mathfrak{c}({H}, {{\mathbf{a}}})}$ is given by a polynomial of the claimed degree.
The uniqueness follows from the fact that the Zariski closure of ${\mathbb{N}}^n$ inside $\mathbb{C}^n$ equals $\mathbb{C}^n$.
\[def:hpoly\] In the situation of , we call $p$ the of $H$ and denote it with ${\mathcal{P}(H)}$. In the degenerate case $n=0$, $H$ is an integer and we set ${\mathcal{P}(H)} = H$.
Which series are Hilbert series?
================================
Let ${\mathbb{K}}$ be a field. We consider the polynomial ring $R={\mathbb{K}}[X_1, \ldots , X_m]$ equipped with a ${\mathbb{Z}}^n$-grading, such that each variable is homogeneous. In this section, we deal with the question of which formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ arise as Hilbert series of finitely generated $R$-modules. There are two obvious necessary conditions:
- $H \geq 0$ coefficientwise, and
- $\prod_{i=1}^n (1-{\bm{t}}^{\deg X_i}) H$ is a Laurent polynomial.
It is a consequence of [@U Theorem 2.1] that in the case of the standard ${\mathbb{N}}$-grading (i.e. $n = 1$), these conditions are already sufficient. In general, this is not true, as we will see below.
In a previous work, the second and third author already obtained a general characterization of Hilbert series, cf. [@MU Corollary 2.2]:
\[thm:hdec\] A formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ is the Hilbert series of a finitely generated $R$-module if and only if it can be written in the form $$\label{eq:hdec}
H = \sum_{I \subseteq [m]} \frac{Q_I}{\prod_{i \in I}(1-{\bm{t}}^{\deg X_i})}$$ for Laurent polynomials $Q_I \in {\mathbb{Z}}[t_1^{\pm 1}, \dotsc, t_n^{\pm 1}]$ and $I \subseteq [m]$ with nonnegative coefficients.
A decomposition as in is called a of $H$. This result is stated and proven in [@MU] for ${\mathbb{Z}}$-gradings only, but the proof given there can be easily extended for multigradings.
\[ex:arith\] Let $R = {\mathbb{K}}[X_1, X_2, X_3]$ with the ${\mathbb{N}}$-grading given by $\deg X_1 = 2, \deg X_2=3$ and $\deg X_3 = 5$. Consider the series $$\begin{aligned}
H &= \frac{1}{2}\left(\frac{t}{1-t^2} + \frac{1}{1-t^3} + \frac{1+t}{1-t^5} + \frac{t^7}{(1-t^3)(1-t^5)}\right) \\
&= \frac{t+t^3}{1-t^6} + \frac{1}{(1-t^5)(1-t^6)}
\end{aligned}$$ One sees immediately that $2H$ has a Hilbert decomposition and therefore it is indeed a Hilbert series of a finitely generated $R$-module. In particular, $2H$ and thus $H$ satisfy the necessary conditions mentioned above. However, $H$ is not a Hilbert series, as it does not admit a Hilbert decomposition.
To see this, note that $H$ has a pole of order $2$ at $t = 1$. Considering the possible summands in , it follows that $Q_{{\left\{\,2,3,5\,\right\}}} = 0$ and that at least one of $Q_{{\left\{\,2,3\,\right\}}}, Q_{{\left\{\,2,5\,\right\}}}$ and $Q_{{\left\{\,3,5\,\right\}}}$ is non-zero. One can compute that the $i$-th coefficient of $H$ is of the order $\frac{i}{30} + O(1)$. On the other hand, the $i$-th coefficient of $1/(1-t^a)(1-t^b)$ is of the order $\frac{1}{ab} + O(1)$ for coprime $a,b \in {\mathbb{N}}$. As $2\cdot3,2\cdot5,3\cdot5 < 30$, the series $H$ does not have a Hilbert decomposition and thus it does not arise as a Hilbert series.
The standard $\mathbb{Z}^n$-grading
-----------------------------------
The criterion of is very useful for showing that a given Laurent series is a Hilbert series: One only needs to construct a Hilbert decomposition. However, it is rather difficult to use this criterion to show that a given series is *not* a Hilbert series. Moreover it does not provide a good insight into the structure of the set of Hilbert series. We would like to have a characterization of the Hilbert series in terms of inequalities. In view of the preceding example there is no hope for such a characterization in full generality.
So we now specialize our considerations to the case that the degree of every variable of $R$ is a unit vector. More precisely, we consider the case that $R = {\mathbb{K}}[X_{ij} {\ \vrule\ }1 \leq i \leq n, 1 \leq j \leq m_i]$, where $n \in {\mathbb{N}}$, ${\mathbf{m}}= (m_1, \dotsc, m_n) \in {\mathbb{N}}^n$ and $\deg X_{ij} = {\mathbf{e}}_i$. In this setting, we give a characterization of the Hilbert series of finitely generated modules over $R$ in terms of certain inequalities. Roughly speaking, this can be seen as an implicitization of the set of Hilbert series. Before we can state our result we need to introduce some notation.
Let $H = \sum_{{\mathbf{a}}\in {\mathbb{Z}}^n} h_{\mathbf{a}}{\bm{t}}^{\mathbf{a}}\in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ be a formal Laurent series. For $I \subseteq [n]$ and ${\mathbf{u}}\in {\mathbb{Z}}^n$ we define $$H|_{I,{\mathbf{u}}} := \sum_{{\mathbf{a}}\in {\mathbb{N}}^I} h_{{\mathbf{u}}+ {\mathbf{a}}} {\bm{t}}^{\mathbf{a}}\in {\mathbb{Z}}(\!(t_i | i \in I)\!),$$ where ${\mathbb{N}}^I := {\left\{\,\sum_{i \in I} c_i{\mathbf{e}}_i {\ \vrule\ }c_i \in {\mathbb{N}}\,\right\}} \subseteq {\mathbb{N}}^n$. We call $H|_{I,{\mathbf{u}}}$ the to ${\mathbf{u}}+ {\mathbb{N}}^I$.
Note that $H|_{\emptyset,{\mathbf{u}}} = {\mathcal{P}(H|_{\emptyset,{\mathbf{u}}})}= {\mathfrak{c}({H}, {{\mathbf{u}}})}$ for ${\mathbf{u}}\in {\mathbb{Z}}^n$. Also, note that the summands of ${\bm{t}}^{\mathbf{u}}H|_{I,{\mathbf{u}}}$ are summands of $H$. It is more convenient to consider $H|_{I,{\mathbf{u}}}$ instead of ${\bm{t}}^{\mathbf{u}}H|_{I,{\mathbf{u}}}$, because the former lives in a smaller ring of Laurent series.
Let $p \in {\mathbb{Q}}[Z_1, \dots, Z_n]$ be a polynomial. We call a monomial ${\bm{Z}}^{\mathbf{r}}$ appearing in $p$ if it does not divide any other monomial of $p$. Moreover, we say that $p$ has if the coefficient of every extremal monomial of $p$ is positive.
The following characterization of Hilbert series of $R$-modules is the main result of this section.
\[thm:hilberts\] The following statements are equivalent for a formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$:
1. There exists a finitely generated graded $R$-module $M$ whose Hilbert series equals $H$.
2. $H$ satisfies the following two conditions:
1. $H \cdot \prod_{i=1}^n (1-t_i)^{m_i}$ is a polynomial, and
2. for every ${\mathbf{u}}\in {\mathbb{Z}}^n$ and every $I \subseteq [n]$, the Hilbert polynomial of the restriction $H|_{I,{\mathbf{u}}}$ of $H$ has positive extremal coefficients.
The condition that $H \geq 0$ is implicit in the last condition of above, because ${\mathcal{P}(H|_{\emptyset, {\mathbf{u}}})}= {\mathfrak{c}({H}, {{\mathbf{u}}})}$ for ${\mathbf{u}}\in {\mathbb{Z}}^n$.
\[ex:nonh\] Let $n = 2$ and ${\mathbf{m}}= (3,3)$. Consider the series $$H := \sum_{i \geq 0}\sum_{j \geq 0} (i-j)^2 t_1^i t_2^j = \frac{t_1t_2^2 + t_1^2t_2 + t_1^2 - 6t_1t_2+t_2^2 + t_1+t_2}{(1-t_1)^3(1-t_2)^3}.$$ Clearly $H \cdot \prod_{i=1}^2 (1-t_i)^{m_i}$ is a polynomial, and it is also clear that $H \geq 0$. So $H$ satisfies the obvious necessary conditions for being a Hilbert series.
Moreover, ${\mathcal{P}(H)} = (i-j)^2 = i^2 -2ij+j^2$. Here, all three monomials are extremal, so in particular ${\mathcal{P}(H)}$ does not have positive extremal coefficients. Hence $H$ does not arise as Hilbert series of a finitely generated $R$-module.
Although this can be obtained from , one way to see this directly is as follows: Assume to the contrary that there $H = H_M$ for a finitely generated $R$-module $M$. We write $\deg_1$ and $\deg_2$ for the first and second component of the degree of an element $m \in M$, respectively. Let $g_1, \dotsc, g_r$ be a set of generators of $M$ and let $\deg g_k = (i_k,j_k)$. If $i_k < j_k$, then $(R g_k)_{(j_k,j_k)} = 0$ and hence $\deg_1 m < j_k$ for any $m \in R g_k$. Similarly, if $i_k > j_k$, then $\deg_2 m < i_k$ for any $m \in R g_k$. Hence, in both cases we have that $\min(\deg_1 m, \deg_2 m) \leq \max(i_k,j_k)$ for all $m \in R g_k$. As $M$ is generated by $g_1, \dotsc, g_r$, it follows that $\min(\deg_1 m, \deg_2 m)\leq \max(i_1,\dotsc,i_r, j_1,\dotsc,j_r)$ for all $m \in M$. This contradicts our assumption that $H_M = H$.
Our next example shows that it is not sufficient to consider only the Hilbert polynomial of $H$. Let $$H := \sum_{i \geq 0}\sum_{j \geq 0}\sum_{j \geq 0} \left( (i-j)^2 + ijk \right) t_1^i t_2^j t_3^k.$$ It holds that ${\mathcal{P}(H)} = (i-j)^2 + ijk$, so all extremal coefficients are nonnegative. On the other hand, for $I = {\left\{\,1,2\,\right\}}, {\mathbf{u}}= 0$ it holds that ${\mathcal{P}(H|_{I,{\mathbf{u}}})} = (i-j)^2$ and this polynomial does not have positive extremal coefficients. Thus $H$ is not a Hilbert series.
One common trait in the theory of Hilbert series is that many properties can be determined by examining only those exponents which are below the exponent ${\mathbf{g}}$ which is the join of the exponents of the numerator. So one might hope to sharpen by showing that one only needs to consider restrictions $H_{I, {\mathbf{u}}}$ for ${\mathbf{u}}\leq {\mathbf{g}}$. However, the next example shows that this does not hold.
For $\lambda \in {\mathbb{N}}$, consider the series $$H := \sum_{i \geq 0}\sum_{j \geq 0}\sum_{k \geq 0} \left( i^2 + j^2 + ij(k - \lambda)(k-\lambda-2) \right) t_1^i t_2^j t_3^k.$$ This series is nonnegative, because for $i,j,k \in {\mathbb{N}}$ it holds that $$i^2 + j^2 + ij(k - \lambda)(k-\lambda-2) \geq i^2 + j^2 + ij\cdot (-1) \geq 0.$$ The Hilbert polynomial clearly has nonnegative extremal coefficients. Moreover, for $I = {\left\{\,2,3\,\right\}}$ and any ${\mathbf{u}}\in {\mathbb{Z}}^n$, the Hilbert polynomial of the restriction is $${\mathcal{P}(H|_{I,{\mathbf{u}}})} = j^2 + jk^2 + \text{ lower terms}$$ so it has nonnegative extremal coefficients. By symmetry, the same holds for $I = {\left\{\,1,3\,\right\}}$. Further, it follows from below that the Hilbert polynomials of restrictions $H|_{I,{\mathbf{u}}}$ with $|I| = 1$ have nonnegative extremal coefficients.
So it remains to consider the case $I = {\left\{\,1,2\,\right\}}$. Let ${\mathbf{u}}= (0,0, \kappa) \in {\mathbb{Z}}^3$. Then $${\mathcal{P}(H|_{I,{\mathbf{u}}})} = i^2 + j^2 + (\kappa - \lambda)(\kappa-\lambda-2)ij$$ and all three terms are extremal. So this restriction has nonnegative extremal coefficients if and only if $\kappa \neq \lambda + 1$. In particular, $H$ is not a Hilbert series.
On the other hand, writing $H$ as a rational function one sees that the degrees of all terms in the numerator are less or equal than ${\mathbf{g}}:= (2,2,2) \in {\mathbb{N}}^n$. Thus for $\lambda \geq 2$ it is not sufficient to consider restrictions $H|_{I,{\mathbf{u}}}$ with ${\mathbf{u}}\leq {\mathbf{g}}$.
We need some preparations before we give the proof of . First, note that Hilbert decompositions are compatible with restrictions in the following sense:
Let $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ be a formal Laurent series. If $H$ has a Hilbert decomposition, then so does every restriction $H|_{I, {\mathbf{u}}}$ of $H$ for ${\mathbf{u}}\in {\mathbb{Z}}^n, I \subseteq[n]$.
If $H$ has a Hilbert decomposition, then there exists a finitely generated graded $R$-module $M = \bigoplus_{\mathbf{a}}M_{\mathbf{a}}$ with $H = H_M$. Set $$M' := \bigoplus_{{\mathbf{a}}\in {\mathbf{u}}+ {\mathbb{N}}^I} M_{\mathbf{a}}.$$ This is a module over $R' := {\mathbb{K}}[X_{ij} {\ \vrule\ }i \in I] \subseteq R$ in a natural way. We give it the structure of an $R$-module by letting the other variables act as zero. Its Hilbert series equals ${\bm{t}}^{\mathbf{u}}H|_{I, {\mathbf{u}}}$, hence this series has a Hilbert decomposition. But then $H|_{I, {\mathbf{u}}}$ clearly has a Hilbert decomposition as well.
Next, we show that polynomials with positive extremal coefficients admit a certain decomposition. This is the key step in our proof of .
\[prop:binomial\] For a polynomial $p \in {\mathbb{Q}}[Z_1, \dots, Z_n]$, the following statements are equivalent:
1. $p$ has positive extremal coefficients.
2. $p$ can be written as follows: $$p = \sum_{({\mathbf{a}}, {\mathbf{r}}) \in \Omega} c_{({\mathbf{a}},{\mathbf{r}})} \binom{Z_1 + r_{1} - a_{1}}{r_{1}}\dotsm \binom{Z_n + r_{n} - a_{n}}{r_{n}}$$ for some finite set $\Omega \subseteq {\mathbb{Z}}^n \times {\mathbb{N}}^n$ and $c_{({\mathbf{a}},{\mathbf{r}})} > 0$ for all $({\mathbf{a}},{\mathbf{r}})\in \Omega$.
If, in addition, there exists a ${\mathbf{u}}\in {\mathbb{Z}}^n$ such that $p({\mathbf{a}}) \in {\mathbb{Z}}$ for all ${\mathbf{a}}\geq {\mathbf{u}}$, then the coefficients $c_{({\mathbf{a}},{\mathbf{r}})}$ can be chosen to be natural numbers.
We start with the implication (1) $\Rightarrow$ (2). Let ${\bm{Z}}^{\mathbf{r}}$, for ${\mathbf{r}}\in {\mathbb{N}}^n$, be an extremal monomial of $p$, and let $c$ be its coefficient. For ${\mathbf{a}}\in {\mathbb{Z}}^n$ consider $$Q := \binom{Z_1+r_1-a_1}{r_1} \dotsm \binom{Z_n+r_n-a_n}{r_n}.$$ It is easy to see that ${\bm{Z}}^{\mathbf{r}}$ is also an extremal monomial of $Q$, and in fact it is its only extremal monomial. Further, its coefficient $1/r_1!\dotsm r_n! > 0$. Hence, the corresponding terms cancel in $$p_1 := p - \frac{c}{r_1!\dotsm r_n!} Q.$$ We show that $p_1$ still satisfies the hypothesis (1), so the claim follows by induction. For this, note that the only possible new extremal monomials of $p_1$ are the monomials ${\bm{Z}}^{\mathbf{r}}/ Z_i$ for $1 \leq i \leq n$, so we need to compute their coefficients in $Q$. We start with one factor of $Q$: $$\begin{aligned}
\binom{Z + r - a}{r} &= \frac{1}{r!}(Z+r-a)(Z-r-a-1)\dotsm(Z+1-a)\\
&= \frac{1}{r!}\left( Z^r + \left(\sum_{\ell=0}^{r-1}r-a-\ell\right) Z^{r-1} + \dotsb \right)\\
&= \frac{1}{r!}\left( Z^r + r\left(\frac{r+1}{2}-a\right) Z^{r-1} + \dotsb \right).
\end{aligned}$$ This implies that $$\begin{split}
Q = \frac{1}{r_1!\dotsm r_n!}\left( {\bm{Z}}^{\mathbf{r}}+ \sum_{i=1}^n r_i\left(\frac{r_i+1}{2}-a_i\right) \frac{{\bm{Z}}^{\mathbf{r}}}{Z_i}
+\text{lower terms}\right).
\end{split}$$ Note that for a sufficiently large choice of $a_1, \dotsc, a_n$, the coefficients of ${\bm{Z}}^{\mathbf{r}}/ Z_i$ become arbitrarily negative. Hence, for large ${\mathbf{a}}$, $p_1$ still satisfies the hypothesis.
For the other implication, note that if $p$ can be written as in (2), then there can be no cancellation between the extremal monomials on the right-hand side. So the coefficients of the extremal monomials of $p$ are (sums of) multiples of the $c_i$, and thus positive.
Finally, assume that $p({\mathbf{a}}) \in {\mathbb{Z}}$ for all ${\mathbf{a}}\in {\mathbb{Z}}^n$ which are greater than some fixed ${\mathbf{u}}$. We first note that this implies that $p({\mathbf{a}}) \in {\mathbb{Z}}$ for all ${\mathbf{a}}\in {\mathbb{Z}}^n$, cf. Corollary I.1.2 and Corollary XI.1.5 in Cahen and Chabert [@CCbook]. So by a classical result of Ostrowski [@ostrowski] (see also [@CCbook Corollary XI.1.11]), $p$ can be written as an integral linear combination of polynomials of the form $$H_{k_1, \dots, k_n} := \binom{Z_1}{k_1}\dotsm\binom{Z_n}{k_n}$$ with $k_1, \dotsc, k_n \in {\mathbb{N}}$. If ${\bm{Z}}^{\mathbf{r}}$ is an extremal monomial of $p$, then only $H_{r_1, \dots, r_n}$ contributes to this term, so its coefficient $c$ is a multiple of the corresponding coefficient in $H_{r_1, \dots, r_n}$, which is $1/r_1!\dotsm r_n!$. It now follows from the construction above that the $c_{({\mathbf{a}},{\mathbf{r}})}$ are positive integers.
We start with the necessity: If $H$ has a Hilbert decomposition, then so does every restriction of $H$. Hence we only need to consider the case $I = [n]$ and thus we need to show that ${\mathcal{P}(H)}$ has positive extremal coefficients. Consider a Hilbert decomposition $$H = \sum_{({\mathbf{a}}, {\mathbf{e}}) \in \Omega} \frac{c_{({\mathbf{a}},{\mathbf{e}})} {\bm{t}}^{\mathbf{a}}}{(1-{\bm{t}})^{\mathbf{e}}}$$ of $H$, where $\Omega \subseteq {\mathbb{Z}}^n \times {\mathbb{N}}^n$ is a suitable finite index set and $c_{({\mathbf{a}},{\mathbf{e}})} \geq 0$ for all $({\mathbf{a}}, {\mathbf{e}}) \in \Omega$. Expanding every summand into a series, it follows that $${\mathcal{P}(H)} = \sum_{({\mathbf{a}}, {\mathbf{e}}) \in \Omega} c_{({\mathbf{a}},{\mathbf{e}})} \binom{Z_1 + e_{1} - a_{1}-1}{e_{1}-1}\dotsm \binom{Z_n + e_{n} - a_{n}-1}{e_{n}-1}.$$ Hence ${\mathcal{P}(H)}$ has positive extremal coefficients by .
Now we turn to the sufficiency: We proceed by induction over the number of variables of $H$, with the base case being trivial. First, assume that ${\mathcal{P}(H)} \neq 0$. By assumption, its extremal coefficients are nonnegative, so yields a decomposition $${\mathcal{P}(H)} = \sum_{({\mathbf{a}},{\mathbf{r}})\in\Omega} c_{({\mathbf{a}},{\mathbf{r}})} \binom{Z_1 + r_{1} - a_{1}}{r_{1}}\dotsm \binom{Z_n + r_{n} - a_{n}}{r_{n}}$$ for a finite set $\Omega \in {\mathbb{Z}}^n\times {\mathbb{N}}^n$ and $c_{({\mathbf{a}},{\mathbf{r}})} > 0$ for all $({\mathbf{a}},{\mathbf{r}})\in\Omega$. Moreover, ${\mathcal{P}(H)}$ takes integer values on large ${\mathbf{a}}\gg 0$, so $c_{({\mathbf{a}},{\mathbf{r}})} \in {\mathbb{N}}$ for all $({\mathbf{a}},{\mathbf{r}})\in\Omega$. Let $$H_1 := \sum_{({\mathbf{a}},{\mathbf{r}})\in\Omega} \frac{c_{({\mathbf{a}},{\mathbf{r}})}{\bm{t}}^{{\mathbf{a}}}}{\prod_i (1-t_i)^{r_i+1}}.$$ It is easy to see that ${\mathcal{P}(H_1)} = {\mathcal{P}(H)}$, so there exists a ${\mathbf{g}}\in {\mathbb{Z}}^n$ such that ${\mathfrak{c}({H}, {{\mathbf{a}}})} = {\mathfrak{c}({H_1}, {{\mathbf{a}}})}$ for all ${\mathbf{a}}\geq {\mathbf{g}}$.
Set $H' := H - {\bm{t}}^{\mathbf{g}}H_1|_{[n], {\mathbf{g}}}$. It holds that ${\mathcal{P}(H')} = 0$ and we claim that $H'$ still satisfies the hypothesis on the extremal coefficients. To see this, consider $I \subseteq [n]$ and ${\mathbf{u}}\in {\mathbb{Z}}^{n}$. If ${\mathbf{u}}+ {\mathbb{N}}^I \cap {\mathbf{g}}+ {\mathbb{N}}^n = \emptyset$, then $H'|_{I,{\mathbf{u}}} = H|_{I,{\mathbf{u}}}$; otherwise, let ${\mathbf{v}}\in {\mathbf{u}}+ {\mathbb{N}}^I \cap {\mathbf{g}}+ {\mathbb{N}}^n$. Then ${\mathfrak{c}({H'}, {{\mathbf{a}}})} = 0$ for all ${\mathbf{a}}\geq {\mathbf{v}}$ and hence ${\mathcal{P}(H'|_{I,{\mathbf{u}}})} = 0$. In both cases, the hypothesis is satisfied.
Next, we consider the case that ${\mathcal{P}(H)} = 0$. In this case, the exponent vectors of the nonzero terms of $H$ are contained in finitely many translates of coordinate hyperplanes. Hence we may decompose $H$ as a sum of series in $n-1$ variables as follows: Choose ${\mathbf{g}}\in {\mathbb{Z}}^n$ such that ${\mathfrak{c}({H}, {{\mathbf{a}}})} = 0$ for all ${\mathbf{a}}\geq {\mathbf{g}}$. For $1 \leq i \leq n$ and $0 \leq j \leq g_i-1$ let ${\mathbf{u}}(i,j) := (g_1, \dotsc, g_{i-1},j,0,\dotsc,0) \in {\mathbb{Z}}^n$. We decompose $H$ as follows: $$H = \sum_{i=1}^n \sum_{j=0}^{g_{i}-1} {\bm{t}}^{{\mathbf{u}}(i,j)} H|_{[n] \setminus \{i\}, {\mathbf{u}}(i,j)}$$ Every restriction of $H$ is a series in at most $n-1$ variables, so the claim follows by induction.
Non-Negative series
-------------------
Our next goal is to clarify the relation between series satisfying the hypothesis of and series which are merely nonnegative. We will need the following convex geometric lemma.
\[lem:polytope\] Let $P \subseteq {\mathbb{R}}^n$ be a polytope and $v \in P$ a vertex, such that $v + u \notin P$ for all $u \in {\mathbb{R}}_{\geq 0}^n, u \neq 0$. Then there exists a linear form $\sigma \in ({\mathbb{R}}^n)^*$ which attains its maximum over $P$ exactly at $v$ and whose coefficients are nonnegative integers.
Let $P' \subseteq P$ be the convex hull of all vertices of $P$ which are different from $v$. Further, let $Q := v + {\mathbb{R}}_{\geq 0}^n$. Both $P'$ and $Q$ are convex sets, and our assumption implies that $P' \cap Q = \emptyset$. Then there exists a separating hyperplane, i.e. a linear form $\sigma \in ({\mathbb{R}}^n)^*$ such that $$\label{eq:sep}
\max(\sigma(p) {\ \vrule\ }p \in P') < \inf(\sigma(q) {\ \vrule\ }q \in Q).$$ We may assume that $\sigma$ has rational coefficients, and after clearing denominators we may even assume that the coefficients of $\sigma$ are integers. We show that $\sigma$ has the claimed properties. It is clear that the maximum of $\sigma$ over $P$ is attained only at $v$. To see that the coefficients of $\sigma$ are nonnegative assume that $\sigma({\mathbf{e}}_i) < 0$ for some unit vector ${\mathbf{e}}_i$. Then $\sigma(v + \lambda {\mathbf{e}}_i)$ can be arbitrarily negative for large $\lambda \gg 0$, contradicting .
The following lemma shows that if $H \geq 0$ then some extremal coefficients are automatically nonnegative.
\[lem:ecken\] Let $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ be a formal Laurent series, such that $H \cdot (1-{\bm{t}})^{\mathbf{m}}$ is a Laurent polynomial for some ${\mathbf{m}}\in {\mathbb{N}}^n$. The following conditions are equivalent:
1. $H \geq 0$.
2. For all ${\mathbf{u}}\in {\mathbb{Z}}^n$ and $I \subseteq [n]$, every extremal monomial of ${\mathcal{P}(H|_{{\mathbf{u}},I})}$ which is also a vertex of its Newton polytope has a positive coefficient.
The implication (2) $\Rightarrow$ (1) is clear, because ${\mathfrak{c}({H}, {{\mathbf{a}}})} = {\mathcal{P}(H|_{{\mathbf{a}}, \emptyset})}$ for all ${\mathbf{a}}\in {\mathbb{Z}}^n$. So we only need to show the other implication.
Let ${\bm{Z}}^{\mathbf{r}}$ be an extremal monomial of $p := {\mathcal{P}(H|_{{\mathbf{u}},I})}$ which is also a vertex of its Newton polytope. By , there exists a linear form $\sigma \in ({\mathbb{R}}^n)^*$ with nonnegative integral coefficients, which attains its maximum over the Newton polytope exactly at ${\mathbf{r}}$. Consider the linear map $\hat{\sigma}: {\mathbb{Z}}[Z_1, \dotsc, Z_n] \to {\mathbb{Z}}[Z]$, given by ${\bm{Z}}^{\mathbf{r}}\mapsto Z^{\sigma({\mathbf{r}})}$. Then $\hat{\sigma}(p)$ is a univariate polynomial, which attains nonnegative values at sufficiently large integers. Hence its leading coefficient is nonnegative. On the other hand, by our choice of $\sigma$, this leading coefficient of $\hat{\sigma}(p)$ equals the coefficient of ${\bm{Z}}^{\mathbf{r}}$ in $p$. So the claim is proven.
The fine graded case
--------------------
If there are at most two variables with the same degree, then the obvious necessary conditions for Hilbert series are also sufficient. This includes in particular the case of the fine graded polynomial ring.
\[cor:e2\] In the situation of , assume that $m_i \leq 2$ for all $i$.
Then the following two statements are equivalent for a formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$:
1. There exists a finitely generated graded $S$-module $M$ whose Hilbert series equals $H$.
2. $H$ satisfies the following two conditions:
1. $H \geq 0$, and
2. $H \cdot \prod_{i=1}^n (1-t_i)^{m_i}$ is a polynomial.
For $I \subseteq [n]$ and ${\mathbf{u}}\in {\mathbb{Z}}^n$ let $p = {\mathcal{P}(H|_{I,{\mathbf{u}}})}$. The hypothesis that $m_i \leq 2$ for all $i$ implies that every monomial of $p$ is squarefree. Hence its Newton polytope is a 0/1-polytope, so every lattice point in it is a vertex. In particular, all extremal monomials of $p$ are vertices of its Newton polytope, so the claim follows from .
General inequalities for the Hilbert depth
==========================================
In this section, we present a class of linear inequalities for the Hilbert series of modules with a given depth. We relax our assumptions on $R$ and allow it to be an arbitrary (commutative) ${\mathbb{K}}$-algebra with a positive ${\mathbb{Z}}^n$-grading, such that $\dim_{\mathbb{K}}R_0 < \infty$. The general idea is to compare the Hilbert series in question with *all* Hilbert series of modules from a certain class. Recall that an $R$-module $N$ is called *$p$-th syzygy module* if it can be realized as the $p$-th syzygy of some $R$-module $N'$. See [@B] for alternative characterizations of syzygy modules.
The following is the main result of this section.
First note that every free $R$-module $F$ satisfies $H_{F \otimes N} = \frac{H_F H_N}{H_R}$. This clearly holds for $F = R(a)$ with $a \in {\mathbb{Z}}^n$, and it is easily seen that the equality is preserved under direct sums.
Consider a free resolution of $M$: $${\mathbb{F}}: 0 \to F_p \to \dotsb \to F_0 \to 0.$$ We compute that $$\sum_{i \geq 0} (-1)^i H_{F_i \otimes N} = \sum_{i \geq 0} (-1)^i \frac{H_{F_i} H_N}{H_R} = \frac{H_N}{H_R} \sum_{i \geq 0} (-1)^i H_{F_i} = \frac{ H_N}{H_R} H_M.$$ On the other hand, it holds that $$\sum_{i \geq 0} (-1)^i H_{F_i \otimes N} = \sum_{i \geq 0} (-1)^i H_{H_i({\mathbb{F}}\otimes N)} = \sum_{i \geq 0} (-1)^i H_{\operatorname{Tor}_i(M,N)}.$$ Let $N'$ be an $R$-module such that $N$ is the $p$-th syzygy module of $N'$. Then it holds for $i > 0$ that $\operatorname{Tor}^R_i(M,N) = \operatorname{Tor}^R_{i+p}(M, N') = 0$ because $\operatorname{pdim}M \leq p$. We conclude that $$\frac{H_M H_N}{H_R} = \sum_{i \geq 0} (-1)^i H_{\operatorname{Tor}_i(M,N)} = H_{M \otimes N} \geq 0. \qedhere$$
Let us consider some extremal cases of this theorem in the case that $R$ is the polynomial ring (with an arbitrary ${\mathbb{Z}}^n$-grading).
(1) If $p = \dim R$, all $p$-th syzygies modules are free. Hence reduces to the statement that $H_M \geq 0$ for every $R$-module $M$.
(2) For $p = 0$, every module with $\operatorname{pdim}M \leq p$ is free. On the other hand, let $M$ be a module satisfying the inequality for all $0$-th syzygy modules $N$. Choosing $N = {\mathbb{K}}= R / (X_1, \dotsc, X_m)$ yields that $H_M$ is of the form $$\frac{Q}{\prod_{i=1}^n (1-t_i)}$$ for some $Q \in {\mathbb{N}}[t_1^{\pm 1}, \dotsc, t_n^{\pm 1}]$. If $Q = \sum_{{\mathbf{a}}\in {\mathbb{Z}}^n} q_{{\mathbf{a}}} {\bm{t}}^{{\mathbf{a}}}$, then the free module $$M' := \bigoplus_{{\mathbf{a}}\in {\mathbb{Z}}^n} R(-{\mathbf{a}})^{q_{{\mathbf{a}}}}$$ has the same Hilbert series as $M$. Hence, exactly describes the Hilbert series of free modules.
In general, the inequalities are not sufficient for a Hilbert series to have a given Hilbert depth. Nevertheless, in the next two sections, we consider two special situations where slightly stronger inequalities are indeed sufficient.
For later use, we also record a useful criterion for Hilbert series of modules of positive depth.
\[prop:hdep\] Let $R$ be a ${\mathbb{Z}}^n$-graded polynomial ring, such that every variable is homogeneous. Then a formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, \dotsc, t_n)\!)$ is the Hilbert series of a finitely generated $R$-module of positive depth if and only if it can be written in the form $$\label{eq:hdec0}
H = \sum_{\substack{I \subseteq [m]\\I \neq \emptyset}} \frac{Q_I}{\prod_{i \in I}(1-{\bm{t}}^{\deg X_i})}$$ for Laurent polynomials $Q_I \in {\mathbb{Z}}[t_1^{\pm 1}, \dotsc, t_n^{\pm 1}], I \subseteq [m]$ with nonnegative coefficients.
The difference to is that there is no term $Q_\emptyset$. We call a Hilbert decomposition as in a . As in , this result is essentially contained in [@MU Prop. 2.4], but there it is stated only for ${\mathbb{Z}}$-graded rings; the proof in our context follows by the same method.
The Bigraded case
=================
In this section we consider the ${\mathbb{Z}}^2$–graded situation. More precisely, let $R=\mathbb{K}[X_1,\ldots , X_m,Y_1,\ldots , Y_{\tilde{m}}]$ be the polynomial ring with a ${\mathbb{Z}}^n$-grading given by $\deg X_i =(1,0)$ for all $i \in [m]$ and $\deg Y_i =(0,1)$ for all $i \in [\tilde{m}]$.
Specializing to this situation we obtain the following characterization of Hilbert series over $R$:
\[cor:bigraded\] For a formal Laurent series $H \in {\mathbb{Z}}(\!(t_1, t_2)\!)$, there exists a finitely generated graded $R$-module $M$ with $H = H_M$ if and only if $H$ satisfies the following conditions:
1. $H \geq 0$,
2. $H\cdot (1-t_1)^{m}\cdot(1-t_2)^{\tilde{m}}$ is a Laurent polynomial, and
3. ${\mathcal{P}(H)}$ has positive extremal coefficients.
The necessity of the conditions is clear from .
For the sufficiency, we show that given conditions imply the hypotheses of . So consider $I \subseteq [2]$ and ${\mathbf{u}}\in {\mathbb{Z}}^2$. If $I = \emptyset$, then ${\mathcal{P}(H|_{I,{\mathbf{u}}})} = {\mathfrak{c}({H}, {{\mathbf{u}}})} \geq 0$, and ${\mathcal{P}(H)}$ has positive extremal coefficients by assumption.
So we only need to consider the case that $|I| = 1$. In this case ${\mathcal{P}(H|_{I,{\mathbf{u}}})}$ is a univariate polynomial, so its only extremal monomial is the leading one, which is also a vertex of its Newton polytope. Thus the claim follows from .
Our next main result will be the characterization of Hilbert series of positive Hilbert depth over $R$. For this we will consider certain pairs of sequences of indices in ${\mathbb{Z}}^2$.
1. A sequence $\left({\mathbf{u}}^{(i)}\right)_{i=1}^p$ in ${\mathbb{Z}}^2$ is called *declining*, if $$u_1^{(i)} < u_1^{(i+1)} \quad \mbox{and} \quad u_2^{(i)} >u_2^{(i+1)} \quad \mbox{for} \quad i=1, \ldots , p-1.$$ The set of all declining sequences will be denoted by ${\mathcal{S}}$.
2. For a sequence $U=\left({\mathbf{u}}^{(i)}\right)_{i=1}^p$ we define ${\mathcal{D}(U)} := ({\mathbf{u}}^{(i)} \wedge {\mathbf{u}}^{(i+1)})_{i=1}^{p-1}$.
The two sequences $U$ and ${\mathcal{D}(U)}$ can be visualized as a [“staircase”]{}, see .
(0,0) grid \[step=1cm\](8,6); (0,0) – (8,0); (0,0) – (0,6); (1,5) – (1,4) – (3,4) – (3,3) – (4,3) – (4,1) – (7,1); (1,5) circle \[radius=0.1\]; (3,4) circle \[radius=0.1\]; (4,3) circle \[radius=0.1\]; (7,1) circle \[radius=0.1\]; (1,4) circle \[radius=0.1\]; (3,3) circle \[radius=0.1\]; (4,1) circle \[radius=0.1\];
\[th:stc\] Let $H \in {\mathbb{Z}}(\!(t_1, t_2)\!)$ be a formal Laurent series, which is the Hilbert series of some finitely generated graded $R$-module. Let further $S := {\mathbb{K}}[X, Y]$ be the standard–${\mathbb{Z}}^2$–graded polynomial ring in two variables. Then the following statements are equivalent:
(a) $H$ has positive Hilbert depth.
(b) For any finitely generated torsionfree $S$-module $N$, it holds that $$\label{eq:HR}
\frac{H\cdot H_N}{H_S} \geq 0.$$
(c) Condition (b) holds for any finitely generated torsionfree $S$ module of rank $1$, i.e. every fractional monomial ideal $I \subseteq {\mathbb{K}}[X^{\pm 1},Y^{\pm 1}]$.
(d) $H= \sum_{i,j} h_{ij} \,t_1^i t_2^j$ satisfies the condition $$\label{eq:st}
\sum_{(i,j) \in {\mathcal{D}(U)}} h_{ij} \leq \sum_{(i,j) \in U} h_{ij} \quad \text{for all} \ \ U \in {\mathcal{S}}. \tag{ST}$$
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1) Note that the torsionfree modules (over a domain) are exactly the first syzygy modules.
2) If $({\mathbf{u}}^{(i)})_{i=1}^1$ is a declining sequence with only one entry, then asserts that $h_{{\mathbf{u}}^{(1)}} \geq 0$. Hence any series satisfying condition is in particular nonnegative.
3) Although condition resembles its counterpart in the non-standard–${\mathbb{Z}}$-graded case, namely the condition of below, there are important differences: The inequalities required by condition only relate coefficients lying within one common period of the module’s Hilbert function, they have the same number of terms on both sides, and this number is bounded above by $$\min \{\deg X, \deg Y \}.$$ By contrast, a declining sequence $U$ may have any number of entries, which may be arbitrarily separated, and the right-hand side of has always one term more than the left-hand side. One might think that it could be possible to weaken condition by restricting it to a subset of ${\mathcal{S}}$ consisting of somehow bounded sequences, but this turns out to be a vain hope. For instance, the examples $$H_k = 1 + \sum_{i=0}^k \frac{t_1^i}{1-t_2}, \quad k \in {\mathbb{N}}$$ show that we cannot afford to restrict condition to those sequences $U=({\mathbf{u}}^{(i)})_{i=0}^p$ where $\max_{i \in [m-1]} \{ u_1^{(i+1)} - u_1^{(i)} \} \leq C$ for some $C \in {\mathbb{N}}$.
Before we present the proof of we give several lemmata. First of all, it turns out to be convenient to consider a slightly larger class of inequalities. For this, we call a sequence $\left({\mathbf{u}}^{(i)} \right)_{i=0}^p$ in ${\mathbb{Z}}^2$ *weakly declining*, if $ u_1^{(i)} \leq u_1^{(i+1)}$ and $u_2^{(i)} \geq u_2^{(i+1)}$ for $i=0, \ldots, p-1$ and let ${{\mathcal{S}}'}$ denote the set of all weakly declining sequences. Moreover, for a formal Laurent series $H = \sum_{{\mathbf{u}}\in {\mathbb{Z}}^2} h_{\mathbf{u}}{\bm{t}}^{\mathbf{u}}\in {\mathbb{Z}}(\!({\bm{t}})\!)$ and a sequence $U$, we set $$\sigma_U(H) := \sum_{{\mathbf{u}}\in U} h_{\mathbf{u}}- \sum_{{\mathbf{u}}\in {\mathcal{D}(U)}} h_{\mathbf{u}}$$
\[lem:weakly\] Let $H\in {\mathbb{Z}}(\!(t_1,t_2)\!)$ be a formal Laurent series. Then $\sigma_U(H) \geq 0$ for all $U \in {\mathcal{S}}$ if and only if $\sigma_U(H) \geq 0$ for all $U \in {{\mathcal{S}}'}$.
One implication is trivial. So assume that $\sigma_U(H) \geq 0$ for all $U \in {\mathcal{S}}$ and consider a weakly declining sequence $U=\left({\mathbf{u}}^{(i)}\right)_{i=0}^p$ in ${\mathbb{Z}}^2$. If $u_1^{(i)} = u_1^{(i+1)}$ for some index $i$, then ${\mathbf{u}}^{(i+1)}= {\mathbf{u}}^{(i)} \wedge {\mathbf{u}}^{(i+1)}$. Hence if $U'$ is the sequence obtained from $U$ by deleting ${\mathbf{u}}^{(i+1)}$, then $\sigma_U(H) = \sigma_{U'}(H)$. Similarly, if $u_1^{(i)} = u_1^{(i+1)}$ then ${\mathbf{u}}^{(i)} ={\mathbf{u}}^{(i)} \wedge {\mathbf{u}}^{(i+1)}$, so we may delete ${\mathbf{u}}^{(i)}$ from $U$. After finitely many such deletions, we obtain a declining sequence $U'' \in {\mathcal{S}}$ with $\sigma_U(H) = \sigma_{U''}(H)$. As $\sigma_{U''}(H) \geq 0$ by assumption, the claim follows.
The following lemma essentially reduces the question to the fine-graded situation.
Recall that ${\mathfrak{c}({H}, {{\mathbf{u}}})}$ denotes the coefficient of ${\bm{t}}^{\mathbf{u}}$ in the series $H$.
\[lem:dec\] Assume that $H \in {\mathbb{N}}(\!(t_1,t_2)\!)$ admits a Hilbert decomposition without polynomial part. Then there exists an ${{\mathbf{g}}}_0 \in {\mathbb{Z}}^2$ with the following property:
For any ${{\mathbf{g}}}\geq {{\mathbf{g}}}_0$, there exists a decomposition $H = H_1 + H_2$ such that
1. $H_1$ is of the form $\frac{Q_1}{1-t_1} + \frac{Q_2}{1-t_2}$ for some $Q_1, Q_2 \in {\mathbb{N}}[t_1^{\pm 1}, t_2^{\pm 1}]$. Moreover, it holds that ${\mathfrak{c}({H_1}, {{{\mathbf{g}}}})} = 0$ and ${\mathfrak{c}({H_1}, {{\mathbf{u}}\wedge {{\mathbf{g}}}})} = {\mathfrak{c}({H_1}, {{\mathbf{u}}})}$ for all ${\mathbf{u}}\in {\mathbb{Z}}^2$.
2. $H_2$ has a Hilbert decomposition without polynomial part and it satisfies ${\mathfrak{c}({H_2}, {{\mathbf{u}}})} = 0$ for all ${\mathbf{u}}< {{\mathbf{g}}}$.
Choose a Hilbert decomposition of $H$ without polynomial part: $$H = \sum_{{\mathbf{a}}\in{\mathbb{N}}^2, {\mathbf{a}}\neq 0} \frac{Q_{\mathbf{a}}}{\prod_{i =1}^2(1-t_i)^{a_i}}.$$ Choose ${{\mathbf{g}}}_0 \in {\mathbb{Z}}^2$ which is strictly larger than the degrees of all monomials in the numerator polynomials $Q_{(1,0)}$ and $Q_{(0,1)}$. By repeatedly using the relation $$\frac{1}{\prod_{i=1}^2(1-t_i)^{a_i}} = t_j\frac{1}{\prod_{i=1}^2(1-t_i)^{a_i}} + (1-t_j) \frac{1}{\prod_{i =1}^2(1-t_i)^{a_i}},$$ we may modify the Hilbert decomposition such that it satisfies the following:
- $Q_{(0,0)} = 0$.
- For every ${\mathbf{a}}\in {\mathbb{N}}^2, {\mathbf{a}}\neq (0,1),(1,0)$ such that $Q_{\mathbf{a}}\neq 0$, it holds that $Q_{\mathbf{a}}$ contains no monomials of degree strictly less than ${{\mathbf{g}}}$.
- The polynomial $(1-t_1) Q_{(0,1)} + (1-t_2)Q_{(1,0)}$ contains only monomials which are strictly less than ${{\mathbf{g}}}$.
We set $$H_1 := \frac{Q_{(1,0)}}{1-t_1} + \frac{Q_{(0,1)}}{1-t_2}$$ and $H_2 := H - H_1$. The claimed properties of $H_1$ and $H_2$ follow readily.
The third lemma is the key step in our proof of . Here we show how to decompose a non-negative Laurent series of the form $Q_0 + Q_1/(1-t_1) + Q_2/(1-t_2)$.
\[l:r1\] Let $$H= \sum_{i=k}^{\infty} \sum_{j= \ell}^{\infty} h_{ij} \, t_1^i t_2^j$$ be a formal Laurent series satisfying condition . Assume further that $h_{p \ell} = 0$ for some $p \geq k$ and let $\mu := \min {\left\{\, h_{i \ell} {\ \vrule\ }i > p \,\right\}}$. Then the series $$H - \frac{1}{1-t_2} \sum_{i=k}^{p-1} h_{i \ell} t_1^i t_2^\ell - \frac{1}{1-t_1} \mu t_1^{p+1} t_2^{\ell}$$ satisfies condition as well.
To prove the lemma, it is enough to show the following two claims:
1. For any $k \leq r < p$ the series $$\tilde{H}:= H - h_{r \ell} \, \frac{t_1^r t_2^{\ell}}{1-t_2}$$ satisfies condition .
2. Further, for $\mu := \min {\left\{\, h_{i \ell} {\ \vrule\ }i > p \,\right\}}$ the series $$\hat{H}:= H - \mu \, \frac{ t_1^{p+1} t_2^{\ell}}{1-t_1}$$ satisfies condition .
We start with the proof of the first claim. For this we have to show that condition is valid for $\tilde{H} =: \sum_{i,j} \tilde{h}_{ij} \, t_1^i t_2^j$. The coefficients $\tilde{h}_{ij}$ and $h_{ij}$ are equal except for $i = r$ and $j \geq \ell$, where $\tilde{h}_{ij} = h_{ij}-h_{r \ell} $ holds. Hence we only have to consider declining sequences $U \subseteq{\mathcal{S}}$ such that $U$ or ${\mathcal{D}(U)}$ intersect $C_{r,\ell}:=\left\{ (r,j) ~|~ j \geq \ell \right\}$. If $ C_{r,\ell}$ intersects both $U$ and ${\mathcal{D}(U)}$, then we have $$\sigma_U(\tilde{H}) = \sigma_U(H) - h_{r\ell} + h_{r\ell} = \sigma_U(H) \geq 0$$ so we may assume that only $U$ intersects with $ C_{r,\ell}$. In this case it is not clear a priori whether the corresponding inequality still holds for $\tilde{H}$, because only the right–hand side of the original inequality is diminished. (A similar problem occurs in [@MU], where the analogous inequalities are called *critical*.)
Let ${\mathbf{u}}^{(e)}=(r, j')$ denote the intersection of $U$ and $ C_{r,\ell}$. Since we assume that ${\mathcal{D}(U)}$ and $ C_{r,\ell}$ do not intersect, the sequence $U$ either ends in ${\mathbf{u}}^{(e)}$ or the point ${\mathbf{u}}^{(e)} \wedge {\mathbf{u}}^{(e+1)}=\left(r,j'' \right)$, and hence all further points of $U$ and ${\mathcal{D}(U)}$ lie in the half–plane $\{(x,y) ~|~ y < \ell \}$. Since all coefficients of $H$ and $\tilde{H}$ in this half–plane vanish, we may assume that the staircase ends in ${\mathbf{u}}^{(e)}$. We amend $U$ by $\tilde{{\mathbf{u}}}^{(e+1)} := (p,\ell)$ to build $\tilde{U}$. Note that $\tilde{U}$ is weakly declining, hence $\sigma_{\tilde{U}}(H) \geq 0$ by . As ${\mathbf{u}}^{(e)} = (r,j')$, it follows that ${\mathbf{u}}^{(e)}\wedge \tilde{{\mathbf{u}}}^{(e+1)} = (r,\ell)$ and thus $$\sigma_{U}(\tilde{H}) = \sigma_{U}(H) - h_{r\ell} = (\sigma_{\tilde{U}}(H) + h_{r\ell} - h_{p\ell}) - h_{r\ell} = \sigma_{\tilde{U}}(H) \geq 0.$$
Now we turn to the proof of the second claim. By the choice of $\mu$ the series $\hat{H} =: \sum_{i,j} \hat{h}_{ij} t_1^i t_2^j$ is nonnegative. Similar to the proof of the first claim the verification of condition for $\hat{H}$ reduces to an inspection of sequences $U\in{\mathcal{S}}$ intersecting $B_{p+1,\ell} := \left\{ (i,\ell) ~|~ i \geq p+1 \right\}$ in $U$, but not in $D$. Let ${\mathbf{u}}^{(e)} = (i',\ell)$ be this intersection; again we may assume that $U$ ends in ${\mathbf{u}}^{(e)}$.
Let $\tilde{U}$ be the new weakly declining sequence obtained from $U$ by replacing the last element ${\mathbf{u}}^{(e)} = (i', \ell)$ by $\tilde{{\mathbf{u}}}^{(e)} := (p,\ell)$. We assumed that ${\mathbf{u}}^{(e-1)} \wedge {\mathbf{u}}^{(e)} \notin B_{p+1,\ell}$, hence it holds that $u_1^{(e-1)} \leq p$ and thus ${\mathbf{u}}^{(e-1)} \wedge {\mathbf{u}}^{(e)} = {\mathbf{u}}^{(e-1)}\wedge \tilde{{\mathbf{u}}}^{(e)}$. It follows that $$\begin{aligned}
\sigma_{U}(\hat{H}) &= \sigma_{\tilde{U}}(\hat{H}) - h_{p\ell} + {\mathfrak{c}({\hat{H}}, {(i',\ell)})} = \sigma_{\tilde{U}}(H) + {\mathfrak{c}({\hat{H}}, {(i',\ell)})} \\
&= \sigma_{\tilde{U}}(H) +(h_{i'\ell} - \mu) \geq 0,\end{aligned}$$ since $h_{i'\ell} - \mu \geq 0$.
Now we are able to prove Theorem \[th:stc\]:
Let $N$ be a finite torsionfree $S$-module. After shifting the degrees, we may assume that all homogeneous components of $N$ have non-negative degrees. Let ${\mathbf{u}}\in {\mathbb{Z}}^2$ be a multidegree. By we can find a decomposition $H = H_1 + H_2$, such that all coefficients of $H_2$ in degrees less or equal than ${\mathbf{u}}$ vanish. Hence $$\mathfrak{c}\left(\frac{H\cdot H_N}{H_S},{\mathbf{u}}\right) = \mathfrak{c}\left(\frac{H_1\cdot H_N}{H_S},{\mathbf{u}}\right ).$$ But $\frac{H_1}{H_S}$ is a sum of terms of the form $(1-t_1){\bm{t}}^{\mathbf{a}}$ or $(1-t_2){\bm{t}}^{\mathbf{a}}$ with ${\mathbf{a}}\in {\mathbb{Z}}^2$ and nonnegative coefficients. Hence it is enough to show that $(1-t_1) H_N \geq 0$ and $(1-t_2) H_N \geq 0$. Note that because $N$ is torsionfree, multiplication by $X$ and $Y$ gives inclusions $N(-(1,0)) \hookrightarrow N$ and $N(-(0,1)) \hookrightarrow N$, thus these claims follow.
This is trivial.
For a given declining sequence $U = ({\mathbf{u}}^{(i)})_{i=1}^p \in {\mathcal{S}}$, consider the fractional ideal $I$ generated by $X^{-u_1^{(1)}}Y^{-u_2^{(1)}}, \dotsc, X^{-u_1^{(p)}}Y^{-u_2^{(p)}}$. Considering the minimal free resolution of $I$ over $S$ yields that $$\frac{H_I}{H_S} = \sum_{i=1}^p {\bm{t}}^{-{\mathbf{u}}^{(i)}} - \sum_{i = 1}^{p-1} {\bm{t}}^{(-{\mathbf{u}}^{(i)}) \vee (-{\mathbf{u}}^{(i+1)})}.$$ This and (c) imply that $$\begin{aligned}
0 \leq \frac{H\cdot H_I}{H_S} &= \left(\sum_{{\mathbf{a}}\in {\mathbb{Z}}^2} h_{{\mathbf{a}}} {\bm{t}}^{\mathbf{a}}\right) \left(\sum_{i=1}^p {\bm{t}}^{-{\mathbf{u}}^{(i)}} - \sum_{i = 1}^{p-1} {\bm{t}}^{(-{\mathbf{u}}^{(i)}) \vee (-{\mathbf{u}}^{(i+1)})}\right)\\
&= \sum_{{\mathbf{a}}\in {\mathbb{Z}}^2} {\bm{t}}^{\mathbf{a}}\left(\sum_{i=1}^p h_{{\mathbf{a}}+ {\mathbf{u}}^{(i)}} - \sum_{i = 1}^{p-1} h_{{\mathbf{a}}- ((-{\mathbf{u}}^{(i)}) \vee (-{\mathbf{u}}^{(i+1)}))}\right) \\
&= \sum_{{\mathbf{a}}\in {\mathbb{Z}}^2} {\bm{t}}^{\mathbf{a}}\left(\sum_{i=1}^p h_{{\mathbf{a}}+ {\mathbf{u}}^{(i)}} - \sum_{i = 1}^{p-1} h_{({\mathbf{a}}+ {\mathbf{u}}^{(i)}) \wedge ({\mathbf{a}}+ {\mathbf{u}}^{(i+1)})}\right).
\end{aligned}$$ Note that the coefficient of ${\bm{t}}^{\mathbf{0}}$ equals $\sigma_U(H)$, hence the latter is nonnegative. So $H$ satisfies .
First, choose a decomposition $H = P + H'$, such that $P$ is a polynomial with non-negative coefficients and $H'$ has a Hilbert decomposition with no polynomial part. Let $\tilde{{{\mathbf{g}}}} \in {\mathbb{Z}}^2$ such that all non-zero coefficients of $P$ lie in degrees below $\tilde{{{\mathbf{g}}}}$. Again, using there is ${{\mathbf{g}}}\geq \tilde{{{\mathbf{g}}}}$ and a decomposition $H' = H_1 + H_2$, such that $H_1$ and $H_2$ satisfy the conditions mentioned above.
We are going to construct a Hilbert decomposition without polynomial part of $H_3 := P + H_1$. As $H_2$ already has such a Hilbert decomposition, this is enough to prove the claim. For this, we first need to show that $H_3$ still satisfies .
Let $U = ({\mathbf{u}}^{(i)})_{i=0}^p \in {\mathcal{S}}$ be a declining sequence. Then $U' := ({\mathbf{u}}^{(i)} \wedge {{\mathbf{g}}})_{i=0}^p$ is a weakly declining sequence. If ${{\mathbf{g}}}\in U'$, then it is easy to see that $$\sigma_{U'}(H_3) = {\mathfrak{c}({H_3}, {{{\mathbf{g}}}})} = 0.$$ Otherwise, our choice of ${{\mathbf{g}}}$ and imply that ${\mathfrak{c}({H_3}, {{\mathbf{a}}\wedge {{\mathbf{g}}}})} = {\mathfrak{c}({H_3}, {{\mathbf{a}}})}$ for all ${\mathbf{a}}\in {\mathbb{Z}}^2$. Hence it follows that $\sigma_{U}(H_3) = \sigma_{U'}(H_3)$. Further, it holds that $$\sigma_{U'}(H_3) = \sigma_{U'}(H_3 + H_2) \geq 0,$$ because the coefficients of $H_2$ vanish below ${{\mathbf{g}}}$ and $H = H_2 + H_3$ satisfies . Hence $H_3$ satisfies as well. The series $H_3$ is of the form $$Q_0 + \frac{Q_1}{1-t_1} + \frac{Q_2}{1-t_2}.$$ We obtain a Hilbert decomposition of $H_3$ without polynomial part by repeatedly applying .
With a little more work, one can show directly that condition (d) implies (c). Indeed, we already showed that every declining sequence gives rise to a fractional ideal. On the other hand, to every fractional ideal one can associate a declining sequence in a natural way, thus proving the equivalence directly.
The non-standard Z-graded case
==============================
Let $R = {\mathbb{K}}[X,Y]$ with the grading given by $\deg X = \alpha$ and $\deg Y = \beta$ for two coprime numbers $\alpha, \beta \in {\mathbb{N}}$. In [@MU], the second and third author characterized the Hilbert series of modules of positive depth over this ring. In this section, we give a reformulation of this result along the lines of the previous results.
First of all, we recall the characterization of Hilbert series of finitely generated graded modules over this ring (cf. the authors in [@KMU Theorem 2.6]): These are exactly the formal Laurent series $H$ with nonnegative integral coefficients, such that $H\cdot (1-t^\alpha)(1-t^\beta)$ is a Laurent polynomial, thus there is no analog of condition (3) of in this setting.
In order to state the characterization of Hilbert series with positive Hilbert depth we need some more notation. Denote by ${\langle \alpha,\beta \rangle}\subseteq {\mathbb{N}}$ the numerical semigroup generated by $\alpha$ and $\beta$. A (with respect to $\alpha$ and $\beta$) is pair $[I,J]$ of two integer sequences $I = (i_k)_{k=0}^m$ and $J = (j_k)_{k=0}^m$ which are subject to the following conditions:
1. $i_0=0$;
2. $i_1, \ldots , i_m, j_1, \ldots , j_{m-1} \in {\mathbb{N}}\setminus{\langle \alpha,\beta \rangle}$ and $j_0,j_m \leq \alpha \beta$;
3. 4. $\begin{array}{lllll}
i_k \equiv j_k &\!\!\!\mod \alpha &~\mbox{and}~& i_k < j_k & ~~\mbox{for} ~~ k= 0, \ldots ,m;\\
j_k \equiv i_{k+1} &\!\!\!\mod \beta &~\mbox{and}~& j_k > i_{k+1} & ~~\mbox{for} ~~ k= 0, \ldots ,m-1;\\
j_m \equiv i_{0} &\!\!\!\mod \beta &~\mbox{and}~& j_m \geq i_{0}. &
\end{array}$
5. 6. $|i_k - i_{\ell}| \in {\mathbb{N}}\setminus{\langle \alpha,\beta \rangle}~~\mbox{for} ~~1 \leq k < \ell \leq m$.
Denote by ${\mathcal{F}_{\alpha,\beta}}$ the set of all fundamental couples.
The characterization of positive Hilbert depth over $R$ can be stated as follows:
\[thm:nonstandart\] Let $H \in {\mathbb{Z}}(\!(t)\!)$ be a formal Laurent series, which is the Hilbert series of some finitely generated graded $R$-module. Let further $S := R / (X^\beta - Y^\alpha)$. Then the following statements are equivalent:
(a) $H$ hat positive Hilbert depth.
(b) For any finitely generated torsionfree $S$-module $N$, it holds that $$\label{eq:HR2}
\frac{H\cdot H_N}{H_R} \geq 0.$$
(c) Condition (b) holds for any finitely generated torsionfree $S$ module of rank $1$.
(d) $H = \sum_{i} h_{i} t^i$ satisfies the condition $$\label{eq:star}
\sum_{i \in I} h_{i+n} \leq \sum_{j \in J} h_{j+n} ~\text{for all }n \in {\mathbb{N}}, [I,J] \in {\mathcal{F}_{\alpha,\beta}}. \tag{$\star$}$$
The equivalence of (a) and (d) is the main theorem of [@MU]. This is also the most difficult part of the result.
Note that the denominator of is $H_R$, in contrast to the $H_S$ in the denominator of . In both cases, it is easy to see that $\frac{H_M H_N}{H_R} \geq 0$ implies that $\frac{H_M H_N}{H_S} \geq 0$. We present two examples to show that this implication is strict in both cases:
1. In the setting of , let $R = {\mathbb{K}}[X_1,X_2,Y_1,Y_2]$, $N = S$ and $H_M = 1 /(1-t_1)$. Then $M$ clearly has Hilbert depth $1$, but $$\frac{H_M H_N }{H_R} =\frac{1}{1-t_1} \cdot\frac{1}{(1-t_1)(1-t_2)} \cdot (1-t_1)^2(1-t_2)^2 = 1-t_2 \ngeq 0.$$
2. In the setting of , let $\alpha = 2, \beta = 3$ and consider the Hilbert series $H_M = 1 + t^3$. The module $M$ has finite length and therefore Hilbert depth $0$, but for any torsionfree $S$-module $N$ it holds that $$\frac{H_M H_N}{H_S} = (1+t^3)\cdot \frac{(1-t^3)(1-t^3)}{1-t^6}H_N = (1-t^2) H_N \geq 0,$$ where for the last inequality, we use that multiplication by $X$ gives an injection $N(-2) \hookrightarrow N$.
We need the following result about the structure of fundamental couples.
\[lemma:fundcouple\] Let $[I,J]$ be a fundamental couple of length $m$. Then there exist two integer sequences $$\begin{aligned}
\beta &> a_0 > a_1 > \dotsb > a_m = 0 \text{ and}\\
0 &= b_0 < b_1 < \dotsb < b_m < \alpha
\end{aligned}$$ such that $$\label{eq:pres}
\begin{aligned}
i_k &= \alpha\beta - a_{k-1} \alpha - b_k \beta &\text{for } 1 \leq k \leq m &\text{ and}\\
j_k &= \alpha\beta - a_k \alpha - b_k \beta &\text{for } 0 \leq k \leq m
\end{aligned}$$
Recall from Rosales, García-Sánchez, García-García and Jiménez-Madrid [@R Lemma 1] that an integer $e \in {\mathbb{Z}}$ is not contained in ${\langle \alpha,\beta \rangle}$ if and only if there are $a,b \in {\mathbb{N}}$ with $a,b \geq 1$ such that $e = \alpha\beta - a \alpha - b \beta$. In this case, $a$ and $b$ are uniquely determined and we denote them by $a(e)$ and $b(e)$, respectively.
Now consider a fundamental couple $[I,J]$. We define $a_{k} := a(j_k)$ and $b_k := b(j_k)$ for $1 \leq k \leq m-1$. Further, $j_0 \equiv 0 \mod{\alpha}$ so it can be written as $j_0 = \alpha\beta - \alpha a_0$ for some $0 \leq a_0 < \beta$. We set $b_0 = 0$, so $j_0 = \alpha\beta - a_{0} \alpha - b_0 \beta$ as required. Similarly, $j_m \equiv 0 \mod{\beta}$, so there exists $0 \leq b_m < \alpha$ such that $j_m = \alpha\beta - a_{m} \alpha - b_m \beta$ with $a_m = 0$.
Next, set $\tilde{\imath}_k := \alpha\beta - a_{k-1} \alpha - b_k \beta$ for $1 \leq k \leq m$. We need to show that $i_k = \tilde{\imath}_k$. For this, note that $i_k \equiv j_k \equiv \tilde{\imath}_k \mod{\alpha}$ and similarly $i_k \equiv \tilde{\imath}_k \mod{\beta}$. Hence $i_k = \tilde{\imath}_k + r \alpha \beta$ for some $r \in {\mathbb{Z}}$. As $\tilde{\imath}_k < \alpha\beta$, it follows that $\tilde{\imath}_k + r\alpha \beta < 0$ for $r < 0$, so we may assume that $r \geq 0$. On the other hand, it $r > 0$, then $$\begin{aligned}
i_k &= \tilde{\imath_k} + r \alpha \beta
= \alpha\beta - a_{k-1} \alpha - b_k \beta + r \alpha \beta\\
&= \alpha\beta - a_{k} \alpha - b_k \beta + \alpha(r \beta - a_{k-1} + a_k) > j_k,
\end{aligned}$$ which is a contradiction. Thus $r = 0$ and so $i_k = \tilde{\imath}_k$ for $1 \leq k \leq m$.
Finally, $i_k < j_k, j_{k-1}$ implies that $a_k < a_{k-1}$ and $b_k > b_{k-1}$ for $1 \leq k \leq m$.
If $H$ has positive Hilbert depth, then it can be written as $$H = \frac{Q_1}{1-t^\alpha} + \frac{Q_2}{1-t^\beta} + \frac{Q_{1,2}}{(1-t^\alpha)(1-t^\beta)}.$$ with $Q_1, Q_2, Q_{1,2} \in {\mathbb{N}}[t^{\pm 1}]$. Using that $N$ is torsionfree, it easily follows that holds for each summand and therefore for $H$.
This is trivial.
Let $[I,J]$ be a fundamental couple. Recall that $S = {\mathbb{K}}[t^\alpha, t^\beta]$ is the monoid algebra of ${\langle \alpha,\beta \rangle}$. Let $N \subseteq {\mathbb{K}}[t]$ be the $S$-module generated by $t^{\alpha\beta - j_0}, \dotsc, t^{\alpha\beta - j_m}$. This module is torsionfree, hence $\frac{H_M H_N}{H_R} \geq 0$ by assumption. To see that this inequality implies , we need to compute the Hilbert series of $N$.
Let $(a_k)_{k=0}^m, (b_k)_{k=0}^m$ be the sequences as in Lemma \[lemma:fundcouple\] and let $$\tilde{N} := (X^{a_0}Y^{b_0}, \dotsc, X^{a_m}Y^{b_m}).$$ It is easy to see that $\tilde{N}$ is the preimage of $N$ under the projection $R \to S$. In particular, note that $X^\beta - Y^\alpha \in \tilde{N}$, because $X^{a_0}, Y^{b_m} \in \tilde{N}$. Hence $N \cong \tilde{N} / (X^\beta - Y^\alpha)$ and thus $H_N = H_{\tilde{N}} - t^{\alpha\beta} H_R$.
By considering the minimal free resolution of $\tilde{N}$, one sees that its syzygies are generated in the degrees $a_{k-1} \alpha + b_{k} \beta$ for $1 \leq k \leq m$. So we can compute the Hilbert series of $N$ as follows: $$\begin{aligned}
\frac{H_N}{H_R} &= \frac{H_{\tilde{N}} - t^{\alpha\beta} H_R}{H_R}
=\sum_{k=0}^m t^{a_k\alpha + b_k\beta} - \sum_{k=1}^m t^{a_{k-1} \alpha + b_k \beta} - t^{\alpha \beta}\\
&=\sum_{k=0}^m t^{\alpha\beta - j_k} - \sum_{k=1}^m t^{\alpha\beta - i_k} - t^{\alpha\beta - i_0}
=t^{\alpha\beta}\left(\sum_{j \in J} t^{-j} - \sum_{i \in I} t^{-i}\right)
\end{aligned}$$ Together with , we obtain the following: $$\begin{aligned}
0 \leq \frac{H\cdot H_N}{H_R} &= (\sum_{n \in {\mathbb{Z}}} h_n t^n) t^{\alpha\beta}\left(\sum_{j \in J} t^{-j} - \sum_{i \in I} t^{-i}\right) \\
&= t^{\alpha\beta} \sum_{n \in {\mathbb{Z}}} t^n \left(\sum_{j \in J} h_{n+j} - \sum_{i \in I} h_{n+i}\right)
\end{aligned}$$ So is satisfied for $[I,J]$.
This is Theorem 3.13 of [@MU].
Acknowledgments {#acknowledgments .unnumbered}
===============
The second author wishes to express his gratitude to the Institute of Mathematics at the University of Osnabrück for kind hospitality.
[^1]: The first author was partially supported by the German Research Council DFG-GRK 1916. The second author was partially supported by the Spanish Government Ministerio de Economía y Competitividad (MINECO), grants MTM2012-36917-C03-03 and MTM2015-65764-C3-2-P, as well as by Universitat Jaume I, grant P1-1B2015-02.
[^2]: In the literature concerned with the Stanley depth, the term “Hilbert depth” refers to a different invariant, see for example Bruns, Krattenthaler and Uliczka [@BKU]. The latter is defined via Hilbert decompositions and so it is sometimes called *decomposition Hilbert depth*. In the standard-${\mathbb{Z}}$-graded setup, these two notions coincide, but in general our Hilbert depth is only an upper bound for the decomposition Hilbert depth.
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author:
- |
Parikshit Gopalan\
VMware Research\
[pgopalan@vmware.com]{}
- |
Roie Levin\
Carnegie Mellon University\
[roiel@cs.cmu.edu]{}[^1]
- |
Udi Wieder\
VMware Research\
[uwieder@vmware.com]{}
bibliography:
- 'refs.bib'
title: Finding Skewed Subcubes Under a Distribution
---
[^1]: Work done while an intern at VMware Research.
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---
abstract: 'We search for signatures of Lorentz and $CPT$ violations in the cosmic microwave background (CMB) temperature and polarization anisotropies by using the WMAP and the 2003 flight of BOOMERANG (B03) data. We note that if the Lorentz and $CPT$ symmetries are broken by a Chern-Simons term in the effective lagrangian, which couples the dual electromagnetic field strength tensor to an external four-vector, the polarization vectors of propagating CMB photons will get rotated. Using the WMAP data alone, one could put an interesting constraint on the size of such a term. Combined with the B03 data, we found that a nonzero rotation angle of the photons is mildly favored: $\Delta \alpha= -6.0^{+4.0}_{-4.0}$ $^{+3.9}_{-3.7}$ deg (1, 2 $\sigma$ ).'
author:
- 'Bo Feng${}^{a,b}$'
- 'Mingzhe Li${}^c$'
- 'Jun-Qing Xia${}^d$'
- 'Xuelei Chen${}^a$'
- 'Xinmin Zhang${}^d$'
title: Searching for CPT Violation with Cosmic Microwave Background Data from WMAP and BOOMERANG
---
After decades of pursuance and many advances in both the theoretical and the observational fronts of cosmological research, a “standard model” of structure formation has been established. It is now possible, with the unprecedented precision of the cosmological observations[@wmap; @sloanx], to have robust tests and effective distinctions of the many theoretical models of new physics.
A possible signature of new physics is the $CPT$ violation. In the standard model of particle physics $CPT$ is an exact symmetry. The detection of $CPT$ violation will reveal new physics beyond the standard model. There have been a number of high precision experimental tests on $CPT$ conservation in the laboratory [@kostelecky]. Now cosmology provides another way to test this important symmetry. Also, breaking of the $CPT$ symmetry may have played an active role in cosmological evolution. For example, $CPT$-violating interactions in the baryons and leptons provide a baryogenesis mechanism where the baryon number asymmetry is produced in thermal equilibrium [@cohenkaplan; @kostelecky2; @li; @steinhardt; @hongli; @alberghi].
In this paper we study the cosmological $CPT$ violation in the photon sector. Phenomenologically, we introduce a Chern-Simons term in the effective lagrangian of the form [@jackiw; @field] \[CFJCS\] [L]{}\_[int]{} \~p\_ A\_\^ , where $p_{\mu}$ is a four vector and $\widetilde{F}^{\mu\nu}=(1/2)\epsilon^{\mu\nu\rho\sigma}
F_{\rho\sigma}$ is the dual of the electromagnetic tensor. The action of (\[CFJCS\]) is gauge invariant if $p_{\mu}$ is a constant and homogeneous vector or the gradient of a scalar. It violates Lorentz and $CPT$ symmetries if the background value of $p_{\mu}$ is non-zero. In the scenario of quintessential baryo/leptogenesis the four vector $p_\mu$ is in the form of the derivative of the quintessence[@quint] scalar, $\partial_{\mu}\phi$. During the evolution of quintessence, the time component of $\partial_{\mu}\phi$ does not vanish, which causes $CPT$ violation. In the scenario of gravitational baryo/leptogenesis [@steinhardt; @hongli], $p_\mu$ is the gradient of a function of Ricci scalar $R$ [@lehnert].
The interaction in (\[CFJCS\]) violates also the $P$ and $CP$ symmetries, as long as $p_{0}$ dose not vanish (e.g., in the case $p_{\mu}$ is the gradient of a time dependent scalar field) [@klinkhamer]. This term can lead to a rotation of the polarization vector of electromagnetic waves when they are propagating over cosmological distances [@jackiw]. This effect is known as “cosmological birefringence”. The change in the position angle of the polarization plane $\Delta\alpha$, which is obtained by observing polarized radiation from distant radio galaxies and quasars, provides a sensitive measure of the strength of the cosmological birefringence, and this has been used to constrain modified electrodynamics [@jackiw; @field; @CF97; @carroll].
In the present paper we use the current CMB polarization data to measure this type of Lorentz- and $CPT$-violating term [*for the first time*]{}. Our results show that the current CMB data provide an interesting indication for a nonzero $p_{\mu}$.
The Stokes parameters Q and U of the CMB polarization can be decomposed into a gradient-like (G) and a curl-like (C) components [@HW97]. If parity is not violated in the temperature/polarization distribution, the TC and GC cross correlation power spectra vanish due to the intrinsic properties of the tensor spherical harmonics. With the presence of cosmological birefringence, the polarization vector of each photon is rotated by an angle $\Delta \alpha$, and one would observe non-zero TC and GC correlations, even if they are zero at the last scattering surface. Denoting the rotated quantities with a prime, one gets [@kamionkowski] $$\label{TC} C_l'^{TC}= C_l^{TG}\sin 2 \Delta
\alpha~$$ and[@FLLZ] $$\label{GC} C_l'^{GC}=
\frac{1}{2}(C_l^{GG}- C_l^{CC})\sin 4 \Delta \alpha ~.$$ On the other hand, the original TG, GG and CC spectra are also modified: $$\begin{aligned}
\label{TG}
C_l'^{TG} &=& C_l^{TG}\cos 2 \Delta \alpha ~,\\
\label{GG}
C_l'^{GG} &=& C_l^{GG}\cos^2 2 \Delta \alpha + C_l^{CC}\sin^2 2 \Delta \alpha~,\\
\label{CC}
C_l'^{CC} &=& C_l^{CC}\cos^2 2 \Delta \alpha + C_l^{GG}\sin^2 2\Delta \alpha~ ~.
\end{aligned}$$
From the above discussion, we see that even with only the TG cross correlation power spectrum (and the TT autocorrelation power spectrum), the Lorentz- and $CPT$-violating term can still be measured. Of course, direct measurements of the TC and GC power spectra would allow more stringent constraints. Indeed, the GC spectrum will be the most sensitive probe of such Lorentz- and $CPT$-violating term [@FLLZ], this is because the GC power spectrum is generated by the rotation of the GG power spectrum, which is a more sensitive probe of the primordial fluctuation than the TT and TG spectra [@kpola].
To break possible degeneracy between this term and the variation of other parameters, we make a global fit to the CMB data with the publicly available Markov Chain Monte Carlo package `cosmomc`[@Lewis:2002ah; @IEMCMC], which has been modified to allow the rotation of the power spectra discussed above, with a new free parameter $\Delta \alpha$. We assume purely adiabatic initial conditions, and impose the flatness condition motivated by inflation. The following set of 8 cosmological parameters are sampled: the Hubble constant $h$, the physical baryon and cold dark matter densities, $\omega_b=\Omega_bh^2$ and $\omega_c=\Omega_ch^2$, the ratio of the sound horizon to the angular diameter distance at decoupling, $\Theta_s$, the scalar spectral index and the overall normalization of the spectrum, $n_s$ and $A_s$, the tensor to scalar ratio of the primordial spectrum $r$, and, finally, the optical depth to reionization, $\tau_r$. We have imposed the Guassian Hubble Space Telescope prior[@freedman]: $h=0.72\pm 0.08$ and also a weak big-bang nucleosynthesis prior[@bbn]: $\Omega_b h^2 = 0.022
\pm 0.002$ (1 $\sigma$). For the other parameters we have adopted flat priors, and in the computation of the CMB spectra we have considered lensing contributions.
When the first version of this paper was being prepared, the available temperature and polarization power spectra included the results from the first-year observation of WMAP [@wmap; @hinshaw; @kogut; @Verde03], and those from the January 2003 Antarctic flight of BOOMERANG (Hereafter B03)[@B03; @B03EE; @TCGC]. The WMAP three year data (WMAP3) has since been released, with significant improvements on the estimate of the TE power at small $\ell$ [@page]. We have repeated our analysis with the new WMAP three year data.
![ (color online). One dimensional constraints on the rotation angle $\Delta \alpha$ from WMAP data alone (Green or light gray line), WMAP and the 2003 flight of BOOMERANG B03 TT, TG, GG and CC(Orange or gray line), and from WMAP and the full B03 observations (TT, TG, GG, CC, TC, GC) (Black line). \[fig:fig1\]](fig1.eps)
In Fig. \[fig:fig1\] we plot our one dimensional constraints on $\Delta \alpha$ from the WMAP data alone, and from the combined WMAP and B03 data. We have assumed that the cosmic rotation is not too large and imposed a flat prior $-\pi/2\le \Delta \alpha \le \pi/2$. The CMB temperature power spectrum remains unchanged with the rotation while the TG power spectrum gets modified, as given by Eq. (\[TG\]). Using the data from WMAP alone, for both the first and three year data set, we obtain a null detection within the error limits. For WMAP3 the 1, 2 $\sigma$ constraints are $\Delta \alpha=
0.0^{+11.6}_{-11.7}$ $^{+5.9}_{-5.9}$ deg. The uncertainty is considerable, as the error bars of the WMAP TG data are relatively large, and TG data are not very sensitive probe. In the likelihood of Fig. \[fig:fig1\] we have gained double peaks, which can be easily understood from Eq.(\[TG\]) due to the symmetry around $\Delta \alpha = 0$.
With the inclusion of the B03 data, the measurement could be improved dramatically. In a first step we also consider the indirect measurements only by including the B03 TT, TG, GG and CC data. We find the constraint on $\Delta \alpha$ becomes a bit more stringent compared with WMAP only, a nonzero $\Delta \alpha$ is slightly favored and the double peaks are still present. When the B03 TC and GC data are also included the degeneracy around $\Delta \alpha = 0$ is broken. We get the 1, 2 $\sigma$ constraints to be $\Delta \alpha=
-6.0^{+4.0}_{-4.0}$ $^{+3.9}_{-3.7}$ deg with WMAP3 and the B03 full data set.
![ (color online). One dimensional constraints on the rotation angle $\Delta \alpha$ from WMAP and the B03 observations, assuming only CMB TC is rotated to be nonzero by $\Delta
\alpha$(Green or gray line) and only GC rotated(Black line). \[fig:fig2\]](fig2.eps)
The covariance matrices of the B03 TC and GC data are correlated. In order to find out what role the TC and GC data play in our fitting respectively, we have made fits with, in one case, only the TC spectrum rotated as Eq. (\[TC\]), and in the other case only the GC spectrum rotated. To make the comparison clear and avoid the problem of convergence we set the flat prior $-1.2 \le \Delta \alpha
\le 0.8$. In Fig. \[fig:fig2\], we plot the resulting one dimensional constraints. In neither case is the likelihood symmetric at $\Delta \alpha = 0$. In the TC rotated only case, the symmetric points are around $\pm \pi/4$, as we can see from Eq.(\[TC\]). Such a symmetry is lost for this narrower prior, but in our global fittings (Fig. \[fig:fig1\]) we have allowed a larger range of $\Delta \alpha$. We find from Fig. \[fig:fig2\] the TC data are very weak in breaking the degeneracy around $\Delta \alpha = 0$, while for GC the rotation is more eminent, where the likelihood in Fig. \[fig:fig2\] is centered around $\Delta \alpha = - \pi/8$. In this fit $\Delta \alpha = 0$ has an excess of $\Delta \chi^2 = 4$, which is disfavored compared with the best fit case.
![ (color online). Joint 2-dimensional posterior probability contour plots in the $\Delta \alpha - \tau_r$ (top left), $\Delta
\alpha - n_S$ (top right), $\Delta \alpha - \log [10^{10} A_S]$ (bottom left) and $\Delta \alpha - r$ (bottom right), showing the $68\%$ and $95\%$ contours from the WMAP + B03 constraints. \[fig:fig3\]](fig3.eps)
The effect of the polarization rotation is degenerate with variation on the amplitude of the primordial spectrum and the tensor to scalar ratio. These parameters are also degenerate with the optical depth of reionization. In Fig. \[fig:fig3\] we plot the joint two-dimensional posterior probability contours of $\Delta \alpha$ with $\tau_r$, $n_S$, $A_S$ and $r$. More precise measurements on these four parameters will help to break the degeneracy on the constraints of cosmic Lorentz and $CPT$ violations discussed here. We have also made fits with a running spectrum index, but found that it does not affect the above results significantly. The inclusion of the matter power spectrum obtained from large scale structure measurements also does not change our constraints on $\Delta \alpha$ significantly.
Previously, the cosmological birefringence effect has been constrained by looking for correlations between the elongation axes and polarization vectors of distant radio galaxies and quasars. The most recent searches yield null results, with an error on $\Delta \alpha$ at the order of $1^{\circ}$ level [@carroll; @field; @CF97]. The typical redshifts of the sources in these searches are of order of unity. Conceivably, for the much greater redshift range between the last scattering surface and the present-day observer, the cumulative effect of cosmological birefringence could be stronger.
It was claimed that the individual B03 CC and CG data are consistent with zero [@B03EE], however, we found that a negative rotation angle is preferred in our combined analysis. It is noteworthy that our result relies mainly on the fact that at $\l\sim 350$, GG power of B03 is positive, CC power is (slightly) positive and GC power is (slightly) negative. Given Eq.(\[GC\]), the GC power spectrum helps to increase the statistical significance on nonzero $\Delta
\alpha$. At present, the only publicly available (polarization) data are the three-year WMAP and the data from a 200-hour flight of the BOOMERANG balloon. In the coming few years, the quantity and quality of the CMB polarization data are likely to be improved rapidly, with the ongoing WMAP observations and many balloon experiments like the BOOMERANG. These would allow better measurements of $\Delta\alpha$.
While nonzero TG and CG power can also be induced by Faraday rotations[@Scannapieco:1997mt] and higher dimensional Lorentz and $CPT$ violating operators[@Gambini:1998it], these are often frequency-dependent, while the effects described here are not[@Pogosian:2001np]. This provides, at least in principle, a way to distinguish between these different effects. The Faraday rotation induced by magnetic field is given by $$\frac{\Delta \alpha}{\rm rad} =8.1\times 10^{5}
\left(\frac{\lambda}{\rm m}\right)^2
\int_{0}^L \left(\frac{B_{\parallel}}{\rm Gs}\right)
\left(\frac{n_e}{\cm^{-3}}\right) \frac{dL}{\rm pc}.$$ If we assume that reionization occurs at $z<20$, then for a global intergalactic magnetic field of $10^{-9}$ Gs, at the frequency of 145 GHz where BOOMERANG operates, the Faraday rotation is only of the order of $10^{-3}$ deg, which is much smaller than the range of $\alpha$ uncertainty distribution and hence insignificant. The apparent rotation might also be due to contamination from foreground emission. In some attempts to obtain CMB temperature and polarization spectra, including those of WMAP, foreground-removing procedure has been applied. For the BOOMERANG experiment, which operates at relatively high frequency, it is believed that the primary CMB polarization signal is dominant, and the contribution of the polarized galactic synchrotron foreground is small [@B03EE], but at present a small contamination can not be ruled out completely. Future multi-wavelength polarization observations would help distinguish this possibility.
We could not yet conclude that $CPT$ is definitely violated if a non-zero $\Delta\alpha$ is detected. However, if such a detection is confirmed, it would certainly raise the possibility of a Lorentz-violating term like that given in Eq.(\[CFJCS\]), or others of the similar form. For example, a term of this form could be due to the interaction between dark energy and the electromagnetic sector, if we take $p_{\mu}$ as $\partial_{\mu}\phi$ with $\phi$ being quintessence field. Thus, the results we obtained can be used to put additional constraints on the behaviors of dynamical dark energy between the redshift range $z\sim 1$ to $z
\sim 1000$.
A Lorentz violation also implies the violation of the equivalence principle. In our case where only a small violation is present, the group velocity of light remains unchanged, and the weak equivalence principle is satisfied. On the other hand, the Einstein equivalence principle is violated, as there would be a split of photon helicities[@Ni:1975vk]. Furthermore, causality is violated for timelike $p_{\mu}$. However this violation is significant only in the regions where the wavelengthes of photons are very large [@causality].
In summary, current cosmological observations have opened a new window for probing new physics. In this paper we show that the current data from WMAP and BOOMERANG might indicate a rotated polarization angle, which can be resulted from the $CPT$ and Lorentz violations. Such a result, if confirmed at greater significance by future observations, would reveal hitherto unknown dynamics of the nature.
We are grateful to T. Montroy for making the B03 TC and GC covariance matrix available [@TCGC] and kind comments. We thank G. Hinshaw and L. Page for helpful correspondences. Our fittings were finished in the Shanghai Supercomputer Center (SSC). We thank K. Ichiki, A. Lewis, H. Peiris, J. Yokoyama and G. Zhao for helpful discussions and F. Klinkhamer and R. Lehnert for comments. This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 10533010, 90303004, 19925523 and by the Ministry of Science and Technology of China under Grant No. NKBRSF G19990754. B. F. is supported by JSPS and M. L. is by the Alexander von Humboldt Foundation. X.C. is supported by the China National Science Fund for Distinguished Scholars (Grant No 10525314).
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abstract: |
The whitespace-discovery problem describes two parties, Alice and Bob, trying to establish a communication channel over one of a given large segment of whitespace channels. Subsets of the channels are occupied in each of the local environments surrounding Alice and Bob, as well as in the global environment between them (Eve). In the absence of a common clock for the two parties, the goal is to devise time-invariant (stationary) strategies minimizing the synchronization time. This emerged from recent applications in discovery of wireless devices.
We model the problem as follows. There are $N$ channels, each of which is open (unoccupied) with probability $p_1,p_2,q$ independently for Alice, Bob and Eve respectively. Further assume that $N \gg 1/(p_1 p_2 q)$ to allow for sufficiently many open channels. Both Alice and Bob can detect which channels are locally open and every time-slot each of them chooses one such channel for an attempted sync. One aims for strategies that, with high probability over the environments, guarantee a shortest possible expected sync time depending only on the $p_i$’s and $q$.
Here we provide a stationary strategy for Alice and Bob with a guaranteed expected sync time of $O(1 / (p_1 p_2 q^2))$ given that each party also has knowledge of $p_1,p_2,q$. When the parties are oblivious of these probabilities, analogous strategies incur a cost of a poly-log factor, i.e. $\tilde{O}(1 / (p_1 p_2 q^2))$. Furthermore, this performance guarantee is essentially optimal as we show that any stationary strategies of Alice and Bob have an expected sync time of at least $\Omega(1/(p_1 p_2 q^2))$.
author:
- '[Yossi Azar[^1]]{}'
title: Optimal whitespace synchronization strategies
---
=1
Introduction {#sec:intro}
============
Consider two parties, Alice and Bob, who wish to establish a communication channel in one out of a segment of $N$ possible channels. Subsets of these channels may already be occupied in the local environments of either Alice or Bob, as well as in the global environment in between them whose users are denoted by Eve. Furthermore, the two parties do not share a common clock and hence one does not know for how long (if at all) the other party has already been trying to communicate. Motivated by applications in discovery of wireless devices, the goal is thus to devise time-invariant strategies that ensure fast synchronization with high probability (w.h.p.) over the environments.
We formalize the above problem as follows. Transmissions between Alice and Bob go over three environments: local ones around Alice and Bob and an additional global one in between them, Eve. Let $A_i,B_i,E_i$ for $i=1,\ldots,N$ be the indicators for whether a given channel is open (unoccupied) in the respective environment. Using local diagnostics Alice knows $A$ yet does not know $B,E$ and analogously Bob knows $B$ but is oblivious of $A,E$. In each time-slot, each party selects a channel to attempt communication on (the environments do not change between time slots). The parties are said to *synchronize* once they select the same channel $i$ that happens to be open in all environments (i.e., $A_i=B_i=E_i=1$). The objective of Alice and Bob is to devise strategies that would minimize their expected synchronization time.
For a concrete setup, let $A_i,B_i,E_i$ be independent Bernoulli variables with probabilities $p_1,p_2,q$ respectively for all $i$, different channels being independent of each other. (In some applications the two parties have knowledge of the environment densities $p_1,p_2,q$ while in others these are unknown.) Alice and Bob then seek strategies whose expected sync time over the environments is minimal.
Suppose that $p_1=p_2=1$ (local environments are fully open) and Alice and Bob use the naive strategy of selecting a channel uniformly over $[N]$ and independently every round. If there are $Q \approx q N$ open channels in the global environment Eve then the probability of syncing in a given round is $Q/N^2 \approx q/N$, implying an expected sync time of about $N / q$ to the very least.
In the above framework it could occur that *all* channels are closed, in which case the parties can never sync; as a result, unless this event is excluded the expected sync time is always infinite. However, since this event has probability at most $(1-p_1 p_2 q)^N \leq \exp(-N p_1 p_2 q)$ it poses no real problem for applications (described in further details later) where $N \gg 1/(p_1 p_2 q)$. In fact, we aim for performance guarantees that depend only on $p_1,p_2,q$ rather than on $N$, hence a natural way to resolve this issue is to extend the set of channels to be infinite, i.e. define $A_i,B_i,E_i$ for every $i\in {\mathbb N}$. (Our results can easily be translated to the finite setting with the appropriate exponential error probabilities.)
A *strategy* is a sequence of probability measures $\{\mu^t\}$ over ${\mathbb N}$, corresponding to a randomized choice of channel for each time-slot $t\geq 1$. Suppose that Alice begins the discovery via the strategy $\mu_a$ whereas Bob begins the synchronization attempt at time $s$ via the strategy $\mu_b$. Let $X_t$ be the indicator for a successful sync at time $t$ and let $X$ be the first time Alice and Bob sync, that is $$\begin{aligned}
{\mathbb{P}}(X_t = 1 \mid A,B,E) &= \sum_j \mu_a^t(j) \mu_b^{s+t}(j) A_j B_j E_j\,,\label{eq-Xt-def}\\
X &= \min\{ t : X_t = 1\}\,.\label{eq-X-def}\end{aligned}$$ The choice of $\mu_a,\mu_b$ aims to minimize ${\mathbb{E}}X$ where the expectation is over $A,B,E$ as well as the randomness of Alice and Bob in applying the strategies $\mu_a,\mu_b$.
Suppose that both Alice and Bob apply the same pair of strategies independently for all rounds, $\mu_a$ and $\mu_b$ respectively. In this special case, given the environments $A,B,E$ the random variable $X$ is geometric with success probability $\sum_j \mu_a(j) \mu_b(j) A_j B_j E_j$, thus the mappings $A\mapsto \mu_a$ and $B\mapsto \mu_b$ should minimize ${\mathbb{E}}X = {\mathbb{E}}\left[\big(\sum_j \mu_a(j) \mu_b(j) A_j B_j E_j\big)^{-1} \right]$.
A crucial fact in our setup is that Alice and Bob have no common clock and no means of telling whether or not their peer is already attempting to communicate (until they eventually synchronize). As such, they are forced to apply a *stationary* strategy, where the law at each time-slot is identical (i.e. $\mu^t \sim \mu^1$ for all $t$). For instance, Alice may choose a single $\mu_a$ and apply it independently in each step (cf. above example). Alternatively, strategies of different time-slots can be highly dependent, e.g. Bob may apply a periodic policy comprising $\mu_b^1,\ldots,\mu_b^n$ and a uniform initial state $s\in[n]$.
The following argument demonstrates that stationary strategies are essentially optimal when there is no common clock between the parties. Suppose that Alice has some finite (arbitrarily long) sequence of strategies $\{\mu_a^t\}_1^{M_a}$ and similarly Bob has a sequence of strategies $\{\mu_b^t\}_1^{M_b}$. With no feedback until any actual synchronization we may assume that the strategies are non-adaptive, i.e. the sequences are determined in advance. Without loss of generality Alice is joining the transmission after Bob has already attempted some $\beta$ rounds of communication, in which case the expected synchronization time is ${\mathbb{E}}_{0,\beta} X$, where ${\mathbb{E}}_{\alpha,\beta} X$ denotes the expectation of $X$ as defined in , using the strategies $\{\mu_a^{t+\alpha}\},\{\mu_b^{t+\beta}\}$. Having no common clock implies that in the worst case scenario (over the state of Bob) the expected time to sync is $\max_{\beta} {\mathbb{E}}_{0,\beta} X$ and it now follows that Bob is better off modifying his strategy into a stationary one by selecting $\beta \in [M_b]$ uniformly at random, leading to an expected synchronization time of $ M_b^{-1} \sum_{\beta}{\mathbb{E}}_{0,\beta} X$.
Optimal synchronization strategies
----------------------------------
Our main result is a recipe for Alice and Bob to devise stationary strategies guaranteeing an optimal expected synchronization time up to an absolute constant factor, assuming they know the environment densities $p_1,p_2,q$ (otherwise the expected sync time is optimal up to a poly-log factor).
\[thm-1\] Consider the synchronization problem with probabilities $p_1,p_2,q$ for the environments $A,B,E$ respectively, and let $X$ denote the expected sync time. The following then holds:
\[thm-1-upper\] There are fixed strategies for Alice and Bob guaranteeing ${\mathbb{E}}X = O(1/(p_1 p_2 q^2))$, namely:
Alice takes $\mu_a \sim {\operatorname{Geom}}(p_2 q/6)$ over her open channels $\{i : A_i=1\}$,
Bob takes $\mu_b \sim {\operatorname{Geom}}(p_1 q/6)$ over his open channels $\{i : B_i = 1\}$.
Furthermore, for any fixed ${\varepsilon}> 0$ there are fixed strategies for Alice and Bob that do not require knowledge of $p_1,p_2,q$ and guarantee ${\mathbb{E}}X = O\Big(\frac{1}{p_1 p_2 q^2}\log^{2+{\varepsilon}}\big(\frac{1}{p_1 p_2 q}\big)\Big) = \tilde{O}\big(\frac{1}{p_1 p_2 q^2}\big)$, obtained by taking $\mu_a(\mbox{$j$-th open $A$ channel}) = \mu_b(\mbox{$j$-th open $B$ channel}) \propto 1/(j\log^{1+{\varepsilon}/2} j)$.
\[thm-1-lower\] The above strategies are essentially optimal as every possible choice of stationary strategies by Alice and Bob satisfies ${\mathbb{E}}X = \Omega(1/(p_1 p_2 q^2))$.
The factor $1/6$ in the parameters of the geometric distributions can be fine-tuned to any smaller (or even slightly larger) fixed $\alpha>0$ affecting the overall expected sync time ${\mathbb{E}}X$ by a multiplicative constant. See Fig. \[fig:sync\] for a numerical evaluation of ${\mathbb{E}}X$ for various values of $\alpha$.
Recall that Alice and Bob must apply stationary strategies in the absence of any common clock or external synchronization device shared by them, a restriction which is essential in many of the applications of wireless discovery protocols. However, whenever a common external clock does happen to be available there may be strategies that achieve improved performance. The next theorem establishes the optimal strategies in this simpler scenario.
\[thm-2\] Consider the synchronization problem with probabilities $p_1,p_2,q$ for the environments $A,B,E$ respectively, and let $X$ denote the expected sync time. If Alice and Bob have access to a common clock then there are non-stationary strategies for them achieving ${\mathbb{E}}X = O(1/(\min\{p_1,p_2\} q))$. Moreover, this is tight as the expected sync time is always $\Omega(1/(\min\{p_1,p_2\}q))$.
Applications in wireless networking and related work
----------------------------------------------------
The motivating application for this work comes from recent developments in wireless networking. In late 2008, the FCC issued a historic ruling permitting the use of unlicensed devices in the unused portions of the UHF spectrum (mainly the part between 512Mhz and 698Mhz), popularly referred to as “whitespaces”. Due to the potential for substantial bandwidth and long transmission ranges, whitespace networks (which are, in a more general context, also frequently referred to as cognitive wireless networks) represent a tremendous opportunity for mobile and wireless communication. One critical equipment imposed by the FCC in its ruling is that whitespace wireless devices must not interfere with incumbents, i.e., the current users of this spectrum (specifically, in the UHF bands, these are TV broadcasters as well as wireless microphones). Hence, these incumbents are considered “primary users” of the spectrum, while whitespace devices are secondary users and are allowed to use the spectrum only opportunistically, whenever no primary user is using it (The FCC mandates whitespace devices to detect the presence of primary users using a combination of sensing techniques and a geo-location database). At any given time, each whitespace device thus has a spectrum map on which some parts are blocked off while others are free to use.
The problem studied in this paper captures exactly the situation in whitespace networks when two nodes $A$ and $B$ seek to discover one another to establish a connection. Each node knows its own free channels on which it can transmit, but it does not know which of these channels may be available at the other node, too. Furthermore, given the larger transmission range in whitespace networks (up to a mile at Wi-Fi transmission power levels), it is likely that the spectrum maps at $A$ and $B$ are similar yet different. For example, a TV broadcast tower is likely to block off a channel for both $A$ and $B$, but a wireless microphone — due to its small transmission power — will prevent only one of the nodes from using a channel.
Thus far, the problem of synchronizing/discovery of whitespace nodes has only been addressed when one of the nodes is a fixed access point (AP) and the other node is a client. Namely, in the framework studied in [@BCMMW] the AP broadcasts on a fixed channel and the client node wishes to scan its local environment and locate this channel efficiently. That setting thus calls for technological solutions (e.g. based on scanning wider channel widths) to allow the client to find the AP channel faster than the approach of searching all possible channels one by one.
To the best of our knowledge, the results in this paper are the first to provide an efficient synchronization scheme in the setting where both nodes are remote clients that may broadcast on any given channel in the whitespace region.
Related work on Rendezvous games
--------------------------------
From a mathematical standpoint, the synchronization problems considered in this paper seem to belong to the field of Rendezvous Search Games. The most familiar problem of this type is known as The Telephone Problem or The Telephone Coordination Game. In the telephone problem each of two players is placed in a distinct room with $n$ telephone lines connecting the rooms. The lines are not labeled and so the players, who wish to communicate with each other, cannot simply use the first line (note that, in comparison, in our setting the channels are labeled and the difficulty in synchronizing is due to the local and global noise).
The optimal strategy in this case, achieving an expectation of $n/2$, is for the first player to pick a random line and continue using it, whereas the second player picks a uniformly random permutation on the lines and try them one by one. However, this strategy requires the players to determine which is the first and which is the second. It is very plausible that such coordination is not possible, in which case we require both players to employ the same strategy.
The obvious solution is for each of them to pick a random line at each turn, which gives an expectation of $n$ turns. It turns out, however, that there are better solutions: Anderson and Weber [@AW] give a solution yielding an expectation of $\approx 0.8288497 n$ and conjecture it’s optimality.
To our knowledge, the two most prominent aspects of our setting, the presence of asymmetric information and the stationarity requirement (stemming from unknown start times) have not been considered in the literature. For example, the Anderson-Weber strategy for the telephone problem is not stationary — it has a period of $n-1$. It would be interesting to see what can be said about the optimal stationary strategies for this and other rendezvous problems. The interested reader is referred to [@ABE; @AG] and the references therein for more information on rendezvous search games.
Analysis of synchronization strategies {#sec:main}
======================================
Proof of Theorem \[thm-1\], upper bound on the sync time
--------------------------------------------------------
Let $\mu_a$ be geometric with mean $(\alpha p_{2}q)^{-1}$ over the open channels for Alice $\{i:A_i=1\}$ and analogously let $\mu_b$ be geometric with mean $(\alpha p_{1}q)^{-1}$ over the open channels for Bob $\{i:B_i=1\}$, where $0<\alpha<1$ will be determined later.
Let $J = \min\{ j : A_j = B_j = E_j = 1\}$ be the minimal channel open in all three environments. Further let $J_a,J_b$ denote the number of locally open channels prior to channel $J$ for Alice and Bob resp., that is $$\begin{aligned}
& J_a = \#\{ j < J : A_j = 1 \}\,,\quad J_b = \#\{ j < J : B_j = 1 \}\,.\end{aligned}$$ Finally, for some integer $k \geq 0$ let $M_k$ denote the event $$\label{eq-max-J0-J1}
k \leq \max\left\{ J_a p_2 q ~,~ J_b p_1 q\right\} < k + 1 \,.$$ Notice that, by definition, Alice gives probability $(1 - \alpha p_2 q)^{j-1} \alpha p_2 q$ to her $j$-th open channel while Bob gives probability $(1 - \alpha p_1 q)^{j-1} \alpha p_1 q$ to his $j$-th open channel. Therefore, on the event $M_k$ we have that in any specific round, channel $J$ is chosen by both players with probability at least $$\begin{aligned}
(1 - \alpha p_1 q)^{\frac{k+1}{p_1 q}} (1- \alpha p_2 q)^{\frac{k+1}{p_2 q}} \alpha^2 p_1 p_2 q^2 \geq
\mathrm{e}^{-4\alpha(k+1)} \alpha^2 p_1 p_2 q^2\,,\end{aligned}$$ where in the last inequality we used the fact that $(1-x)\geq \exp(-2x)$ for all $0 \leq x\leq \frac12$, which will be justified by later choosing $\alpha < \frac12$. Therefore, if $X$ denotes the expected number of rounds required for synchronization, then $$\begin{aligned}
\label{eq-exp-on-Mk}
{\mathbb{E}}[ X \mid M_k ] \leq \mathrm{e}^{4\alpha(k+1)} (\alpha^2 p_1 p_2 q^2)^{-1}\,.\end{aligned}$$ On the other hand, $J_a$ is precisely a geometric variable with the rule ${\mathbb{P}}(J_a=j) = (1-p_2 q)^j p_2 q$ and similarly ${\mathbb{P}}(J_b=j) = (1-p_1 q)^j p_1 q$. Hence, $${\mathbb{P}}(M_k) \leq (1-p_2 q)^{k/(p_2 q)} + (1-p_1 q)^{k/(p_1 q)} \leq 2\mathrm{e}^{-k}\,.$$ Combining this with we deduce that $$\label{eq-Ex-upper-bound}
{\mathbb{E}}X \leq 2 \sum_k \mathrm{e}^{-k} {\mathbb{E}}[X \mid M_k] \leq 2\mathrm{e}^{4\alpha} (\alpha^2 p_1 p_2 q^2)^{-1} \sum_k
\mathrm{e}^{(4\alpha-1)k} \leq \frac{2\mathrm{e}}{\alpha^2\left(\mathrm{e}^{1-4\alpha}-1\right)}\left(p_1 p_2 q^2\right)^{-1}$$ where the last inequality holds for any fixed $\alpha < \frac14$. In particular, a choice of $\alpha = \frac16$ implies that ${\mathbb{E}}X \leq 500/ \left(p_1 p_2 q^2\right)$, as required.
\[rem-p1=p2\] In the special case where $p_1=p_2$ (denoting this probability simply by $p$) one can optimize the choice of constants in the proof above to obtain an upper bound of ${\mathbb{E}}X \leq 27 / (p q)^2$.
Strategies oblivious of the environment densities
-------------------------------------------------
Observe first that if Alice and Bob multiply the parameters of their geometric distributions (as specified in Part of Theorem \[thm-1\]) by some absolute constant $0 < c < 1$ then the upper bound on ${\mathbb{E}}X$ will increase by a factor of at most some absolute $C > 0$.
With this in mind, fix ${\varepsilon}> 0$ and consider the following strategies. Every round, Alice guesses $p_2 q$ to be $\exp(-i)$ for $i=1,2,\ldots$ with probability proportional to $i^{-(1+{\varepsilon}/2)}$, while Bob does the same for $p_1 q$. This way, both Alice and Bob successfully guess these parameters (within a factor of $\mathrm{e}$) with probability at least $$c \log^{-(1+{\varepsilon}/2)}\Big(\frac{1}{p_1 q}\Big) \log^{-(1+{\varepsilon}/2)}\Big(\frac{1}{p_2 q}\Big) \geq c \log^{-(2+{\varepsilon})} \Big(\frac{1}{p_1 p_2 q}\Big)$$ where $c > 0$ is some absolute constant. Hence, restricting our analysis to these rounds only we obtain an expected sync time of at most $O(1/(p_1 p_2 q^2))$ such rounds and overall $${\mathbb{E}}X = O\Big(\frac{1}{p_1 p_2 q^2} \log^{2+{\varepsilon}}\Big(\frac{1}{p_1 p_2 q}\Big)\Big) = \tilde{O}\Big(\frac{1}{p_1 p_2 q^2}\Big)\,.$$
An elementary calculation shows that the above strategy is equivalent to having both Alice and Bob choose their corresponding $j$-th open channel ($j=1,2,\ldots$) with probability proportional to $j \log^{1+{\varepsilon}/2}j $. Indeed, one may repeat the arguments from the previous section directly for these strategies and obtain that (in the original notation) for any given round $${\mathbb{P}}(\mbox{Alice and Bob select $J$}\mid M_k) \geq c p_1 p_2 q^2 \left[ (k+1)^2 \log^{1+{\varepsilon}/2}\Big(\frac{k+1}{p_1 q}\Big)\log^{1+{\varepsilon}/2}\Big(\frac{k+1}{p_2 q}\Big) \right]^{-1}$$ for some absolute constant $c>0$. From this we then infer that $$\begin{aligned}
{\mathbb{E}}X &\leq O\left(1/(p_1 p_2 q^2)\right) \sum_k \mathrm{e}^{-k} (k+1)^2 \log^{1+{\varepsilon}/2}\Big(\frac{k+1}{p_1 q}\Big)\log^{1+{\varepsilon}/2}\Big(\frac{k+1}{p_2 q}\Big) \\
&= O\Big(\frac{1}{p_1 p_2 q^2} \log^{2+{\varepsilon}}\Big(\frac{1}{p_1 p_2 q}\Big)\Big)\,,\end{aligned}$$ as argued above.
Proof of Theorem \[thm-1\], lower bound on the sync time
--------------------------------------------------------
\[thm-R-lower-bound\] Let $\mu_a,\mu_b$ be the stationary distribution of the strategies of Alice and Bob resp., and let $R=\sum_j \mu_a(j) \mu_b(j) A_j B_j E_j$ be the probability of successfully syncing in any specific round. Then there exists some absolute constant $C>0$ such that ${\mathbb{P}}(R<C p_0 p_1 q^2)\ge \frac12$.
Given the environments $A,B$ define $$\begin{aligned}
S^a_k = \{j \,:\, 2^{-k} < \mu_a(j) \le 2^{-k+1} \} \,,\quad
S^b_k = \{j \,:\, 2^{-k} < \mu_b(j) \le 2^{-k+1} \} \,.\end{aligned}$$ Notice that the variables $S^a_k$ are a function of the strategy of Alice which in turn depends on her local environment $A$ (an analogous statement holds for $S^b_k$ and $B$). Further note that clearly $$|S^a_k| < 2^k\quad\mbox{ and }\quad |S^b_k| < 2^k\quad\mbox{ for any $k$}\,.$$ Let $T^a_k$ denote all the channels where the environments excluding Alice’s (i.e., both of the other environments $B,E$) are open, and similarly let $T^b_k$ denote the analogous quantity for Bob: $$T^a_k = \{j\in S^a_k \,:\, B_j=E_j=1 \}\,,\quad
T^b_k = \{j\in S^b_k \,:\, A_j=E_j=1 \}\,.$$ Obviously, ${\mathbb{E}}|T^a_k| < 2^k p_2 q$ and ${\mathbb{E}}|T^b_k| < 2^k p_1 q$.
Since $\{B_j\}_{j\in {\mathbb N}}$ and $\{E_j\}_{j\in {\mathbb N}}$ are independent of $S^a_k$ (and of each other), for any $\beta>0$ we can use the Chernoff bound (see, e.g., [@JLR]\*[Theorem 2.1]{} and [@AS]\*[Appendix A]{}) with a deviation of $t = (\beta-1)2^k p_2 q$ from the expectation to get $${\mathbb{P}}\left(|T^a_k|>\beta 2^k p_2 q \right) < \exp\bigg(- \frac32 \frac{(\beta-1)^2}{\beta+2} 2^k p_2 q\bigg) \,,$$ and analogously for Bob we have $${\mathbb{P}}\left(|T^b_k|>\beta 2^k p_1 q \right) < \exp\bigg(- \frac32 \frac{(\beta-1)^2}{\beta+2} 2^k p_1 q\bigg) \,.$$ Clearly, setting $$K_a = \log_2(1/(p_2 q))-3 \,,\quad K_b = \log_2(1/(p_1 q))-3$$ and taking $\beta$ large enough (e.g., $\beta=20$ would suffice) we get $$\begin{aligned}
\label{eq-Tk-large1} {\mathbb{P}}\left( \mbox{$\bigcup$}_{k \geq K_a}\left\{ |T^a_k|> \beta 2^k p_2 q\right\} \right)
\leq 2\, {\mathbb{P}}\left(|T^a_{K_a}|>\beta 2^{K_a} p_2 q \right) < \frac18\end{aligned}$$ and $$\begin{aligned}
\label{eq-Tk-large2} {\mathbb{P}}\left( \mbox{$\bigcup$}_{k \geq K_b}\left\{ |T^b_k|> \beta 2^k p_1 q\right\} \right) < \frac18\,.
\end{aligned}$$ Also, since $\sum_{k < K_a} |S^a_k| < 2^{K_a} \le (8 p_2 q)^{-1}$ and similarly $\sum_{k < K_b} |S^b_k| < 2^{K_b} \le (8p_1 q)^{-1}$, we have by Markov’s inequality that $$\label{eq-Tk-empty1}
{\mathbb{P}}\left( \mbox{$\bigcup$}_{k < K_a}\left\{ |T^a_k| > 0\right\} \right) \leq
\sum_{k<K_a} {\mathbb{E}}|T_k^a| = p_2 q \sum_{k<K_a} {\mathbb{E}}|S_k^a| < \frac18$$ and similarly $$\label{eq-Tk-empty2}
{\mathbb{P}}\left( \mbox{$\bigcup$}_{k < K_b}\left\{ |T^b_k| > 0\right\} \right) < \frac18\,.$$ Putting together ,,,, with probability at least $\frac12$ the following holds: $$\label{eq-control-Tk}
|T_k^a| \leq \left\{\begin{array}{ll}
\beta 2^k p_2 q & k \geq K_a \\
0 & k < K_a\end{array}\right.\,,\quad
|T_k^b| \leq \left\{\begin{array}{ll}
\beta 2^k p_1 q & k \geq K_b \\
0 & k < K_b\end{array}\right.
\quad\mbox{ for all $k$.}$$ When holds we can bound $R$ as follows: $$\begin{aligned}
R&=\sum_j \mu_a(j) \mu_b(j) A_j B_j E_j =\sum_k \sum_\ell \sum_{j\in T^a_k \cap T^b_\ell} \mu_a(j) \mu_b(j)
\le \sum_k \sum_\ell |T^a_k \cap T^b_\ell| 2^{-k+1} 2^{-\ell+1} \\
&\le \sum_k \sum_\ell \sqrt{|T^a_k|\, |T^b_\ell|} 2^{-k+1} 2^{-\ell+1} = 4\bigg(\sum_k \sqrt{|T_k^a|}2^{-k}\bigg) \bigg(\sum_\ell \sqrt{|T_\ell^b|}2^{-\ell}\bigg) \\
&\leq 4 \beta (p_1 p_2)^{1/2} q \bigg(\sum_{k\ge K_a} 2^{-k/2}\bigg) \bigg(\sum_{\ell\geq K_b} 2^{-\ell/2}\bigg) \,,\end{aligned}$$ where the second inequality used the fact that $|F_1 \cap F_2| \leq \min\{|F_1|,|F_2|\}\leq \sqrt{|F_1||F_2|}$ for any two finite sets $F_1,F_2$ and the last inequality applied . From here the proof is concluded by observing that $$\begin{aligned}
R &\leq 16 (p_1 p_2)^{1/2} q 2^{-K_a/2} 2^{-K_b/2} = 128 \beta p_1 p_2 q^2\,.\qedhere\end{aligned}$$
\[cor-stat-lower-bound\] There exists some absolute $c>0$ such that for any pair of stationary strategies, the expected number of rounds required for a successful synchronization is at least $c/(p_1 p_2 q^2)$.
Recall that the success probability in any specific round, given the environments and strategies, is precisely $R$. Hence, conditioned on the value of $R$, the probability of synchronizing in one of the first $1/(2 R)$ rounds is at most $\frac12$. Theorem \[thm-R-lower-bound\] established that with probability at least $\frac12$ we have $R < C p_1 p_2 q^2$, therefore altogether with probability at least $\frac14$ there is no synchronizing before time $(2Cp_1 p_2 q^2)^{-1}$. We conclude that the statement of the corollary holds with $c = 1/(8C)$.
Proof of Theorem \[thm-2\]
--------------------------
Partition the channels into infinitely many sets of infinitely many channels each. On the $i$-th round, Alice chooses the first channel in the $i$-th set which is open in her environment. Bob does likewise.
Consider the probability that both parties choose the same channel: Each channel has probability $1-(1-p_1)(1-p_2)=p_1+p_2-p_1 p_2$ of being open to Alice or Bob and probability $p_1 p_2$ of being open to both. Hence, the probability that the first channel open to either is open to both is $$\frac{p_1 p_2}{p_1+p_2 - p_1 p_2} \ge \min\{p_1, p_2\}/2 \, .$$
If indeed both players chose the same channel at some round, it is necessarily open for both of them and with probability $q$ it is also open in the global environment. Hence, the probability of success at each round is at least $\min\{p_1,p_2\}q/2$. Different rounds use disjoint sets of channels, so the event of success at different rounds are independent and the number of round to success has a geometric distribution with expectation ${\mathbb{E}}X \leq 2/(\min\{p_1,p_2\}q)$.
For a lower bound of matching order observe the following: For any strategy Alice might employ and at any given round, the probability that the channel she chooses is open for both Bob and Eve is $p_2 q$, and this is an upper bound for the probability of success. Similar argument for Bob yields that the probability of success at any given round is at most $\min\{p_1,p_2\}q$. A straightforward first moment argument (as in the proof of Corollary \[cor-stat-lower-bound\]) now implies that ${\mathbb{E}}X\ge 1/(4\min\{p_1,p_2\}q)$.
This completes the proof.
[^1]: Department of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email address: [azar@tau.ac.il]{}.
|
---
author:
- 'L. Decin[^1]'
- 'B. Vandenbussche'
- 'C. Waelkens'
- 'K. Eriksson'
- 'B. Gustafsson'
- 'B. Plez'
- 'A.J. Sauval'
- 'K. Hinkle'
date: 'Received data; accepted date'
subtitle: 'II. General results'
title: 'ISO-SWS calibration and the accurate modelling of cool-star atmospheres [^2]'
---
Introduction
============
For the astronomical community analysing ISO-SWS data , a first point to assess when judging and qualifying their data concerns the flux calibration accuracy. Since the calibration process is not straightforward, knowledge on the [*[full]{}*]{} calibration process and on the still remaining calibration problems is crucial when processing the data.
One way to detect calibration problems is by comparing observed data with theoretical predictions of a whole sample of standard calibration sources. But, as explained in (hereafter referred to as Paper I) a full exploitation of the ISO-SWS data may only result from an iterative process in which both new theoretical developments on the computation of stellar spectra — based on the MARCS and Turbospectrum code , version May 1998 — and more accurate instrumental calibration are involved.
Precisely because this research entails an iterative process, one has to be extremely careful not to confuse technical detector problems with astrophysical issues. Therefore, the analysis in its entirety encloses several steps. Some steps have already been demonstrated in the case of $\alpha$ Tau in . They will be summarised in Sect. \[summary\]. Other points will be introduced in Sect.\[summary\] and will be elaborated on in the first sections of this article (Sect. \[sample\] – \[highres\]). Having described the method of analysis, the general discrepancies between observed and synthetic spectra are subjected to a careful scrutiny in order to elucidate their origin. At this point, a distinction can be made between discrepancies typically for [*[warm]{}*]{} stars and those typical for [*[cool]{}*]{} stars. For this research, [*[warm]{}*]{} stars are defined as being hotter than the Sun (T$_{\mathrm{eff},\odot} = 5770$ K) and their infrared spectra are mainly dominated by atomic lines, while molecular lines are characteristic of cool star spectra. A description on the general trends in the discrepancies for [*[warm]{}*]{} and [*[cool]{}*]{} stars will be made in this paper, while each star of the sample will be discussed individually in two forthcoming papers in which also an overview of other published stellar parameters will be given.
As stated in , the detailed spectroscopic analysis of the ISO-SWS data has till now been restricted to the wavelength region from 2.38 to 12[$\mu$m]{}. So, if not specified, the wavelength range under research is limited to band 1 (2.38 – 4.08[$\mu$m]{}) and band 2 (4.08 – 12.00[$\mu$m]{}). Band 3 (12.00 – 29.00[$\mu$m]{}) will be elaborated on by Van Malderen (Van Malderen et al., 2001, in prep.).
This paper is organised as follows: in Sect. \[summary\] the general method of analysis is summarised. The sample of ISO-SWS observations is described in Sect. \[sample\], while the data reduction procedure is discussed in Sect. \[datareduction\]. The observations of two independent instruments are introduced in Sect. \[highres\]. In Sect. \[results\], the results are elaborated on. In the last section, Sect. \[impact\], the impact on the calibration of ISO-SWS and on the theoretical modelling is given.
The appendix of this article is published electronically. Most of the grey-scale plots in the article are printed in colour in the appendix, in order to better distinguish the different spectra.
Description of the strategy {#summary}
===========================
Since this research includes an iterative process, one has to be very careful not to introduce (and so to propagate) any errors. The strategy in its entirety therefore encompasses a number of steps, including (1.) a spectral coverage of standard infrared sources from A0 to M8, (2.) a homogeneous data reduction, (3.) a detailed literature study, (4.) a detailed knowledge of the impact of the various parameters on the spectral signature, (5.) a statistical method to test the goodness-of-fit (Kolmogorov-Smirnov test) and (6.) high-resolution observations with two independent instruments. Some points (in particular, points 4 and 5) have already been demonstrated in the case of $\alpha$ Tau in Paper I. Points 1, 2 and 6 will be elaborated on in Sect. \[sample\], Sect. \[datareduction\] and Sect.\[highres\] respectively.
In its totality, the general method of analysis — based on these 6 points — may be summarised as follows:\
a large set of standard stars (A0 – M8) has been observed with ISO-SWS (Sect. \[sample\]). The observational data first have been subjected to a homogeneous data-reduction procedure (Sect.\[datareduction\]). Thereafter, the carefully reduced ISO-SWS data of one [*warm*]{} and one [*cool*]{} star were compared with the observational data of two independent instruments (FTS-KP and FTS-ATMOS, see Sect. \[highres\]). This step is very crucial, since this is the only secure and decisive way to point out calibration problems with the detectors of ISO-SWS. The complete observational data-set, covering a broad parameter space, was then compared with theoretical predictions. By knowing already some problematic points in the calibration of ISO-SWS, these comparisons led both to a refinement of our knowledge on the calibration problems and to a determination of theoretical modelling problems (Sect.\[results\]).
The knowledge on the relative importance of the different molecules [[^3]]{} — displaying their characteristic absorption pattern somewhere in the broad ISO-SWS wavelength-range — and on the impact of the various stellar parameters on the infrared spectrum enabled us also to determine the fundamental stellar parameters for the cool giants in our sample . Due to severe calibration problems with the band-2 data (see Sect.\[datareduction\]), only band-1 data were used for this part of the process. Once a high-level of agreement between observed and synthetic data was reached, a statistical test was needed to objectively judge on the different synthetic spectra. A choice was made for the Kolmogorov-Smirnov test . This statistical test [*[globally]{}*]{} checks the goodness-of-fit of the observed and synthetic spectra by computing a deviation estimating parameter $\beta$ . The lower the $\beta$-value, the better the accordance between the observed data and the synthetic spectrum.
Using this method, the effective temperature, gravity, metallicity, microturbulent velocity together with the abundance of C, N and O and the [$\mathrm{^{12}C/^{13}C}$]{}-ratio were estimated for the cool stars. From the energy distribution of the synthetic spectrum between 2.38 and 4.08[$\mu$m]{} and the absolute flux-values in this wavelength range of the ISO-SWS spectrum, the angular diameter was deduced. We therefore have minimised the residual sum of squares $$\sum\limits_{i=1}^{n} (f(i) - g(i))^2\,,
\nonumber$$ with $f(i)$ and $g(i)$ representing respectively an observational and a synthetic data point at the $i$th wavelength point.
The error bars on the atmospheric parameters were estimated from 1. the intrinsic uncertainty on the synthetic spectrum (i.e. the possibility to distinguish different synthetic spectra at a specific resolution, i.e. there should be a significant difference in $\beta$-values) which is thus dependent on both the resolving power of the observation and the specific values of the fundamental parameters, 2. the uncertainty on the ISO-SWS spectrum which is directly related to the S/N of the ISO-SWS observation, 3. the value of the $\beta$-parameters in the Kolmogorov-Smirnov test and 4.the still remaining discrepancies between observed and synthetic spectra.
It should be noted that an error on the effective temperature introduces an error on the angular diameter. The IR flux of the [*cool*]{} giants does not follow the Rayleigh-Jeans law for a black-body, and we can write ${\cal{F}}_{\mathrm{obs}}^{\lambda} \propto \mathrm{T_{eff}}^q \cdot
\theta_d^2$. Thus, with $\theta_d \propto
{{\cal{F}}_{\mathrm{obs}}^{\lambda}}^{1/2} \cdot \mathrm{T_{eff}}^{-q/2}$ and $\sigma_{<{\cal{F}}>} = 0.10 {\cal{F}}$, one obtains $$\frac{\sigma_{<\theta_d>}}{\theta_d} = \sqrt{\frac{1}{4} (0.10)^2 +
\frac{q^2}{4}
\frac{\sigma^2_{<\mathrm{T_{eff}}>}}{\mathrm{T_{eff}}^2} +
\sigma^2_{\mathrm{int}}}\,,
\label{angdiamq}$$ with $\sigma^2_{\mathrm{int}}$ being the intrinsic uncertainty on the angular diameter (i.e. the amount by which one may change the angular diameter without significant difference in the residual sum of squares). The uncertainty on the angular diameter (Eq. (\[angdiamq\])) is mainly determined by the uncertainty in the absolute flux (the first term in Eq. (\[angdiamq\])), resulting in almost the same percentage errors in [$\theta_d$]{}. Using the ISO-SWS observational data of these cool giants, we could demonstrate that for 3600K $\le$ [T$_{\mathrm{eff}}$]{}$\le$ 4400K [@Decinthesis]. With $q \approx
1.3$, $\sigma_{<\mathrm{T_{eff}}>} = 70$K and [T$_{\mathrm{eff}}$]{} = 3850K, the uncertainty increases with a factor 1.008 compared to the $q = 1$ situation.
From the angular diameter and the parallax measurements (mas) from Hipparcos , the stellar radius was derived. This radius, together with the gravity — determined from the ISO-SWS spectrum — then yielded the gravity-inferred mass. From the radius and the effective temperature, the stellar luminosity could be extracted.
This method of analysis could however not be applied to the [*[warm]{}*]{} stars of the sample. Absorption by atoms determines the spectrum of these stars. It turned out to be unfeasible to determine the effective temperature, gravity, microturbulence, metallicity and abundances of the chemical elements from the ISO-SWS spectra of these [*[warm]{}*]{} stars, due to
- [problems with atomic oscillator strengths in the infrared (see Sect. \[hot\]),]{}
- [the small dependence of the IR continuum on the fundamental parameters (when changed within their uncertainty),]{}
- [the small dependence of the observed atomic line strength on the fundamental parameters, and]{}
- [the absence of molecules, which are each of them specifically dependent on the various stellar parameters.]{}
Therefore, good-quality published stellar parameters were used to compute the theoretical model and corresponding synthetic spectrum. The angular diameter was deduced directly from the ISO-SWS spectrum, which then yielded — in conjunction with the assumed parallax, gravity and effective temperature — the stellar radius, mass and luminosity.
Description of the sample {#sample}
=========================
It seems important to cover a broad parameter space in order to be able to distinguish between calibration problems and problems related to the computation of a theoretical model and/or to the generation of a synthetic spectrum. Therefore, stellar standard candles spanning the spectral types A0 – M8 were observed. The observations were obtained in the context of two ISO-SWS proposals. Some calibration data were also provided by the SIDT (SWS Instrument Dedicated Team) in the framework of a quick-feedback refining of the model SEDs used for the SWS calibration. In total, full-resolution SWS scans of 20 standard stars, covering the full A0 – M8 spectral range were obtained. Stars with an earlier spectral type were included in the proposals of S. Price. An overview of the objects, observing dates and integration times ($t_{\mathrm{int}}$) can be found in Tables \[obs\] – \[obs2\]. Stars indicated by a ‘$\bullet$’ are stars which have been scrutinised more carefully than stars indicated by a ‘$\circ$’. We first focused on $\alpha$ Bootes (K2 IIIp), because of its well-known stellar parameters and the high quality of the ISO-SWS data for this object [@Decin1997fiso.work..185D]. From then on, we have gone, step by step, towards both higher and lower temperatures. Why the two warmest stars in our sample (being Vega and Sirius) have been considered, but have not been studied into all detail, will be explained in Sect. \[hot\]. The other ‘$\circ$’-stars in Table \[obs2\] are stars cooler than an M2 giant — i.e. cooler than $\sim 3500$ K. Calibration problems with $\gamma$ Cru, variability [@Monnier1998ApJ...502..833M], the possible presence of a circumstellar envelope, stellar winds or a warm molecular envelope above the photosphere made the use of hydrostatic models for these stars inappropriate and have led to the decision to postpone the modelling of the coolest stars in the sample.
[lrlrclll]{}
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name & HD & HR & HIC & Spectral & AOT mode & $t_{\mathrm{int}}$ & revo-\
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& & & & Type & (speed) & \[sec\] & lution\
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$\circ$ $\alpha$ Lyr & 172167 & 7001 & 91262 & A0V & AOT01 (3) & 3462 & 178\
Vega & & & & & AOT06 & 5642 & 650$^{*1}$\
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& & & & & AOT06 & 4354 & 678$^{*2}$\
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$\circ$ $\alpha$ CMa & 48915 & 2491 & 32349 & A1V & AOT01 (4) & 6538 & 689\
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Sirius & & & & & AOT01 (1) & 1140 & 868\
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$\bullet$ $\beta$ Leo & 102647 & 4534 & 57632 & A3Vv & AOT01 (3) & 3462 & 189\
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Denebola & & & & & AOT01 (1) & 1096 & 040\
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$\bullet$ $\alpha$ Car & & & & & & &\
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Canopus & [\[0pt\][45348]{}]{} & [\[0pt\][2326]{}]{} & [\[0pt\][30438]{}]{} & [\[0pt\][F0II]{}]{} & [\[0pt\][AOT01 (4)]{}]{} & [\[0pt\][6538]{}]{} & [\[0pt\][729]{}]{}\
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$\bullet$ $\alpha$ Cen A & 128620 & 5459 & 71683 & G2V & AOT01 (4) & 6538 & 607\
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& & & & & AOT01 (1) & 1140 & 294\
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$\bullet$ $\delta$ Dra & 180711 & 7310 & 94376 & G9III & AOT1 (4) & 6538 & 206\
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& & & & & AOT01 (4) & 6528 & 072\
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$\bullet$ $\xi$ Dra & 163588 & 6688 & 87585 & K2III & AOT01 (3) & 3454 & 314\
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& & & & & AOT01 (1) & 1044 & 068\
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$\bullet$ $\alpha$ Boo & 124897 & 5340 & 69673 & K2IIIp & AOT01 (4) & 6538 & 452\
Arcturus & & & & & AOT01 (4) & 6528 & 071\
& & & & & AOT01 (1) & 1140 & 275\
& & & & & AOT01 (1) & 1094 & 056\
& & & & & AOT06 & 3904 & 583$^{*3}$\
& & & & & AOT06 & 4720 & 601$^{*4}$\
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& & & & & AOT06 & 4510 & 608$^{*5}$\
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$\bullet$ $\alpha$ Tuc & 211416 & 8502 & 110130 & K3III & AOT01 (4) & 6539 & 866\
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$\bullet$ $\beta$ UMi & 131873 & 5563 & 72607 & K4III & AOT01 (4) & 6546 & 182\
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Kochab & & & & & AOT01 (2) & 1816 & 079\
[lrrrclll]{}
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name & HD & HR & HIC & Spectral & AOT mode & $T_{\mathrm{int}}$ & revo-\
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& & & & Type & (speed) & \[sec\] & lution\
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$\bullet$ $\gamma$ Dra & 164058 & 6705 & 87833 & K5III & AOT01 (4) & 6538 & 377\
Etamin & & & & & AOT01 (4) & 6542 & 040\
& & & & & AOT01 (2) & 1912 & 811\
& & & & & AOT01 (1) & 1140 & 496\
& & & & & AOT01 (1) & 1062 & 126\
& & & & & AOT06 & 4568 & 501$^{*6}$\
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& & & & & AOT06 & 5676 & 559$^{*7}$\
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$\bullet$ $\alpha$ Tau & 29139 & 1457 & 21421 & K5III & AOT01(4) & 6538 & 636\
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Aldebaran & & & & & AOT06$^{*8}$ & 3700 & 681\
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$\bullet$ H Sco & 149447 & 6166 & 81304 & K6III & AOT01(4) & 6538 & 847\
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$\bullet$ $\beta$ And & 6860 & 337 & 5447 & M0III& AOT01 (4) & 6538 & 795\
Mirach & & & & & AOT01 (3) & 3454 & 440\
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& & & & & AOT01 (2) & 1912 & 423\
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$\bullet$ $\alpha$ Cet & 18884 & 911 & 14135 & M2III& AOT01 (4) & 6538 & 797\
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Menkar & & & & & AOT01 (4) & 6538 & 806\
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$\bullet$ $\beta$ Peg & 217906 & 8775 & 113881 & M2.5III & AOT01(4) & 6538 & 551\
Scheat & & & & & AOT01 (3) & 3454 & 206\
& & & & & AOT01 (1) & 1140 & 206\
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& & & & & AOT01 (1) & 1096 & 056\
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$\circ$ $\gamma$ Cru & 108903 & 4763 & 61084 & M4III & AOT01(4) & 6538 & 609\
& & & & & AOT01 (2) & 1912 & 258\
& & & & & AOT01 (2) & 1816 & 079\
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& & & & & AOT06 & 5630 & 643$^{*9}$\
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$\circ$ $\beta$ Gru & 214952 & 8636 & 112122 & M5III & AOT01 (3) & 3454 & 538\
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$\circ$ g Her & 148783 & 6146 & 80704 & M6III & AOT01 (4) & 6538 & 800\
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$\circ$ T Mic & 194676 & & 100935 & M7/8III & AOT01 (4) & 6538 & 872\
Data reduction {#datareduction}
==============
In order to reveal calibration problems, the ISO-SWS data have to be reduced in a homogeneous way. For all the stars in our sample, at least one AOT01 observation[[^4]]{} (AOT=Astronomical Observation Template; AOT01= a single up-down scan for each aperture with four possible scan speeds at degraded resolution) is available, some stars have also been observed using the AOT06 mode (=long up-down scan at full instrumental resolution). Since these AOT01 observations form a complete and consistent set, they were used as the basis for the research. In order to check potential calibration problems, the AOT06 data are used. The scanner speed of the highest-quality AOT01 observations was 3 or 4, resulting in a resolving power $\simeq$ 870 or $\simeq$ 1500, respectively [@Leech2002]. The appropriate resolving power of each sub-band was taken to be the most conservative theoretical resolving power as determined by Lorente in @Leech2002, with the exception being band 1A for which this value has been changed from 1500 to 1300, as will be discussed in Sect.\[cool\].
The ISO-SWS data were processed to a calibrated spectrum by using the same procedure as described in using the calibration files available in OLP6.0.
The band 2 (Si:Ga) detectors used in SWS ‘remember’ their previous illumination history. Going from low to high illumination, or vice versa, results in detectors asymptotically reaching their new output value. These are referred to as memory effects or transients. For sources with fluxes greater than about 100Jy, memory effects cause the up and down scans in the SPD (=Standard Processed Data) to differ in response by up to 20% in band 2. Since an adapted version of the Fouks-Schubert model to correct for these memory effects in band 2 was still in development [@Leech2002], this method could not be applied during our reduction procedure. Instead, we have used the down-scan data of our observation as a reference to do a correction of the flux level of the first scan (up-scan). This is justified since the memory effects appear to be less severe in the down-scan measurements, suggesting a more stabilised response to the flux level for the down-scan data.
Also the band 2 dark current subtraction is closely tied to the band 2 memory effect correction. The memory effect for Si:Ga detectors as described by the Fouks-Schubert model is an additive effect. As such, its proper correction will take place during the dark current subtraction. Since this correction tool was still not available, all dark currents were checked individually. When a dark current was corrupted too much by memory effects, its value was replaced by the value of a preceding or following dark-current not being affected. In this way, a small error can occur, which is, however, negligible due to the high flux level of our stellar sources.
[ccccccccrrr]{}
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& 1A & 1B & 1D & 1E & 2A & 2B & 2C & rev. & dy & dz\
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$\lambda / \Delta \lambda$ & 1300 & 1200 & 1500 & 1000 & 1200 & 800 & 800 & & &\
$\lambda_b$ \[[$\mu$m]{}\] & 2.38 & 2.60 & 3.02 & 3.52 & 4.08 & 5.30 & 7.00 & & &\
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$\lambda_e$ \[[$\mu$m]{}\] & 2.60 & 3.02 & 3.52 & 4.08 & 5.30 & 7.00 & 12.00 & & &\
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$\alpha$ Lyr & 1.06 & 1.06 & 1.00 & 1.00 & 1.00 & 0.97 & [*[+12Jy]{}*]{} & 178 & $-$0.608 & $-$1.179\
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$\alpha$ CMa & 1.00 & 1.00 & 1.00 & 0.995 & 1.12 & 1.23 &1.00 & 689 & 0.034 & 0.003\
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$\beta$ Leo & 0.99 & 0.99 & 1.00 & 1.00 & 1.16 & 1.27 &[*[+3.5Jy]{}*]{} & 189 & 0.478 & 0.556\
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$\alpha$ Car & 0.97 & 0.98 & 1.00 & 1.00 & 0.98 & 1.10 & 0.91 & 729 & 0.024 & 0.072\
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$\alpha$ Cen A & 1.01 & 1.02 & 1.00 & 1.01 & 0.985 & 1.06 & 0.91 & 607 & 0.000 & 0.000\
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$\delta$ Dra & 0.97 & 0.98 & 1.00 & 1.015 & 1.03 & 1.02 & 1.10 & 206 & -0.422 & 1.480\
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$\xi$ Dra & 0.99 & 0.99 & 1.00 & 0.99 & 1.12 & 1.15 & 1.05 & 314 & 1.286 & $-0.282$\
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$\alpha$ Boo & 0.995 & 1.01 & 1.00 & 1.005 & 0.95 & 1.05 & 1.00 & 452 & 0.000 & 0.000\
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$\alpha$ Tuc & 1.005 & 1.02 & 1.00 & 1.01 & 1.05 & 1.00 & 1.00 & 866 & 0.000 & 0.000\
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$\beta$ UMi & 1.00 & 1.015 & 1.00 & 1.01 & 0.91 & 0.885 & 1.00 & 182 & $-1.062$ & 0.045\
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$\gamma$ Dra & 0.995 & 1.005 & 1.00 & 1.005 & 0.935 & 0.98 & 0.91 & 377 & $-0.304$ & 0.181\
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$\alpha$ Tau & 1.00 & 1.01 & 1.00 & 1.00 & 1.00 & 1.045 & 1.00 & 636 & 0.000 & 0.000\
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H Sco & 1.00 & 1.015 & 1.00 & 1.00 & 1.13 & 1.05 & 1.15 & 847 & 0.000 & 0.000\
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$\beta$ And & 1.00 & 1.00 & 1.00 & 1.005 & 1.00 & 1.10 & 0.95 & 795 & 0.000 & 0.000\
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$\alpha$ Cet & 0.985 & 1.00 & 1.00 & 1.01 & 0.935 & 1.03 & 0.91 & 797& 0.000 & 0.000\
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$\beta$ Peg & 1.00 & 1.015 & 1.00 & 1.005 & 0.935 & 1.03 & 0.935 & 551 & 0.017 & 0.095\
Since the different sub-band spectra can show jumps in flux at the band-edges, several sub-bands had to be multiplied by a small factor to construct a smooth spectrum. Three causes for the observed shift factors between different sub-bands of an observation and between different observations of a given stellar source can be reported: 1. pointing errors, 2. problems with the RSRF correction, and 3. a problematic dark current subtraction, from which the pointing errors are believed to have the largest impact. The pointing errors as well as the RSRF correction causes a decrease in flux by a gain factor, while the dark current subtraction can lower the flux level by an offset. As the effects of the pointing errors are estimated to have the biggest effect, and since the stars in our sample have a high flux level so that the dark current subtraction only plays a marginal role, the individual sub-bands were multiplied with a factor — rather than shifted with an offset — in order to obtain a smooth spectrum. These factors (see Table \[factors\]) were determined by using the overlap regions of the different sub-bands and by studying the other SWS observations. The band-1D data were taken as reference data, due to the absence of strong molecular absorption in this wavelength range which may cause a higher standard deviation in the bins obtained when rebinning the oversampled spectrum, and — most importantly — due to the low systematic errors in this band, caused by e.g. errors in the curve of the RSRF, detector noise, uncertainties in the conversion factors from $\mu$V/s to Jy, ... [@Leech2002]. Using the total absolute uncertainty values — which have accumulated factors from each of the calibration steps plus estimated contributions from processes which were unprobed or uncorrected — as given in Table 5.3 in @Leech2002, the estimated 1$\sigma$ uncertainty on these factors is 10%. As is clearly visible from Table \[factors\], these factors do not show any trend with spectral type or flux-level. This is displayed in Fig.\[ratiosfig\], where the band-border ratios between 1A-1B, 1B-1D and 1D-1E are plotted in function of the flux at 2.60, 3.02 and 3.52 $\mu$m respectively. For this plot, all the observations of the cool stars in our sample, discussed in the Appendix of Paper IV of this series, are used. In band 1, the band-border ratios of 1A-1B and 1D-1E are from bands within the same aperture. Going from band 1B to band 1D, the aperture changes. Satellite mispointings can have a pernicious impact on this band-border ratio: the mean deviation of the band-border ratios w.r.t. 1 is significantly larger for 1B-1D ($=0.015$) than for 1A-1B and 1D-1E (being respectively 0.009 and 0.005). Due to the problems with memory effects in band 2 (4.08 – 12[$\mu$m]{}), the factors of each sub-band of band 2 were determined by use of the corresponding spectral data of Cohen [@Cohen1992AJ....104.2030C; @Cohen1995AJ....110..275C; @Cohen1996AJ....112.2274C; @Witteborn1999AJ....117.2552W]: for Vega and Sirius Cohen has constructed a calibrated model spectrum; a composite spectrum (i.e. various observed spectra have been spliced to each other using photometric data) is available for $\alpha$ Cen A, $\alpha$ Boo, $\gamma$ Dra, $\alpha$ Tau, $\beta$ And, $\alpha$ Cet, and $\beta$ Peg; a template spectrum (i.e. a spectrum made by using photometric data of the star itself and the shape of a ‘template’ star) is built for $\delta$ Dra (template: $\beta$ Gem: K0 III), $\xi$ Dra (template: $\alpha$ Boo: K2 IIIp), $\alpha$ Tuc (template: $\alpha$ Hya: K3 II-III) and H Sco (template: $\alpha$ Tau: K5 III). When no template was available (for $\beta$ Leo, $\alpha$ Car and $\beta$ UMi), the synthetic spectrum showing the best agreement with the band-1 data was used as reference. This does not imply that we are trapped in a circular argument, since the stellar parameters for the synthetic spectrum were determined from the band-1 data only. Moreover, the maximum difference in the correction factors for band 2 obtained when using the synthetic spectra instead of a Cohen template for the 13 stars common in the sample is 7%, which is well within the photometric absolute flux uncertainties claimed by @Leech2002. Note that all shift factors are in within the AOT01 band border ratios as derived in Fig. 5.33 and Fig. 5.34 in @Leech2002. Using the overlap regions in band 2 can have quite a big effect on the final composed spectrum: focussing on $\beta$ UMi, we note that by using these overlap regions band 2A (and consequently bands 2B and 2C) should be shifted downwards by a factor 1.04; in order then to match the shifted band 2B and band 2C, band 2C should be once more shifted downwards by a factor 1.12. In general, the error in the absolute flux could increase to $\sim 20$% at the end of band 2C when this method would be used.
For a more elaborate discussion on the SWS error budget, we would like to refer to @Leech2002.
Verification of the results with FTS-KP and FTS-ATMOS spectra {#highres}
=============================================================
To test our findings — indicating either a calibration problem or a theoretical modelling problem — with data taken with an independent instrument, a high-resolution observation of both one [*[warm]{}*]{} and one [*[cool]{}*]{} star were included. The high-resolution Fourier Transform Spectrometer (FTS) spectrum of $\alpha$ Boo [@Hinkle1995PASP..107.1042H] and the Atmospheric Trace Molecule Spectroscopy (ATMOS) spectrum of the Sun [@Farmer1989QB551.F37......; @Geller1992] are used as external control to the process.
The Arcturus observations were obtained with the FTS at the Kitt Peak (KP) 4m Mayall telescope mainly in 1993 and 1994 and are described in @Hinkle1995 [@Hinkle1995PASP..107.1042H]. The entirety of the 1 to 5 $\mu$m Arcturus spectrum detectable from the ground was observed twice at opposite heliocentric velocities, allowing the removal of many telluric lines. We refer to these two spectra as the winter and summer observations. The data were obtained at a resolving power ($\lambda$/$\Delta\lambda$) near 100000. Spectra of the Earth’s atmosphere, derived from high resolution solar spectra, have been ratioed to the Arcturus spectra to largely remove the telluric lines. Most telluric lines less than 30% deep are cleanly removed but as the optical depth of the telluric lines increases more information is lost and lines over 50% deep can not be removed. Gaps in the plots appear at these spectral regions. A typical example of the Arcturus spectra is shown in Fig. \[FTS\] on page . In the upper panel of Fig.\[FTS\], some CO lines of the first overtone are plotted. In the lower panel of Fig. \[FTS\], one can clearly distinguish atomic and OH lines.
During the period from November 3 to November 14, 1994, the Atmospheric Trace Molecule Spectroscopy (ATMOS) experiment operated as part of the ATLAS-3 payload of the shuttle Atlantis [@Gunson1996GeoRL..23.2333G]. The principal purpose of this experiment was to study the distribution of the atmosphere’s trace molecular components. The instrument, a modified Michelson interferometer, covering the frequency range from 625 to 4800cm$^{-1}$ at a spectral resolution of 0.01cm$^{-1}$, also recorded high-resolution infrared spectra of the Sun. A small part of the intensity spectrum is shown in Fig. \[sun\]. The Holweger-Müller model [@HolwegerMuller1974SoPh...39...19H] was used as input for the computation of the synthetic spectrum of the Sun. A microturbulent velocity of 1 was assumed.
In the case of $\alpha$ Boo, a high-resolution synthetic spectrum was first generated with the stellar parameters of as input parameters ([T$_{\mathrm{eff}}$]{} = 4350K, $\log$ g = 1.80, \[Fe/H\] = $-0.50$, $\varepsilon$(C) = 7.89, $\varepsilon$(N) = 7.61, $\varepsilon$(O) = 8.68). For the microturbulence [$\xi_t$]{} a velocity of 1.7 was assumed as being the mean for red giants . Since both and @Bell1985MNRAS.212..497B found a Si and Mg abundance depleted with respect to the solar values, $\varepsilon$(Mg) was taken to be 7.33 and $\varepsilon$(Si) to be 7.20. For the anisotropic macroturbulence $\Gamma_t$, a radial-tangential profile was assumed [@Gray1975ApJ...202..148G] with FWHM of 3. This synthetic spectrum was then convolved with the beam-profile function (i.e. a sinc-function). Both observed FTS-KP and synthetic spectrum were rebinned to a resolving power of 60000, since the resolving power of the FTS-KP was not constant over the whole wavelength range. Using this set of input-parameters, the high-excitation CO ($\Delta v = 1$ and $\Delta v = 2$) and SiO ($\Delta v = 2$) lines were predicted as being too weak. This was confirmed from both the FTS-KP and the ISO-SWS spectrum. Especially from the ISO-SWS spectrum — and more specifically from the slope in bands 1B and 1D — it became clear that the gravity should be lowered. Using then the FTS-KP-spectra, the input parameters were improved until an optimal fit was obtained (see Fig. \[FTS\] on page ). This resulted in the following parameters: [T$_{\mathrm{eff}}$]{} = 4350K, $\log$ g = 1.50, \[Fe/H\] = $-0.50$, [$\xi_t$]{} = 1.7, [$\mathrm{^{12}C/^{13}C}$]{} = 7, $\varepsilon$(C) = 7.96, $\varepsilon$(N) = 7.61, $\varepsilon$(O) = 8.68, $\varepsilon$(Mg) = 7.33 and $\varepsilon$(Si) = 7.20.
Discrepancies appearing from the confrontation of the synthetic spectrum with both the FTS-KP and the ISO-SWS spectra, may be clearly attributed to problems in constructing the model or generating the synthetic spectrum. Other discrepancies in the ISO-SWS versus synthetic spectrum are the ones caused by calibration problems.
For the [*[warm]{}*]{} stars in the sample, an analogous comparison was performed with the help of the FTS-ATMOS intensity spectrum of the Sun. Discrepancies between the ISO-SWS and synthetic spectra of $\alpha$ Cen A (G2V) were compared with the discrepancies found in the comparison between the FTS-ATMOS spectrum of the Sun and its synthetic spectrum (Fig. \[sun\]).
Inspecting Fig. \[sun\] and Fig. \[FTS\], some discrepancies become immediately visible, e.g. the CO 2-0 and 3-1 are predicted too weak in both figures. The reason for this and other discrepancies will be explained in the next section.
Results
=======
Computing synthetic spectra is one step, distilling useful information from it is a second — and far more difficult — one. Fundamental stellar parameters for this sample of bright stars are a first direct result which can be deduced from this comparison between ISO-SWS data and synthetic spectra. In papers III and IV of this series, these parameters will be discussed and confronted with other published stellar parameters.
A typical example of both a [*[warm]{}*]{} and [*[cool]{}*]{} star is given in Fig. \[acenarrow\] and Fig. \[gamdra\] respectively. Different types of discrepancies do emerge. The size of the discrepancies between ISO-SWS observations and theoretical predictions varies a lot. Both for the [*[warm]{}*]{} and for the [*[cool]{}*]{} stars we see a general good agreement in shape between the ISO-SWS and theoretical data in band 1, with however (local) error peaks up to $\sim 8\,\%$. An exception is $\beta$ Peg for which the shape is wrong by $\sim 6\,\%$ and local error peaks may go up to $\sim 15\,\%$. The agreement between observational and synthetic data is worse in band 2: a general mismatch by up to $\sim 15\,\%$ may occur.
By scrutinising carefully the various discrepancies between the ISO-SWS data and the synthetic spectra of the standard stars in our sample, the origin of the different discrepancies was elucidated. First of all, a description on the general trends in discrepancies for the [*[warm]{}*]{} stars will be made, after which the [*[cool]{}*]{} stars will be discussed.
Warm stars: A0 – G2 {#hot}
-------------------
1. Especially when concentrating on $\alpha$ Cen A (G2 V), one notifies quite a few spectral features which appear in the ISO-SWS spectrum, but are absent in the synthetic spectrum. By comparing the spectra of the other warm stars ($\beta$ Leo, $\alpha$ Car, $\alpha$ Lyr and $\alpha$ CMa) with each other, corresponding spectral features can be recognised in their ISO-SWS spectra, although this is somewhat more difficult for $\beta$ Leo due to the lower resolving power and lower signal-to-noise ratio. Also for the cooler stars in the sample, these spectral features are (weakly) present. Some of the most prominent ones are indicated by arrows in the spectrum of $\alpha$ Cen A in Fig.\[acenarrow\]. Molecular absorption can not be a possible cause of/ contribution to e.g. the 3.9 $\mu$m feature, since this feature is seen for the warm stars in the sample, where no molecular absorption occurs at these wavelength ranges. The solar FTS-ATMOS spectrum proved to be extremely useful for the determination of the origin of these features. All spectral features, indicated by an arrow in Fig. \[acenarrow\], turned out to be caused by — strong — atomic lines (Mg, Si, Fe, Al, C, ...). This is illustrated for the wavelength range from 3.85 to 3.92 [$\mu$m]{} in Fig. \[atomic\]. In panel (a) of Fig.\[atomic\], the high-resolution FTS-ATMOS spectrum of the Sun is compared with its synthetic spectrum in the wavelength range from 3.853 to 3.917 [$\mu$m]{}. The strongest lines are identified by using the line list of @Geller1992. At the ISO resolving power of 1000 in band 1E, these atomic lines are reduced to the features indicated in panel (b). The same features can be recognised in the ISO-SWS spectrum of $\alpha$ Cen A in panel (c). It is clear that these atomic features are not well calculated for the synthetic spectra of the Sun and $\alpha$ Cen A.
For Fig. \[atomic\], the atomic line list of @Hirhor95 was used to generate the synthetic spectrum. This line list has as starting files the compilation by @Kurucz1975SAOSR.362.....K and @Kurucz1989 who lists the semi-empirical $gf$-values for many ions. Energy levels were adopted from recent compilations [@Sugar1985aeli.book.....S] or individual works. The line list of @Kelly1983 was merged into the file. The $gf$-values are taken from several compilations [e.g. @Fuhr1988atpi.book.....F; @Fuhr1988atps.book.....F; @Reader1980wtpa.book.....R; @Wiese1966atp..book.....W; @Wiese1969atp..book.....W; @Morton1991ApJS...77..119M; @Morton1992ApJS...81..883M]. Published results of the Opacity Project [@Seaton1995QB809.O63......] were also included. Comparing the atomic line list of @Hirhor95 with the identifications as given by @Geller1992 made clear that quite some lines are misidentified or not included in the IR atomic line list of @Hirhor95.
The usage of VALD and the line list of @Vanhoof1998sese.conf...67V did not solve the problem. Obviously, the oscillator strengths of the atomic lines in the infrared are not known sufficiently well.
In order to test this hypothesis, Sauval (2002, [*[priv.comm.]{}*]{}) has constructed a new atomic line list by deducing new oscillator strengths from the high-resolution ATMOS spectra of the Sun (625 – 4800 cm$^{-1}$). A preliminary comparison using this new linelist is given in Fig. \[atomicsauv\] (which should be compared with Fig. \[atomic\]). The contents, nature, limitations, uncertainties, … of this new line list will be discussed in a forthcoming paper. But it is already obvious from a confrontation between Fig. \[atomic\] and Fig.\[atomicsauv\] that these new oscillator strengths from Sauval are more accurate.
It is plausible that the same reason, i.e. wrong and missing oscillator strengths of atomic lines in the infrared, together with noise, is the origin of the observed discrepancies between the ISO-SWS and synthetic spectra for the other warm stars in the sample, because
- [incomplete atomic data already caused a problem for the synthetic spectrum of $\alpha$ Cen A, and]{}
- [the stars concerned are hotter than $\alpha$ Cen A, so atoms (maybe of higher excitation or ionisation) will determine even more the spectral signature of these stars.]{}
The lack of reliable atomic data rendered the determination of the fundamental stellar parameters for the [*[warm]{}*]{} stars from the ISO-SWS data impossible. Another consequence concerns the continuum, which was very difficult to determine. Therefore, the uncertainty on the angular diameter of these [*[warm]{}*]{} stars is more pronounced.
2. Secondly, the hydrogen lines are conspicuous. For example, the synthetic hydrogen Pfund lines are almost always predicted as being too strong for main-sequence stars, while they are predicted as being too weak for the supergiant $\alpha$ Car. This indicates a problem with the generation of the synthetic hydrogen lines, which is corroborated when the high-resolution FTS-ATMOS spectrum of the Sun is compared with its synthetic spectrum (see Fig. \[sunHlines\]). In the TurboSpectrum program the hydrogen line-profile calculation, adopted from the SYNTHE code of Kurucz, includes the Stark, the Doppler, the van der Waals and the resonance broadening. For the first four lines in every series, the fine structure is also included for the core calculations. The problematic computation of the self broadening of hydrogen lines and the unproper treatment of the convection by using the mixing-length theory (see, e.g., where realistic ab-initio 3D radiative-hydrodynamical convection simulations have been used) may account for part of this discrepancy.
3. Compared to the ISO-SWS data, the synthetic spectra of [*[warm]{}*]{} stars display a higher synthetic flux between the H5-9 and H5-8 hydrogen line (see Fig. \[humphreys\]). From other SWS observations available in the ISO data-archive, we could deduce that this ‘pseudo-continuum’ starts arising for stars hotter than K2 ($\sim
4500$K). Since this effect is [*[systematically]{}*]{} seen for [*[all]{}*]{} [*[warm]{}*]{} stars in the sample, the origin of this discrepancy can not be a wrong multiplication factor. Note that band 1D and band 1E overlap each other over quite a large wavelength range, so that the multiplication factor of band 1E can be accurately determined. Moreover such an effect was not seen for the cooler K and M giants, the spectrum of which is dominated by OH features in this wavelength range, so we could reduce the problem as having an atomic origin. A scrutiny on the hydrogen lines shows that the high-excitation Humphreys-lines (from H6-18 on) — and Pfund-lines — are always calculated as too weak. Moreover, the Humphreys ionisation edge occurs at 3.2823[$\mu$m]{}. Since the discrepancy does not appear above the limit (i.e. at shorter wavelengths) and is disappearing beyond the Brackett-$\alpha$ line, the conclusion is reached that a problem with the computation of the complex line profile of the crowded Humphreys hydrogen lines towards the series limit causes this discrepancy. The higher the transition, i.e. going from H6-18 to H6-93, the more the $\log$ gf-value should be increased in order to get a good match between observations and theoretical computations: increasing $\log$ gf with $+0.3$dex gives us a good match for the H6-18 line in $\alpha$ Car, while we should increase $\log$ gf with $+1.0$dex for the H6-16 line in $\alpha$ Car. We may conclude that a complex problem with the computation of the pressure broadening will be on the origin of the discussed discrepancy.
4. [From 3.48[$\mu$m]{} on, fringes at the end of band 1D affect the ISO-SWS spectra of almost all stars in the sample.]{}
5. A clear discrepancy is visible at the beginning of band 1A. For the [*[warm]{}*]{} stars, the H5-22 and H5-23 lines emerge in that part of the spectrum. An analogous discrepancy is also seen for the [*[cool]{}*]{} stars, though it is somewhat more difficult to recognise due to the presence of many CO features (Fig.\[dis1a\]). Being present in the continuum of both [*[warm]{}*]{} and [*[cool]{}*]{} stars, this discrepancy is attributed to problems with the Relative Spectral Response Function (RSRF). A broad-band correction was already applied at the short-wavelength edge of band 1A [@Vandenbussche2001clim.confE..97V], but the problem seems not to be fully removed. At the band edges, the system response is always small. Since the data are divided by the RSRF, a small problem with the RSRF at these places may introduce a pronounced error at the band edge.
6. Memory effects of the detectors make the calibration of band 2 for all the stars very difficult. These memory effects are more severe for the [*[cool]{}*]{} stars, since the CO and SiO absorptions cause a steep increase (decrease) in flux for the up (down) scan (Fig.\[band2\]). Band-border ratios are determined from the composite SEDs by Cohen or from the synthetic spectrum. The RSRFs for the sub-bands will therefore only be modelled well once there is a fool-proof method to correct SWS data for detector memory effects. Two other examples of this problematic RSRF modelling are visible in band 2B and band 2C. For almost all [*[warm]{}*]{} and [*[cool]{}*]{} stars, there is a ‘dip’ around 6[$\mu$m]{} and a jump in flux level around 9.3[$\mu$m]{} (Fig. \[band2\]). This latter problem is a residue of the correction for an instrumental absorption feature, documented by @Vandenbussche2001clim.confE..97V. The short-wavelength part of band 2A (where the CO $\Delta v = 1$ absorption starts) and of band 2C (with SiO $\Delta v = 1$) are thus useless for passing a quantitative judgement upon the parameters influencing the observed spectrum in this wavelength range, like [T$_{\mathrm{eff}}$]{}, $\varepsilon$(C), $\varepsilon$(O), ... From the longer wavelength parts of these same bands, one will not be able to estimate directly fundamental stellar parameters, but these data enable us to see if there are no contradictions between this part of the ISO-SWS spectrum and the synthetic spectrum generated using the parameters adopted from literature or estimated from the ISO-SWS data in band 1.
Cool stars: G2-M2 {#cool}
-----------------
1. [The situation changes completely when going to the [*[cool]{}*]{} stars of the sample. While the spectrum of the [*[warm]{}*]{} stars is dominated by atomic-line features, molecules determine the spectral signature of the [*[cool]{}*]{} stars. A few of the — problematic — atomic features (see Sect. \[hot\]) can still be identified in these cool stars, e.g. the Mg-Si-Al-Ti-Fe spectral feature around 3.97[$\mu$m]{} (Fig. \[atomic\]) remains visible for the whole sample, even in $\beta$ Peg.]{}
2. One of the most prominent molecular features in band 1 is the first-overtone band of carbon monoxide (CO, $\Delta v = 2$) around 2.4[$\mu$m]{}. Already from $\alpha$ Cen A on, CO lines emerge in band 1A, although the atomic features are still more dominant for this star. It is striking that for all stars in the sample with spectral type later than G2, the strongest CO features (= band heads of $^{12}{\rm{CO}}$ 5-3, 6-4, 7-5, 8-6, and of $^{13}{\rm{CO}}$ 4-2, 5-3, 6-4 and 7-5) are always predicted too strong compared to the ISO-SWS observation (see, e.g., band 1A of $\gamma$ Dra in Fig. \[restoohigh\], where both SWS and theoretical data are rebinned to a resolving power of 1500, being the most conservative theoretical resolution for band 1A [@Leech2002]). The only exception is seen for the band head of $^{13}{\rm{CO}}$ 4-2 since the problem with the RSRF at the beginning of band 1A (see point 5. in previous section) dominates in this wavelength range.
In these oxygen-rich stars, the CO lines are a direct measure of the C abundance. From the present spectra, this carbon abundance can be estimated in two ways:
- [the strength of the CO spectral features is directly related to $\varepsilon$(C);]{}
- [the presence of many CO absorption lines causes a global depression in the spectrum up to $\sim$ 2.9[$\mu$m]{} (see Fig. 4 in Paper I), which may be used to determine $\varepsilon$(C).]{}
Computing a synthetic spectrum with the carbon abundance determined from this last criterion, results however in the (strongest) computed CO spectral features being too strong compared to the ISO-SWS observation, most visible in the strong bandheads (2 – 4%). It has to be noted that this mismatch occurs in band 1A, where the standard deviation of the rebinned spectrum is larger than for the other sub-bands (see Fig. 6 in ) and that the error is within the quoted accuracy of ISO-SWS in band 1A [@Leech2002]. However, this mismatch is not random, in the sense that the observed CO features are [*[always]{}*]{} weaker than the synthetic ones.
A first step towards solving this problem was the use of the high-resolution FTS-KP spectrum of $\alpha$ Boo (see Section \[highres\]). As was explained in Sect. \[highres\] a high-resolution synthetic spectrum was generated for $\alpha$ Boo and rebinned to a resolution of 60000. The agreement between the observed FTS-KP and synthetic spectrum is extremely good! One example was already shown in Fig. \[FTS\], another one is depicted in Fig. \[COfts\] on page . In this latter figure, the problematic subtraction of the atmospheric contribution causes the spurious features around 2.46700[$\mu$m]{}for the summer FTS-KP spectrum and around 2.46735[$\mu$m]{} for the winter FTS-KP spectrum.[[^5]]{} When scrutinising carefully the first-overtone CO lines in the FTS-KP spectrum, it is obvious that all the $^{12}$CO 2-0 lines, and almost all the $^{12}$CO 3-1 lines, are predicted as too [*[weak]{}*]{} (by 1 – 2%) and not as too [*strong*]{} as was suggested in previous paragraph from the comparison between ISO-SWS and synthetic spectra! Also the fundamental CO lines are predicted as being a few percent too weak. For a better judgement of the KP-SWS-synthetic correspondence, the FTS-KP spectrum was rebinned to the ISO resolving power. This was not so straightforward due to the presence of the spurious features originating from the problematic subtraction of the atmospheric contribution. In order to conserve the flux, these spurious features were replaced by the flux value of the synthetic spectrum. This adaptation is acceptable for the following reasons:
- [most of the changes were made in regions that are not strongly contributing to the most prominent features;]{}
- [if for a particular feature the first argument does not apply, the change mentioned above does still not introduce additional errors. The FTS-KP-spectrum will be locally shifted towards the ISO spectrum, since the synthetic predictions are (in this particular case) systematically in between the ISO-SWS and FTS-KP spectrum. Thus, if statistically relevant differences are still detected, they will always correspond to an underestimate of the real differences between the ISO-SWS and FTS-KP spectra.]{}
Fig. \[FTSISOsynCO\] on page shows the comparison between the ISO-SWS, the FTS-KP and the synthetic spectrum for $\alpha$ Boo at a resolving power of 1500. It is clear that the differences between the strength of the CO features in the ISO-SWS and FTS-KP spectrum are significant. This is a strong indication for problems in the [*calibration*]{} process. The question now arises where this calibration problem originates from.
Firstly, it has to be noted that the flux values are unreliable in the wavelength region from 2.38 to 2.40[$\mu$m]{} due to problems with the RSRF of band 1A; main features here include the CO 2-0 P18 and the CO 2-0 P21 lines.
Secondly, no correlation is found with the local minima and maxima in the RSRF of band 1A.
Defining the spectral resolution and instrumental profile for the ISO-SWS grating spectrometers, is not straightforward [@Lorente1998; @Lutz1999]. Only for an AOT02 observation has the instrumental profile now been derived quite accurately [@Lutz1999]. The AOT01 mode introduces an additional smoothing which, due to the intricacies of SWS-data acquisition, is different from a simple boxcar smooth. The simulation of SWS-AOT01 scans gives non-gaussian profiles, as can be seen in Fig. 6 by @Lorente1998. Nevertheless, the theoretical profile of a speed-4 observation approximates a gaussian profile very closely. Therefore, since the instrumental profile of an AOT01 is still not exactly known, the synthetic data were convolved with a gaussian with FWHM=$\lambda$/resolution. This incorrect gaussian instrumental profile introduces an error which will be most visible on the strongest lines.
Only for $\gamma$ Dra an AOT06 observation, scanning this wavelength range, was available in the ISO data-archive. The comparison between the AOT01 speed-4 and AOT06 observation, both rebinned to a resolving power of 1500, is an indication that the resolving power of an AOT01 speed-4 observation in band 1A is lower than 1500 (Fig. \[res1500\]). The theoretical resolving power for an AOT01 speed-4 observation in band 1A is $\ge 1500$. Fig. 5 in @Lorente1998 shows, however, a large deviation from this value, attributed to the fainter continuum of the source used for measuring the instrumental profile and the less accurate fitting in this band. The use of a resolving power of 1300 — instead of the theoretical resolving power of 1500 — for band 1A yields a) almost no difference from the AOT01 SWS data at a resolving power of 1500 and b) a better match between the SWS and synthetic data for the strongest CO features (Fig.\[resolution\]). This observational resolving power of 1300 is also in good agreement with the observational value given by @Lorente1998 in her Fig. 5.
The incorrect use of a gaussian instrumental profile, together with too high a — theoretical — resolving power of 1500 form the origin of the discrepancy seen for the strongest CO features. Therefore, the resolving power of band 1A was taken to be 1300 instead of 1500. Part of the other discrepancies seen in band 1A may be explained by problems with 1. the accuracy of the oscillator strengths of atomic transitions in the near-IR (see point 1. in this section) or 2. the RSRF in the beginning of band 1A (see point 5. in previous section).
3. Concentrating in the OH-lines, we see that for both the high-resolution FTS-KP (Fig. \[FTSOH\], Fig. \[FTS\]) and the medium-resolution SWS spectra (Fig. \[FTSISOsynOH\]), the strongest lines (OH 1-0 and OH 2-1 lines) are predicted as too weak, while the other OH lines match very well. Since the same effect occurs for these two different observations, it is plausible to assume that the origin of the problem is situated in the theoretical model or in the synthetic-spectrum computation. Wrong oscillator strengths for the OH lines could, e.g., cause that kind of problems. Being based on different electric dipole moment functions (EDMFs), the OH-line lists of Sauval [@Melen1995], @Partridge+Schwenke1995 and @Goldman1998, do however all show the same trend. The difference in $\log gf$-value for the main branches is small ($< 0.01$) between Goldman and Sauval, and is somewhat larger ($< 0.05)$ between Schwenke and Sauval. Thus, no systematic error seems to occur in the oscillator strenghts. Since a similar discrepancy was also noted for the low-excitation CO lines, some assumptions on which the models are based, e.g. homogeneity, hydrostatic equilibrium, are cleary questionable for these stars.
4. [From 3.48[$\mu$m]{} on, fringes at the end of band 1D affect the ISO-SWS spectra of almost all stars in the sample.]{}
5. [Also for the [*[cool]{}*]{} stars, the same remark as given in point 6 for the [*[warm]{}*]{} stars concerning the memory effects applies. Clearly, the fundamental bands of CO and SiO are present for almost all [*[cool]{}*]{} stars, though no accurate quantitative scientific interpretation can be made due to these memory effects.]{}
Implications on calibration and modelling {#impact}
=========================================
The results of this detailed comparison between observed ISO-SWS data and synthetic spectra have an impact both on the calibration of the ISO-SWS data and on the theoretical description of stellar atmospheres.
From the calibration point of view, a first conclusion is reached that the broad-band shape of the relative spectral response function is at the moment already quite accurate, although some improvements can be made at the beginning of band 1A (see Fig.\[vglbroadrsrf\]) and band 2. Also, a fringe pattern is recognised at the end of band 1D. Inaccuracies in the adopted instrumental profile used to convolve the synthetic spectra with, together with too high a resolving power, may cause the strongest CO $\Delta v = 2$ band heads in the observed ISO-SWS spectrum to be weaker than the predicted line strength.
Concerning the modelling part, problems with the construction of the theoretical model and the computation of the synthetic spectra are pointed out. The comparison between the high-resolution FTS-KP spectrum of $\alpha$ Boo and the corresponding synthetic spectrum revealed that the low-excitation first-overtone (and fundamental) CO lines and fundamental OH lines are predicted as being a few percent too weak, indicating that some assumptions, on which the models are based, are questionable for cool stars. This is not surprising: in the MARCS-code radiative equilibrium is assumed, also for the outermost layers, implying that a temperature bifurcation, caused by e.g. effects of convection and convective overshoot with inhomogeneities in the upper photosphere, can not be allowed for. Consequently, the cores of e.g. CO lines — or the saturated CO lines in general — are not described with full success. Noting however the high level of accordance between observations and theoretical predictions for many molecular lines, we may conclude that the oscillator strengths for these molecular transitions are now already accurate enough in order to use these lines to test some of the assumptions made in the mathematical stellar atmosphere code: e.g. the temperature distribution can be disturbed in order to simulate a chromosphere, convection, a change in opacity, ... The complex computation of the hydrogen lines, together with the inaccurate atomic oscillator strengths in the infrared rendered the computation of the synthetic spectra for [*[warm]{}*]{} stars difficult. Therefore, one of us (J. S.) has derived empirical oscillator strengths from the high-resolution FTS-ATMOS spectrum of the Sun [for more details, we refer to Paper V of this series and to @Sauval2000].
Although it was impossible to perform a detailed comparison between observed and synthetic data in band 2, we may conclude that the continua of the synthetic spectra in this wavelength range are reliable, but that strong molecular lines (e.g. band heads of CO $\Delta v = 1$ and SiO $\Delta v = 1$) may be predicted a few percent too weak. Nevertheless, combining the synthetic spectra of both warm and cool stars, allowed us to test the recently developed method for memory effect correction [@Kester2001clim.confE..46K] and to re-derive the relative spectral response function for bands 1 and 2 for OLP10 [@Vandenbussche2001clim.confE..97V]. The results of these tests will be described in Paper V of this series (Decin et al., 2002, in preparation). In conjunction with photometric data, this same input data-set was used for the re-calibration of the absolute flux-level of the spectra observed with ISO-SWS [@Shipman2001clim.confE..18S]. In this way, both consistency and integrity were implemented. Moreover, the synthetic spectra of the standard sources of our sample are not only used to improve the flux calibration of the observations taken during the nominal phase, but they are also an excellent tool to characterise instabilities of the SWS spectrometers during the post-helium mission.
Although we have mainly concentrated on the discrepancies between the ISO-SWS and synthetic spectra — since this was the main task of this research — we would like to emphasise the good agreement between observed ISO-SWS data and theoretical spectra. The small discrepancies still remaining in band 1 are at the 1 – 2% level for the giants, proving not only that the calibration of the (high-flux) sources has already reached a good level of accuracy, but also that the description of cool star atmospheres and molecular line lists is quite accurate.
LD acknowledges support from the Science Foundation of Flanders. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France and of the VALD database, operated at Vienna, Austria. It is a pleasure to thank the referees, J. Hron and F.Kupka, for their careful reading of the manuscript and for their valuable suggestions.
[^1]: *Postdoctoral Fellow of the Fund for Scientific Research, Flanders*
[^2]: Based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA.
[^3]: As for atoms, laboratory data for molecules are generally more accurate than computed data, but much too sparse to serve as the sole source for the computation of photospheric models or to represent the myriads of weak lines in a cool star spectrum. An overview of existing databases can be found in . Some of these, useful for our research, are described in @Decinthesis. A copy of this description can be retrieved from http://www.ster.kuleuven.ac.be/\~$\!$leen. The line lists used for this research are enumerated in .
[^4]: Each observation is determined uniquely by its observation number (8 digits), in which the first three digits represent the revolution number. The observing data can be calculated from the revolution number which is the number of days after 17 November 1995.
[^5]: The colour plots showing the comparison between FTS-KP and synthetic spectrum for the wavelength range 2.38 – 4.08[$\mu$m]{}, may be obtained at www.ster.kuleuven.ac.be/\~leen/artikels/ISO2/FTS/. Spectral ranges, for which the atmospheric contribution causes too much trouble, are omitted.
|
Introduction
============
The dimer model is extremely simple to define. We take a rectangular grid ${\mathcal{L}}$ with $M$ rows and $N$ columns, and consider all arrangements of dimers (dominoes) so that all sites of ${\mathcal{L}}$ are covered by exactly one dimer. An example of such an arrangement for a $9 \times 18$ grid is shown in Fig. 1a. In case both $M$ and $N$ are odd, one site has to be removed from the grid; this site is conventionally to be one of the four corners.
One can be more general and introduce monomers (vacancies), namely fixed sites which cannot be covered by dimers. A dimer arrangement is possible in presence of $m$ monomers provided the numbers of sites which must be covered by dimers and belonging to the even sublattice and to the odd sublattice (sum of coordinates even or odd resp.) are equal. In particular, $MN-m$ must be even. A dimer configuration on a $9 \times 18$ grid with four monomers is shown in Fig. 1b. The case $m=0$ (or $m=1$ if $MN$ is odd) is referred to as the close-packed limit of the dimer model. Corresponding to these arrangements of monomers and dimers, we introduce the partition function Z(x,y|z\_1,…,z\_m) = \_[coverings]{} x\^[n\_h]{} y\^[n\_v]{}. \[pf\] It counts the number of dimer coverings in presence of $m$ monomers located at positions $z_1, \ldots,z_m$, in the bulk or on boundaries, with respective weights $x$ and $y$ assigned to horizontal and vertical dimers. As the number $n_h + n_v$ of dimers is fixed, the partition function essentially depends on $x$, $y$ through the ratio $x/y$ only
There are obvious generalizations (other types of lattices, coverings by other lattice animals), but the basic questions remain the same. Depending on the values of the parameters (here, $x/y$), can the system become critical and does it exhibit phase transitions? What is the probability that certain chosen bonds be covered by dimers? How do monomers affect the number of dimer coverings? More importantly for our purpose here, what are the finite-size correction terms in the partition function, and, in case the system is critical, is its scaling regime conformally invariant?
The dimer model has been originally introduced to model physical adsorption of diatomic molecules on crystal surfaces [@foru]. The first studies of the dimer model were conducted in the early sixties, with pioneering works by Kasteleyn [@kast1; @kast2], Fisher [@fish], and Fisher and Temperley [@fite]. Soon after that, correlations between dimers and monomers on the square lattice have been studied in [@fist; @hart], and have been revisited more recently for the triangular lattice in [@fend; @baeh]. The effects caused by the insertion of monomers have also been reconsidered recently [@tzwu; @kong; @wu]. Finite-size effects also have a long history, starting in [@fish; @ferd], with many subsequent works [@coywu; @bhana; @brpr; @luwu1; @luwu2; @ivash; @izm1; @izm2; @kong].
Many critical systems have been shown to have a local scale invariance, so that their scaling limit can be described by a conformal field theory. Such a theory is primarily (but non-exclusively) parametrized by the value of its central charge $c$, which itself is related to the finite-size corrections to the critical free energy.
The calculation of the central charge based on the finite-size corrections has led to some confusion in the literature. Indeed it is well-known since [@ferd] that the finite-size corrections depend on the parities of $M$ and $N$. This has prompted several authors to claim that correspondingly the central charge was also dependent of the parities of $M$ and $N$ [@tzwu].
Here we reanalyze this question. The finite-size corrections determine univoquely the effective central charge $c_{\rm eff} =
c - 24h_{\rm min}$ [@itz]. To extract the central charge itself, one needs to determine the groundstate energy $h_{\rm
min}$ of the Hamiltonian with prescribed boundary conditions. So because of $h_{\rm min}$ the effective central charge depends on the boundary conditions. We show, by changing variables from dimer coverings to spanning trees (and more generally, to arrow configurations), that a change of parity of $M$ or $N$ has precisely the effect of changing the boundary conditions. The known values of $h_{\rm min}$ and the values of $c_{\rm eff}$ as computed from the finite-size corrections enable us to obtain a consistent value for the central charge, here equal to $c=-2$. Surprisingly, this is not the only consistent value of $c$ that can be used to describe the dimer model, since it has been shown in that a dimer configuration can be encoded in a height function [@bh], which converges in the scaling limit to a Gaussian field, with $c=1$ [@ken]. We will discuss this peculiar situation in the Conclusion.
In what follows, we consider the dimer model on a square lattice in the close-packed limit, and for two different boundary conditions, free and periodic. In Section 2, we collect the known results for the finite-size corrections, and the way these relate to the central charge. In Sections 3 and 4, we compute the central charge from the finite-size analysis respectively on a strip with free boundary conditions, and on a cylinder (periodic boundary conditions). The last section summarizes our point of view on the description in terms of $c=-2$ versus the one based on $c=1$, relating this issue to the more general monomer-dimer problem. The present results have been reported in [@izm2], but we take here the opportunity to give more details.
Finite-size analysis
====================
The partition function (\[pf\]), in the close-packed case, has been first computed by Kasteleyn as the Pfaffian of a certain matrix [@kast1]. Since then it has reproduced by a variety of methods [@baxt], including that based on spanning trees [@temp; @priez] which we will use later on.
The partition function, up to an irrelevant factor, depends on $x/y$ only. In the infinite volume limit, the dimer model on the square lattice becomes critical but does not show a phase transition (unlike on other lattices). The critical properties do not depend on the values of $x/y$ (provided it is neither zero nor infinite), so we set $x=y=1$ in the following. The partition function, $$Z(M,N) = \# \hbox{ dimer coverings},$$ simply counts the number of ways the grid ${\mathcal{L}}$ (minus a corner if $MN$ is odd) can be fully covered by dimers. The topology of ${\mathcal{L}}$ is fixed by the boundary conditions: it forms a rectangle if free boundary conditions are imposed in two directions (like in Fig. 1a), or a cylinder if periodic boundary condition is chosen in the horizontal direction.
We are especially interested in the free energy $F_N$ per unit of height for an $\infty \times N$ lattice, $$F_N = -\lim_{M \to \infty} {1 \over M} \log{Z(M,N)}.$$ The free energy depends on the boundary conditions, free or periodic, and on the parity of $N$. This leads to four quantities $F_{N,\rm even}^{\rm free}$, $F_{N,\rm odd}^{\rm free}$, $F_{N,\rm
even}^{\rm per}$ and $F_{N,\rm odd}^{\rm per}$.
The partition functions with these boundary conditions can be expressed in terms of $Z_{\alpha,\beta}$ for $\alpha,\beta = 0,
\frac{1}{2}$ [@izm1], with $$Z^2_{\alpha,\beta}(1,M,N) =
\prod_{n=0}^{N-1}\prod_{m=0}^{M-1}4\left[ \sin^2{\frac{\pi
(n+\alpha)}{N}} + \sin^2{\frac{\pi (m+\beta)}{M}}\right].$$
The full expansion of $Z_{\alpha,\beta}$ for large $M$, $N$ has been obtained in [@ivash]. Using these results, one can easily obtain the dominant finite-size corrections terms for the above four free energy densities, as power series in $N$.
For an infinitely long strip of width $N$ with free boundary conditions, one obtains $$\begin{gathered}
F_{N,\rm odd}^{\rm free} = -\frac{G}{\pi}N - \frac{G}{\pi} + {1
\over 2}\log{(1+\sqrt{2})} + \frac{\pi}{12}\frac{1}{N} + \cdots,
\label{fodd} \\
F_{N,\rm even}^{\rm free} = -\frac{G}{\pi}N - \frac{G}{\pi} + {1
\over 2}\log{(1+\sqrt{2})} - \frac{\pi}{24}\frac{1}{N} + \cdots ,
\label{feven}\end{gathered}$$ where $G = 0.915965$ is the Catalan constant.
The analogous results for the periodic case, i.e. an infinite cylinder of perimeter $N$, read [@ferd] F\_[N,odd]{}\^[per]{} = - N + + , F\_[N,even]{}\^[per]{} = - N - + . \[p\]
The dominant term in these expressions yields the exponential growth of the number of dimer coverings of a $M \times N$ grid, namely $Z(M,N) \simeq (e^{G/\pi})^{MN} = (1.79162)^{MN}$, for both types of boundary conditions.
The general form of the free energy per unit length of an infinitely long strip of finite width $N$ at criticality is F\_N = f\_[bulk]{} N + 2f\_[surf]{} + + , \[free\] where $f_{\rm bulk}$ is the free energy per bulk site, and $f_{\rm bulk} + f_{\rm surf}$ is the free energy per boundary site (assuming identical boundary conditions on the two sides of the strip), and $A$ is a constant. Unlike the free energy densities $f_{\rm bulk}$ and $f_{\rm surf}$, the constant $A$ is universal. The value of $A$ is related to the central charge $c$ of the underlying conformal theory, and depends on the boundary conditions in the transversal direction. The result is that $A$ is proportional to the effective central charge $c_{\rm eff} = c -
24h_{\rm min}$ [@blote; @aff; @cardy], $$\begin{gathered}
A = -{\pi \over 24}\, c_{\rm eff} = \pi \left(h_{\rm
min}-\frac{c}{24}\right)
\;\; \mbox{on a strip}, \label{Astrip} \\
\noalign{\medskip} A = -{\pi \over 6}\, c_{\rm eff} = 4\pi
\left(h_{\rm min}-\frac{c}{24}\right) \;\; \mbox{on a cylinder}.
\label{Acyl}\end{gathered}$$ The number $h_{\rm min}$ is the smallest conformal weight in the spectrum of the Hamiltonian with the given boundary conditions (for the cylinder, we assumed that this operator is scalar, $h_{\rm min} = \bar h_{\rm min}$).
The values of $A$ given above are most easily understood from the transfer matrix formalism for calculating the lattice partition function, even though the result seems generally more valid (i.e.when there is no transfer matrix). Let us first consider a cylinder of perimeter $N$ and height $M$, with certain boundary conditions $|b\rangle$ and $|t\rangle$ on the bottom and top edges[^1]. The transfer matrix $\cal T_{\rm cyl}$ is labelled by the degrees of freedom living on a horizontal circle, in terms of which the partition function is equal to $Z(M,N) = \langle b|{\cal T}_{\rm
cyl}^M|t\rangle$. In the thermodynamic limit, the vector space spanned by the degrees of freedom living on a circle (space coordinate) goes over to an infinite Hilbert space ${\cal
H}_{\rm cyl}$, and the transfer matrix, which can be viewed as a unit translation operator in the vertical direction (time coordinate), can be written as ${\cal T}_{\rm cyl} = {\rm
e}^{-H_{\rm cyl}}$, in terms of a Hamiltonian $H_{\rm cyl}$. In the large $M$ limit, the partition function $Z(M,N) = \langle
b|{\rm e}^{-MH_{\rm cyl}}|t \rangle$ will be dominated by ground-state $E_0$ of $H_{\rm cyl}$, so that $F_N =
-\lim\limits_{M \to \infty} {1 \over M} \log{Z(M,N)} = E_0$.
The Hamiltonian is the charge associated to the time-time component of the stress-energy tensor $H_{\rm cyl} = {1 \over 2\pi} \int_0^N {\rm d}u T_{00}$. If the system is critical and conformally invariant, the Fourier modes of the stress-energy tensor are the left and right Virasoro modes $L_n$, $\bar L_n$, and the Hamiltonian is simply given by the zero-th moded Virasoro generators as $H_{\rm cyl} = {2\pi \over N} (L_0 + \bar L_0 - {c \over 12})$. This last formula assumes a normalization where the ground-state energy vanishes in the thermodynamic limit $N \to \infty$, and therefore ignores the bulk term in (\[free\]) (in the periodic geometry we consider here, there is no surface and therefore no surface term in (\[free\])). In addition the Hilbert space decomposes into representations of the left and right Virasoro algebras as ${\cal H}_{\rm cyl}
= \oplus_{h,\bar h} \; N_{h,\bar h} \; {\cal R}_h \otimes {\cal R}_{\bar h}$, in which the $N_{h,\bar h}$ are integer multiplicities. The representations ${\cal R}_h \otimes {\cal R}_{\bar h}$ are highest weight representations, meaning that they are built from a highest weight state $|h\rangle \otimes |\bar h \rangle$ by applying all Virasoro modes. All states of the representation are eigenvectors of $L_0 + \bar L_0$, but the state with the smallest eigenvalue is the highest weight state, $(L_0 + \bar L_0 - h - \bar h)|h \rangle \otimes |\bar h \rangle = 0$. Putting all together, one obtains that the ground-state of $H_{\rm cyl}$ in ${\cal H}_{\rm cyl}$ is equal to $$E_0 = {2\pi \over N} \left(h_{\rm min} + \bar h_{\rm min} - {c
\over 12}\right),$$ where $h_{\rm min}$ and $\bar h_{\rm min}$ label the representation in ${\cal H}_{\rm cyl}$ with the smallest $(L_0 +
\bar L_0)$ eigenvalue. Assuming the equality $h_{\rm min} = \bar h_{\rm min}$ yields the result quoted above.
For the strip, a first change is that one has to specify the boundary conditions $a$, $b$ on the edges on the strip. The lattice transfer matrix depends on $a$, $b$, as do the Hilbert space ${\cal H}_{a,b}$ and the Hamitonian $H_{a,b}$ in the thermodynamic limit, but the formula $F_N = E_0^{a,b}$ still holds.
The second change is that there is only one copy of the Virasoro algebra on the strip. It implies that the Hamiltonian is now equal to $H_{a,b} = {\pi \over N} (L_0 - {c \over 24})$, and the Hilbert space decomposition reads ${\cal H}_{a,b} = \oplus_h
N_h^{a,b}
{\cal R}_h$, leading directly to the value of $A$ given in (\[Astrip\]). Again the normalization means that the quantum Hamiltonian ignores the bulk and surface terms in (\[free\]).
The finite-size corrections computed above for $F^{\rm free}$ and $F^{\rm per}$ have the correct form (\[free\]), but at first sight look paradoxical. Comparing them with (\[Astrip\]) and (\[Acyl\]), we see that the effective central charge depends on the parity of $N$. The boundary conditions do not seem to change with the parity of $N$. This would imply that $h_{\rm
min}$ does not change either, that the central charge itself has to change. This would be most peculiar since the central charge also controls the bulk conformal theory.
The only way out is to accept that the boundary conditions change with the parity of $N$, although this is not apparent in the dimer variables. We will show in the following that the effective central charge changes with the parity of $N$, $c_{\rm eff}=-2$ for $N$ odd and $c_{\rm eff}=1$ for $N$ even, not because the central charge changes, but because the value of $h_{\rm min}$ changes, due to the fact that a change in the parity of $N$ effectively changes the boundary condition. On the strip, this effect has been already noted in [@brank].
To understand this peculiarity of the dimer model, we consider a change of variables, namely we replace dimer configurations by arrow configurations on a sublattice. On the strip, the arrow configurations define spanning trees, so that the dimer model is mapped to the spanning tree model [@priez; @brank] or, equivalently, the Abelian sandpile model [@sand]. The case of the cylinder is slightly more complicated because the arrow configurations do not always define spanning trees. The analysis of this case is however similar.
Dimers on a strip
=================
Let us consider first the dimer model on the rectangular lattice ${\mathcal{L}}$ of size $M \times N$ with free boundary conditions. Since we are interested in the limit $M \to \infty$, the parity of $M$ will not matter here. For simplicity, we take $M$ odd, and discuss successively the cases $N$ odd and $N$ even.
When $M$ and $N$ are both odd, there is a well-known bijection [@temp] between close-packed dimer coverings of ${\mathcal{L}}$ with one corner removed and spanning trees on the odd-odd sublattice $\G
\subset {\mathcal{L}}$ (thus $\G$ contains the sites whose coordinates, counted from the lower left corner, are both odd).
A dimer containing a site of $\G$, in red in Fig. 2, can be represented as an arrow directed along the dimer from this site to a nearest neighbour site in $\G$. It is easy to prove that the resulting set of arrows generates a uniquely defined spanning tree; all the arrows point to a unique root, located at the corner which had been removed from ${\mathcal{L}}$ (see Fig. 2). Since the dimers which do not contain a site of $\G$ are completely fixed by the others, one has a one-to-one correspondence between dimer coverings on ${\mathcal{L}}$ minus a corner and spanning trees on $\G$, rooted at the removed corner. The Kirchhoff theorem then expresses the number of dimer configurations as $Z =\det
\Delta$, where $\Delta$ is the Laplacian matrix on $\G$ with appropriate boundary conditions, see Section IV for a proof of this result. Viewing $\G$ as a graph, one sets $\Delta_{ij} = -1$ if sites $i$ and $j$ are connected in $\G$, and $\Delta_{ii}$ is equal to the number of sites in $\G$ that $i$ is connected to, plus 1 if $i$ is connected to the root. As shown in [@sand], spanning trees on $\G$, rooted at a corner, are in bijection with the configurations of the Abelian sandpile model (ASM) on $\G$, with closed boundary conditions on the four boundaries, the only sink (dissipative) site being the root of the trees.
When $M \to \infty$, the lattice $\G$ becomes an infinitely long strip of width $N$. The root is sent to infinity so that no boundary site (at finite distance) is connected to the root. In the ASM language, this means that the boundaries are not dissipative and hence there is no out-flow of sand at the boundaries; such boundaries conditions have been called closed boundary conditions.
There are now very strong arguments to believe that the (scaling limit of the) ASM on a square lattice is described by a logarithmic conformal field theory with a central charge $c=-2$ [@sand; @ruelle; @piru; @piru1; @piru2]. In particular, the spectrum of the ASM Hamiltonian on a slice of the strip with closed boundary conditions at the two ends has been computed in [@ruelle]. There are two ground-states, the identity operator and its logarithmic partner, both of conformal weight 0, so that $h_{\rm min}=0$. The effective central charge in this sector is therefore $c_{\rm eff} = -2$, and the general formula (\[Astrip\]) reproduces the finite-size corrections (\[fodd\]).
When $M$ is odd and $N$ is even, dimer coverings exist without the need to remove a corner. In this case, the above construction leads to a set of spanning trees on the odd-odd sublattice $\G$, where certain arrows may point out of the lattice from the right vertical side, see Fig. 3. Viewing this vertical boundary of $\G$ as roots for the spanning trees, we see that dimer coverings on ${\mathcal{L}}$ map onto spanning trees on $\G$ which can grow from any site of the right side. The sites on this boundary, being connected to roots, are dissipative in the ASM language, and form an open boundary. Thus the spanning trees map onto the ASM configurations with one vertical open, dissipative boundary, and the three other closed.
In the limit $M \to \infty$, the lattice becomes an infinite strip with open and closed boundary conditions on the two sides. In this case, the ground-state of the Hamiltonian with such boundary conditions is a primary field of conformal weight $h_{\rm min}=-1/8$ [@ruelle]. With $c=-2$, this yields $c_{\rm
eff} = 1$ and again the formula (\[Astrip\]) gives the correct result (\[feven\]).
Thus the leading finite-size corrections for an infinitely long strip of width $N$ agree with the prediction of a $c=-2$ conformal field theory, provided one realizes that changing the parity of $N$ genuinely changes the boundary conditions, an effect due to the strong non-locality of the dimer model [@izm2]. The change of boundary conditions is not apparent in the dimer model itself, but is manifest when one maps it onto the spanning tree model or the sandpile model.
Interestingly the bijection between the dimer coverings and the spanning trees holds if we use the even-even, even-odd or odd-even sublattice. The boundary conditions however change, but one can easily see what the changes are. If, instead of choosing the odd-odd sublattice, one selects the sites whose horizontal (resp. vertical) coordinates are even, the left and right (resp. bottom and top) boundaries change from closed to open, and from open to closed. So if $N$ is odd, the vertical sides become open rather than closed. The spectrum of the corresponding Hamiltonian changes, with a non-degenerate ground-state being the identity operator [@ruelle], so that the value $h_{\rm min}=0$ remains. If $N$ is even, the left and right boundaries, previously closed and open respectively, become open and closed respectively, so the Hamiltonian remains the same, $h_{\rm min}=-1/8$. Fig. 4 shows the spanning tree associated to the dimer covering in Fig. 3 if we choose the even-even sublattice.
In fact the odd-odd and even-even sublattices are dual, as are the corresponding spanning trees, see Fig. 4b. The same is true of the odd-even and even-odd sublattices. Under this duality, the boundary conditions open and closed are exchanged.
Dimers on a cylinder
====================
We consider here an $M \times N$ rectangular lattice ${\mathcal{L}}$ with periodic boundary condition in the horizontal direction, so that ${\mathcal{L}}$ is a cylinder of perimeter $N$ and height $M$. As before, we will eventually take $M$ to infinity, which makes its parity irrelevant. For convenience we choose $M$ even. We discuss the cases $N$ odd and $N$ even separately.
If $N$ is odd, we select again the odd-odd sublattice $\G$. It is easy to see that two columns of $\G$ will be neighbours in $\G$ and in ${\mathcal{L}}$ (connected by horizontal bonds). Therefore a dimer may cover zero, one or two sites of $\G$. As before the dimers covering no site of $\G$ are completely fixed by the others and play no role. For the others, we do the same construction as in the previous section. If a dimer touches only one site of $\G$, we draw an arrow directed along the dimer from that site to the nearest neighbouring site of $\G$. However, for a dimer laid on two sites of $\G$, the two arrows would point from either site to the other, ruining the spanning tree picture. It can nevertheless be restored in the following way.
Instead of seeing the two arrows as pointing from one site to its neighbour, we say that they point towards roots inserted between the neighbour sites, thus replacing the arrows by . This in effect amounts to opening the cylinder by removing the horizontal bonds of ${\mathcal{L}}$ which connect sites of $\G$, unwrapping it into a strip, and to adding columns of roots on the left and on the right side of the strip. The new arrow configurations define spanning trees, rooted anywhere on the left and right boundaries. So dimer coverings on the original cylinder are mapped to spanning trees on a strip, with open upper and closed lower horizontal boundaries (by the arguments of the previous section, since we chose $M$ even), and open vertical boundaries.
However these two vertical boundaries are not independent. If a tree grows from a site $i$ on the left boundary, then a tree must grow from the site $j$ on the right boundary, where $j$ is the site which were neighbour of $i$ in $\G$ before the opening of the cylinder. Conversely if no tree grows from $i$, then no tree can grow from $j$. Thus the growth patterns on the left and on the right boundaries must exactly match.
For finite $N$, this correlation between the two boundaries will be felt throughout the lattice. As $N$ (and $M$) increase, the generic properties of the spanning trees will not change if the trees are allowed to grow from slightly different boundary sites. Thus if the density of roots on the left and on the right boundaries is finite, it will make no difference whether the tree growth patterns are identical or random on each boundary.
Following this argument, when $M$ goes to infinity, the lattice becomes an infinite strip with open boundary condition on either side. As mentioned above, the ground-state of the Hamiltonian is the identity, of weight $h_{\rm min} = 0$, leading to an effective central charge $c_{\rm eff} = -2$. The general formula (\[Astrip\]) for the strip gives the correct result (\[p\]).
This is a very unusual situation. Although the dimer model is originally defined on a cylinder, it shows the finite-size corrections expected on a strip, and must really be viewed as a model on a strip.
For $N$ even, the problem of having a dimer occupying two neighbouring sites of $\G$ does not arise. Hence the arrows point from one site of $\G$ to a neighbouring site of $\G$ and they never overlap. However the arrow configurations one obtains do not define spanning trees because there can be sequences of arrows looping around the cylinder. In general the arrows form a combination of oriented loops wrapped around the cylinder with trees growing from the loops. The one-to-one correspondence between the oriented loops combined with tree branches on one side, and dimer configurations on the other side can be established as above. The enumeration of the loop-tree configurations requires a slight generalization of Kirchhoff’s theorem. In order to appreciate it, we make a little digression, to recall the combinatorial content of the usual Kirchhoff theorem (see for instance [@priez]), as used in Section 3.
Define the Laplacian of an unoriented graph $\G$ as the symmetric matrix $\Delta$ with entries $\Delta_{i,j}=-1$ if there is a bond connecting site $i$ and site $j \neq i$, and $z_i = \Delta_{i,i}$ equal to the number of sites to which $i$ is connected, including the root(s). Thus $\sum_j \, \Delta_{i,j}$ is the number of connections from $i$ to the root(s). Let us now place, at each site $i$ of $\G$, an arrow pointing to any one of the $z_i$ neighbours of $i$ (possibly to the root if $i$ has a connection to it). Such a set of arrows defines an arrow configuration on $\G$, and we want to show that the determinant of $\Delta$ precisely counts those arrow configurations which contain no loop.
The determinant is a sum over the permutations in $S_N$ with $N=|\G|$ the number of sites of $\G$, = \_[S\_N]{} \_ \_[1,(1)]{} \_[2,(2)]{} \_[N,(N)]{}. Writing a permutation as a product of cycles of lengths $\ell_m$, and using $\varepsilon_\sigma = (-1)^{N + \#\,{\rm cycles}}$, it is not difficult to see that each term in the sum carries a sign equal to $(-1)^{\#\,{\rm proper\ cycles}}$ where by proper cycle we mean a cycle of length strictly larger than 1. Up to this sign, a permutation $\sigma$ with exactly $k$ proper cycles brings a contribution equal to $\prod_i z_i$ where the product is over those sites left invariant by the permutation. Combinatorially this number counts the arrangements of arrows containing $k$ loops whose locations in the graph are completely specified by the cycle structure of $\sigma$; on the other hand, the arrows coming out of the sites fixed by $\sigma$ are free to point in any of the allowed directions, and may therefore form themselves other loops.
The sum over the permutations can now be reorganized as a sum over the number $k$ of proper cycles, = \_[k=0]{}\^[\[N/2\]]{} (-1)\^k \_[ k[proper cycles]{}]{} |\_[1,(1)]{} \_[N,(N)]{}|. \[altern\]
The term for $k=0$, equal to $\prod_{i \in \G} z_i$ counts the total number of unconstrained arrow configurations. The term $k=1$, up to the minus sign, counts the arrow configurations which have at least one loop of a fixed type, and sums over the possible loops. It therefore overcounts the number of arrow configurations with at least one loop by the number of arrow configurations which have at least two loops. This is taken care of by the $k=2$ term, which however overcounts it by a amount equal to the number of configurations with at least three loops, and so on. By the inclusion-exclusion principle, the alternating sum (\[altern\]) exactly counts the configurations of arrows with no loop at all.
Let us now come back to the problem of counting the loop-tree configurations of arrows, in which the only loops we allow must wrap around the cylinder (i.e. be non-contractible). Because they cannot intersect themselves, the loops can wrap only once around the cylinder. For this, we modify the previous Laplacian $\Delta$ by changing from $-1$ to $+1$ the entries $\Delta_{i,j}$ for all pairs $i,j$ such that the bond $i-j$ crosses a line going from one boundary of the cylinder to the other boundary, which we will call a defect line. The resulting still symmetric matrix $\Delta_A$ can be viewed as the antiperiodic Laplacian (the lower and upper boundaries are closed and open, or vice-versa, since we have taken $M$ even). We show that $Z = \det{\Delta_A}$ precisely counts the loop-tree configurations we need.
The key observation is that a contractible loop on the cylinder crosses the defect line an even number of times, whereas a non-contractible loop crosses it an odd number of times. As a consequence, and with respect to the previous situation, there is an extra minus sign for each proper cycle giving rise to a non-contractible loop of arrows. One thus obtain $$\det \Delta_A = \sum_{k=0}^{[N/2]} (-1)^k \sum_{\sigma {\rm \ has\
}k{\rm \ proper\ cycles}}
(-1)^{\#\,{\rm non-contr.}} \:|\Delta_{1,\sigma(1)} \cdots \Delta_{N,\sigma(N)}|.$$ If we write $k=p+q$, where $p$ is the number of non-contractible loops (NCL) and $q$ is the number of contractible loops (CL), the summation over $k$ is replaced by two summations over $p$ and $q$, $$\det \Delta_A = \sum_{p \geq 0} \; \sum_{q \geq 0} (-1)^q
\sum_{\sigma {\rm \ has\ }p{\rm \ NCL}, q{\rm \ CL}} |\Delta_{1,\sigma(1)} \cdots \Delta_{N,\sigma(N)}|.$$ One sees that the arrow configurations with a fixed number $p$ of non-contractible loops, i.e. the terms with $q=0$, are all counted positively. Moreover, for each $p$, the inner summation over $q$ is an alternating sum which implements the inclusion-exclusion principle and removes all contractible loops. Therefore $\det \Delta_A$ exactly counts the arrow configurations with no contractible loops, as claimed.
In the continuum limit, it becomes the partition function of a free theory of antiperiodic Grassmannian fields which, in turn, gives $c=-2$ and $h_{\rm min}=-1/8$ [@ruelle]. This is again consistent, since the finite size correction (\[p\]) together with the general formula (\[Acyl\]) for the cylinder yield $c_{\rm eff}=1$.
Conclusion
==========
By analyzing the finite-size effects in terms of the effective central charge $c_{\rm eff} = c - 24h_{\rm min}$, we have shown that the non-local boundary effects in the close-packed dimer model can be consistently accounted for by a single conformal theory having central charge $c=-2$. We have provided a consistent framework for understanding the dependence of the finite-size effects upon the boundary conditions. However this should not be taken as a proof that $c$ must be equal to $-2$. Indeed since the effective central charge merely determines some combination of $c$ and $h_{\rm min}$, one cannot obtain the values of both without some assumption about one of them. This assumption can be a posteriori justified if the conformal description obtained from it is fully consistent.
It turns out in this case that another consistent conformal description exists, with $c=1$ [@bh; @ken], although a detailed analysis of boundary conditions and parity dependence effects has not been carried out in this context. Our explanation for this curious fact is that the $c=1$ theory not only describes the close-packed dimer model, but the general monomer-dimer model (it has been suggested in [@gdj] that trimers would require $c=2$).
On one hand, this has been illustrated for instance in [@fend] where the monomer 2-point correlation function [@fist] has been interpreted in terms of two uncoupled Majorana fermions. The same interpretation can be made for the general $n$-point function for monomers on a boundary [@prru]. On the other hand, the spanning tree description leading to the value $c=-2$ cannot describe dimers with monomers in generic positions, for the basic reason that they are defined on a sublattice and therefore cannot keep track of the positions of all the monomers.
Thus the conformal theory with $c=-2$ must be viewed as a subtheory of that with $c=1$, as it is able to describe the degrees of freedom corresponding to dimer coverings but not general monomer insertions. For those degrees of freedom, the two descriptions should be equivalent.
Acknowledgements {#acknowledgements .unnumbered}
----------------
P.R. is financially supported by the Belgian Fonds National de la Recherche Scientifique (FNRS).
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[^1]: In the limit $M \to \infty$, we expect that the specific boundary conditions we choose do not matter if the theory is local, since the boundaries are sent off to infinity. We could as well choose periodic boundary condition in the vertical direction, closing the cylinder into a torus.
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files/rila0.tex files/rila1.tex files/rila15.tex files/rila2.tex files/rila3.tex files/rilabib.tex
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---
abstract: 'Fuzzy Description Logics (FDLs) are logic-based formalisms used to represent and reason with vague or imprecise knowledge. It has been recently shown that reasoning in most FDLs using truth values from the interval $[0,1]$ becomes undecidable in the presence of a negation constructor and general concept inclusion axioms. One exception to this negative result are FDLs whose semantics is based on the infinitely valued Gödel t-norm ([$\mathsf{G}$]{}). In this paper, we extend previous decidability results for [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em\ALC$]{}to deal also with qualified number restrictions. Our novel approach is based on a combination of the known crispification technique for finitely valued FDLs and the automata-based procedure originally developed for reasoning in . The proposed approach combines the advantages of these two methods, while removing their respective drawbacks.'
author:
- |
Stefan Borgwardt\
Theoretical Computer Science,\
TU Dresden, Germany\
[Stefan.Borgwardt@tu-dresden.de](Stefan.Borgwardt@tu-dresden.de) Rafael Pe[ñ]{}aloza\
KRDB Research Centre,\
Free University of Bozen-Bolzano, Italy\
[rafael.penaloza@unibz.it](rafael.penaloza@unibz.it)
bibliography:
- 'medium-string.bib'
- 'GoedelSROIQ.bib'
title: 'Reasoning in Infinitely Valued [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}'
---
Introduction
============
It is well-known that one of the main requirements for the development of an intelligent application is a formalism capable of representing and handling knowledge without ambiguity. Description Logics (DLs) are a well-studied family of knowledge representation formalisms [@BCM+07]. They constitute the logical backbone of the standard Semantic Web ontology language OWL2,[^1] and its profiles, and have been successfully applied to represent the knowledge of many and diverse application domains, particularly in the bio-medical sciences.
DLs describe the domain knowledge using *concepts* (such as ) that represent sets of individuals, and *roles* () that represent connections between individuals. *Ontologies* are collections of axioms formulated over these concepts and roles, which restrict their possible interpretations. The typical axioms considered in DLs are *assertions*, like ${\ensuremath{\textsf{bob}\hspace{0.08em}{:}\hspace{0.05em}\textsf{Patient}}\xspace}$, providing knowledge about specific individuals; and *general concept inclusions (GCIs)*, such as $\textsf{Patient}\sqsubseteq\textsf{Human}$, which express general relations between concepts. Different DLs are characterized by the constructors allowed to generate complex concepts and roles from atomic ones. [@ScSm-AI91] is a prototypical DL of intermediate expressivity that contains the concept constructors conjunction ($C\sqcap D$), negation ($\lnot C$), and existential restriction ($\exists r.C$ for a role $r$). If additionally qualified number restrictions (${\ensuremath{{{\geqslant}}n\,r.C}\xspace}$ for $n\in\naturals$) are allowed, the resulting logic is denoted by [$\ALC\kern-0.16em\Qmc$]{}. Common reasoning problems in [$\ALC\kern-0.16em\Qmc$]{}, such as consistency of ontologies or subsumption between concepts, are known to be -complete [@Schi-IJCAI91; @Tobi-01].
Fuzzy Description Logics (FDLs) have been introduced as extensions of classical DLs to represent and reason with vague knowledge. The main idea is to consider all the truth values from the interval $[0,1]$ instead of only *true* and *false*. In this way, it is possible give a more fine-grained semantics to inherently vague concepts like $\textsf{LowFrequency}$ or $\textsf{HighConcentration}$, which can be found in biomedical ontologies like [SnomedCT]{},[^2] and Galen.[^3] The different members of the family of FDLs are characterized not only by the constructors they allow, but also by the way these constructors are interpreted.
To interpret conjunction in complex concepts like $$\begin{aligned}
\exists\textsf{hasHeartRate}.\textsf{LowFrequency}\sqcap {} \qquad \qquad \qquad \\
\exists\textsf{hasBloodAlcohol}.\textsf{HighConcentration},\end{aligned}$$ a popular approach is to use so-called *t-norms* [@KlMP-00]. The semantics of the other logical constructors can then be derived from these t-norms in a principled way, as suggested by @Haje-01 . Following the principles of mathematical fuzzy logic, existential restrictions are interpreted as suprema of truth values. However, to avoid problems with infinitely many truth values, reasoning in fuzzy DLs is often restricted to so-called *witnessed models* [@Haje-FSS05], in which these suprema are required to be maxima; i.e., the degree is witnessed by at least one domain element.
Unfortunately, reasoning in most FDLs becomes undecidable when the logic allows to use GCIs and negation under witnessed model semantics [@BaPe-FroCoS11; @CeSt-ISc13; @BoDP-AI15]. One of the few exceptions known are FDLs using the *Gödel* t-norm defined as $\min\{x,y\}$ to interpret conjunctions [@BoDP-KR14]. Despite not being as well-behaved as finitely valued FDLs, which use a finite total order of truth values instead of the infinite interval $[0,1]$ [@BoPe-JoDS13], it has been shown using an automata-based approach that reasoning in Gödel extensions of exhibits the same complexity as in the classical case, [i.e. ]{}it is -complete. A major drawback of this approach is that it always has an exponential runtime, even when the input ontology has a simple form.
In this paper, we extend the results of [@BoDP-KR14] to deal with qualified number restrictions, showing again that the complexity of reasoning remains the same as for the classical case; i.e., it is -complete. To this end, we focus only on the problem of *local consistency*, which is a generalization of the classical concept satisfiability problem. We follow a more practical approach that combines the automata-based construction from [@BoDP-KR14] with reduction techniques developed for finitely valued FDLs [@Stra-JELIA04; @BDGS-IJAR09; @BoSt-13]. We exploit the forest model property of classical [$\ALC\kern-0.16em\Qmc$]{} [@Kaza-JELIA04] to encode order relationships between concepts in a fuzzy interpretation in a manner similar to the Hintikka trees from [@BoDP-KR14]. However, instead of using automata to determine the existence of such trees, we reduce the fuzzy ontology directly into a classical [$\ALC\kern-0.16em\Qmc$]{}ontology whose local consistency is equivalent to that of the original ontology. This enables us to use optimized reasoners for classical DLs. In addition to the *cut-concepts* of the form ${\scalebox{0.93}{\ensuremath{\fbox{$C{\geqslant}q$}}}\xspace}$ for a fuzzy concept $C$ and a value $q$, which are used in the reductions for finitely valued DLs [@Stra-JELIA04; @BDGS-IJAR09; @BoSt-13], we employ *order concepts* ${\scalebox{0.93}{\ensuremath{\fbox{$C{\leqslant}D$}}}\xspace}$ expressing relationships between fuzzy concepts. Contrary to the reductions for finitely valued Gödel FDLs presented by @BDGS-IJAR09 , our reduction produces a classical ontology whose size is polynomial in the size of the input fuzzy ontology. Thus, we obtain tight complexity bounds for reasoning in this FDL [@Tobi-01]. An extended version of this paper appears in [@BoPe-FroCoS15].
Preliminaries
=============
For the rest of this paper, we focus solely on vague statements that take truth degrees from the infinite interval $[0,1]$, where the *Gödel t-norm*, defined by $\min\{x,y\}$, is used to interpret logical conjunction. The semantics of implications is given by the *residuum* of this t-norm; that is, $$x{\ensuremath{\Rightarrow}\xspace}y:=\begin{cases}
1 &\text{if $x{\leqslant}y$,} \\
y &\text{otherwise.}
\end{cases}$$ We use both the *residual negation* $\ominus x:=x{\ensuremath{\Rightarrow}\xspace}0$ and the *involutive negation* ${\sim}x:=1-x$ in the rest of this paper.
We first recall some basic definitions from [@BoDP-KR14], which will be used extensively in the proofs throughout this work. An *order structure* $S$ is a finite set containing at least the numbers $0$, $0.5$, and $1$, together with an involutive unary operation ${\ensuremath{\mathsf{inv}}\xspace}\colon S\to S$ such that ${\ensuremath{\mathsf{inv}}\xspace}(x)=1-x$ for all $x\in S\cap[0,1]$. A *total preorder* over $S$ is a transitive and total binary relation ${{\ensuremath{\preccurlyeq}}}\subseteq S\times S$. For $x,y\in S$, we write $x\equiv y$ if $x{\ensuremath{\preccurlyeq}}y$ and $y{\ensuremath{\preccurlyeq}}x$. Notice that $\equiv$ is an equivalence relation on $S$. The total preorders considered in [@BoDP-KR14] have to satisfy additional properties; for instance, that $0$ and $1$ are always the least and greatest elements, respectively. These properties can be found in our reduction in the axioms of ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$ (see Section \[sec:local-consistency\] for more details).
The syntax of the FDL [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}is the same as that of classical [$\ALC\kern-0.16em\Qmc$]{}, with the addition of the implication constructor (denoted by the use of at the beginning of the name). This constructor is often added to FDLs, as the residuum cannot, in general, be expressed using only the tnorm and negation operators, in contrast to the classical semantics. In particular, this holds for the Gödel t-norm and its residuum, which is the focus of this work. Let now , , and be mutually disjoint sets of *concept*, *role*, and *individual names*, respectively. *Concepts* of [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}are built using the syntax rule $$C,D::=\top\mid A\mid \lnot C\mid C\sqcap D\mid C\to D\mid\forall r.C\mid
{\ensuremath{{{\geqslant}}n\,r.C}\xspace},$$ where $A\in\NC$, $r\in\NR$, $C,D$ are concepts, and $n\in\naturals$. We use the abbreviations $$\begin{aligned}
\bot:={} & \lnot\top, \\
C\sqcup D:= {} & \lnot(\lnot C\sqcap\lnot D), \\
\exists r.C:= {} & {\ensuremath{{{\geqslant}}1\,r.C}\xspace}, \text{\quad \quad \quad and} \\
{\ensuremath{{{\leqslant}}n\,r.C}\xspace}:= {} & \lnot({\ensuremath{{{\geqslant}}(n+1)\,r.C}\xspace})\end{aligned}$$ Notice that @BDGS-IJUF12 consider a different definition of *at-most restrictions*, which uses the residual negation; that is, they define ${\ensuremath{{{\leqslant}}n\,r.C}\xspace}:=({\ensuremath{{{\geqslant}}(n+1)\,r.C}\xspace})\to\bot$ . This has the strange side effect that the value of ${\ensuremath{{{\leqslant}}n\,r.C}\xspace}$ is always either $0$ or $1$ (see the semantics below). However, this discrepancy in definitions is not an issue since our algorithm can handle both cases.
The semantics of this logic is based on interpretations. A *[$\mathsf{G}$]{}-interpretation* is a pair $\Imc=(\Delta^\Imc,\cdot^\Imc)$, where $\Delta^\Imc$ is a non-empty set called the *domain*, and $\cdot^\Imc$ is the *interpretation function* that assigns to each individual name $a\in\NI$ an element $a^\Imc\in\Delta^\Imc$, to each concept name $A\in\NC$ a fuzzy set $A^\Imc\colon\Delta^\Imc\to[0,1]$, and to each role name $r\in\NR$ a fuzzy binary relation $r^\Imc\colon\Delta^\Imc\times\Delta^\Imc\to[0,1]$. The interpretation of complex concepts is obtained from the semantics of first-order fuzzy logics via the well-known translation from DL concepts to first-order logic [@Stra-JAIR01; @BDGS-IJUF12], [i.e. ]{}for all $d\in\Delta^\Imc$, $$\begin{aligned}
\top^\Imc(d) &:= 1 \\
(\lnot C)^\Imc(d) &:= 1-C^\Imc(d) \\
(C\sqcap D)^\Imc(d) &:= \min\{C^\Imc(d),D^\Imc(d)\} \\
(C\to D)^\Imc(d) &:= C^\Imc(d){\ensuremath{\Rightarrow}\xspace}D^\Imc(d) \\
(\forall r.C)^\Imc(d) &:= \inf_{e\in\Delta^\Imc}r^\Imc(d,e){\ensuremath{\Rightarrow}\xspace}C^\Imc(e) \\
({\ensuremath{{{\geqslant}}n\,r.C}\xspace})^\Imc(d) &:=
\sup_{\substack{e_1,\dots,e_n\in\Delta^\Imc\\\text{pairwise different}}}
\min_{i=1}^n\min\{r^\Imc(d,e_i),C^\Imc(e_i)\}
$$ Recall that the usual duality between existential and value restrictions that appears in classical DLs does not hold in .
A *classical interpretation* is defined similarly, with the set of truth values restricted to $0$ and $1$. In this case, the semantics of a concept $C$ is commonly viewed as a set $C^\Imc\subseteq\Delta^\Imc$ instead of the characteristic function $C^\Imc\colon\Delta^\Imc\to\{0,1\}$.
In the following, we restrict all reasoning problems to so-called *witnessed* [$\mathsf{G}$]{}-interpretations [@Haje-FSS05], which intuitively require the suprema and infima in the semantics to be maxima and minima, respectively. More formally, the is *witnessed* if, for every $d\in\Delta^\Imc$, $n{\geqslant}0$, $r\in\NR$, and concept $C$, there exist $e,e_1,\dots,e_n\in\Delta^\Imc$ (where $e_1,\dots,e_n$ are pairwise different) such that $$\begin{aligned}
(\forall r.C)^\Imc(d) &= r^\Imc(d,e){\ensuremath{\Rightarrow}\xspace}C^\Imc(e)\text{ and} \\
({\ensuremath{{{\geqslant}}n\,r.C}\xspace})^\Imc(d) &= \min_{i=1}^n\min\{r^\Imc(d,e_i),C^\Imc(e_i)\}.\end{aligned}$$
The axioms of [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}extend classical axioms by allowing to compare degrees of arbitrary assertions in so-called *ordered ABoxes* [@BoDP-KR14], and to state inclusions relationships between fuzzy concepts that hold to a certain degree, instead of only $1$. A *classical assertion* is an expression of the form ${\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}$ or ${\ensuremath{(a,b)\hspace{0.08em}{:}\hspace{0.05em}r}\xspace}$ for $a,b\in\NI$, $r\in\NR$, and a concept $C$. An *order assertion* is of the form ${\ensuremath{\langle \alpha\bowtie q\rangle}\xspace}$ or ${\ensuremath{\langle \alpha\bowtie\beta\rangle}\xspace}$ where ${\bowtie}\in\{<,{\leqslant},=,{\geqslant},>\}$, $\alpha,\beta$ are classical assertions, and $q\in[0,1]$. A *(fuzzy) general concept inclusion axiom (GCI)* is of the form ${\ensuremath{\langle C\sqsubseteq D{\geqslant}q\rangle}\xspace}$ for concepts $C,D$ and $q\in[0,1]$. An *ordered ABox* is a finite set of order assertions, a *TBox* is a finite set of GCIs, and an *ontology* $\Omc=(\Amc,\Tmc)$ consists of an ordered ABox and a TBox . A [$\mathsf{G}$]{}-interpretation *satisfies* (or is a *model* of) an order assertion ${\ensuremath{\langle \alpha\bowtie\beta\rangle}\xspace}$ if $\alpha^\Imc\bowtie\beta^\Imc$ (where $({\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace})^\Imc:=C^\Imc(a^\Imc)$, $({\ensuremath{(a,b)\hspace{0.08em}{:}\hspace{0.05em}r}\xspace})^\Imc:=r^\Imc(a^\Imc,b^\Imc)$, and $q^\Imc:=q$); it *satisfies* a GCI ${\ensuremath{\langle C\sqsubseteq D{\geqslant}q\rangle}\xspace}$ if $C^\Imc(d){\ensuremath{\Rightarrow}\xspace}D^\Imc(d){\geqslant}q$ holds for all $d\in\Delta^\Imc$; and it *satisfies* an ordered ABox, TBox, or ontology if it satisfies all its axioms. An ontology is *consistent* if it has a (witnessed) model.
Given an ontology , we denote by ${\ensuremath{\mathsf{rol}}\xspace}(\Omc)$ the set of all role names occurring in and by $\sub(\Omc)$ the closure under negation of the set of all subconcepts occurring in . We consider the concepts $\lnot\lnot C$ and $C$ to be equal, and thus the latter set is of quadratic size in the size of . Moreover, we denote by [$\Vmc_\Omc$]{}the closure under the involutive negation $x\mapsto 1-x$ of the set of all truth degrees appearing in , together with $0$, $0.5$, and $1$. This set is of size linear on the size of . We sometimes denote the elements of ${\ensuremath{\Vmc_\Omc}\xspace}\subseteq[0,1]$ as $0=q_0<q_1<\dots<q_{k-1}<q_k=1$.
We stress that we do not consider the general consistency problem in this paper, but only a restricted version that uses only one individual name. An ordered ABox is *local* if it contains no role assertions ${\ensuremath{(a,b)\hspace{0.08em}{:}\hspace{0.05em}r}\xspace}$ and there is a single individual name $a\in\NI$ such that all order assertions in only use $a$. The *local consistency* problem, [i.e. ]{}deciding whether an ontology $(\Amc,\Tmc)$ with a local ordered ABox is consistent, can be seen as a generalization of the classical concept satisfiability problem [@BoPe-JoDS13].
Other common reasoning problems for FDLs, such as concept satisfiability and subsumption can be reduced to local consistency [@BoDP-KR14]: the subsumption between $C$ and $D$ to degree $q$ [w.r.t. ]{}a TBox is equivalent to the (local) inconsistency of $(\{{\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C\to D}\xspace}<q\rangle}\xspace}\},\Tmc)$, and the satisfiability of $C$ to degree $q$ w.r.t. is equivalent to the (local) consistency of $(\{{\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}{\geqslant}q\rangle}\xspace}\},\Tmc)$.
In the following section we show how to decide local consistency of a [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}ontology through a reduction to classical ontology consistency.
Deciding Local Consistency {#sec:local-consistency}
==========================
Let $\Omc=(\Amc,\Tmc)$ be a [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}ontology where is a local ordered ABox that uses only the individual name $a$. The main ideas behind the reduction to classical [$\ALC\kern-0.16em\Qmc$]{}are that it suffices to consider tree-shaped interpretations, where each domain element has a unique role predecessor, and that we only have to consider the *order* between values of concepts, instead of their precise values. This insight allows us to consider only finitely many different cases [@BoDP-KR14].
To compare the values of the elements of $\sub(\Omc)$ at different domain elements, we use the order structure $$\Umc :=
{\ensuremath{\Vmc_\Omc}\xspace}\cup\sub(\Omc)\cup\sub_\uparrow(\Omc)\cup\{\lambda,\lnot\lambda\},$$ where $\sub_\uparrow(\Omc):=\{{\ensuremath{\langleC\rangle_\uparrow}\xspace} \mid C\in\sub(\Omc)\}$, ${\ensuremath{\mathsf{inv}}\xspace}(\lambda):=\lnot\lambda$, ${\ensuremath{\mathsf{inv}}\xspace}(C):=\lnot C$, and ${\ensuremath{\mathsf{inv}}\xspace}({\ensuremath{\langleC\rangle_\uparrow}\xspace}):={\ensuremath{\langle\lnot C\rangle_\uparrow}\xspace}$, for all concepts $C\in\sub(\Omc)$. The idea is that total preorders over describe the relationships between the values of $\sub(\Omc)$ and [$\Vmc_\Omc$]{}at a single domain element. The elements of $\sub_\uparrow(\Omc)$ allow us to additionally refer to the relevant values at the unique role predecessor of the current domain element (in a tree-shaped interpretation). The value $\lambda$ represents the value of the role connection from this predecessor. For convenience, we define ${\ensuremath{\langleq\rangle_\uparrow}\xspace}:=q$ for all $q\in{\ensuremath{\Vmc_\Omc}\xspace}$.
In order to describe such total preorders in a classical [$\ALC\kern-0.16em\Qmc$]{}ontology, we employ special concept names of the form ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}$ for $\alpha,\beta\in\Umc$. This differs from previous reductions for finitely valued FDLs [@Stra-JELIA04; @BoSt-ISC11; @BDGS-IJUF12] in that we not only consider *cut-concepts* of the form ${\scalebox{0.93}{\ensuremath{\fbox{$q{\leqslant}\alpha$}}}\xspace}$ with $q\in{\ensuremath{\Vmc_\Omc}\xspace}$, but also relationships between different concepts.[^4] For convenience, we introduce the abbreviations ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\beta$}}}\xspace}:={\scalebox{0.93}{\ensuremath{\fbox{$\beta{\leqslant}\alpha$}}}\xspace}$, ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha<\beta$}}}\xspace}:=\lnot{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\beta$}}}\xspace}$, and similarly for $=$ and $>$. Furthermore, we define the complex expressions
- ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\min\{\beta,\gamma\}$}}}\xspace} :=
{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\beta$}}}\xspace}\sqcup{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\gamma$}}}\xspace}$,
- ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\min\{\beta,\gamma\}$}}}\xspace} :=
{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}\sqcap{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\gamma$}}}\xspace}$,
- ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\beta{\ensuremath{\Rightarrow}\xspace}\gamma$}}}\xspace} :=
({\scalebox{0.93}{\ensuremath{\fbox{$\beta{\leqslant}\gamma$}}}\xspace}\to{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}1$}}}\xspace})\sqcap
({\scalebox{0.93}{\ensuremath{\fbox{$\beta>\gamma$}}}\xspace}\to{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\geqslant}\gamma$}}}\xspace})$,
- ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta{\ensuremath{\Rightarrow}\xspace}\gamma$}}}\xspace} :=
{\scalebox{0.93}{\ensuremath{\fbox{$\beta{\leqslant}\gamma$}}}\xspace}\sqcup{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\gamma$}}}\xspace}$,
and extend these notions to the expressions ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha\bowtie\beta{\ensuremath{\Rightarrow}\xspace}\gamma$}}}\xspace}$ etc., for ${\bowtie}\in\{<,=,>\}$, analogously.
For each concept $C\in\sub(\Omc)$, we now define the classical [$\ALC\kern-0.16em\Qmc$]{}TBox ${\ensuremath{\mathsf{red}}\xspace}(C)$, depending on the form of $C$, as follows. $$\begin{aligned}
{\ensuremath{\mathsf{red}}\xspace}(\top) :={}
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$\top{\geqslant}1$}}}\xspace}\} \\
{\ensuremath{\mathsf{red}}\xspace}(\lnot C) := {}
& \emptyset \\
{\ensuremath{\mathsf{red}}\xspace}(C\sqcap D) :={}
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$C\sqcap D=\min\{C,D\}$}}}\xspace}\} \\
{\ensuremath{\mathsf{red}}\xspace}(C\to D) :={}
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$C\to D=C{\ensuremath{\Rightarrow}\xspace}D$}}}\xspace}\} \displaybreak[0] \\
{\ensuremath{\mathsf{red}}\xspace}(\forall r.C) :={}
& \{\top\sqsubseteq
\exists r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\geqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}\sqcap {} \\
& \phantom{\{\top\sqsubseteq{}}
\forall r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\leqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}\} \displaybreak[0] \\
{\ensuremath{\mathsf{red}}\xspace}({\ensuremath{{{\geqslant}}n\,r.C}\xspace}) :={}
& \{\top\sqsubseteq
{\ensuremath{{{\geqslant}}n\,r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}{\leqslant}\min\{\lambda,C\}$}}}\xspace}}\xspace}\sqcap{} \\
& \phantom{\{\top\sqsubseteq{}}
\lnot{\ensuremath{{{\geqslant}}n\,r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}<\min\{\lambda,C\}$}}}\xspace}}\xspace}\}
$$ Intuitively, ${\ensuremath{\mathsf{red}}\xspace}(C)$ describes the semantics of $C$ in terms of its order relationships to other elements of . Note that the semantics of the involutive negation $\lnot C={\ensuremath{\mathsf{inv}}\xspace}(C)$ is already handled by the operator [$\mathsf{inv}$]{}(see also the last line of the definition of ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$ below).
The reduced classical [$\ALC\kern-0.16em\Qmc$]{}ontology ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$ is defined as follows: $$\begin{aligned}
{\ensuremath{\mathsf{red}}\xspace}(\Omc) :={}
& ({\ensuremath{\mathsf{red}}\xspace}(\Amc),{\ensuremath{\mathsf{red}}\xspace}(\Umc)\cup{\ensuremath{\mathsf{red}}\xspace}(\uparrow)\cup{\ensuremath{\mathsf{red}}\xspace}(\Tmc)), \\
{\ensuremath{\mathsf{red}}\xspace}(\Amc) :={}
& \{{\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}{\scalebox{0.93}{\ensuremath{\fbox{$C\bowtie q$}}}\xspace}}\xspace}\mid{\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie
q\rangle}\xspace}\in\Amc\}\cup {} \\
& \{{\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}{\scalebox{0.93}{\ensuremath{\fbox{$C\bowtie D$}}}\xspace}}\xspace}\mid
{\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie{\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}D}\xspace}\rangle}\xspace}\in\Amc\}, \displaybreak[0] \\
{\ensuremath{\mathsf{red}}\xspace}(\Umc) :={}
& \{{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}\sqcap{\scalebox{0.93}{\ensuremath{\fbox{$\beta{\leqslant}\gamma$}}}\xspace}\sqsubseteq
{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\gamma$}}}\xspace} \mid
\alpha,\beta,\gamma\in\Umc\} \cup{} \\
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}\sqcup{\scalebox{0.93}{\ensuremath{\fbox{$\beta{\leqslant}\alpha$}}}\xspace} \mid
\alpha,\beta\in\Umc\} \cup{} \\
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$0{\leqslant}\alpha$}}}\xspace}\sqcap{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}1$}}}\xspace} \mid
\alpha\in\Umc\} \cup{} \\
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$\alpha\bowtie\beta$}}}\xspace} \mid
\alpha,\beta\in{\ensuremath{\Vmc_\Omc}\xspace},\ \alpha\bowtie\beta\} \cup{} \\
& \{{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\mathsf{inv}}\xspace}(\beta){\leqslant}{\ensuremath{\mathsf{inv}}\xspace}(\alpha)$}}}\xspace} \mid
\alpha,\beta\in\Umc\}, \displaybreak[0] \\
{\ensuremath{\mathsf{red}}\xspace}(\uparrow) := {}
& \{{\scalebox{0.93}{\ensuremath{\fbox{$\alpha\bowtie\beta$}}}\xspace}\sqsubseteq
\forall r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\alpha\rangle_\uparrow}\xspace}\bowtie{\ensuremath{\langle\beta\rangle_\uparrow}\xspace}$}}}\xspace} \mid{} \\
&\quad\quad\quad\quad
\alpha,\beta\in{\ensuremath{\Vmc_\Omc}\xspace}\cup\sub(\Omc),\ r\in{\ensuremath{\mathsf{rol}}\xspace}(\Omc)\}, \\
{\ensuremath{\mathsf{red}}\xspace}(\Tmc) :={}
& \{\top\sqsubseteq{\scalebox{0.93}{\ensuremath{\fbox{$q{\leqslant}C{\ensuremath{\Rightarrow}\xspace}D$}}}\xspace} \mid
{\ensuremath{\langle C\sqsubseteq D{\geqslant}q\rangle}\xspace}\in\Tmc\} \cup {} \\
& \bigcup_{C\in\sub(\Omc)}{\ensuremath{\mathsf{red}}\xspace}(C).\end{aligned}$$ We briefly explain this construction. The reductions of the order assertions and fuzzy GCIs in are straightforward; the former expresses that the individual $a$ must belong to the corresponding order concept $C\bowtie q$ or $C\bowtie D$, while the latter expresses that every element of the domain must satisfy the restriction provided by the fuzzy GCI. The axioms of ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$ intuitively ensure that the relation “${\leqslant}$” forms a total preorder that is compatible with all the values in [$\Vmc_\Omc$]{}, and that ${\ensuremath{\mathsf{inv}}\xspace}$ is an antitone operator. Finally, the TBox ${\ensuremath{\mathsf{red}}\xspace}(\uparrow)$ expresses a connection between the orders of a domain element and those of its role successors.
The following lemmata show that this reduction is correct; i.e., that it preserves local consistency.
\[lem:soundness\] If ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$ has a classical model, then has a [$\mathsf{G}$]{}-model.
By [@Kaza-JELIA04], ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$ must have a *tree model* , [i.e. ]{} we can assume that $\Delta^\Imc$ is a prefix-closed subset of $\naturals^*$, $a^\Imc={{\varepsilon}}$, for all $n_1,\dots,n_k\in\naturals$, $k{\geqslant}1$, with $u:=n_1\dots n_k\in\Delta^\Imc$, the element ${\ensuremath{u_\uparrow}\xspace}:=n_1\dots n_{k-1}\in\Delta^\Imc$ is an $r$-predecessor of $u$ for some $r\in{\ensuremath{\mathsf{rol}}\xspace}(\Omc)$, and there are no other role connections. For any $u\in\Delta^{\Imc}$, we denote by ${\ensuremath{\preccurlyeq}}_u$ the corresponding total preorder on , that is, we define $\alpha{\ensuremath{\preccurlyeq}}_u\beta$ iff $u\in{\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}^\Imc$, and by $\equiv_u$ the induced equivalence relation. As a first step in the construction of a [$\mathsf{G}$]{}-model of , we define the auxiliary function $v\colon\Umc\times\Delta^\Imc\to[0,1]$ that satisfies the following conditions for all $u\in\Delta^\Imc$:
1. \[p1\] for all $q\in{\ensuremath{\Vmc_\Omc}\xspace}$, we have $v(q,u)=q$,
2. \[p2\] for all $\alpha,\beta\in\Umc$, we have $v(\alpha,u){\leqslant}v(\beta,u)$ iff $\alpha{\ensuremath{\preccurlyeq}}_u\beta$,
3. \[p3\] for all $\alpha\in\Umc$, we have $v({\ensuremath{\mathsf{inv}}\xspace}(\alpha),u)=1-v(\alpha,u)$,
4. \[p4\] if $u\neq{{\varepsilon}}$, then for all $C\in\sub(\Omc)$ it holds that $v(C,{\ensuremath{u_\uparrow}\xspace})=v({\ensuremath{\langleC\rangle_\uparrow}\xspace},u)$.
We define $v$ by induction on the structure of $\Delta^\Imc$ starting with ${{\varepsilon}}$. Let $\Umc/{\equiv_{{\varepsilon}}}$ be the set of all equivalence classes of $\equiv_{{\varepsilon}}$. Then ${\ensuremath{\preccurlyeq}}_{{\varepsilon}}$ yields a total order ${\leqslant}_{{\varepsilon}}$ on $\Umc/{\equiv_{{\varepsilon}}}$. Since satisfies ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$, we have $$[0]_{{\varepsilon}}<_{{\varepsilon}}[q_1]_{{\varepsilon}}<_{{\varepsilon}}\dots<_{{\varepsilon}}[q_{k-1}]_{{\varepsilon}}<_{{\varepsilon}}[1]_{{\varepsilon}}$$ [w.r.t. ]{} this order. For every $[\alpha]_{\varepsilon}\in\Umc/{\equiv_{{\varepsilon}}}$, we now set ${\ensuremath{\mathsf{inv}}\xspace}([\alpha]_{{\varepsilon}}):=[{\ensuremath{\mathsf{inv}}\xspace}(\alpha)]_{{\varepsilon}}$. This function is well-defined by the axioms in ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$. On all $\alpha\in[q]_{{\varepsilon}}$ for $q\in{\ensuremath{\Vmc_\Omc}\xspace}$, we now define $v(\alpha,{{\varepsilon}}):=q$, which ensures that \[p1\] holds. For the equivalence classes that do not contain a value from [$\Vmc_\Omc$]{}, note that by ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$, every such class must be strictly between $[q_i]_{{\varepsilon}}$ and $[q_{i+1}]_{{\varepsilon}}$ for $q_i,q_{i+1}\in{\ensuremath{\Vmc_\Omc}\xspace}$. We denote the $n_i$ equivalence classes between $[q_i]_{{\varepsilon}}$ and $[q_{i+1}]_{{\varepsilon}}$ as follows: $$[q_i]_{{\varepsilon}}<_{{\varepsilon}}E_1^i<_{{\varepsilon}}\dots<_{{\varepsilon}}E_{n_i}^i<_{{\varepsilon}}[q_{i+1}]_{{\varepsilon}}.$$ For every $\alpha\in E_j^i$, we set $v(\alpha,{{\varepsilon}}):=q_i+\tfrac{j}{n_i+1}(q_{i+1}-q_i)$, which ensures that \[p2\] is also satisfied. Furthermore, observe that $1-q_{i+1}$ and $1-q_i$ are also adjacent in [$\Vmc_\Omc$]{}and we have $$[1-q_{i+1}]_{{\varepsilon}}<_{{\varepsilon}}{\ensuremath{\mathsf{inv}}\xspace}(E_{n_i}^i)<_{{\varepsilon}}\dots<_{{\varepsilon}}{\ensuremath{\mathsf{inv}}\xspace}(E_1^i)<_{{\varepsilon}}[1-q_i]_{{\varepsilon}}$$ by the axioms in ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$. Hence, it follows from the definition of $v(\alpha,{{\varepsilon}})$ that \[p3\] holds.
Let now $u\in\Delta^\Imc$ be such that the function $v$, satisfying the properties \[p1\]–\[p4\], has already been defined for ${\ensuremath{u_\uparrow}\xspace}$. Since is a tree model, there must be an $r\in\NR$ such that $({\ensuremath{u_\uparrow}\xspace},u)\in r^\Imc$. We again consider the set of equivalence classes $\Umc/{\equiv_u}$ and set $v(\alpha,u):=q$ for all $q\in{\ensuremath{\Vmc_\Omc}\xspace}$ and $\alpha\in[q]_u$, and $v(\alpha,u):=v(C,{\ensuremath{u_\uparrow}\xspace})$ for all $C\in\sub(\Omc)$ and $\alpha\in[{\ensuremath{\langleC\rangle_\uparrow}\xspace}]_u$. To see that this is well-defined, consider the case that $[{\ensuremath{\langleC\rangle_\uparrow}\xspace}]_u=[{\ensuremath{\langleD\rangle_\uparrow}\xspace}]_u$, [i.e. ]{}$u\in{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langleC\rangle_\uparrow}\xspace}={\ensuremath{\langleD\rangle_\uparrow}\xspace}$}}}\xspace}^\Imc$. From the axioms in ${\ensuremath{\mathsf{red}}\xspace}(\uparrow)$ and the fact that $({\ensuremath{u_\uparrow}\xspace},u)\in r^\Imc$, it follows that ${\ensuremath{u_\uparrow}\xspace}\in{\scalebox{0.93}{\ensuremath{\fbox{$C=D$}}}\xspace}^\Imc$, and thus $[C]_{{\ensuremath{u_\uparrow}\xspace}}=[D]_{{\ensuremath{u_\uparrow}\xspace}}$. Since \[p2\] is satisfied for ${\ensuremath{u_\uparrow}\xspace}$, we get $v(C,{\ensuremath{u_\uparrow}\xspace})=v(D,{\ensuremath{u_\uparrow}\xspace})$. The same argument shows that $[q]_u=[{\ensuremath{\langleq\rangle_\uparrow}\xspace}]_u=[{\ensuremath{\langleC\rangle_\uparrow}\xspace}]_u$ implies $v(q,{\ensuremath{u_\uparrow}\xspace})=v(C,{\ensuremath{u_\uparrow}\xspace})$. For the remaining equivalence classes, we can use a construction analogous to the case for ${{\varepsilon}}$ by considering the two unique neighboring equivalence classes that contain an element of ${\ensuremath{\Vmc_\Omc}\xspace}\cup\sub(\Omc)$ (for which $v$ has already been defined). This construction ensures that \[p1\]–\[p4\] hold for $u$.
Based on the function $v$, we define the [$\mathsf{G}$]{}-interpretation $\Imc_{\fsf}$ over the domain $\Delta^{\Imc_{\fsf}}:=\Delta^\Imc$, where $a^{\Imc_{\fsf}}:=a^\Imc={{\varepsilon}}$; $$\begin{aligned}
A^{\Imc_{\fsf}}(u) &:= \begin{cases}
v(A,u) &\text{if $A\in\sub(\Omc)$,} \\
0 &\text{otherwise; \quad and}
\end{cases} \\
r^{\Imc_{\fsf}}(u,w) &:= \begin{cases}
v(\lambda,w) &\text{if $(u,w)\in r^\Imc$,} \\
0 &\text{otherwise.}
\end{cases}
\end{aligned}$$ We show by induction on the structure of $C$ that $$\label{If-correct}
C^{\Imc_{\fsf}}(u)=v(C,u)
\text{ for all }C\in\sub(\Omc)\text{ and }u\in\Delta^\Imc.$$ For concept names, this holds by the definition of $\Imc_{\fsf}$. For $\top$, we know that $\top^{\Imc_{\fsf}}(u)=1=v(\top,u)$ by the definition of ${\ensuremath{\mathsf{red}}\xspace}(\top)$ and \[p2\]. For $\lnot C$, we have $$(\lnot C)^{\Imc_{\fsf}}(u)=1-C^{\Imc_{\fsf}}(u)=1-v(C,u)=v(\lnot C,u)$$ by the induction hypothesis and \[p3\]. For conjunctions $C\sqcap D$, we know that $$\begin{aligned}
(C\sqcap D)^{\Imc_{\fsf}}(u) &
{} = \min\{C^{\Imc_{\fsf}}(u),D^{\Imc_{\fsf}}(u)\} \\ &
{} = \min\{v(C,u),v(D,u)\} \\ &
{} = v(C\sqcap D,u)
\end{aligned}$$ by the definition of ${\ensuremath{\mathsf{red}}\xspace}(C\sqcap D)$ and \[p2\]. Implications can be treated similarly.
Consider a value restriction $\forall r.C\in\sub(\Omc)$. For every $w\in\Delta^\Imc$ with $(u,w)\in r^\Imc$, we have $w\in{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\leqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}^\Imc$ since satisfies ${\ensuremath{\mathsf{red}}\xspace}(\forall r.C)$. By the induction hypothesis, the fact that ${\ensuremath{w_\uparrow}\xspace}=u$, \[p2\], and \[p4\], this implies that $v(\forall r.C,u)
{\leqslant}v(\lambda,w){\ensuremath{\Rightarrow}\xspace}v(C,w)
= r^{\Imc_{\fsf}}(u,w){\ensuremath{\Rightarrow}\xspace}C^{\Imc_{\fsf}}(w)$, and thus $$\begin{aligned}
(\forall r.C)^{\Imc_{\fsf}}(u)
& {} = \inf\limits_{\mathclap{\phantom{======} w\in\Delta^{\Imc},\ (u,w)\in r^\Imc}} \quad
r^{\Imc_{\fsf}}(u,w){\ensuremath{\Rightarrow}\xspace}C^{\Imc_{\fsf}}(w) \\
& {} {\geqslant}v(\forall r.C,u).
\end{aligned}$$ Furthermore, by the existential restriction introduced in ${\ensuremath{\mathsf{red}}\xspace}(\forall r.C)$, we know that there exists a $w_0\in\Delta^\Imc$ such that $(u,w_0)\in r^\Imc$ and $w_0\in{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\geqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}^\Imc$. By the same arguments as above, we get $$\begin{aligned}
v(\forall r.C,u)
{} & {\geqslant}r^{\Imc_{\fsf}}(u,w_0){\ensuremath{\Rightarrow}\xspace}C^{\Imc_{\fsf}}(w_0) \\
{} & {\geqslant}(\forall r.C)^{\Imc_{\fsf}}(u),
\end{aligned}$$ which concludes the proof of for $\forall r.C$. As a by-product, we have found in the element $w_0$ the witness required for satisfying the concept $\forall r.C$ at $u$.
Consider now ${\ensuremath{{{\geqslant}}n\,r.C}\xspace}\in\sub(\Omc)$. For any $n$tuple $(w_1,\dots,w_n)$ of different domain elements with $(u,w_1),\dots,(u,w_n)\in r^\Imc$, by ${\ensuremath{\mathsf{red}}\xspace}({\ensuremath{{{\geqslant}}n\,r.C}\xspace})$ there must be an index $i$, $1{\leqslant}i{\leqslant}n$, such that $w_i\notin{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}<\min\{\lambda,C\}$}}}\xspace}^\Imc$. Using arguments similar to those introduced above, we obtain that $$\begin{aligned}
v({\ensuremath{{{\geqslant}}n\,r.C}\xspace},u) &
{} {\geqslant}\min\{r^{\Imc_{\fsf}}(u,w_i),C^{\Imc_{\fsf}}(w_i) \\ &
{} {\geqslant}\min_{j=1}^n\min\{r^{\Imc_{\fsf}}(u,w_j),C^{\Imc_{\fsf}}(w_j)\}.
\end{aligned}$$ On the other hand, we know that there are $n$ different elements $w_1^0,\dots,w_n^0\in\Delta^\Imc$ such that $(u,w_j^0)\in r^\Imc$ and $w_j\in{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}{\leqslant}\min\{\lambda,C\}$}}}\xspace}^\Imc$ for all $j$, $1{\leqslant}j{\leqslant}n$. As in the case of $\forall r.C$ above, we conclude that $$\begin{aligned}
v({\ensuremath{{{\geqslant}}n\,r.C}\xspace},u) &
{} {\leqslant}\min_{j=1}^n\min\{r^{\Imc_{\fsf}}(u,w_j^0),C^{\Imc_{\fsf}}(w_j^0)\} \\ &
{} {\leqslant}({\ensuremath{{{\geqslant}}n\,r.C}\xspace})^{\Imc_{\fsf}}(u)
{\leqslant}v({\ensuremath{{{\geqslant}}n\,r.C}\xspace},u),
\end{aligned}$$ as required. Furthermore, $w_1^0,\dots,w_n^0$ are the required witnesses for ${\ensuremath{{{\geqslant}}n\,r.C}\xspace}$ at $u$. This concludes the proof of .
It remains to be shown that $\Imc_{\fsf}$ is a model of . For every ${\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie q\rangle}\xspace}\in\Amc$, we have $a^\Imc={{\varepsilon}}\in[C\bowtie q]^\Imc$, and thus $C^{\Imc_{\fsf}}(a^{\Imc_{\fsf}})=v(C,{{\varepsilon}})\bowtie v(q,{{\varepsilon}})=q$ by , \[p1\], and \[p2\]. A similar argument works for handling order assertions of the form ${\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie{\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}D}\xspace}\rangle}\xspace}$. To conclude, consider an arbitrary GCI ${\ensuremath{\langle C\sqsubseteq D{\geqslant}q\rangle}\xspace}\in\Tmc$ and $u\in\Delta^\Imc$. By the definition of ${\ensuremath{\mathsf{red}}\xspace}(\Tmc)$ and \[p1\], we have $v(q,u){\leqslant}v(C,u){\ensuremath{\Rightarrow}\xspace}v(D,u)$. Thus, and \[p2\] yield $C^{\Imc_{\fsf}}(u){\ensuremath{\Rightarrow}\xspace}D^{\Imc_{\fsf}}(u){\geqslant}q$. Thus, $\Imc_{\fsf}$ satisfies all the axioms in , which concludes the proof.
For the converse direction, we now show that it is possible to unravel every [$\mathsf{G}$]{}-model of into a classical tree model of ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$.
\[lem:completeness\] If has a [$\mathsf{G}$]{}-model, then ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$ has a classical model.
Given a [$\mathsf{G}$]{}-model of , we define a classical interpretation $\Imc_{\csf}$ over the domain $\Delta^{\Imc_{\csf}}$ of all paths of the form ${\varrho}=r_1d_1\dots r_md_m$ with $r_i\in\NR$, $d_i\in\Delta^\Imc$, $m{\geqslant}0$. We set $a^{\Imc_{\csf}}:={{\varepsilon}}$ and $$r^{\Imc_{\csf}}:=\{({\varrho},{\varrho}rd) \mid
{\varrho}\in\Delta^{\Imc_{\csf}},\ d\in\Delta^\Imc\}$$ for all $r\in\NR$. We denote by ${\ensuremath{\mathsf{tail}}\xspace}(r_1d_1\dots r_md_m)$ the element $d_m$ if $m>0$, and $a^\Imc$ if $m=0$. Similarly, we set ${\ensuremath{\mathsf{prev}}\xspace}(r_1d_1\dots r_md_m)$ to $d_{m-1}$ if $m>1$, and to $a^\Imc$ if $m=1$. Finally, ${\ensuremath{\mathsf{role}}\xspace}(r_1d_1\dots r_md_m)$ denotes $r_m$ whenever $m>0$. For any $\alpha\in\Umc$ and ${\varrho}\in\Delta^{\Imc_{\csf}}$, we define $\alpha^\Imc({\varrho})$ as $$\begin{aligned}
C^\Imc({\ensuremath{\mathsf{tail}}\xspace}({\varrho})) &\text{ if $\alpha=C\in\sub(\Omc)$;} \\
C^\Imc({\ensuremath{\mathsf{prev}}\xspace}({\varrho})) &\text{ if $\alpha={\ensuremath{\langleC\rangle_\uparrow}\xspace}, C\in\sub(\Omc)$;} \\
q &\text{ if $\alpha=q\in{\ensuremath{\Vmc_\Omc}\xspace}$;} \\
{\ensuremath{\mathsf{role}}\xspace}({\varrho})^\Imc({\ensuremath{\mathsf{prev}}\xspace}({\varrho}),{\ensuremath{\mathsf{tail}}\xspace}({\varrho})) &\text{\ if $\alpha=\lambda$;}\\
1-{\ensuremath{\mathsf{role}}\xspace}({\varrho})^\Imc({\ensuremath{\mathsf{prev}}\xspace}({\varrho}),{\ensuremath{\mathsf{tail}}\xspace}({\varrho}))
&\text{\ if $\alpha=\lnot\lambda$.}
\end{aligned}$$ Note that for ${\varrho}={{\varepsilon}}$ this expression is only defined for $\alpha\in{\ensuremath{\Vmc_\Omc}\xspace}\cup\sub(\Omc)$. We fix the value of $\alpha^\Imc({{\varepsilon}})$ for all other $\alpha$ arbitrarily, in such a way that for all $\alpha,\beta\in\Umc$ we have $\alpha^\Imc({{\varepsilon}}){\leqslant}\beta^\Imc({{\varepsilon}})$ iff ${\ensuremath{\mathsf{inv}}\xspace}(\beta)^\Imc({{\varepsilon}}){\leqslant}{\ensuremath{\mathsf{inv}}\xspace}(\alpha)^\Imc({{\varepsilon}})$. We can now define the interpretation of all concept names ${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}$ with $\alpha,\beta\in\Umc$ as $${\scalebox{0.93}{\ensuremath{\fbox{$\alpha{\leqslant}\beta$}}}\xspace}^{\Imc_{\csf}}:=
\{{\varrho}\mid \alpha^\Imc({\varrho}){\leqslant}\beta^\Imc({\varrho})\}.$$ It is easy to see that we have ${\varrho}\in{\scalebox{0.93}{\ensuremath{\fbox{$\alpha\bowtie\beta$}}}\xspace}^{\Imc_{\csf}}$ iff $\alpha^\Imc({\varrho})\bowtie\beta^\Imc({\varrho})$ also for all other order expressions $\bowtie$, and that $\Imc_{\csf}$ satisfies ${\ensuremath{\mathsf{red}}\xspace}(\Umc)$. We now show that $\Imc_{\csf}$ satisfies the remaining parts of ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$.
For any order assertion ${\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie{\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}D}\xspace}\rangle}\xspace}\in\Amc$ we have $C^\Imc(a^\Imc)\bowtie D^\Imc(a^\Imc)$. This implies that $C^\Imc({{\varepsilon}})\bowtie D^\Imc({{\varepsilon}})$, and thus $a^\Imc={{\varepsilon}}\in{\scalebox{0.93}{\ensuremath{\fbox{$C\bowtie D$}}}\xspace}^{\Imc_{\csf}}$, as required. A similar argument works for assertions of the form ${\ensuremath{\langle {\ensuremath{a\hspace{0.08em}{:}\hspace{0.05em}C}\xspace}\bowtie q\rangle}\xspace}$. Consider now a GCI ${\ensuremath{\langle C\sqsubseteq D{\geqslant}q\rangle}\xspace}\in\Tmc$ and any ${\varrho}\in\Delta^{\Imc_{\csf}}$. We know that $C^\Imc({\ensuremath{\mathsf{tail}}\xspace}({\varrho})){\ensuremath{\Rightarrow}\xspace}D^\Imc({\ensuremath{\mathsf{tail}}\xspace}({\varrho})){\geqslant}q$, and thus ${\varrho}\in{\scalebox{0.93}{\ensuremath{\fbox{$q{\leqslant}C{\ensuremath{\Rightarrow}\xspace}D$}}}\xspace}^{\Imc_{\csf}}$.
For ${\ensuremath{\mathsf{red}}\xspace}(\uparrow)$, consider any $\alpha,\beta\in{\ensuremath{\Vmc_\Omc}\xspace}\cup\sub(\Omc)$, $r\in{\ensuremath{\mathsf{rol}}\xspace}(\Omc)$, and ${\varrho}\in{\scalebox{0.93}{\ensuremath{\fbox{$\alpha\bowtie\beta$}}}\xspace}^{\Imc_{\csf}}$. Thus, it holds that $\alpha^\Imc({\varrho})\bowtie\beta^\Imc({\varrho})$. Every $r$-successor of ${\varrho}$ in $\Imc_{\csf}$ must be of the form ${\varrho}rd$. Since ${\ensuremath{\langle\alpha\rangle_\uparrow}\xspace}^\Imc({\varrho}rd)=\alpha^\Imc({\varrho})\bowtie\beta^\Imc({\varrho})=
{\ensuremath{\langle\beta\rangle_\uparrow}\xspace}^\Imc({\varrho}rd)$, we know that all $r$-successors of ${\varrho}$ satisfy ${\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\alpha\rangle_\uparrow}\xspace}\bowtie{\ensuremath{\langle\beta\rangle_\uparrow}\xspace}$}}}\xspace}$.
It remains to be shown that $\Imc_{\csf}$ satisfies ${\ensuremath{\mathsf{red}}\xspace}(C)$ for all concepts $C\in\sub(\Omc)$. For $C=\top$, the claim follows from the fact that $\top^\Imc({\varrho})=\top^\Imc({\ensuremath{\mathsf{tail}}\xspace}({\varrho}))=1$. For $\lnot C$, the result is trivial, and for conjunctions and implications, it follows from the semantics of $\sqcap$ and $\to$ and the properties of $\min$ and ${\ensuremath{\Rightarrow}\xspace}$, respectively.
Consider the case of $\forall r.C$ and an arbitrary domain element ${\varrho}\in\Delta^{\Imc_{\csf}}$, and set $d:={\ensuremath{\mathsf{tail}}\xspace}({\varrho})$. Since is witnessed, there must be an $e\in\Delta^\Imc$ such that $$\begin{aligned}
{\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}^\Imc({\varrho}re)
& {} = (\forall r.C)^\Imc(d) \\
& {} = r^\Imc(d,e){\ensuremath{\Rightarrow}\xspace}C^\Imc(e) \\
& {} = \lambda^\Imc({\varrho}re){\ensuremath{\Rightarrow}\xspace}C^\Imc({\varrho}re).
\end{aligned}$$ Since $({\varrho},{\varrho}re)\in r^{\Imc_{\csf}}$, this shows that $\exists r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\geqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}$ is satisfied by ${\varrho}$ in $\Imc_{\csf}$. Additionally, for any $r$-successor ${\varrho}re$ of ${\varrho}$ we have $$\begin{aligned}
{\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}^\Imc({\varrho}re)
& {} = (\forall r.C)^\Imc(d) \\
& {} {\leqslant}r^\Imc(d,e){\ensuremath{\Rightarrow}\xspace}C^\Imc(e) \\
& {} = \lambda^\Imc({\varrho}re){\ensuremath{\Rightarrow}\xspace}C^\Imc({\varrho}re),
\end{aligned}$$ and thus $\forall r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle\forall r.C\rangle_\uparrow}\xspace}{\leqslant}\lambda{\ensuremath{\Rightarrow}\xspace}C$}}}\xspace}$ is also satisfied.
For at-least restrictions ${\ensuremath{{{\geqslant}}n\,r.C}\xspace}$, we similarly know that there are $n$ different elements $e_1,\dots,e_n$ such that, for all $i$, $1{\leqslant}i{\leqslant}n$, $$\begin{aligned}
{\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}^\Imc({\varrho}re_i)
& {} = ({\ensuremath{{{\geqslant}}n\,r.C}\xspace})^\Imc(d) \\
& {} = \min_{j=1}^n\min\{r^\Imc(d,e_j),C^\Imc(e_j)\} \\
& {} {\leqslant}\min\{r^\Imc(d,e_i),C^\Imc(e_i)\} \\
& {} = \min\{\lambda^\Imc({\varrho}re_i),C^\Imc({\varrho}re_i)\}.
\end{aligned}$$ Since also the elements ${\varrho}re_1$, …, ${\varrho}re_n$ are different, this shows that the at-least restriction ${\ensuremath{{{\geqslant}}n\,r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}{\leqslant}\min\{\lambda,C\}$}}}\xspace}}\xspace}$ is satisfied by $\Imc_{\csf}$ at ${\varrho}$. On the other hand, for *all* $n$-tuples $({\varrho}re_1,\dots,{\varrho}r e_n)$ of different $r$-successors of ${\varrho}$ and all $i$, $1{\leqslant}i{\leqslant}n$, we must have $$\begin{aligned}
{\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}^\Imc({\varrho}re_i)
& {} = ({\ensuremath{{{\geqslant}}n\,r.C}\xspace})^\Imc(d) \\
& {} {\geqslant}\min_{j=1}^n\min\{r^\Imc(d,e_j),C^\Imc(e_j)\} \\
& {} = \min_{j=1}^n\min\{\lambda^\Imc({\varrho}re_j),C^\Imc({\varrho}re_j)\},
\end{aligned}$$ and thus there must be at least one $j$, $1{\leqslant}j{\leqslant}n$, such that $${\varrho}re_j\in
{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}{\geqslant}\min\{\lambda,C\}$}}}\xspace}^{\Imc_{\csf}}.$$ In other words, there can be no $n$ different elements of the form ${\varrho}re$ that satisfy ${\varrho}re\in
{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}<\min\{\lambda,C\}$}}}\xspace}^{\Imc_{\csf}}$, [i.e. ]{}${\varrho}\notin
{\ensuremath{{{\geqslant}}n\,r.{\scalebox{0.93}{\ensuremath{\fbox{${\ensuremath{\langle{\ensuremath{{{\geqslant}}n\,r.C}\xspace}\rangle_\uparrow}\xspace}<\min\{\lambda,C\}$}}}\xspace}}\xspace}^{\Imc_{\csf}}$.
In contrast to the reductions for finitely valued Gödel FDLs [@BDGS-IJAR09; @BDGS-IJUF12], the size of ${\ensuremath{\mathsf{red}}\xspace}(\Omc)$ is always polynomial in the size of . The reason is that we do not translate the concepts occurring in the ontology recursively, but rather introduce a polynomial-sized subontology ${\ensuremath{\mathsf{red}}\xspace}(C)$ for each relevant subconcept $C$. Moreover, we do not need to introduce role hierarchies for our reduction, since the value of role connections is expressed using the special element $\lambda$. -completeness of concept satisfiability in classical [$\ALC\kern-0.16em\Qmc$]{} [@Schi-IJCAI91; @Tobi-01] now yields the following result.
\[thm:local-consistency\] Local consistency in [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}is -complete.
Conclusions
===========
Using a combination of techniques developed for infinitely valued Gödel extensions of [@BoDP-KR14] and for finitely valued Gödel extensions of [$\Smc\kern-0.05em\Rmc\kern-0.05em\Omc\kern-0.03em\Imc\kern-0.18em\Qmc$]{} [@BDGS-IJAR09; @BDGS-IJUF12], we have shown that local consistency in infinitely valued [$\mathsf{G}$]{}-[$\mathfrak{I}\kern-0.14em{\ensuremath{\ALC\kern-0.16em\Qmc}\xspace}$]{}is -complete. Our reduction is more practical than the automata-based approach proposed by @BoDP-KR14 and does not exhibit the exponential blowup of the reductions developed by @BDGS-IJAR09 . Beyond the complexity results, an important benefit of our approach is that it does not need the development of a specialized fuzzy DL reasoner, but can use any state-of-the-art reasoner for classical [$\ALC\kern-0.16em\Qmc$]{}without modifications. For that reason, this new reduction aids to shorten the gap between efficient classical and fuzzy DL reasoners.
In future work, we want to extend this result to full consistency, possibly using the notion of a *pre-completion* as introduced in [@BoDP-KR14]. Our ultimate goal is to provide methods for reasoning efficiently in infinitely valued Gödel extensions of the very expressive DL [$\Smc\kern-0.05em\Rmc\kern-0.05em\Omc\kern-0.03em\Imc\kern-0.18em\Qmc$]{}, underlying OWL2DL. We believe that it is possible to treat transitive roles, inverse roles, role hierarchies, and nominals using the extensions of the automata-based approach developed originally for finitely valued FDLs in [@BoPe-JoDS13; @BoPe-14; @Borg-DL14].
As done previously in [@BDGS-IJUF12], we can also combine our reduction with the one for infinitely-valued Zadeh semantics. Although Zadeh semantics is not based on t-norms, it nevertheless is important to handle it correctly, as it is one of the most widely used semantics for fuzzy applications. It also has some properties that make it closer to the classical semantics, and hence become a natural choice for simple applications.
A different direction for future research would be to integrate our reduction directly into a classical tableaux reasoner. Observe that the definition of ${\ensuremath{\mathsf{red}}\xspace}(C)$ is already very close to the rules employed in (classical and fuzzy) tableaux algorithms (see, e.g. [@BaSa-SL01; @BoSt-FSS09]). However, the tableaux procedure would need to deal with total preorders in each node, possibly using an external solver.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by the German Research Foundation (DFG) under the grant BA 1122/17-1 (FuzzyDL) and within the Cluster of Excellence ‘cfAED.’ Most of this work was developed while R. Peñaloza was still affiliated with TU Dresden and the Center for Advancing Electronics Dresden, Germany.
[^1]: <http://www.w3.org/TR/owl2-overview/>
[^2]: <http://www.ihtsdo.org/snomed-ct/>
[^3]: <http://www.opengalen.org/>
[^4]: For the rest of this paper, expressions of the form denote (classical) concept names.
|
---
abstract: 'We study the problem of the upper critical field ($H_{c2}$) for tight–binding electrons in a two–dimensional lattice. The external magnetic field is introduced into the model Hamiltonian both via the Peierls substitution and the Zeeman term. Carrying out calculations for finite systems we analyze the influence of the external field in the commensurable and incommensurable case on an equal footing. The upper critical field has been discussed for intrasite as well as anisotropic intersite pairing that, in the absence of magnetic field, has a $d_{x^2-y^2}$ symmetry. A comparison of $H_{c2}$ determined for different symmetries shows that the on–site pairing is more affected by the external field i.e., the critical temperature for the on–site pairing decreases with the increase of the magnetic field faster than in the anisotropic case. Moreover, we have shown that the tight–binding form of the Bloch energy can lead to the upward curvature of $H_{c2}$, provided that the Fermi level is close enough to the van Hove singularity.'
address: ' Department of Theoretical Physics, Institute of Physics, University of Silesia, 40-007 Katowice, Poland'
author:
- 'Maciej M. Maśka and Marcin Mierzejewski[^1]'
title: 'Upper critical field for anisotropic superconductivity. A tight–binding approach.'
---
Introduction
============
One of many striking properties of high–temperature superconductors is related to the field–induced transition from superconducting to normal state. Magnetic properties of high–$T_c$ compounds give rise to both quantitative and qualitative differences with respect to the conventional superconductors. The systems under consideration are characterized by extremely high values of the upper critical and its unusual temperature dependence. For optimally doped samples experimental investigation of the critical field is limited only to temperatures close to $T_c$ [@osofsky], whereas at lower temperatures the magnitude of $H_{c2}$ is far beyond the reach of laboratory magnetic fields. The measurements carried out in a wide range of temperature for underdoped superconductors clearly indicate the positive curvature of $H_{c2}(T)$ even at genuinely low temperatures [@osofsky; @mackenzie; @Iye; @Joy]. Theoretical approaches do not provide a unique, complete description of these phenomena. The most of unconventional properties of high-temperature superconductors, like narrow quasiparticle bands, lifetime effects of states close to the Fermi level and linear temperature dependence of the normal–state resistivity are usually attributed to strong Coulomb correlations. However, upward curvature of the upper critical field is observed also in overdoped compounds, where the temperature dependence of resistivity changes gradually from linear to quadratic behavior [@kubo; @suzuki]. This feature suggests that the positive curvature of $H_{c2}(T)$ could originate from e.g., symmetry of the superconducting order parameter or details of the density of states and may be explained without a sophisticated treatment of the most difficult problem that is related to the presence of strong electronic correlations.
It is believed that the symmetry of the superconducting state can be close related to the pairing mechanism. There is a lot of node–sensitive experiments, based on the angle resolved photoemission spectroscopy [@arpes], London penetration depth [@london], NMR [@nmr] and quasiparticle tunneling [@tunel], which indicate that the energy gap is strongly anisotropic and vanishes in particular directions in the Brillouin zone. Moreover, the phase–sensitive superconducting interference device experiments [@squid] demonstrated the sign change of the order parameter between the $x$ and $y$ directions. Generally, these results are consistent with the $d_{x^2-y^2}$ pairing scenario. On the other hand there are experimental indications, which had questioned the pure $d_{x^2-y^2}$ symmetry of the energy gap and suggest mixed pairing symmetry with a dominant $d$–wave component (e.g., $d\pm s$ or $d\pm is$) [@kouznetsov; @krishan; @sun; @ma].
The measurement of the upper critical field can give insight into the microscopic parameters of a relevant model. For example, the coherence length $\xi$ is usually derived indirectly from the expression $H_{c2}(0)=\phi_0/2\pi\xi^2$ [@hidaka], where $H_{c2}(0)$ is the upper critical field determined at $T=0$, and $\phi_0$ is the magnetic flux quantum. The theoretical investigation of the upper critical field for different pairing symmetries is predominantly based on the Ginzburg–Landau (GL) [@ginzburg] theory or the Lawrence-Doniach [@doniach] approach in case of layered superconductors. With the help of linearized GL equations Won and Maki[@Won] have shown that $H_{c2}$ in a model with repulsive on–site interaction depends linearly on temperature near $T_c$ and saturates at $T\rightarrow 0$. They have not found any sign of the upward behavior. There are also calculations for $H_{c2}$ in systems with mixed symmetries, especially for superconductors in which the dominant $d$–wave order parameter coexists with a subdominant $s$–wave component. However, in the most of these approaches $H_{c2}(T)$ exhibits negative curvature. On the other hand, results obtained in Ref. [@dai] suggest that the upward curvature of the critical field could be a characteristic feature of a $d$–wave superconductor. The positive curvature of $H_{c2}(T)$ can also originate from the presence of magnetic impurities [@kresin1; @kresin2].
A separate problem, that is usually neglected in the above approaches, is the influence of the periodic lattice potential on the upper critical field [@my]. Application of magnetic field to the two–dimensional (2D) electron system in a tight–binding approximation leads to a fractal energy spectrum known as Hofstadter’s butterfly, where even very small changes in magnetic field can result in a drastic changes of the spectrum [@harper; @langbein; @hofstadter]. In this paper we investigate the upper critical field for electrons described by the two–dimensional tight–binding model with intra– and intersite pairing. We show that anisotropic superconductivity is less affected by the external magnetic field then the isotropic one. We also demonstrate that the lattice effects can give rise to important corrections with respect to Helfand–Werthamer [@hw; @kresin1] solution of the Gor’kov equations [@gorkov]. This effect is of particular importance in the vicinity of the van Hove singularity and at low temperatures.
Gap equation close to $H_{c2}$
==============================
We consider a two–dimensional square lattice immersed in a uniform, perpendicular, magnetic field. The BCS–type Hamiltonian is of the form $$\begin{aligned}
\hat{H} &=& \hat{H}_{\rm kin} + \hat{H}_V
-\mu \sum_{i,\sigma} c^{\dagger}_{i\sigma}c_{i\sigma} \nonumber \\
&& -g\mu_{B}H_{z}\sum_{i}
\left(c^{\dagger}_{i\uparrow}c_{i\uparrow}
-c^{\dagger}_{i\downarrow}c_{i\downarrow} \right),\end{aligned}$$ where $c^{\dagger}_{i \sigma}$ ($c_{i \sigma}$) creates (annihilates) an electron with spin $\sigma$ on site $i$. The chemical potential $\mu$ is introduced in order to control the doping level. The last term in the above Hamiltonian describes the paramagnetic Pauli coupling to the external field. Here, $g$ stands for the gyromagnetic ratio, $\mu_{B}$ is the Bohr magneton and $H_z$ is the $z$–component of the external field. The first ($\hat{H}_{\rm kin}$) and the second ($\hat{H}_V$) term in the Hamiltonian represents the kinetic energy and the pairing interaction, respectively. Within the tight–binding approach $$\hat{H}_{\rm kin} = \sum_{<ij>,\sigma} t_{ij}\left({\mbox{{\bf A}}}\right)
c^{\dagger}_{i\sigma}c_{j\sigma}.$$ The electrons are gauge–invariantly coupled with local $U(1)$ gauge field by a phase–factor in the kinetic–energy hopping term. According to the Peierls substitution [@peierls] in the presence of magnetic field the original hopping integral between sites $i$ and $j$, $t_{ij}$ acquires an additional factor $$t_{ij}\left({\mbox{{\bf A}}}\right)= t_{ij}
\exp\left(\frac{ie}{\hbar c} \int^{{\mbox{{\bf R}}}_{i}}_{{\mbox{{\bf R}}}_{j}}
{\mbox{{\bf A}}}\cdot d{\mbox{{\bf l}}}\right).$$ In the case of the on–site pairing, which leads to isotropic order parameter, the BCS–type interaction takes on the form $$\hat{H}_V=-\:
V\sum_{i}\left(c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}\Delta_{i}
+c_{i\downarrow}c_{i\uparrow} \Delta^{\star}_{i} \right).$$ Here, we have introduced local superconducting order parameter, $\Delta_{i}=\langle c_{i\downarrow}c_{i\uparrow}\rangle$, which in the presence of the magnetic field can change from site to site [@my]. We also consider anisotropic superconductivity with the intersite pairing interaction given by $$\hat{H}_V=-\:
V\sum_{<ij>}\left(c^{\dagger}_{i\uparrow}c^{\dagger}_{j\downarrow}
\Delta_{ij}
+c_{i\downarrow}c_{j\uparrow} \Delta^{\star}_{ij} \right).$$ For the sake of simplicity we restrict our considerations only to the nearest–neighbor coupling with the singlet order parameter $\Delta_{ij}=\langle c_{i\downarrow}c_{j\uparrow}
-c_{i\uparrow}c_{j\downarrow}\rangle$.
We start with the discussion of the normal state properties. Similarly to Ref. [@my] we make use of an unitary transformation $U$ that diagonalizes the kinetic part of the Hamiltonian $$U^\dagger \hat{H}_{\rm kin} U = {\cal H}_{\rm kin}.$$ This transformation defines a new set of fermionic operators $a_{n\sigma}=\sum_i U^\dagger_{ni} c_{i\sigma}$, in which the Hamiltonian in the normal state takes on the diagonal form $${\cal H} = \sum_{n\sigma} \left( E_{n} - \mu -\sigma g \mu_B H_z \right)
a^\dagger_{n\sigma} a_{n\sigma}.$$ In the absence of the magnetic field $U$ represents transformation from the Wannier to the Bloch representation. For finite magnetic field and general gauge the quantum number $n$ enumerates eigenstates, although does not represent a reciprocal lattice vector. In order to simplify further discussion we restrict our considerations only to the nearest neighbor hopping with $t_{\langle ij \rangle} \equiv -t$. We also assume the type–II limit of superconductors where the magnetic field can be regarded as a spatially uniform object. Choosing the Landau gauge ${\bf A} = H_z(0,\,x,\, 0)$ the hopping integral depends explicitly only on $x$ and the momentum in $y$ direction $p_y$ remains a good quantum number. Due to the plane–wave behavior in $y$ direction the unitary matrix $U$ takes on the form $$U_{i\left(\bar{p}_{x},p_{y}\right)}=
U_{\left(x,y\right)\left(\bar{p}_{x},p_{y}\right)}=
N^{-1/4}\;e^{\displaystyle ip_{y}ya}
g\left(\bar{p}_{x},p_{y},x\right),$$ where $(ax,ay)$ is the position of the $i$–th site and $\left(\bar{p}_x,p_y\right)$ represents $n$–th eigenstate of the Hamiltonian (7). Straightforward calculations [@hofstadter] show that the $x$–dependent part of the wave function $g\left(\bar{p}_{x},p_{y},x\right)$ fulfills a one–dimensional difference equation $$\begin{aligned}
&& g\left(\bar{p}_{x},p_{y},x+1\right)+
2\cos\left(h x -p_{y}a\right)g\left(\bar{p}_{x},p_{y},x\right) \nonumber \\
&& + g\left(\bar{p}_{x},p_{y},x-1\right)
=t^{-1}E\left(\bar{p}_{x},p_{y}\right)
g\left(\bar{p}_{x},p_{y},x\right),\end{aligned}$$ where we have introduced a reduced dimensionless magnetic field $h=e a^{2}H_{z}/(\hbar c)$. This quantity can be expressed with the help of magnetic flux $\phi$ through lattice cell and flux quantum ($h=2 \pi \phi/\phi_0$). Equation (9) is known as the Harper equation [@harper] and has extensively been studied [@harpeq; @hasegawa]. The Harper equation, derived here within a tight-binding approximation, can be also obtained in a case of weak perturbation of a Landau–quantized two–dimensional electron system[@langbein; @thouless]. Now, let us take into account the pairing potential $H_V$. In order to investigate the transition from the superconducting to the normal state we make use of equation of motion for the anomalous Green function. In the case of the isotropic on–site pairing one obtains $$\begin{aligned}
&&\left[\omega-E\left(\bar{p}_{x},p_{y}\right)
+\mu+g\mu_BH_z \right]
\langle\langle a_{\left(\bar{p}_{x},p_{y} \right) \uparrow}\mid
a_{\left(\bar{k}_{x},k_{y}\right) \downarrow} \rangle\rangle \nonumber \\
&&= -V \sum_{i,\bar{k}^{\prime}_{x},k^{\prime}_{y} } \Delta_{i}
U^{\star}_{i\left(\bar{p}_{x},p_{y} \right)}
U^{\star}_{i\left(\bar{k}^{\prime}_{x},k^{\prime}_{y}\right) }
\langle\langle
a^{\dagger}_{\left(\bar{k}^{\prime}_{x},k^{\prime}_{y}\right) \downarrow}\mid
a_{\left(\bar{k}_{x},k_{y}\right) \downarrow}
\rangle\rangle. \nonumber \\\end{aligned}$$ As far as we are close to the phase transition we make use of a linearized gap equation i.e., we calculate the propagator $\langle\langle
a^{\dagger}_{\left(\bar{k}^{\prime}_{x},k^{\prime}_{y}\right) \downarrow}\mid
a_{\left(\bar{k}_{x},k_{y}\right) \downarrow}
\rangle\rangle$ in the normal state. Similarly to the standard BCS theory, such approach allows one to determine the critical temperature or, in our case, the upper critical field. However, it is irrelevant for calculations below $T_c$.
The choice of the Landau gauge implies that the isotropic order parameter does not depend on $y$: $\Delta_i \equiv \Delta_{(x,y)} = \Delta_x$. Then, the linearized gap equation reads $$\vec{\Delta} = {\cal M}\vec{\Delta},$$ where $\vec{\Delta} = (\Delta_1,\ \Delta_2,\ \Delta_3, ...)$ and $$\begin{aligned}
{\cal M}(x,x') & = & \frac{V}{\sqrt{N}} \sum_{\bar{p}_x,p_y,\bar{k}_x}
g(\bar{p}_x,p_y,x) \: g(\bar{k}_x,-p_y,x) \nonumber \\
&\times& g(\bar{k}_x,-p_y,x')\: g(\bar{p}_x,p_y,x') \: \chi(\bar{p}_x,p_y;
\bar{k}_x,-p_y). \nonumber \\ \end{aligned}$$ In the presence of the magnetic field the Cooper pair susceptibility is given by $$\begin{aligned}
\chi(\bar{p}_x,p_y;\bar{k}_x,k_y)& =& \left[\tanh\frac{E\left(\bar{p}_{x},
p_{y}\right)-\mu-g\mu_BH_z}{2k_{B}T}
\right. \nonumber \\
&&\left. +\tanh\frac{E\left(\bar{k}_{x},k_{y}
\right)-\mu+g\mu_BH_z }{2k_{B}T}
\right] \nonumber \\
&& \times
\left[2\left(E\left(\bar{p}_{x},p_{y}\right)+E\left(\bar{k}_{x},
k_{y}\right)-2\mu\right) \right]^{-1}. \nonumber \\\end{aligned}$$
In the case of the nearest–neighbor pairing we obtain the gap equation analogous to Eq. (11). Similarly to the isotropic pairing $\Delta_{ij}$ does not depend explicitly on $y$. However, there are two types of order parameter at each site: $\Delta^{(x)}_{x}$ when sites $i$ and $j$ lay along the $x$ axis, and $\Delta^{(y)}_{x}$ when sites $i$ and $j$ lay along the $y$ axis. Close to the upper critical field the gap equation for anisotropic superconductivity can be written in a matrix form $$\left(\begin{array}{c} \vec{\Delta}^{(x)} \\ \vec{\Delta}^{(y)} \end{array}
\right)=\left(\begin{array}{cc}{\cal M}^{\left(x,x\right)}&{\cal M}^{\left(x,y\right)}
\\
{\cal M}^{\left(y,x\right)}&{\cal M}^{\left(y,y\right)}\end{array}\right)
\left(\begin{array}{c} \vec{\Delta}^{(x)} \\ \vec{\Delta}^{(y)} \end{array}
\right),$$ where $$\begin{aligned}
{\cal M}^{\left(\alpha,\beta\right)}(x,x^{\prime})
& = & \frac{V}{\sqrt{N}} \sum_{\bar{p}_x,p_y,\bar{k}_x}
\chi(\bar{p}_x,p_y;\bar{k}_x,-p_y) \nonumber \\
&& \times A^{\left(\alpha\right)}\left(\bar{p}_x,\bar{k}_x,p_y,x\right)
A^{\left(\beta\right)}\left(\bar{p}_x,\bar{k}_x,p_y,x^{\prime}\right), \nonumber \\\end{aligned}$$ and $$\begin{aligned}
A^{\left(x\right)}\left(\bar{p}_x,\bar{k}_x,p_y,x\right)&=&
g(\bar{p}_x,p_y,x) g(\bar{k}_x,-p_y,x+1) \nonumber \\
&+& g(\bar{p}_x,p_y,x+1) g(\bar{k}_x,-p_y,x), \nonumber \\
&& \\
A^{\left(y\right)}\left(\bar{p}_x,\bar{k}_x,p_y,x\right)&=& 2 \cos( p_y a) \nonumber \\
&\times& g(\bar{p}_x,p_y,x) g(\bar{k}_x,-p_y,x).\end{aligned}$$
Equations (11) and (14) constitute a system of linear equations for the order parameters and the condition for existence of a non–zero solution can be written as $$\det \left({\cal M} - I\right) = 0$$ in the case of isotropic pairing, and $$\det \left(\begin{array}{cc}{\cal M}^{\left(x,x\right)} - I&
{\cal M}^{\left(x,y\right)}\\
{\cal M}^{\left(y,x\right)}&{\cal M}^{\left(y,y\right)} - I\end{array}\right) = 0$$ for anisotropic superconductivity, where $I$ is the unit matrix. These equations allow one to obtain the magnitude of the upper critical field perpendicular to the plane. For the two–dimensional square lattice the size of matrices which enter Eqs. (18) and (19) is proportional to the square root of the number of the lattice sites. Analytical solutions of the Harper equation (9) are known only in a few cases of commensurable field[@hasegawa] (in our notation $h=2\pi p/q$, where $p$ and $q$ are relative prime integers), which correspond to unphysically high magnetic field. Therefore, in order to investigate $H_{c2}$ we restrict our considerations to a finite lattice, for which we are able to analyze numerically the commensurable and incommensurable magnetic field on an equal footing.
Discussion of results
=====================
We consider square $M\times M$ cluster with periodic boundary conditions (bc) along the $y$ axis. As the Landau gauge breaks the translation invariance along $x$ axis we use fixed bc in this direction. An additional advantage originating from such a mixed bc is the absence of the unphysical degeneracy of states at the Fermi level, which occurs for the half-filled band in cluster calculations with fixed or periodic bc taken in both directions [@jokojama]. In order to estimate the finite size effects we have carried out numerical calculations for clusters of different sizes. We have found that in the case of the isotropic pairing and small concentration of holes ($\delta < 0.2$) there are no significant differences between results obtained on $150 \times 150$ and $200 \times 200$ clusters. For anisotropic pairing already $120 \times 120$ clusters give convergent results.
Figures 1. and 2. show the reduced critical field, $h_{c2}=e a^{2}H_{c2}/(\hbar c)$, for different concentrations of holes. Independently on the symmetry of the superconducting order parameter i.e., for isotropic (Fig. 1.) as well as anisotropic pairing (Fig. 2.), the slope of $H_{c2}(T)$ strongly decreases with increasing doping. Note that our cluster results exactly reproduce the BCS transition temperature when the magnetic field tends to zero. In the case of intersite pairing the arrows indicate BCS solutions for $d_{x^2-y^2}$ superconductivity. However, the external magnetic field affects the relative phases of the order parameter in the $x$ and $y$ directions, which can change from site to site. Therefore, it is impossible to determine globally the type of the symmetry of the energy gap in the presence of magnetic field.
Contrary to the conclusion presented in Ref. [@dai], our results (Figs. 1. and 2.) do not indicate that the upward curvature of $H_{c2}(T)$ can emerge as a direct consequence of the symmetry of superconducting state. However, the anisotropy of the order parameter can significantly influence the magnitude of the upper critical field. In order to investigate this relationship we have directly compared results obtained for on– and intersite pairing for isotropic and anisotropic superconductivity. We have chosen the magnitudes of the pairing potentials $V$, which, in the absence of magnetic field, lead to the same superconducting transition temperatures for isotropic and anisotropic superconductivity. Fig 3. shows the temperature dependence of the upper critical field obtained for the half-filled case. One can see that the anisotropic superconductivity is less affected by the external field than the isotropic one.
An important observation is that this result depends neither on the magnitude of the pairing potential nor on the concentration of holes (see the inset in Fig. 3). Therefore, it can be considered as a characteristic feature of the two–dimensional lattice gas.
In the absence of magnetic field there is a van Hove singularity in the middle of the band. Although, the external field results in a splitting of the Bloch band into a huge number of subbands, the presence of the original van Hove singularity is reflected in the Hofstadter spectrum [@hofstadter]. In contradistinction to the structure of Landau levels, the Hofstadter spectrum does not consist of uniformly distributed energy levels. In particular, the average distance between the energy levels close to the Fermi energy achieves its minimum when the chemical potential is in the middle of the Bloch band. It can be considered as a remnant of the original van Hove singularity.
The question which arises concerns the impact of this feature on the upper critical field. In order to analyze this problem we have fitted $H_{c2}(T)$ obtained for isotropic superconductivity to the results obtained for the two–dimensional version [@kresin1] of the Helfand–Werthamer approach to the Gor’kov equations. Fig. 4 shows the numerical results. Away from the half–filled case the qualitative temperature dependence of the upper critical field can be very well approximated by the solution of the Gor’kov equations. It suggests, that the complicated Hofstadter spectrum does not influence the temperature dependence of the critical field, provided that the Fermi level is far enough from the original van Hove singularity. However, in the vicinity of the van Hove singularity the second derivative of $H_{c2}(T)$ is significantly enhanced, when compared to the results obtained from the Gor’kov equations. It is of particular importance for small values of the pairing potential, when the system remains in superconducting state only at relatively low temperatures and the Cooper–pair susceptibility is strongly peaked at the Fermi level. Then, the curvature of $H_{c2}(T)$ can gradually change from negative to positive, as depicted in the inset in Fig. 4. This effect takes place for isotropic as well as for anisotropic pairing. Similar results have been reported in Ref. [@dias].
Concluding remarks
==================
We have investigated the temperature dependence of the upper critical field for the two–dimensional lattice gas. With the help of unitary transformation we have obtained a diagonal form of the Hamiltonian in the normal state and derived gap equations both for isotropic and anisotropic superconductivity. We have discussed influence of the symmetry of the superconducting state and the van Hove singularity on the upper critical field. Our results clearly indicate that the symmetry of the superconducting order parameter itself can not lead to upward curvature of $H_{c2}(T)$. However, quite pronounced tendency can be observed for the half-filled case, when the Fermi energy is close to the original van Hove singularity. In the absence of the external field this singularity occurs in the middle of the band. The enhancement of curvature of $H_{c2}(T)$ takes place for isotropic as well as anisotropic superconductivity and is of particular importance for small values of the pairing potential. Then, the curvature can gradually change from negative to positive. This effect smears out for larger doping where the temperature dependence of the upper critical field can be rendered very well when solving the Gor’kov equations. We have found that in the case of anisotropic pairing the upper critical field exceeds the critical field obtained for isotropic superconductivity. It takes place for small doping ($\delta < 0.2$) and arbitrary magnitude of the pairing potential. These results suggest that in the two–dimensional lattice gas anisotropic superconductivity is less affected by the external field than the isotropic one.
The proposed method allows one to derive the gap equation in the same way as the standard BCS approach. The only differences are related to the fact that the diagonal form of the normal–state Hamiltonian is obtained numerically and the superconducting order parameter can be a site-dependent quantity. The similarity between our method and the BCS approach allows for straightforward incorporation of the local Coulomb repulsion within any standard approximation. Here, one may expect destructive influence of correlations, in particular in the isotropic channel. This originates from the fact that local repulsion always acts to the detriment of the formation of local Cooper pairs. The impact of Coulomb, Hubbard–like, correlations on anisotropic superconductivity seems to depend on the approximation scheme. This problem is under our current investigation.
This work has been supported by the Polish State Committee for Scientific Research. We acknowledge a fruitful discussion with Janusz Zieli[ń]{}ski.
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[^1]: marcin@phys.us.edu.pl
|
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author:
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Giuseppe GAETA\
Dipartimento di Matematica, Università di Milano,\
v. Saldini 50, I–20133 Milano (Italy)\
[*gaeta@mat.unimi.it*]{}
title: Symmetry of stochastic equations
---
Introduction {#introduction .unnumbered}
============
Symmetry methods are by now recognized as one of the main tools to attack [*deterministic*]{} differential equations (both ODEs and PDEs); see e.g. [@VArn; @BlK; @Gaeb; @Ibr; @Olv1; @Olv2; @Ste; @Win]. The situation is quite different for what concerns [*stochastic*]{} differential equations [@LArn; @Gar; @Gue; @Kam; @Oks]: here, symmetry considerations are of course quite widely used by theoretical physicists (see e.g. [@Bar; @HHZ; @KPZ] in the context of KPZ theory), but a rigorous and general theory comparable to the one developed for deterministic equation is still lacking, and the attention of the community working on symmetry methods should maybe be called to this.
We would like to quote here early work by Misawa [@Mis], mainly concerned with conservation laws for stochastic systems, and on the work by Arnold and Imkeller [@LArn; @ArI] for normal forms of stochatic equations.
In the following I will report on some work I have done – to a large extent in collaboration with N. Rodr iguez Quintero – on symmetries of stochastic (Ito) equations, and how these compare with the symmetries of the associated diffusion (Fokker-Planck) equations [@CicVit; @Fin; @Ris; @StSt]; further details can be found in [@SDE1; @SDE2]. Part of this was concerned with providing a suitable definition for symmetries of a stochastic equation, so I will also show that the symmetries at the basis of the construction of the KPZ equation are actually recovered within the frame developed here.
Projectable symmetries
======================
We assume the reader to be fairly familiar with the concept of symmetry of a deterministic differential equation, and the methods to characterize these; so we will just recall some formula of use in the following, mainly to fix notation, identify the class of transformations we are going to consider, and compare with the situation for stochastic equations.
Let us first consider the case of systems of first order ODEs in $R^n$, $${\dot x}^i \ - \ f^i (x,t) \ = \ 0 \ . \eqno(1.1)$$
The general form of Lie-point vector fields is in this case, writing $\pa_i = \pa / \pa x^i$, $ X = \tau (x,t) \pa_t + \xi^i (x,t) \pa_i$. Such a vector field, however, is somehow too general: indeed, we would like the change on $t$ not to depend on the value assumed by the solution $x(t)$; this is particularly relevant when we think of the ODE as describing not the motion of a single particle in $R^n$, but that of an assembly of particles. We would thus like to keep the special role of time, i.e. require $\tau = \tau (t)$; in other words, we will consider $$X_0 \ = \ \tau (x,t) \pa_t + \xi^i (x,t) \pa_i \ . \eqno(1.2)$$
A similar discussion holds for symmetries of PDEs, first or higher order. Here we will deal only with scalar PDEs, i.e. equations for a single dependent variable $u = u(x,t)$. In this case we would write in general $ X = \tau (x,t,u) \pa_t + \xi^{i} (x,t,u) \pa_i + \phi (x,t,u) \pa_u$. On physical terms, it is preferable to consider only transformations such that the independent variables are transformed independently of the values assumed by the dependent ones, i.e. such that $\tau$ and the $\xi^i$ do not depend on $u$; also, here we are primarily interested in evolution equations, and in order to guarantee the time keeps its distinguished role we should also ask that $\tau$ does not depend on $x$. Thus, in the end we want to consider transformations of the form $$X \ = \ \tau (t) {\pa \over \pa t} + \xi^{i} (x,t) {\pa \over \pa x^{i} } + \phi (x,t,u) {\pa \over \pa u } \ . \eqno(1.3)$$
[**Definition.**]{} [*The vector fields – and in particular symmetry generators – of the form (1.2), (1.3) will be called “fiber-preserving”, or simply [**projectable**]{}.* ]{}
Let us now consider in particular a system of ODEs, which we write in the form (1.1). The determining equations for the symmetry generators of this are easily determined (these should be compared to the determining equations for stochastic differential equations discussed in the next section). We will use, here and below, the notation $$\{ f , \xi \}^i \ := \ (f^j \cdot \pa_j)
\xi^i - (\xi^j \cdot \pa_j) f^i \ . \eqno(1.4)$$
[**Lemma 1.**]{} [*The projectable symmetries of (1.9) are given by vector fields $X_0$ as above with coefficients satisfying $$\pa_t (\xi^i - \tau f^i ) + \{ f , \xi \}^i \ = \ 0 \ . \eqno(1.5)$$*]{}
For given functions $f^{i}(x,t)$ the partial differential equations (1.16) always [@Olv1; @Ste] have non-trivial solutions $\tau(t)$ and $\xi^{i}(x,t)$, for example we can fix $\tau(t)$ or $\xi^{i}(x,t)$ and compute $\xi^{i}(x,t)$ or $\tau(t)$. Although an infinite number of symmetries of (1.16) exist, in general there is no constructive way to find them.
For further detail on symmetries of (deterministic) ODEs and PDEs, and applications, the reader is referred to [@VArn; @BlK; @Gaeb; @Ibr; @Olv1; @Olv2; @Ste; @Win].
Stochastic differential equations
=================================
Let us turn to stochastic equations [@LArn; @DWM; @Gar; @Gue; @Kam; @Oks], and let us consider an Ito equation[^1] $$dx^i \ = \ f^i (x,t) \, dt \ + \ \s^i_k (x,t) \, d w^k \ , \eqno(2.1)$$ where $f$ and $\s$ are smooth functions, $\sigma(x,t)$ is a nonzero matrix and the $w^k$ are independent homogeneous standard Wiener processes, so that $$\< | w^i (t) - w^j (s) |^2 \> = \delta^{ij} (t-s) \ .$$ The equation (2.1) should be seen as a map from the vector Wiener process ${\bf{w}}(t) $ $ = $ $ \{ w^1 (t) , $ $... ,$ $w^n (t) \}$ to the stochastic process undergone by $\{ x^1 (t) , ... , x^n (t) \}$, and its meaning is precisely that of defining the vector stochastic process ${\bf x} (t)$.
If we consider a function $y = \Phi (x)$, its evolution under the Ito equation (2.1) is described (again in terms of a stochastic equation) by the [*Ito formula*]{}: $$\begin{array}{rl}
dy^i = & {\pa \Phi^i \over \pa x^j }\, d x^j + {1 \over 2}
{\pa^2 \Phi^i \over \pa x^j \pa x^k} \, dx^j dx^k \\
& = \[ f^{j} \pa_j \Phi^i +
(1/2) \pa^2_{jm} \Phi^i (\s \s^T)^{jm} \] \, dt \ + \ \[ (\pa_j \phi^i )
\s^{jk} \] dw^k \ . \end{array} \eqno(2.2)$$
To the Ito equation (2.1) is associated the corresponding Fokker-Planck equation $$\pa_t u \ = \ - \, \pa_i \, (f^i u ) \ + \ {1 \over 2} \,
\pa_{ij}^2 \, \[ (\s \s^T )^{ij} \, u \] \ , \eqno(2.3)$$ describing the evolution of the probability measure $\rho = u (x,t)$ under the stochastic process described by (2.1). Equations (2.1) and (2.3) contains the same [*statistical*]{} information [@LArn; @Gar; @Gue; @Kam; @Oks] as far as one-particle processes are concerned (but not if we think the Ito equation as describing a random dynamical system, see [@LArn; @SDE2]), provided $\s$ satisfies the [**non-degeneracy condition** ]{} $$A \ := \ {1 \over 2} \ \s \, \s^T \ \not= \ 0 \,$$ which we will assume throughout this note.
We stress that we are interested in equation (2.3) as far as it describes the time evolution of the probability measure $\rho = u(x,t)$ under the stochastic process (2.1). It is obvious that for this interpretation $u(x,t)$ should be non-negative and subject to the condition $$\int_{-\infty}^{+\infty} \, u (x,t) \, dx^1...dx^n \ = \ 1 \ . \eqno(2.4)$$
This is relevant in connection with the allowed transformations in the $(x,t;u)$ space: only transformations preserving this normalization do represent symmetries of the Fokker-Planck equation compatible with its probabilistic interpretation[^2], and one should expect a correspondence between symmetries of the Ito equation and this subclass – rather than all – of the symmetries of the Fokker-Planck equation.
Another simple but relevant point which should be stressed is the following: different Ito equations which have the same $f$ but different matrices $\s$ can provide the same term $\s \s^T$ and thus the same Fokker-Planck equation.
A simple example is provided e.g. by $\s$ orthogonal: in this case we have by definition $\s \s^T
= I$, so that all the Ito equations with the same $f$ and any orthogonal matrix $\s$ give the same Fokker-Planck equation[^3]. Similarly, $\s$ and $\~\s = \s B$, with $B$ any orthogonal matrix, will give the same Fokker-Planck equation (and conversely $\s$ and $\~\s$ give the same Fokker-Planck – with the same $f$ – only if there is an orthogonal matrix $B$ such that the above relation is satisfied.
Thus if we consider a continuous variation of $\s$, say $\s + \eps \gamma$, in the Ito equation, the associated Fokker-Planck equation remains unchanged provided $(\s + \eps \gamma ) (\s + \eps \gamma )^T = \s
\s^T$, which at order $\eps$ is simply $$\s \gamma^T + \gamma \s^T \ = \ 0 \ . \eqno(2.5)$$
Symmetries of stochastic ODEs
=============================
Spatial symmetries
------------------
We will at first consider symmetries not acting on the time variable $t$, i.e. $X_0 = \xi^i (x,t) \pa_i$. Thus, we consider a near-identity change of coordinates, passing from $x$ to $y$ via $$x^i \to y^i = x^i + \eps \xi^i (x,t) \ . \eqno(3.1)$$ Using the Ito formula, we easily check that at first order in $\eps$, this transformation maps the Ito equation (2.1) into a new Ito equation $$d y^i \ = \ \~f^i (y,t) \, dt \ + \ \~\s^i_k (y,t) dw^k \ , \eqno(3.2)$$ where we have explicitly $$\begin{array}{rl}
\~f^i = \ & f^i \ + \ \eps \, [ \pa_t \xi^i + f^j \pa_j \xi^i - \xi^j
\pa_j f^i + (1/2) (\s \s^T)^{jk} \pa^2_{jk} \xi^i ] \ \\
\~\s^i_k = \ & \s^i_k \ + \ \eps \, [ \s^j_k \pa_j \xi^i - \xi^j
\pa_j \s^i_k ] \ . \end{array} \eqno(3.3)$$
When (3.1) maps (2.1) into itself, i.e. when (3.2) coincides with (2.1) \[up to terms $o (\eps )$\], we say that $X_0 = \xi^i (x,t) \pa_i$ is a (Lie-point) spatial [**symmetry**]{} of (2.1). Thus, Lie-point spatial symmetries of (2.1) are identified by the vanishing of terms $O (\eps )$ in (3.3). In the following statement, we use the notations introduced above for $A$ and $\{.,.\}$.
[**Proposition 1.**]{} [*The (generators of) Lie-point spatial symmetries of the Ito equation (2.1) are given by vector fields $X_0 = \xi^i (x,t) \pa_x^i$ with coefficients $\xi^i$ satisfying the [determining equations for spatial symmetries]{} of an Ito equation: $$\begin{array}{l}
\pa_t \xi^i + \{ f,\xi \}^i + (1/2) (\s \s^T )^{jk} \pa^2_{jk} \xi^i = 0 \\
\{ \s_k , \xi \}^i = 0 \ . \end{array} \eqno(3.4)$$*]{}
These are $n+n^2$ equations for the $n+1$ functions $\xi^i , \tau$; thus for $n>1$ they have some solution only in very exceptional cases. This should not be surprising, as symmetry is a non-generic property.
Note that if $\s = \s (t)$ does not depend on the spatial variables, the second of (3.4) reduces to $\s^j_k \pa_j \xi^i = 0$, which in turns imply the vanishing of the term $(\s \s^T )^{jk}
\pa^2_{jk} \xi^i$, which in this case can be rewritten as $\s^k_p \pa_k
(\s^j_p \pa_j \xi^i)$. Thus, (3.4) are in this case equivalent to the determining equations for symmetries of the deterministic part of the Ito equation \[see (1.5) and recall now we are assuming $\tau = 0$\] with the additional condition $\s^j_k \pa_j \xi^i = 0$.
Spatio-temporal symmetries.
---------------------------
We could of course consider transformations acting also on $t$: in this case the $t$ transformation would also affect the processes $w (t)$, and some extra care should be paid. We will not go over the details (which consist in studying how $w (t)$ is changed and include this in the change undergone by $\s$), and just report the result obtained in [@SDE1], with the same notation as above.
[**Proposition 2.**]{} [*The projectable vector field $X_0 = \tau (t)
\pa_t + \xi^i (x,t) \pa_x^i$ is a symmetry generator for the Ito equation (2.1) if and only if the coefficients $\{\tau, \xi^i\} $ satisfy the [full determining equations]{} for projectable symmetries of an Ito equation: $$\begin{array}{l}
\pa_t \( \xi^i - \tau f^i \) + \{f,\xi\}^i - A^{jk}
\pa^2_{jk} \xi^i \ = \ 0 \\
\{ \s_k , \xi \}^i - \tau \pa_t \s^i_k - (1/2) \s^i_k \pa_t \tau \
= \ 0 \ . \end{array} \eqno(3.5)$$*]{}
Note that the symmetries which are linear in $x$, i.e. such that $\xi^i (t;{\bf x}) = M^i_j (t) x^j$, are given by the same equations as for symmetries of the deterministic part of the Ito equation (2.1), i.e. (1.5), plus the additional condition $ M^i_j \s^j_k = M^j_p x^p \pa_j \s^i_k + \tau \pa_t \s^i_k + (1/2)
\s^i_k \pa_t \tau$.
A relevant case in applications is the one where $f^i =
f^i(\xb)$ and $\s^i_k = \s^i_k (t)$ or even $\s^i_k = {\rm const} =
S^i_k$; this correspond to an autonomous dynamical system subject to a noise which depends only on $t$ or even a constant noise. In this case (3.4) can be discussed quite completely. Indeed, with $\s$ independent of the spatial coordinates, the second of these reads $ 2 \s^j_k \pa_j \xi^i = 2 \tau \pa_t \s^i_k + \s^i_k \pa_t \tau $. As the r.h.s. only depends on $t$, by differentiating with respect to $x^m$ we get the equation $ \s^j_k (\pa^2 \xi^i / \pa x^j \pa x^m) = 0$: i.e., the (symmetric) matrix of second derivatives of $\xi^i$, $H^i_{jm} $ must be such that $\s H^i = H^i \s = 0$. Notice that in particular, if $\s^i_k (t) = \la_{(i)} (t) \delta^i_k $ with all $\la_{(i)} \not\equiv 0$ (or however if $\s^{-1}$ exists), this means that $\xi$ can be at most linear in the $x$.
Symmetries of the Fokker-Planck equation.
=========================================
We will now discuss the determining equations for projectable symmetry generators of the Fokker-Planck equation (2.3) in arbitrary spatial dimensions; general symmetries of the Fokker-Planck equations (with some limitations on $\s$) in one and two space dimensions have been completely classified [@CicVit; @Fin; @StSt].
It will be convenient to rewrite the Fokker-Planck equation as $$u_t \ + \ A^{ij} \, \pa^{2}_{ij} u \ + \ B^i \, \pa_{i} u \ + \ C
u \ = \ 0 \ ;
\eqno(4.1)$$ the coefficients $A,B,C$ depend on $x$ and $t$ only and are given explicitly by[^4] $$A = -(1/2) \s \s^T \ ; \
B^i = f^i + 2 \pa_j A^{ij} \ ; \
C (x,t) = (\pa_i \cdot f^i ) + \pa^2_{ij} A^{ij} \ . \eqno(4.2)$$
We consider a projectable vector field of the form (1.3). Computing the second prolongation of this and applying it to the equation, we obtain the determining equations for symmetries of the Fokker-Planck equation.
It follows by general results on Lie-point symmetries of linear equations [@BlK; @Gaeb; @Olv1], or by an elementary explicit computation, that:
[**Lemma 2.**]{} [*The projectable symmetries of the Fokker-Planck equation (4.1) are given by vector fields in the form (1.8) with $\phi = \a (x,t) + \b (x,t) u$.*]{}
Again by linearity, we will have “trivial” symmetries $X_\a = \a (x,t) \pa_u$, with $\a (x,t)$ an arbitrary solution to the FP equation itself; this is just expressing the linear superposition principle [@Olv1]. Note that in order to preserve the normalization (2.4), we should moreover require $\int \a (x,t) d x = 0$, which rules out nontrivial non-negative solutions.
We will thus from now on focus on other, i.e. “nontrivial”, symmetries only, and take $\a (x,t) = 0$.
[**Proposition 3.**]{} [*The nontrivial projectable symmetries of the Fokker-Planck equation (4.1) are given by vector fields in the form (1.3) with $\phi = \b (x,t) u$, where $\tau$, $\xi^i$, $\b$ satisfy the [determining equations]{} $$\begin{array}{l}
\pa_t (\tau A^{ik} ) + (\xi^m \pa_{m} A^{ik} - A^{im} \pa_{m}
\xi^k - A^{mk} \pa_{m} \xi^i ) = 0 \\
\pa_t (\tau B^i ) - \[ \xi^i_t + B^m \pa_{m} \xi^i - \xi^m \pa_{m} B^i \] + ( A^{ik} \pa_{k} \b + A^{mi} \pa_{m} \b ) - A^{mk} \pa_{mk}^{2} \xi^i = 0 \\
\pa_t (\tau C ) + \b_t + A^{ik} \pa_{ik}^{2} \b + B^i \pa_{i} \b + \xi^m
\pa_{m} C = 0 \ . \end{array} \eqno(4.3)$$*]{}
It should be stressed that in the computations leading to (4.3) we have not required preservation of the normalization $\int u \ d x = 1$; when we require this, it turns out that
In other words, this lemma guarantees that if we are looking for the (nontrivial) symmetries of a Fokker-Planck equation compatible with its probabilistic interpretation, we do not have to deal with the general form (1.3) of the symmetry vector field, but we can use instead the ansatz $$X \ = \ \tau (t) \pa_t \, + \, \xi^i (x,t)
\pa_i \, - \, [{\rm div} (\xi) u] \pa_u \ ; \eqno (4.4)$$ this substantially simplifies the analysis of symmetries.
Symmetries of the Ito versus symmetries of the associated FP equation.
======================================================================
We are specially interested in discussing how the symmetries of the partial differential equation (4.1) and those of the symmetries of the system of stochastic ODEs (2.1) are related.
In order to do this, we should express the coefficients $A,B,C$ in terms of $f$ and $\s$; indeed $A = - (1/2) \s \s^T$, $ B^i = f^i + 2 \pa_k A^{ik}$, and $C = \pa_i f^i + \pa^2_{ik} A^{ik}$. Using the latter two of these and with some manipulations, see [@SDE1], (4.3) reads $$\begin{array}{l}
\pa_t (\tau A^{ik} ) + \( \xi^m \pa_{m} A^{ik} - A^{im} \pa_{m} \xi^k -
A^{km} \pa_{m} \xi^i \) \ = \ 0 \\
\[ \pa_t ( \xi^i - \tau f^i ) + \{ f, \xi \}^i - A^{mk} \pa^{2}_{mk}
\xi^i \] - 2 \[ A^{ik} \pa_{k} \b + \ A^{im} \pa_{mk}^{2} \xi^k
\] \ = \ 0 \\
\[ \pa_t + f^i \pa_{i} - A^{ik} \pa^{2}_{ik} \] \[ \b + \pa_{m} \xi^m
\] \ = \ 0 \ . \end{array} \eqno(5.1)$$
It is convenient to introduce the shorthand notations $$\begin{array}{l}
\Gamma^k_j \ = \ \s^m_j \pa_m \xi^k - \xi^m \pa_m \s^k_j -
\tau \pa_t \s^k_j - (1/2) \s^k_j \pa_t \tau \ , \\
\Lambda^i \ = \ - \, \[ \pa_t ( \xi^i - \tau f^i ) + \{ f, \xi \}^i - A^{mk} \pa^{2}_{mk}
\xi^i \] \ . \end{array} \eqno(5.2)$$ With these, (5.1) is rewritten as $$\begin{array}{l}
\s^i_j \Gamma^k_s \delta^{js} + \s^k_j \Gamma^i_s \delta^{js} = 0 \\
\Lambda^i + 2 \[ A^{ik} \pa_{k} \b + \ A^{im} \pa_{mk}^{2} \xi^k \] \\
\[ \pa_t + f^i \pa_{i} - A^{ik} \pa^{2}_{ik} \] \[ \b + \pa_{m} \xi^m
\] \ = \ 0 \ . \end{array} \eqno(5.3)$$
It should be stressed that with the notation just introduced, the determining equations (3.5) for symmetries of the Ito equation (2.1) read simply $$\Lambda^i \ = \ 0 \ \ ; \ \ \Gamma^i_j \ = \ 0 \ . \eqno(5.4)$$
With this notation, the comparison of symmetries for the Ito and the associated Fokker-Planck equation is quite a simple matter.
From the Ito to the FP symmetries
---------------------------------
We will first investigate if symmetries of an Ito equation result in symmetries of the associated Fokker-Planck equation.
We first note that for $\Gamma = 0$, the first of (5.3) is satisfied. As for the second of (5.3), when $\Lambda^i = 0$ this reduces to $$A^{ik}\pa_k \b = - A^{im} \pa^2_{mk} \xi^k \ . \eqno(5.5)$$
Obviously $\b$ is not present in symmetries of the Ito equation, so we can choose it as to satisfy the third of (5.3) and (5.5): for this it suffices to choose $$\b \ = \ - \pa_m \xi^m + c_0 \ = \ - {\rm div} (\xi) + c_0 \ .
\eqno(5.6)$$ We know from lemma 2 that actually we have to take $c_0 = 0$, and that this is actually the only choice of $\b$ providing us with a normalization-preserving symmetry of the FP. Our findings are summarized as follows:
From the FP to the Ito symmetries
---------------------------------
Let us now consider the converse question, i.e. if and when a symmetry of the FP equation associated to an Ito equation can be projected to a symmetry of the Ito equation itself.
As the probabilistic interpretation of the FP requires the normalization condition (2.4), we expect only symmetries identified by lemma 2 can provide symmetries of related Ito equations.
Moreover, as recalled above, there are transformations which map an Ito equation into a different one with the same statistical properties and thus the same associated FP equation: these would be symmetries of the FP but not of the Ito equation. Thus, not all the symmetries of the FP should be expected to produce symmetries of the Ito equation.
By lemma 2, we assume $\b = - {\rm div} (\xi)$; this guarantees that the third of (5.3) holds, and also that the second of (5.3) reduces to $\Lambda^i = 0$. Hence we only have to discuss the relation between the first of (5.3) and the second of (5.4). These two equations can be rewritten, respectively, as $$\s \Gamma^T + \Gamma \s^T = 0 \ \ {\rm and} \ \ \Gamma = 0 \ . \eqno(5.7)$$ The second of these imply the first, but the converse is not true: hence there is [*not*]{} a complete equivalence.
This just corresponds to a given FP equation corresponding to different Ito equations: the transformations with $\Gamma \not= 0$ but $\s \Gamma^T + \Gamma \s^T = 0$ will be precisely those which map an Ito equation $E_0$ into a different Ito equation $E_1$ which has the [*same*]{} Fokker-Planck associated equation, see (2.5).
We may note that from the above system (5.4), restricting to the case where $f(t,x)=f(x)$ and $\s(t,x)=\s(x)$, we recover the results of Cicogna and Vitali [@CicVit] for the one-dimensional setting. From (5.2) it is also possible to recover the results of Shtelen and Stogny [@StSt] for the two-dimensional Kramers equation, as well as the results of Finkel [@Fin] for certain two-dimensional FP equations.
Examples
--------
[**Example 1.**]{} As the first example, we consider the case $f(t,x) = 0$, $\s(t,x) = \s_0 = {\rm const} \not=0$, i.e. the equation $$dx = \s_0 \, dw(t) \eqno(5.8)$$ which represents a free particle subject to constant noise. The corresponding Fokker-Planck equation is simply the heat equation $ u_t = (\s_0^2 / 2) u_{xx}$. The symmetries of the heat equation (other than the trivial ones $v_\a = \a (x,t) \pa_u$ with $\a_t = \a_{xx}$) are well known to be [@Olv1; @Ste] $$\begin{array}{l}
v_1 = \pa_t \ ; \
v_2 = \pa_x \ ; \
v_3 = u \pa_u \ ; \
v_4 = \s_0^2 t \pa_x - \s_0 x u \pa_u \ ; \\
v_5 = 2 t \pa_t + x \pa_x \ ; \
v_6 = t^2 \pa_t + x t \pa_x - (1/2) (t + x^2 / \s_0^2 ) u \pa_u \end{array} \eqno(6.2)$$ Of these, $v_1 , v_2$ and $v_5$ (which do not act on $u$) are also symmetries of the Ito equation (5.8), as is easily checked using (3.4). Notice that (4.4) is satisfied for these, and is not satisfied for $v_3,v_4$ and $v_6$. The vector fields $v_1 , v_2$ and $v_5$ do actually span the symmetry algebra of (5.8).
[**Example 2.**]{} As an example in two space dimensions \[with coordinates $(x_1,x_2) = (x,y)$\], we choose $$\begin{array}{rl}
dx \ = \ & y \, dt \\
dy \ = \ & - k^2 y \, dt \ + \
\sqrt{2 k^2} \, dw(t) \ \end{array} \eqno(5.9)$$ with $k^2$ a positive constant.
The corresponding Fokker-Planck equation is the Kramers equation $$u_t \ = \ k^2 u_{yy} - y u_x + k^2 y u_y +
k^2 u \ ; \eqno(5.10)$$ the symmetries of this were studied in [@StSt] and, apart from the trivial ones $v_\a$, are $$\begin{array}{l}
v_1 \ = \pa_t \ ; \
v_2 \ = \pa_x \ ; \
v_3 \ = e^{-k^2 t} \, [ k^{-2} \pa_x - \pa_y ] \ ; \
v_4 \ = u \pa_u \ ; \\
v_5 \ = t \pa_x + \pa_y - (1/2) (y + k^2 x) u \pa_u \ ; \
v_6 \ = e^{k^2 t} \, [ k^{-2} \pa_x + \pa_y - yu \pa_u ] \end{array}$$ Here $v_1 , v_2$ and $v_3$ satisfy (4.4), while for $v_4 , v_5$ and $v_6$ this is violated. According to our definition, the symmetries of the equations (5.9) are again $v_1$, $v_2$ and $v_3$, as easily checked.
[**Example 3.**]{} We consider now an example where the correspondence between normalization-preserving symmetries of the FP equation and symmetries of the Ito equation is not complete, i.e. the two-dimensional Ito system (with zero drift) $$\begin{array}{rl}
dx^1 \ = & \ \cos (t) dw^1 \, - \, \sin (t) d w^2 \\
dx^2 \ = & \ \sin (t) dw^1 \, + \, \cos (t) dw^2 \ ; \end{array} \eqno(5.11)$$ the corresponding Fokker-Planck equation is now just the two dimensional heat equation $u_t = (1/2) \triangle u$.
It is now immediate to check that the vector field $X_0 = \pa_t$ is a symmetry of the FP equation, but [*not* ]{} a symmetry of (5.11). Obviously, the case of any orthogonal $\s$ with $\pa_t \s^i_k \not\equiv 0$ will be exactly the same.
[**Example 4.**]{} Consider $n$ uncoupled equations for equal “Langevin harmonic oscillators” subject to independent stochastic noises [@Kam; @Ris]; this system is described by the Ito system $$dx^{i} \ = \ - x^{i} \, dt \, + \, \sqrt{2 s_i} \, dw^i \ ,
\ \ i=1, 2, ..., n \, \eqno(5.12)$$ (no sum on $i$) where we assume all the $s_i$ are strictly positive; the corresponding Fokker-Planck equation is $$\pa_{t} u \ = \ \sum_{i=1}^n \, [ s_i \pa^{2}_{ii} u + x^{i} \pa_{i} u + u ] \ .$$ In this case we get the symmetries ($i=1,2,...,n$) $$v_1 = \pa_t \ , \
v_{2} = e^{-2 t} [ \pa_t - \sum_{i=1}^{n} x^{i} \pa_i + n u \pa_u ] \ , \
v_{q_i} = e^{-t} \pa_i \ .$$
[**Example 5.**]{} Finally, we consider now the n-dimensional nonlinear case $$dx^{i} = - (1 - \lambda \| x \|^2) x^{i} \, dt + dw^{i} \ , \eqno(5.13)$$ where $ \| x \|$ is the norm of the vector ${\bf x}$ and $\lambda \not= 0$. By inserting $f^{i}= - x^{i}(1 - \lambda \| x \|^2)$ and $\s_{j}^{i} = \delta_{ij}$ in (5.4) we obtain, from the second of these, $\xi^i = h^i (t) + (\tau_t/2) x^i$. Inserting this into the first of (5.4) and isolating the coefficients of different powers of $x$, we get that the only symmetry is given by $v = \pa_t$.
Note that we got no rotation symmetry; this is because within the class of transformations we are considering we can rotate the vector $x$, but [*not*]{} the vector Wiener process $w(t)$; transformations allowing to rotate $w(t)$ as well will be considered in the next section.
W-symmetries
============
We will consider symmetries involving not only the spatial and time variables $(x , t)$, but also the vector Wiener processes $w (t)$ entering in the $n$-dimensional Ito equation (2.1). We will specifically consider infinitesimal transformations of the form $$\begin{array}{l}
x^i \to y^i \ = x^i + \eps \xi^i (x,t) \\
t \to s \ = t + \eps \tau (t) \\
w^i \to z^i \ = w^i + \eps \mu^i (w,t) \ . \end{array} \eqno(6.1)$$ We also call symmetry generators of this form, “[*W-symmetries*]{}”.
Note that this is not the most general possible form of a transformation for the variables involved in (1.1); some words on this restriction are in order here.
The restriction on $\tau$ and $\xi$ is the same as considered above. Moreover, we only allow “internal” transformations of $w$, i.e. they cannot depend on $x(t)$. As we think of the stochastic process $w (t)$ to be independent of the evolution of the $x (t)$, we like its transformation not to depend on the latter. Finally, we have allowed the transformation on the spatial coordinates $x(t)$ to depend on $x$ and $t$, but not on the $w$; this means that we do not want to consider transformations of the spatial coordinates which depend on the realization of the stochastic process $w (t)$. This again is somewhat a natural requirement in physical terms[^5].
In order to compute how (6.1) acts on the Ito equation (2.1), we use the Ito formula (2.2). Moreover, due to the form of (6.1), we can adopt a “two-steps procedure”, see [@SDE2]: the transformations of the $(x,t)$ variables and of the stochastic process $w(t)$ do not interfere with each other. It turns out that some strong limitation on the functions $\mu^i$ arise.
[**Lemma 4.**]{} [*The infinitesimal transformation $w \to z = w + \eps \mu (w,t)$ maps the vector Wiener process $w(t)$ into a vector Wiener process $z(t)$ if and only if $\mu = B w$, with $B$ a real antisymmetric matrix. Equivalently, if and only if $z = M w$ with $M$ orthogonal.*]{}
Note that as $B$ is antisymmetric matrix, it vanishes in dimension one: for one-dimensional Ito equations we have no new symmetries with respect to those discussed above.
[**Proposition 6.**]{} [*Under the infinitesimal transformation (6.1), with $B$ a real antisymmetric matrix, the Ito equation (2.1) is changed into a (generally, different) Ito equation. The Ito equation (2.1) is invariant under (6.1) if and only if $\tau,\xi,\mu$ satisfy the [*determining equations for W-symmetries*]{}: $$\begin{array}{l}
\pa_t \xi^i \, + \, \{ f , \xi \}^i \, - \,
\pa_t (\tau f^i) \, + \, A^{jk} \pa^2_{jk} \xi^i \ = \ 0 \\
\{ \s_k , \xi \}^i - \tau \pa_t \s^i_k - (1/2)
(\pa_t \tau) \s^i_k - \s^i_p B^p_k \ = \ 0 \ . \end{array} \eqno(6.2)$$*]{}
If we just consider a transformation $w \to z = M w$, with $M$ orthogonal as required by lemma 4, the Fokker-Planck equations associated to the Ito equation and to the transformed one will be the same. Thus all the discussion conducted in section 5 about the relations betwen symmetries of an Ito equation and of the associated FP equation also applies to transformations of the form (6.1).[^6]
[**Example 1.**]{} Let us consider again the two-dimensional Ito equation (5.9) If we look at the second set of determining equations and single out the one for $i=2$ and $k=1$, we have immediately that $B=0$, i.e. in this case we have no new symmetry by allowing transformation of $w$.
[**Example 2.**]{} Let us consider again the two-dimensional Ito system (5.11) We will write $\xi^1 = \a$, $\xi^2 = \b$, and $B_{12}=-B_{21} = b$. The most general solution to the determining equations turns out to be $$\begin{array}{l}
\a \ = \ [\tau_t / 2 - \sin (2t) \tau ] x + [ b + \cos (2t) \tau ] y +
\a_0 (t) \\
\b \ = \ - [b + \tau ] x + [\tau_t / 2] y + \b_0 (t) \ ; \end{array}$$ By setting $\tau = \a_0 = \b_0 = 0$, we obtain a new symmetry, corresponding to $\a = b y$ and $\b = - b x$ with $b$ arbitrary, i.e. given by $$X \ = \ \( y {\pa \over \pa x} \, - \, x {\pa \over \pa y} \) \ + \
\( w^2 {\pa \over \pa w^1} \, - \, w^1 {\pa \over \pa w^2} \) \ .$$ Note this is nothing else than a simultaneous (and identical) rotation in the $(x,y)$ and in the $(w_1 , w_2 )$ planes.
[Example 3.]{} We will now consider again the case of $n$ uncoupled equal “Langevin harmonic oscillators”, see (5.12) above.
If we discard symmetries with $B=0$ (i.e. those obtained above), we are still left with a nontrivial possibility, i.e. $B$ an arbitrary (real, antisymmetric) constant matrix and $\xi^i = C^i_{~k} x^k$, where $C^i_{~k} = (s_i / s_k ) B^i_k$ (no sum on $i,k$ here).
The meaning of this result is obvious: we can act on this system by an arbitrary $SO(n)$ rotation in the $w$ space and a related rotation in the $x$ space.
If we assume $s_1 = s_2 = ... = s_n = s$, we have indeed $C=B$ and the rotations in $w$ and $x$ spaces do just coincide. For general $s_1
, ... , s_n$, the $(s_i/s_k)$ factors relating the $C$ and $B$ matrices are also easily understood: we could rescale each of the $x^i$ by a factor $\sqrt{2 s_i}$, $x^i = \sqrt{2 s_i} y^i$ (no sum on $i$), arriving at the manifestly rotationally invariant $n$-dimensional Ito equation $d y^i = - y^i d t + d w^i$.
[**Example 4.**]{} Let us consider again (5.13). We easily check that it is rotationally invariant with a W-symmetry: take $\tau = 0$ and $\xi^i = B^i_k x^k$, so that $$\{ \xi , f \}^i \ = \ \{ B x , (1 - \lambda ||x||^2 ) x \} \ =
\ - 2 \lambda x^i \( x^j B^j_k x^k \) \ = \ 0 \ ,$$ the last equality following from $B=-B^T$. That is, (5.13) is indeed symmetric under simultaneous identical rotations of the $x$ and $w$ vectors.
Discrete symmetries
===================
We will now briefly consider, for the sake of completeness, [*discrete*]{} symmetries of stochastic differential equations. Similarly to what happens in the deterministic case (see e.g. [@GRo]), the resulting determining equations are in general too difficult to be attacked except for very simple classes of transformations (e.g. reflections, or however linear ones). On the other side, they can be used to check if a given discrete transformation is a symmetry of a given stochastic differential equation; and/or to determine the stochastic differential equations which admit a given discrete transformation as a symmetry.
We consider again an Ito equation of the form (2.1); here we will not consider transformations acting on the $t$ coordinate, and will thus limit to consider the change of coordinates in the $x$ and $w$ spaces given by $$y^i = \phi^i (x,t) \ \ , \ \ w^k (t) = R^k_{~p} (t) z^p (t) \ .
\eqno(7.1)$$ As required by lemma 4, we take $R \in O (n)$. We write as usual $A = - (1/2) \s \s^T$.
With the Ito formula we have at once that $y$ obeys the Ito equation $$d y^i \ = \ \[ {\pa \phi^i \over \pa x^j} f^j \,
- \, A^{jk} {\pa^2 \phi^i \over \pa x^j \pa x^k} \, + \, {\pa \phi^i
\over \pa t} \] \, d t \ + \ \[ {\pa \phi^i \over \pa x^j}
\s^j_{~p} R^p_{~k} \] \, d z^k \ . \eqno(7.2)$$
[**Proposition.**]{} [*The transformation (6.1) is a symmetry of the Ito equation (2.1) if and only if $\phi , R$ satisfy the [determining equations for discrete symmetries]{} of an Ito equation: $$\begin{array}{l}
{\pa \phi^i (x,t) \over \pa x^j} \, f^j (x,t) \, + \, A^{jk} (x,t)
\, {\pa^2 \phi^i (x,t) \over \pa x^j \pa x^k} \, + \, {\pa \phi^i (x,t)
\over \pa t}\ = \ f^i \( \phi (x,t),t \) \ , \\
{\pa \phi^i (x,t) \over \pa x^j} \, \s^j_{~p} (x,t) \, R^p_{~k} \
= \ \s^i_{~k} \( \phi (x,t) , t \) \ . \end{array} \eqno(7.3)$$*]{}
[**Example 1.**]{} The simplest case of discrete transformation is provided by $\phi^i (x,t) = - x^i$, $R= \pm I$. In these cases, (7.3) reduce to $$f^i (x,t) = - f^i (-x,t) \ \ ; \ \ \s (x,t) = \mp \s (-x , t) \ .
\eqno(7.4)$$
[**Example 2.**]{} For the $n$ independent “Langevin oscillators”, see (5.12), it is immediate to check that (7.4) is satisfied (for $R=-I$). More generally, (7.3) read now (no sum over $i$, sum over $j$ when repeated) $$a_{(j)} {\pa \phi^i \over \pa x_j} - {s_{(j)}^2 \over 2} {\pa^2
\phi^i \over \pa x^j \pa x^j} \ = \ a_{(i)} \phi^i \ \ ; \ \
s_{(j)} {\pa \phi^i \over \pa x_j} R^j_{~k} \ = \ s_{(i)}
\de^i_{~k} \ .$$
[**Example 4.**]{} For equations with $\s^i_{~k} (x,t) = s_0 \de^i_{~k} $ (with $s_0$ a real constant), the second of (7.3) always reduces to $ (\pa \phi^i / \pa x^j) = (R^{-1} )^i_{~j}$. But this implies $\phi$ must be a linear function of the $x$, $\phi^i (x,t) = L^i_{~j} (t) x^j$ with $L$ an orthogonal matrix ($L = R^{-1}$). Thus in this case we are always reduced to the simple equation $$L^i_{~j} \, f^j (x,t) \ = \ f^i (Lx,t) \ \ \ [L \in O(n)] \ .$$
Symmetries of the discrete KPZ equation
=======================================
The KPZ (Kardar-Parisi-Zhang) equation is written, in physicists’ notation, as $${\pa h \over \pa t} \ = \ \nu \lapl h \, + \, {\la \over 2} \, \(
\grad h \)^2 \, + \, \eta \ . \eqno(8.1)$$ Here $h = h(y,t)$ is a scalar variable, representing the height of a growing (non-overhanging) interface at point $y \in R^n$ at time $t$, the operators $\grad$ and $\lapl$ represents the gradient and laplacian with respect to the $y$ variables, $\nu > 0$ and $\la \ge 0$ are real numbers, and $\eta = \eta (y,t)$ is a random field with $\delta$ spatial correlation function and the statistical properties of a white noise at each given point $y$. When $\la = 0$ the KPZ reduce to its linear part, which is also known as the Edwards-Wilkinsons equation.
The KPZ is a universal equation describing, under general hypotheses and up to an asymptotic procedure, the time evolution of a growing interface; see [@Bar; @HHZ; @KPZ] for details.
We will limit ourselves to the one-dimensional case $n=1$; moreover, as our discussion deals only with stochastic ODEs (as opposed to stochastic PDEs), we consider a discretization of the KPZ, obtained by considering a uniform chain of intersite distance, $(\de y)$; in other words, we consider a discrete set of points $y_k = k (\de y)$. Thus $h(y,t)$ is now replaced by a vector variable $x (t)$ with components $x^i (t)$ ($i \in {\bf Z}$), where $x^i (t) \approx h (y_i , t)$.
The gradient and Laplacian terms in the KPZ will then be replaced by $$(\grad h ) (y_i,t) \approx {x^{i+1} - x^{i-1} \over 2 (\de y) } \ ;
\ (\lapl h) (y_i ,t) \approx {x^{i+1} - 2 x^i + x^{i-1} \over (\de
y)^2} \ . \eqno(8.2)$$ As for $\eta (y,t)$, its discretization will be a standard vector Wiener process $w (t)$ with component $w^i (t)$ at the point $y_i$. We also write $\a = \nu [2 (\de y)]^{-1}$ and $\b = \la [2 (\de
y)^2]^{-1}$.
In this way, we replace (8.1) by the Ito equation $$\d x^i \ = \ \( \a \[ x^{i+1} - 2 x^i + x^{i-1} \] \ + \ \b \[
x^{i+1} - x^{i-1} \]^2 \) \ \d t \ + \ d w^i (t) \ . \eqno(8.3)$$
It will be convenient to rewrite this in a more abstract notation as $$\d x^i \ = \ \[ M^i_{~k} x^k + \Ga^i_{~jk} x^j x^k \] \, \d t \ + \
\d w^i (t) \ \eqno(8.4)$$ (note we have $\s = I$); by comparing (8.3) and (8.4) we have $$\begin{array}{l}
M^i_{~k} \ = \ \a \[ \de^i_{~k+1} - 2 \de^i_{~k} + \de^i_{~k-1} \] \\
\Ga^i_{~jk} \ = \ \b \[ \de^i_{~j-1} \de^i_{~k-1} + \de^i_{~j+1}
\de^i_{~k+1} - \de^i_{~j-1} \de^i_{~k+1} - \de^i_{~j+1} \de^i_{~k-1}
\] \ . \end{array} \eqno(8.5)$$ Note that $\Ga^i_{jk} = \Ga^i_{kj}$; it will be convenient to also write $(G_{(j)})^i_k = \Ga^i_{jk}$.
Physically an equation describing a growing interface is required to possess the following symmetries [@Bar; @HHZ; @KPZ]: (1) time translation; (2) translation in the $h$ direction; (3) space translations; (4) spatial inversion across any plane and (for $n>1$) rotations around an axis perpendicular to the $\xb$ plane. Moreover, at the linear level we should also require invariance under: (5) inversion in the $h$ direction.[^7]
It is well known that the KPZ is the simplest nonlinear (stochastic PDE) equation possessing these symmetries, and actually it was determined essentially on the basis of this consideration.
We will now check that with our definition of symmetry for a stochastic differential equation, the discretized KPZ (8.3) does indeed admit these symmetries. Note that in the discretized case, (1) and (2) remain continuous symmetries, while (3), (4) and (5) are discrete ones. In particular, (3) is replaced by the shift of one site along the chain, and (4) reduces to spatial inversion.
Continuos symmetries
--------------------
We start by analyzing continuos symmetries. The second of the determining equations (6.2) provides, for $\s = I$, $$\xi^i = (1/2) \tau_t x^i - B^i_k x^k + \a^i (t) \ ,$$ which we also write as $\xi^i = \La^i_k x^k + \a^i$. Plugging this into the first of (6.2), and separating terms homogeneous of different degree in the $x$, we get (using also symmetry of the $\Ga$ and our shorthand notation) $$\begin{array}{l}
\a^i_t - M^i_j \a^j \ = \ 0 \\
{1 \over 2} \[ \tau_{tt} \de^i_j - (B_t)^i_j \] - \tau_t M^i_j + [ \La , M ]^i_j - 2 \a^p \Gj^i_p \ = \ 0 \\
\tau_t \Gj^i_k - \La^i_m \Gj^m_k + \Gj^i_p \La^p_k \ = \ 0 \end{array} \eqno(8.6)$$
We will not try to determine the most general solution to these, but just check that (1)-(5) are indeed solutions to these equations, i.e. that our definitions correctly identify them as symmetries of the (discretized) KPZ equation.
As for (1), in this case $\tau = 1$, while $\xi^i = 0$, which of course implies $\a$ and $\Gamma$ are also zero, and $B = 0$ as well. It is immediate to check that (8.6) are indeed satisfied in this case.
For the transformation (2) we have $\tau=0$, $B=0$, $\xi^i = 1$; the latter means $\Lambda = 0$, $\a^i = 1$. The first and third of (8.6) are immediately seen to hold, while the second reduces to $\sum_j \Gamma^i_{jk} = 0$; with the explicit expression for $\Gamma$ this is also easily seen to hold, as it reads $$\sum_{j=-\infty}^{+\infty} \ \[ \de^{j-1}_{k-1} + \de^{j+1}_{k+1} - \de^{j-1}_{k+1} - \de^{j+1}_{k-1} \] \ .$$
Discrete symmetries
-------------------
In the case of discrete symmetries (3), (4), (5), we deal with linear transformations, i.e. $x \to y = Fx$, $w \to z = R w$. The relevant equations (7.3) are then $$F^i_j f^j \ = \ f^i (Fx) \ \ ; \ \ F^i_j \s^j_p (R^{-1})^p_k \ = \ \s^i_k \ . \eqno(8.7)$$ As $\s = \de$, the second of these actually reads $R = F$. Writing $f$ as in (8.4) and separating linear and quadratic terms in the $x$, the first of (8.7) reads $$\begin{array}{l}
\[ F , M \] \ = \ 0 \\
F^i_j \, \Ga^j_{k m} \ = \ \Ga^i_{pq} \, F^p_k \, F^q_m
\end{array} \eqno(8.8)$$ We thus have to check these are satisfied, with $M$ and $\Lambda$ provided by (8.5), for (3),(4),(5).
In the case of translations, (3), we have $F^i_j = \de^i_{j+1}$. Hence $$(F M)^i_k = \de^{i-1}_{k+1} - 2 \de^{i-1}_k + \de^{i-1}_{k-1} \ , \ (M F)^i_k = \de^i_{k+2} - 2 \de^i_{k+1} + \de^i_k \ ,$$ and the first of (8.8) is satisfied. Similarly, $$\begin{array}{l}
F^i_m \Gamma^m_{j k} \ = \ \de^{i-1}_{j-1} + \de^{i-1}_{k-1} + \de^{i-1}_{j+1} \de^{i-1}_{k+1} - \de^{i-1}_{j-1} - \de^{i-1}_{j+1} \de^{i-1}_{k-1} \ , \\
\Gamma^i_{p q} F^p_j F^q_k \ = \ \de^i_j \de^i_k + \de^i_{j+2} \de^i_{k+2} - \de^i_j \de^i_{k+2} - \de^i_{j+2} \de^i_k \ : \end{array}$$ and the second of (8.8) is satisfied as well.
Let us come to (4), i.e. spatial inversion with respect to any site $m$: this is described by $F^i_k = \de^{i-m}_{m-k}$ (no sum on $m$ in this and the following formulas). In this case we have $$(FM)^i_j = \de^{i-m}_{m-j-1} - 2 \de^{i-m}_{m-j} + \de^{i-m}_{m-j+1} \ ; \
(MF)^i_j = \de^{i-m-1}_{m-j} - 2 \de^{i-m}_{m-j} + \de^{i-m+1}_{m-j} \ .$$ This shows that the first of (8.8) holds. As for the second, it also holds as it reads $$\begin{array}{l}
\de^{i-m}_{m-j+1} \de^{i-m}_{m-k+1} + \de^{i-m}_{m-j-1} \de^{i-m}_{m-k-1} - \de^{i-m}_{m-j+1} \de^{i-m}_{m-k-1} - \de^{i-m}_{m-j-1} \de^{i-m}_{m-k+1} \ = \\
= \ \de^{i-m+1}_{m-j} \de^{i-m+1}_{m-k} + \de^{i-m-1}_{m-j} \de^{i-m-1}_{m-k} -
\de^{i-m+1}_{m-j} \de^{i-m-1}_{m-k} - \de^{i-m-1}_{m-j} \de^{i-m+1}_{m-k} \ . \end{array}$$
Finally, let us consider (5): this corresponds to $F = - I$, so obviously $[F,M]=0$. The second of (8.8) reads in this case $\Gamma = 0$, which only holds for $\la = 0$, i.e. in the linear (Edwards-Wilkinson) approximation.
We conclude that our definition correctly captures the symmetry properties of teh KPZ equation.
Acknowledgement {#acknowledgement .unnumbered}
===============
This paper was written while the author was a guest at the DFG-Graduiertenkolleg “Hierarchie und Symmetrie in mathematischen Modellen” at RWTH Aachen. He would like to thank RWTH Aachen, and in particular prof. S. Walcher, for their hospitality and support.
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[^1]: It is well known that any Ito equation is equivalent to a Stratonovich equation; for the Ito equation (2.1), the associated Stratonovich equation is $ dx^i = b^i (x,t) dt + \s^i_k (x,t) \circ dw^k$, with $ b^i = f^i - (1/2) [ \s (\pa \s^T / \pa x)]$. We will to the Ito formulation in all of this note.
[^2]: In other words, we should set the FP equation in the function space $\F$ defined by (2.4), and thus accept only transformations mapping $\F$ to itself.
[^3]: The one-point stochastic processes described by two different Ito equations having the same associated Fokker-Planck equation have the same statistical properties (the probability measures evolve in the same way), but [*are* ]{} different: the same realization of the Wiener process ${\bf w}(t)$ leads to different sample paths.
[^4]: Recall that our non-degeneracy assumption guarantees $A \not= 0$.
[^5]: It should be however stressed that in this way we are also discarding the transformations needed to obtain [*normal forms*]{} of stochastic differential equations, see [@LArn; @ArI] and references therein.
[^6]: Needless to say, for more general transformations, for which the transformations of the $(x,t)$ variables and of the vector Wiener process $w(t)$ do interfere with each other, this would not be the case.
[^7]: Checking that this is the case for the KPZ equation is immediate.
|
---
abstract: 'A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.'
address: 'Department of Mathematics and Statistics, The University of Melbourne, Vic 3010, Australia'
author:
- 'S. Ole Warnaar'
title: 'Partial-sum analogues of the Rogers–Ramanujan identities'
---
[^1]
Introduction
============
The famous Rogers–Ramanujan identities are given by [@Rogers94] $$\label{RR1}
1+\sum_{n=1}^{\infty} \frac{q^{n^2}}{(1-q)(1-q^2)\cdots(1-q^n)}=
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+1})(1-q^{5j+4})}$$ and $$\label{RR2}
1+\sum_{n=1}^{\infty} \frac{q^{n(n+1)}}{(1-q)(1-q^2)\cdots(1-q^n)}
=\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}$$ for $|q|<1$. In one of his two proofs of these identities, Schur [@Schur17] introduced two sequences of polynomials $(e_n)_{n\geq 2}$ and $(d_n)_{n\geq 2}$, where $e_n$ ($d_n$) is the generating function of partitions with difference between parts at least $2$ (and no part equal to $1$), and largest part at most $n-2$. The partitions $\{\emptyset,(1),(2),(3),(4),(3,1),(4,1),(4,2)\}$, for example, contribute to $e_6=1+q+q^2+q^3+2q^4+q^5+q^6$ and the partitions $\{\emptyset,(2),(3),(4),(4,2)\}$ contribute to $d_6=1+q^2+q^3+q^4+q^6$.
By standard combinatorial arguments, see e.g., [@MacMahon16; @GR90], it follows that $e_{\infty}:=\lim_{n\to\infty} e_n
=\text{LHS}\eqref{RR1}$ and $d_{\infty}:=\lim_{n\to\infty} d_n=\text{LHS}\eqref{RR2}$. Schur proved the Rogers–Ramanujan identities by showing that these limits also hold when $\text{LHS}$ is replaced by $\text{RHS}$. This he achieved by showing that both $e_n$ and $d_n$ satisfy the recurrence $$\label{xrec}
x_{n+2}=x_{n+1}+q^n x_n,$$ and by solving this recurrence subject to the initial conditions $d_1=0$, $e_1=e_2=d_2=1$ (consistent with the combinatorial definition of $e_n$ and $d_n$ for $n\geq 2$). Specifically, Schur’s solution to reads
\[edbos\] $$\begin{aligned}
e_n&=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}
{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor{(n-5j-1)/2}\rfloor}} \\
d_n&=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+3)/2}
{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor{(n-5j-2)/2}\rfloor}}\end{aligned}$$
for $n\geq 1$ and $\lfloor x \rfloor$ denoting the integer part of $x$. Here the $q$-binomial coefficients are given by ${{\textstyle\genfrac{[}{]}{0pt}{}{n}{m}}}=(q;q)_n/(q;q)_m(q;q)_{n-m}$ for $0\leq m\leq n$ and zero otherwise, where $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$.
Employing the notation $(a_1,\dots,a_k;q)_n=
(a_1;q)_n\cdots(a_k;q)_n$ and recalling to the Jacobi triple product identity [@GR90 Eq. (II.28)] $$\label{tpi}
\sum_{n=-\infty}^{\infty}(-1)^n a^n q^{\binom{n}{2}}=(a,q/a,q;q)_{\infty}$$ it is now easy to obtain the desired limits; $$\begin{aligned}
e_{\infty}&=\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}=\frac{1}{(q,q^4;q^5)_{\infty}}
=\text{RHS}\eqref{RR1} \\
d_{\infty}&=\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+3)/2}=\frac{1}{(q^2,q^3;q^5)_{\infty}}
=\text{RHS}\eqref{RR2}.\end{aligned}$$
Representations for the Schur polynomials similar to the left sides of the Rogers–Ramanujan identities are also known [@MacMahon16 §286 and §289], $$\label{edfer}
e_n=\sum_{r=0}^{\infty}q^{r^2}{\genfrac{[}{]}{0pt}{}{n-r-1}{r}} \quad\text{and}\quad
d_n=\sum_{r=0}^{\infty}q^{r(r+1)}{\genfrac{[}{]}{0pt}{}{n-r-2}{r}}.$$ Equating this with yields the following polynomial analogue of the Rogers–Ramanujan identities [@Andrews70]: $$\label{RRpoly}
\sum_{r=0}^{\infty}q^{r(r+a)}{\genfrac{[}{]}{0pt}{}{n-r-a}{r}}=
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+2a+1)/2}
{\genfrac{[}{]}{0pt}{}{n}{\lfloor{(n-5j-a)/2}\rfloor}}$$ for $n\geq 0$ and $a\in\{0,1\}$.
Recently there has been renewed interest in the Schur polynomials [@GIS99; @AKP00; @IPS00; @BP01; @W01] sparked by the following nice generalization of the Rogers–Ramanujan identities due to Garrett, Ismail and Stanton [@GIS99 Eq. (3.5)] $$\label{GIS}
\sum_{r=0}^{\infty}\frac{q^{r(r+m)}}{(q;q)_r}=
\frac{(-1)^m q^{-\binom{m}{2}}d_m}{(q,q^4;q^5)_{\infty}}-
\frac{(-1)^m q^{-\binom{m}{2}}e_m}{(q^2,q^3;q^5)_{\infty}},$$ where $m$ is a nonnegative integer and $e_0=0$ and $d_0=1$ consistent with .
In this paper we show that may be used to prove new polynomial analogues of the Rogers–Ramanujan identities involving the Schur polynomials. These polynomial identities are fundamentally different from in that the product-side is replaced by a partial theta series.
\[thm\] For $k\in\{0,1\}$ and $n\geq 0$ there holds $$\begin{aligned}
\notag
\sum_{j=-n-k}^n&(-1)^j q^{j(5j+1)/2} \\
&=\sum_{r=0}^n e_{2r+k+2}(-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}
\label{RRP1}\\
\intertext{and}
\notag
\sum_{j=-n-k}^n&(-1)^j q^{j(5j+3)/2} \\
&=\sum_{r=0}^n d_{2r+k+2}(-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}.
\label{RRP2}\end{aligned}$$
Partial theta-sum identities of this type were first discovered by Shanks [@Shanks51].
When $n$ tends to infinity (for $|q|<1$) only the term with $r=n$ contributes to the sums on the right. Hence the first identity of the theorem implies $$\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}=
(q;q)_{\infty}e_{\infty}=(q;q)_{\infty}
\sum_{n=0}^{\infty} \frac{q^{n^2}}{(q;q)_n},$$ which is transformed into by the triple product identity. Likewise, arises as the the large $n$ limit of the second identity of the theorem.
Polynomial analogues of the Rogers–Ramanujan strikingly similar to those of Theorem \[thm\] have previously been discovered by Andrews [@Andrews83]. For $n\geq 0$ let $K_n(x)$ denote the Szegö polynomial [@Szego26] $$K_n(x)=\sum_{r=0}^n x^r q^{r(r+1)}{\genfrac{[}{]}{0pt}{}{n}{r}}.$$ Then Andrews posed in the problems section of SIAM Review [@Andrews83] the problem of showing that $$\begin{gathered}
\label{AJ}
\sum_{j=-n-k}^n(-1)^j q^{j(5j+2k+1)/2} \\
=\sum_{r=0}^n K_r(q^{2n-2r+k-1})(-1)^{n-r}
q^{(n-r)(5n-3r+4k+1)/2}\frac{(q;q)_{n+k}}{(q;q)_{n-r}}\end{gathered}$$ for $k=\{0,1\}$ and $n\geq 0$. Note here that the left side of coincides with the left side of () when $k=0$ ($k=1$).
The remainder of this paper is divided in two parts with section \[secp\] containing a proof and section \[secd\] a discussion of Theorem \[thm\]. In the first part of this discussion we examine two simple proofs of found by Jordan and Andrews and indicate our failure in generalizing these to a proof Theorem \[thm\]. The second part of our discussion focuses on some of the combinatorial aspects of Theorem \[thm\].
Proof of Theorem \[thm\] {#secp}
========================
A more general identity
-----------------------
Key to the proof of Theorem \[thm\] is the following proposition.
\[prop2\] For $k\in\{0,1\}$ and $|a|,|q|<1$ there holds $$\begin{gathered}
\label{key}
\sum_{n=0}^{\infty}\frac{a^{2n} q^{n(n+k)}}{(q;q)_n}=
\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^3}\sum_{j=1}^{\infty}
(q^{2j-k},q^{5+k-2j},q^5;q^5)_{\infty} \\
\times \sum_{r=0}^{\infty}\frac{(-1)^{j+r+1}
q^{\binom{j+r}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}{(a;q)_{r+k+1}}.\end{gathered}$$
It is perhaps not immediately clear that the sums on the right converge, but inspection of the potentially problematic terms shows that for $k\in\{0,1\}$ and $j\geq 1$, $$\begin{gathered}
O\Bigl(q^{\binom{j+r}{2}}(q^{5+k-2j};q^5)_{\infty}(aq^{-r};q)_r\Bigr) \\
=\begin{cases}
q^{(j-1)(j+10r+4k+6)/10} & \text{$j\equiv 1,k+4 \pmod{5}$} \\
q^{(j-2)(j+10r+4k+7)/10+r+1} & \text{$j\equiv 2,k+3 \pmod{5}$,}
\end{cases}\end{gathered}$$ which shows that both sums on the right converge and that their order is irrelevant.
Before proving Proposition \[prop2\] we will show how it implies Theorem \[thm\]. Starting point is the observation that $$\begin{gathered}
\label{inffin}
\sum_{j=1}^{\infty}(-1)^j q^{\binom{j+r}{2}}
\frac{(q^{2j-k},q^{5+k-2j},q^5;q^5)_{\infty}}{(q;q)_{\infty}} \\
=\sum_{i=1}^2 \frac{(-1)^{i+r}q^{-\binom{2r+k+2}{2}}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}
\sum_{j=-r-k}^r (-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ To prove this we use that for $f_j$ such that $f_j=0$ if $j\equiv 3k \pmod{5}$ there holds $$\sum_{j=1}^{\infty}f_j=\sum_{i=1}^2\sum_{j=0}^{\infty}
(f_{5j+i}+f_{5j+5-i+k}).$$ This, together with the simple to verify identities $$\begin{gathered}
(q^{m+5n},q^{5-5n-m};q^5)_{\infty}=
(q^m,q^{5-m};q^5)_{\infty}(-1)^n q^{-nm-5\binom{n}{2}} \\[2mm]
\frac{(q^{2i-k},q^{5+k-2i},q^5;q^5)_{\infty}}{(q;q)_{\infty}}=
\frac{1}{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}, \;
i,k+1\in\{1,2\} \\[2mm]
\binom{i+r}{2}-(r+1)(5r+2i+4k)/2=-\binom{2r+k+2}{2}, \;
i,k+1\in\{1,2\} \\[2mm]
\sum_{i=1}^2 (-1)^i=0\end{gathered}$$ and the Jacobi triple product identity , yields $$\begin{aligned}
&\text{LHS}\eqref{inffin} \\
&\; = \sum_{i=1}^2 \frac{(-1)^i q^{\binom{i+r}{2}}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}
\biggl(\sum_{j=-\infty}^{-2r-k-2}\!+\sum_{j=0}^{\infty}\,\biggr)
(-1)^j q^{j(5j+10r+2i+4k+5)/2} \\
&\; =\sum_{i=1}^2 \frac{(-1)^{i+r} q^{\binom{i+r}{2}-(r+1)(5r+2i+4k)/2}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}} \\
&\qquad \times\biggl[\,\sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}
-(q^{i+2k},q^{5-i-2k},q^5;q^5)_{\infty}\biggr] \\
&\; =\text{RHS}\eqref{inffin}.\end{aligned}$$ After substituting in we obtain $$\begin{gathered}
\sum_{n=0}^{\infty}\frac{a^{2n} q^{n(n+k)}}{(q;q)_n} \\
=\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^2}\sum_{r=0}^{\infty} \sum_{i=1}^2
\frac{(-1)^{i+1}q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}(a;q)_{r+k+1}} \\
\times \sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ Here the reader is warned that the order of the sums over $r$ and $i$ must be strictly adhered to. Indeed, our earlier considerations about convergence and the fact that is true, guarantee the not so obvious fact that after summing over $i$ the sum over $r$ converges.
Our next step removes any further convergence issues as we now specialize $a=q^{m+1}$ with $m$ a nonnegative integer. The sum over $r$ then terminates, yielding $$\begin{gathered}
\label{am}
\sum_{n=0}^{\infty}\frac{q^{n(n+2m+k+2)}}{(q;q)_n}
=\sum_{i=1}^2 \frac{(-1)^{i+1}}{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}\\
\times \sum_{r=0}^m
\frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}{(q;q)_{m-r}(q;q)_{m+r+k+1}}
\sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ Rewriting the left-hand side using the Garrett–Ismail–Stanton identity gives $$\frac{d_{2m+k+2}}{(q,q^4;q^5)_{\infty}}-
\frac{e_{2m+k+2}}{(q^2,q^3;q^5)_{\infty}}
=(-1)^k q^{\binom{2m+k+2}{2}}\text{RHS}\eqref{am}.$$ Multiplying both sides by $(q;q)_{2m+k+1}$ this is of the form $$\frac{P(q)}{(q^2,q^3;q^5)_{\infty}}=\frac{Q(q)}{(q,q^4;q^5)_{\infty}}$$ with $P(q)$ and $Q(q)$ polynomials. An identity of this type can only be true if $P(q)=Q(q)=0$, and we infer $$\begin{aligned}
e_{2m+k+2}&=q^{\binom{2m+k+2}{2}}
\sum_{r=0}^m \frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}
{(q;q)_{m-r}(q;q)_{m+r+k+1}}\sum_{j=-r-k}^r (-1)^j q^{j(5j+1)/2} \\
d_{2m+k+2}&=q^{\binom{2m+k+2}{2}}
\sum_{r=0}^m \frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}
{(q;q)_{m-r}(q;q)_{m+r+k+1}}\sum_{j=-r-k}^r (-1)^j q^{j(5j+3)/2}\end{aligned}$$ for $m\geq 0$. All that remains is to invert these new representations of the Schur polynomials. This is easily done recalling the Bailey transform [@Andrews84], which states that if $$\label{ba}
\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$ then $$\label{ab}
\alpha_n=(1-aq^{2n})\sum_{r=0}^n\frac{(-1)^{n-r}q^{\binom{n-r}{2}}
(aq;q)_{n+r-1}}{(q;q)_{n-r}}\,\beta_r.$$ For later reference we remark that a pair of sequences $(\alpha,\beta)$ that satisfies (or, equivalently, ) is called a Bailey pair relative to $a$.
Since our expressions for the Schur polynomials take the form with $a=q^{k+1}$, we may invoke to find the identity claimed in Theorem \[thm\].
Proof of Proposition \[prop2\]
------------------------------
Our proof relies on the following lemma.
\[lem1\] For $k\in\{0,1\}$ and $M$ and $n$ integers there holds $$\begin{gathered}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=0}^M (-1)^{M+j+r+1}
q^{\binom{j+r}{2}+\binom{M-l}{2}+\binom{l}{2}+l(r+k+1)} \\
\times q^{-r(M-l)-n(2j-k)}
\frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{M-l}(q;q)_l} \\[2mm]
=\begin{cases}\displaystyle
\frac{q^{m(m+k)}}{(q;q)_{m-n}(q;q)_{m+n+1}} & \text{$M=2m$} \\[4mm]
0 &\text{$M=2m+1$.}
\end{cases}\end{gathered}$$
Here our earlier definition of $(a;q)_n$ is extended to all integers $n$ by $(a;q)_n=(a;q)_{\infty}/(aq^n;q)_{\infty}$. Note in particular that $1/(q;q)_n=0$ for $n<0$.
Given the triple sum on the left, Lemma \[lem1\] perhaps appears complicated and not readily applicable. However, in view of it is in fact quite useful, and if we multiply boths sides by $\alpha_n$ and then sum $n$ over the nonnegative integers we get $$\begin{gathered}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{n=0}^{\infty}\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}
\sum_{l=0}^M \alpha_n (-1)^{M+j+r+1}
q^{\binom{j+r}{2}+\binom{M-l}{2}+\binom{l}{2}+l(r+k+1)} \\
\times q^{-r(M-l)-n(2j-k)}
\frac{(1-q)(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{M-l}(q;q)_l} \\[2mm]
=\begin{cases}
q^{m(m+k)}\beta_m & \text{$M=2m$} \\[2mm]
0 &\text{$M=2m+1$,}
\end{cases}\end{gathered}$$ where $(\alpha,\beta)$ is a Bailey pair relative to $q$.
Next we multiply both sides by $a^M$ and sum over $M$. If on the left we interchange the sums over $M$ and $l$, shift $M\to M+l$ and then sum over $l$ and $M$ using Euler’s $q$-exponential sum [@GR90 Eq. (II.2)] $$\sum_{n=0}^{\infty}\frac{(-1)^n a^n q^{\binom{n}{2}}}{(q;q)_n}=(a;q)_{\infty}$$ this yields $$\begin{gathered}
\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^3}
\sum_{n=0}^{\infty}\sum_{j=1}^{\infty}\alpha_n q^{n(n-2j+k+2)}
\frac{(1-q)(1-q^{(2j-k)(2n+1)})}{(1-q^{2n+1})} \\
\times \sum_{r=0}^{\infty} \frac{(-1)^{j+r+1}
q^{\binom{j+r}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}{(a;q)_{r+k+1}}
=\sum_{n=0}^{\infty} a^{2n} q^{n(n+k)}\beta_n.\end{gathered}$$ We have nearly arrived at . All that is needed is the following Bailey pair relative to $q$ due to Rogers [@Rogers17]: $$\alpha_n=(-1)^n q^{n(3n+1)/2}\frac{(1-q^{2n+1})}{(1-q)}
\quad\text{and}\quad \beta_n=\frac{1}{(q;q)_n}.$$ Substituting this, interchanging the sum over $n$ and $j$ (with the above choice for $\alpha_n$ this may indeed be done) and using the triple product identity gives .
Replacing $M$ by $2m+i$ where $i\in\{0,1\}$, and shifting $l\to l+m+i$ leads to $$\begin{gathered}
\label{lquad}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=-m-i}^m (-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)+l(l+2r+k+i+1)} \\
\times \frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{m-l}(q;q)_{m+l+i}}
=\frac{\delta_{i,0}}{(q;q)_{m-n}(q;q)_{m+n+1}}.\end{gathered}$$ By the $q$-Chu–Vandermonde summation [@GR90 Eq. (II.6)] $$\label{qCV}
\sum_{j=0}^n \frac{(a,q^{-n};q)_j \,q^j}{(q,c;q)_j}=
\frac{(c/a;q)_n}{(c;q)_n}\,a^n$$ this follows from the simpler to prove identity $$\begin{gathered}
\label{llin}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=-m-i}^m (-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)+l(2r+k+1)} \\
\times \frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{m-l}(q;q)_{m+l+i}}
=\frac{\delta_{i,0}q^{m-n}}{(q;q)_{m-n}(q;q)_{m+n+1}}.\end{gathered}$$ Indeed, if we multiply both sides of by $q^{m(m+i)}/(q;q)_{M-m}$, the resulting identity can be summed over $m$ by the $c=0$ instance of (after first replacing $m\to M-m$). On the right we of course only need to do this sum when $i=0$. Replacing $M$ by $m$ then gives . Those familiar with the concept of a Bailey chain [@Andrews84] will have recognized that the reduction of to corresponds to a simplifying (i.e., backwards) iteration along a Bailey chain relative to $q^i$.
Since is of the form with $a=q^i$ we can use to invert. Hence $$\begin{gathered}
\label{inverted}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)-(m+i)(2r+k+1)} \\
\times \bigl(1-q^{(2n+1)(2j-k)}\bigr)\bigl(1-q^{2r+k+1}\bigr)
\bigl(1+q^{(2m+i)(2r+k+1)}\bigr) \\
=\delta_{i,0}(1-q^{2m})(1-q^{2n+1})
\sum_{r=0}^m\frac{(-1)^{m-r}q^{\binom{m-r}{2}+r-n}(q;q)_{r+m-1}}
{(q;q)_{m-r}(q;q)_{r-n}(q;q)_{r+n+1}},\end{gathered}$$ with the convention that $(1-q^{2m})(q^m;q)_{r+m-1}=2$ for $m=r=0$ in accordance with $(1-q^{2m})(q;q)_{m-1}=(1+q^m)(q;q)_m$. The sum over $r$ on the right may be carried out by the $q$-Chu–Vandermonde sum , leading to $$\begin{gathered}
\text{RHS}\eqref{inverted} \\ =
\delta_{i,0}\frac{(-1)^{m+n}(1-q^{2m})(1-q^{2\max\{n,-n-1\}+1})
(q;q)_{m+\max\{n,-n-1\}-1}}{(q;q)_{m-n}(q;q)_{m+n+1}
(q^2;q)_{\max\{n,-n-1\}-m}},\end{gathered}$$ which is nonzero for $n=\pm m$ and $n=\pm m-1$ only. If we also multiply by $q^{i(k+1)/2}$ and note that on the right this may again be dropped, we obtain $$\begin{gathered}
\label{del}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+1}
q^{\binom{j+r}{2}-n(2j-k)-(2m+i)(2r+k+1)/2} \\
\times \bigl(1-q^{(2n+1)(2j-k)}\bigr)\bigl(1-q^{2r+k+1}\bigr)
\bigl(1+q^{(2m+i)(2r+k+1)}\bigr) \\[1mm]
=\delta_{i,0}\bigl(\delta_{m,n}+\delta_{-m,n}-
\delta_{m-1,n}-\delta_{-m-1,n}\bigr).\end{gathered}$$ Since both sides are invariant under the substitution $m\to -m-i$ this must hold for all $m,n\in{\mathbb{Z}}$ and $i,k\in\{0,1\}$.
Next we observe that is a consequence of the stronger result $$\begin{gathered}
\label{symm}
\frac{q^{\binom{n}{2}+(n-m)/2}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)(2j-k)/2-(m-1)(2r+k)/2} \\
\times \bigl(1-q^{n(2j-k)}\bigr)\bigl(1-q^{m(2r+k+1)}\bigr)
=\delta_{m,n}-\delta_{-m,n},\end{gathered}$$ for $m,n\in{\mathbb{Z}}$ and $k\in\{0,1\}$. If we denote the above two identities by $\eqref{del}|_{m,n}$ and $\eqref{symm}|_{m,n}$ and note that $\delta_{2m+i\pm 1,2n+1}=
\delta_{i,0}\delta_{m-(1\mp 1)/2,n}$, then $\eqref{del}|_{m,n}=
\eqref{symm}|_{2m+i+1,2n+1}-\eqref{symm}|_{2m+i-1,2n+1}$.
Before proving let us point out that without loss of generality we may fix $k=0$. For, if we take with $k=1$, replace $r\leftrightarrow j-1$, and multiply the result by $(-1)^{m-n}q^{(m^2-n^2)/2}$ we find $\eqref{symm}|_{m,n;k=1}=\eqref{symm}|_{n,m;k=0}$. Equation for $k=0$ is a linear combination of yet another identity, given by $$\begin{gathered}
\label{symm2}
\frac{q^{\binom{n}{2}}}{(q;q)_{\infty}^3}
\sum_{j=0}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)j-(m-1)r} \\
+\frac{q^{\binom{n}{2}}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)j-(m-1)r+2nj+m(2r+1)}
=\delta_{m,n}.\end{gathered}$$ Here it should be noted that the first sum over $j$ now includes the term $j=0$. It is easily seen that this extra term is cancelled out in the following linear combination, and that $\eqref{symm2}|_{m,n}-q^{-n}\eqref{symm2}|_{m,-n}=
q^{(m-n)/2}\eqref{symm}|_{m,n;k=0}$.
After this string of reductive steps we are finally in a position to carry out a proof. Replacing $m$ by $m+n$ in and changing the summation variable $j\to n-j$ ($j\to j-n)$ in the first (second) double sum gives $$\begin{gathered}
\sum_{j=-\infty}^n\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j-r}{2}-mr}
+\sum_{j=n+1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j+r+1}{2}+m(r+1)} \\
=(q;q)_{\infty}^3 \delta_{m,0}.\end{gathered}$$ In the second term on the left we rewrite the sum over $r$ using $$\label{zero1}
\sum_{r=0}^{\infty}(-1)^r q^{\binom{r+1}{2}+a(r+1)}
=\sum_{r=0}^{\infty}(-1)^r q^{\binom{r+1}{2}-ar}$$ as follows from $$\label{zero2}
\sum_{r=-\infty}^{\infty}(-1)^r q^{\binom{r+1}{2}-ar}=0.$$ (To prove replace $r\to 2a-1-r$.) As a result we are left with $$\label{fin}
\sum_{j=-\infty}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j-r}{2}-mr}=(q;q)_{\infty}^3\delta_{m,0}.$$ Using on the sum over $r$ and negating $j$ yields $\eqref{fin}|_m=q^m \eqref{fin}|_{-m}$, so that we may assume $m\leq 0$ when proving . If $m<0$ the order of the sums may be interchanged. By this completes the proof. If $m=0$ we need $$\sum_{r=0}^{2j-1}(-1)^r q^{\binom{j-r}{2}}=0$$ for $j\geq 0$ (to prove this replace $r\to 2j-1-r$), and Jacobi’s identity [@Jacobi29 §66, (5.)] $$\sum_{i=0}^{\infty}(-1)^i (2i+1) q^{\binom{i+1}{2}}=(q;q)_{\infty}^3.$$ Equipped with these the rest is easy; $$\begin{aligned}
\sum_{j=-\infty}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}q^{\binom{j-r}{2}}
&=\Bigl\{\sum_{j=0}^{\infty}\sum_{r=2j}^{\infty}+
\sum_{j=-\infty}^{-1}\sum_{r=0}^{\infty}\Bigr\}
(-1)^{j+r} q^{\binom{j-r}{2}} \\
&=\Bigl\{\sum_{j=0}^{\infty}
+\sum_{j=1}^{\infty}\Bigr\} \sum_{r=j}^{\infty}
(-1)^r q^{\binom{r+1}{2}}\\
&=\sum_{r=0}^{\infty}(-1)^r (2r+1) q^{\binom{r+1}{2}}
=(q;q)_{\infty}^3. \qquad \qed\end{aligned}$$
Discussion {#secd}
==========
Eqs and versus
----------------
The proof of Theorem \[thm\] as given in the previous section is very lengthy and complicated, and, as a result, not very illuminating. Here we briefly discuss the proofs of as found by Jordan and Andrews as we hold some hope that at least one of these may be generalized to also prove and .
Perhaps simplest is Jordan’s proof [@Jordan84]. Denoting the right side of by $f_{n;k}$ and the summand on the right of by $f_{n,r;k}$, it is not difficult to show that the functional equation $$(1-x q^{n+2})K_{n+1}(x)=K_n(x)-x^2 q^{n+4}K_n(x q^2)$$ satisfied by the Szegö polynomials implies the recurrence $$\label{frec}
\sum_{r=m+1}^n (f_{n,r;k}-f_{n-1,r-1;k})=
-\frac{1-q^{m-n}}{1-q^{n+k}}f_{n,m;k}.$$ By the $m=0$ instance hereof it is found that $$\begin{aligned}
f_{n;k}-f_{n-1;k}&=f_{n,0;k}+\sum_{r=1}^n(f_{n,r;k}-f_{n-1,r-1;k}) \\
&=q^{-n}\frac{1-q^{2n+k}}{1-q^{n+k}}f_{n,0;k} \\[2mm]
&=(-1)^n q^{n(5n+2k+1)/2}+(-1)^{n+k}q^{(n+k)(5(n+k)-2k-1)/2},\end{aligned}$$ from which follows by induction. Unfortunately, at present we have been unable to find an analogue of for the summands on the right of Theorem \[thm\].
Andrews’ proof of relies on the following multiple series generalization of Watson’s $q$-Whipple transform [@Andrews75 Thm. 4; $k=3$]: $$\begin{gathered}
\label{k3}
{_{10}}W_9(a;b,c,d,e,f,g,q^{-n};q,a^3 q^{n+3}/bcdefg;q,q) \\
=\frac{(aq,aq/fg;q)_n}{(aq/f,aq/g;q)_n}
\sum_{j=0}^n \sum_{k=0}^{n-j}
\frac{(aq/bc,d,e;q)_j}{(q,aq/b,aq/c;q)_j}
\frac{(aq/de;q)_k}{(q;q)_k} \\ \times
\frac{(f,g,q^{-n};q)_{j+k}}{(aq/d,aq/e,fgq^{-n}/a)_{j+k}}
\Bigl(\frac{aq^2}{de}\Bigr)^j q^k.\end{gathered}$$ Here and in the following we employ standard notation for basic hypergeometric series, see e.g., [@GR90]. Taking $b=aq^{n+1}$ and letting $c,d,e,f,g$ tend to infinity yields with $k=0$ if $a=1$ ($k=1$ if $a=q$).
Now if we apply Sears’ $_{4}\phi_3$ transformation [@GR90 Eq. (III.15)] to Watson’s Watson’s $q$-Whipple transform [@GR90 Eq. (III.18)] we readily obtain $$\begin{gathered}
\label{WS}
{_8W_7}(a;b,c,d,e,q^{-n};q,a^2 q^{n+2}/bcde;q,q) \\
=\frac{(aq,b,a^2 q^2/bcde;q)_n}{(aq/c,aq/d,aq/e;q)_n}
{_4\phi_3}\Bigl[\genfrac{}{}{0pt}{}
{aq/bc,aq/bd,aq/be,q^{-n}}{aq/b,a^2q^2/bcde,q^{1-n}/b};q,q\Bigr].\end{gathered}$$ Taking $b=aq^{n+1}$ and letting $c,d,e,f,g$ this simplifies to $$\label{WSlim}
\sum_{j=0}^n \frac{1-aq^{2j}}{1-a}
\frac{(a;q)_j(-1)^j q^{3\binom{j}{2}}(aq)^j}
{(q;q)_j}=(aq;q)_{2n}\sum_{j=0}^n \frac{q^j}{(q,q^{-2n}/a;q)_j}.$$ Chosing $a=q^k$ for $k=0,1$ and making the variable change $j\to n-r$ on the right we obtain the following polynomial analogue of Euler’s identity $$\label{Eulerpol}
\sum_{j=-n-k}^n(-1)^j q^{j(3j+1)/2}=\sum_{r=0}^n (-1)^{n-r}
q^{(n-r)(3n+r+2k+3)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}.$$ (Incidentally, this identity is very similar and can easily be transformed into a polynomial version of Euler’s identity due to Shanks [@Shanks51].) Given the similarity between and the identities and , and given Andrews’ proof of by means of it seems very natural to ask for a proof of Theorem \[thm\] by means of a multiple series generalization of the transformation . If we take with $k=0$ and with $k=1$ and replace $r\to n-j$ in the sums on the right we find that the resulting identities are the $a=1$ and $a=q$ instances of $$\begin{gathered}
\sum_{j=0}^n \frac{1-aq^{2j}}{1-a}
\frac{(a;q)_j(-1)^j q^{5\binom{j}{2}}(aq)^{2j}}{(q;q)_j} \\
=(aq;q)_{2n}\sum_{j=0}^n\sum_{k=0}^{n-j}
\frac{(-1)^k a^{j+k} q^{j^2-\binom{2j+k}{2}+(2n+1)(j+k)}
(q^{-2n-1};q)_{2j+2k}}
{(q,q^{-2n}/a;q)_j(q;q)_k(q^{-2n-1};q)_{2j+k}}\end{gathered}$$ This is to be compared with . Despite numerous attempts we failed to extend this to a multiple series transformation similar to and generalizing . Of course one can try to prove the above by equating coefficients of $a^m$, but the resulting identity $$\begin{gathered}
\sum_{j=0}^n \frac{(-1)^j q^{\binom{j}{2}+j(4j-2m+1)}}{(q;q)_j}
\biggl({\genfrac{[}{]}{0pt}{}{j}{m-2j}}+q^{4j-m+1}{\genfrac{[}{]}{0pt}{}{j}{m-2j-1}}\biggr) \\
=\sum_{r=0}^n\sum_{s=0}^{n-r}
\frac{(-1)^{r+s}q^{\binom{r+s}{2}+r(4n-2m+4)+s(s+r-m+1)}}{(q;q)_r}
\\ \times
{\genfrac{[}{]}{0pt}{}{2n+1-r}{m-2r-s}}{\genfrac{[}{]}{0pt}{}{2n-2r-s+1}{s}}\end{gathered}$$ for $n\geq 0$ and $0\leq m\leq 3n+1$ is not particularly simple (and would only prove half of Theorem \[thm\]).
Some combinatorics related to Theorem \[thm\]
---------------------------------------------
In order to discuss some of the combinatorics of Theorem \[thm\] we need to review several standard results from partition theory [@Andrews76].
Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ be a partition, defined as a weakly decreasing sequence of positive integers $\lambda_j$ (the parts of $\lambda$). The weight $|\lambda|$ of $\lambda$ is given by the sum of its parts. We say that $\lambda$ is a partition of $l$ if $|\lambda|=l$. The Ferrers graph of $\lambda$ is the graph obtained by drawing $r$ left-aligned rows of dots with the $j$th row containing $\lambda_j$ dots. The conjugate $\lambda'$ of $\lambda$ is obtained by transposing its Ferrers graph. The number $d(\lambda)$ is the number of rows in the maximal square of dots of the Ferrers graph of $\lambda$. An alternative way to represent a partition $\lambda$ is as a two-rowed matrix of $d(\lambda)=d$ columns $\bigl(\begin{smallmatrix} t_1 t_2 \dots t_d \\ b_1 b_2 \dots b_d
\end{smallmatrix}\bigr)$, where $t_j=\lambda_j-j$ and $b_j=\lambda_j'-j$, so that, in particular, $t_j>t_{j+1}$ and $b_j>b_{j+1}$. Conversely, any such matrix (also called Frobenius symbol) corresponds to the unique partition $\lambda$ by $\lambda_j=t_j+b_j+1$. We will in the following identify the standard and Frobenius notations for partitions. Note that $|\lambda|=d+\sum_{j=1}^d(t_j+b_j)$. The rank of a partition $\lambda$ is defined as its largest part minus its number of parts, i.e., as $\lambda_1-\lambda'_1=t_1-b_1$. More generally, the $i$th successive rank of $\lambda$ is given by $t_i-b_i$, and $r(\lambda)=(t_1-b_1,t_2-b_2,\dots,t_d-b_d)$ denotes the sequence of successive ranks of $\lambda$. For example, if $\lambda=(7,7,5,3,3,1,1,1)$, then $|\lambda|=28$, $\lambda'=(8,5,5,3,3,2,2)$, $d(\lambda)=3$, $\lambda=\bigl(\begin{smallmatrix} 6 5 2 \\ 7 3 2 \end{smallmatrix}\bigr)$, $\lambda'=\bigl(\begin{smallmatrix} 7 3 2 \\ 6 5 2 \end{smallmatrix}\bigr)$, and $r(\lambda)=(-1,2,0)$.
Now let $b_2(l,n)$ denote the set of all partitions of $l$, with largest part at most $n-2$ and difference between parts at least $2$, and let $B_2(l,n)$ be its cardinality. Then $e_n=\sum_{l\geq 0}B_2(l,n) q^l$. Given a partition $\lambda\in b_2(l,n)$ with exactly $r$ parts, one can form a new partition $\mu$ as follows [@Andrews76 §9.3]: $\mu=\bigl(\begin{smallmatrix} s_1,\dots,s_r \\ c_1,\dots,c_r
\end{smallmatrix}\bigr)$, where $s_j=\lfloor \lambda_j/2 \rfloor$ and $c_j=\lfloor (\lambda_j-1)/2 \rfloor$. Because of the difference-$2$ condition one indeed has $s_j>s_{j+1}$ and $c_j>c_{j+1}$. Since (for $n\in{\mathbb{Z}}$) $\lfloor n/2 \rfloor +\lfloor (n-1)/2 \rfloor=n-1$ one finds that $|\mu|=r+\sum_{j=1}^r (s_j+c_j)=|\lambda|=l$. Furthermore, the restriction that $\lambda_j-\lambda_{j+1}\geq 2$ translates into the fact that the successive ranks of $\mu$ must take the values $0$ and $1$ only. Finally the restriction that $\lambda_1\leq n-2$ implies that $s_1+c_1+1\leq n-2$. Since $s_1-c_1\in\{0,1\}$ this is equivalent to requiring that $\mu_1\leq\lfloor n/2\rfloor$ and $\mu'_1\leq\lfloor (n-1)/2\rfloor$. If we denote the set of all partitions of $l$ with successive ranks in $\{0,1\}$, largest part not exceeding $\lfloor n/2 \rfloor$ and number of parts not exceeding $\lfloor (n-1)/2\rfloor$ by $q_2(l,n)$ (with cardinality $Q_2(l,n)$) then clearly each partition $\mu\in q_2(l,n)$ can also be mapped back onto a partition in $b_2(l,n)$. Specifically, if $\mu\in q_2(l,n)$ has Frobenius symbol $\bigl(\begin{smallmatrix} s_1,\dots,s_r \\ c_1,\dots,c_r
\end{smallmatrix}\bigr)$, then $\lambda=(s_1+c_1+1,\dots,s_r+c_r+1)\in b_2(l,n)$, since $\lambda_j-\lambda_{j+1}=s_j+c_j-s_{j+1}-c_{j+1}\geq 2$ and $\lambda_1=s_1+c_1+1=\mu_1+\mu'_1-1\leq n-2$. Hence $Q_2(l,n)=B_2(l,n)$ and $e_n=\sum_{l\geq 0}Q_2(l,n) q^l$. For example, $\cup_{l\geq 0}q_2(l,n)=
\{\emptyset,(1),(2),(2,1),(3,1),(2,2),(3,2),(3,3)\}$ so that $e_6=1+q+q^2+q^3+2q^4+q^5+q^6$.
The above discussion can be repeated for the Schur polynomial $d_n$ and we define $b_1(l,n)$ as the subset of $b_2(l,n)$ obtained by removing all partitions which have a part equal to $1$. Hence $d_n=\sum_{l\geq 0}B_1(l,n)q^l$. If we also define $q_1(m,n)$ as the set of partitions with successive ranks in $\{1,2\}$, largest part not exceeding $\lfloor (n+1)/2 \rfloor$ and number of parts not exceeding $\lfloor (n-2)/2\rfloor$, then it is not hard to show that $Q_1(l,n)=B_1(l,n)$ so that $d_n=\sum_{l\geq 0} Q_1(l,n) q^l$. For example, $\cup_{l\geq 0} q_1(l,n)=\{\emptyset,(2),(3),(3,1),(3,3)\}$ so that $d_6=1+q^2+q^3+q^4+q^6$.
So far, we have given a combinatorial interpretation of the Schur polynomials $e_n$ and $d_n$ in terms of partitions with restrictions on their size and successive ranks. Next we will discuss the combinatorial interpretation of the partial theta sum $\sum_{j=-n+k}^n (-1)^j q^{j(5j-2i+5)/2}$ in terms of successive ranks.
First we recall some further known properties of $Q_i(l,n)$ [@Andrews72; @Andrews76]. Let $\lambda$ be a partition and $r(\lambda)$ its sequence of successive ranks. The length of the largest subsequence $r'$ of $r(\lambda)$ such that the odd (even) elements of $r'$ are at least $4-i$ and the even (odd) elements of $r'$ are at most $1-i$, is called the $(2,i)$-positive ($(2,i)$-negative) oscillation of $\lambda$. The number of partitions of $l$ that have at most $b$ parts, largest part not exceeding $a$ and $(2,i)$-positive ($(2,i)$-negative) oscillation at least $j$ is denoted by $p_i(a,b;j;l)$ ($m_i(a,b;j;l)$). By inclusion-exclusion arguments it then follows that $$Q_i(l,n)=\sum_{j=0}^{\infty}(-1)^j p_i(\bar{a},\bar{b};j,l)
+\sum_{j=1}^{\infty}(-1)^j m_i(\bar{a},\bar{b};j,l),$$ with $\bar{a}=\bar{a}(n,i)=\lfloor (n-i+2)/2\rfloor$ and $\bar{b}=\bar{b}(n,i)=\lfloor (n+i-3)/2\rfloor$. Furthermore, $$q^{j(5j-2i+5)/2}{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor \frac{n+i-5j-3}{2}\rfloor}}=
\begin{cases}
\sum_{l=0}^{\infty} p_i(\bar{a},\bar{b};-j,l)q^l
&\text{$j\leq 0$, $j$ even} \\[1mm]
\sum_{l=0}^{\infty} m_i(\bar{a},\bar{b};-j,l)q^l
&\text{$j\leq 0$, $j$ odd} \\[1mm]
\sum_{l=0}^{\infty} p_i(\bar{a},\bar{b};j,l)q^l
&\text{$j\geq 0$, $j$ odd} \\[1mm]
\sum_{l=0}^{\infty} m_i(\bar{a},\bar{b};j,l)q^l
&\text{$j\geq 0$, $j$ even,}
\end{cases}$$ from which immediately follows. But now it is also clear what our partial theta sums represent. If we denote by $\lambda^{\pm}_{i,j}$ the (unique) partition of minimal weight that has a positive/negative $(2,i)$-oscillation $j$ and by $M_i$ the set of all such minimal partitions, i.e., $M_i=\{\lambda^{\sigma}_{i,j}\}_{j\geq 0;\sigma\in\{0,1\}}$, then (for $k\in\{0,1\}$ and $i\in\{1,2\}$) $$\sum_{j=-n-k}^n (-1)^j q^{j(5j-2i+5)/2}=
\sum_{\substack{\lambda\in M_i \\
\lambda_1\leq \lfloor (5n+2ki-2i+5)/2\rfloor \\
\lambda'_1\leq \lfloor (5n+2ki)/2\rfloor}}
(-1)^{d(\lambda)} q^{|\lambda|}.$$ One can in fact easily find the partition $\lambda^{\pm}_{i,j}$. For example, using the Frobenius notation it follows immediately that for $j$ even $$\lambda^{+}_{i,j}=
\begin{pmatrix} 5j/2-1,&5j/2-5,&\dots,& 9,& 5, &4, & 0 \\
5j/2+i-5,&5j/2+i-6,& \dots,& i+5,& i+4,& i,& i-1\end{pmatrix}.$$ When converted into standard notation this gives $$\lambda^{+}_{i,j}=
(5j/2,(5j/2-3)^2,\dots,(j+6)^2,(j+3)^2,j^i,(j-2)^3,\dots,4^3,2^3)$$ where $p^f$ stands for $f$ parts of size $p$. Calculating the weight of this partition gives $$|\lambda^{+}_{i,j}|=
5j/2+2\sum_{k=1}^{j/2-1}(j+3k)+ij+3\sum_{k=1}^{j/2-1}(2k)=j(5j+2i-5)/2,
\quad \text{$j$ even}$$ as it should. Similarly one can use the Frobenius notation to find $$\lambda^{-}_{i,j}=
((5j-3)/2)^2,\dots,(j+6)^2,(j+3)^2,j^i,(j-2)^3,\dots,3^3,1^3),$$ $(\lambda^{+}_{i,j})'=\lambda^{-}_{5-i,j}$ both for $j$ odd, and $(\lambda^{-}_{i,j})'=\lambda^{+}_{5-i,j}$ for $j$ even, and thus $|\lambda^{\pm}_{i,j}|=j(5j\mp 2i \pm 5)/2$ for odd $j$ and $|\lambda^{-}_{i,j}|=j(5j-2i+5)/2$ for even $j$.
Summarizing, we have the following remarkable situation. The Schur polynomials, which are the generating functions of certain size and successive rank restricted partitions, can be expressed as an alternating sum over the generating functions of partitions with certain restrictions on their $(2,i)$-oscillations. This well-known fact [@Andrews72; @Andrews76] provides a combinatorial explanation of Schur’s result . But now we see that according to the Theorem \[thm\] there is another side to the coin; the alternating sum over the generating function of a very special subset of partitions with certain restrictions on their $(2,i)$-oscillations can in its turn be expressed as a weighted sum over Schur polynomials. However, by no means is this an example of a trivial (or nontrivial but known) inversion result. Indeed, naively one might think that if we substitute in Theorem \[thm\] to get $$\begin{gathered}
\sum_{j=-n-k}^n(-1)^j q^{j(5j-2i+5)/2}=
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j-2i+5)/2} \\
\times \sum_{r=0}^n (-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}
{{\textstyle\genfrac{[}{]}{0pt}{}{2r+k+1}{\lfloor{(2r+i+k-5j-1)/2}\rfloor}}},\end{gathered}$$ that this is just a consequence of the second line being $\chi(-n-k\leq j\leq n)$ with $\chi(\text{true})=1$ and $\chi(\text{false})=0$. However, it is readily checked that this is only correct when $n=k=0$. It thus seems an extremely challenging problem to find a combinatorial proof of Theorem \[thm\], especially since our analytic proof provides so little insight as to why this theorem is true.
To conclude we remark that the previous discussion has a representation theoretic counterpart. As is well-known, $$\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j-2i+5)/2}$$ is the (normalized) character of the $c=-22/5$ Virasoro algebra corresponding to the highest weight vector $v_{h_i}$ of weight $h_i=(1-i)/5$. According to the Feigin–Fuchs construction [@FF84] the above character can be constructed from the Verma module $V(c,h_i)$ by eliminating submodules generated by singular or null vectors. Because of the embedded structure of these submodules this leads to an inclusion-exclusion type of sum. Specifically, the character corresponding to the submodule $V(c,h_i')$ with singular vector of weight $h_i'$ is given by $q^{h_i'-h_i}/(q;q)_{\infty}$, with the set of weights of singular vectors (including $v_{h_i}$) given by $h_i'=h_i+j(j-2i+5)/2$ for $j\in{\mathbb{Z}}$. Therefore, if we denote by $V_s(c,h_i;N)$ the set comprising of the $N$ singular vectors of $V(c,h_i)$ of smallest weight, and if we denote by $d(v)+1$ the number of (sub)modules $V(c,h_i')$ that contain the singular vector $v$ (so that $d(v)=0$ iff $v=v_{h_i}$), then $$\sum_{j=-n-k}^n (-1)^j q^{j(5j-2i+5)/2}=
\sum_{v\in V_s(c,h_i;2n+k+1)} (-1)^{d(v)} q^{|v|-h_i},$$ where $|v|$ is the weight of $v$, $c=-22/5$ and $h_i=(1-i)/5$. Again it is a challenge to explain Theorem \[thm\] from the above representation theoretic point of view.
Note added {#note-added .unnumbered}
----------
Robin Chapman has informed me that he has found a combinatorial proof of Theorem \[thm\] using Schur’s involution.
[99]{}
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[^1]: Work supported by the Australian Research Council
|
---
abstract: 'By combining test-particle and self-consistent techniques, we have developed a method to rapidly explore the parameter space of galactic encounters. Our method, implemented in an interactive graphics program[^1], can be used to find the parameters required to reproduce the observed morphology and kinematics of interacting disk galaxies. We test this system on an artificial data-set of $36$ equal-mass merging encounters, and show that it is usually possible to reproduce the morphology and kinematics of these encounters and that a good match strongly constrains the encounter parameters.'
author:
- 'Joshua E. Barnes'
- 'John E. Hibbard'
title: 'Identikit 1: A Modeling Tool for Interacting Disk Galaxies'
---
Introduction
============
The diverse morphological and kinematic features of interacting disk galaxies have a simple dynamical explanation: galactic bridges and tails [@TT72], rings [@LT76; @TS77] and related structures result when ordinary galactic disks experience strong tides in close encounters. Tides also cause interacting galaxies to merge by inexorably transferring energy and momentum from relative motion to internal degrees of freedom [@T77; @W78; @B88]. With such a straightforward physical basis, one might expect that dynamical modeling of interacting galaxies would be relatively easy. However, it’s very time-consuming to explore the large parameter space required to describe a galaxy collision and find a good match to the kinematics and morphology of a specific system. In addition, it’s never been entirely clear that a good match yields a unique or physically meaningful model.
In this paper we develop and test an efficient methodology to model the observable morphology and kinematics of pairs of interacting disk galaxies. Empirical tests show that the resulting models can be used to make strong inferences about the systems they match. While we do not consider minor mergers and interactions in this paper, our methodology can easily be extended to treat such encounters.
At first glance, it seems all too easy to model interacting galaxies – and impossible to do so with any degree of confidence. The dynamical state of a galactic collision is described by a phase-space [*distribution function*]{}, $f({\mathbf{r}},{\mathbf{v}})$, which gives the mass density at position ${\mathbf{r}}$ and velocity ${\mathbf{v}}$. In contrast, observations of a specific component $c$ (e.g., neutral hydrogen) yield a [*data cube*]{} $F_c(X,Y,V)$, which represents the distribution of that component at each point $(X,Y)$ on the plane of the sky as a function of line-of-sight velocity $V$. Since $f({\mathbf{r}},{\mathbf{v}})$ depends on [*six*]{} variables, while $F_c(X,Y,V)$ depends on only [*three*]{}, it appears that observations do not provide enough information. Put simply, there are an infinite number of different 6-D distribution functions consistent with any given 3-D data cube.
On further reflection the problem is not quite as hopeless as it appears. This is because a typical galaxy merger begin with a tidal encounter between two normal, fairly symmetric spirals. Galaxies are scrambled as they merge, but the stars and dark matter which constitute most of their mass evolve collisionlessly. The fundamental dynamical equation, $${\partial f \over \partial t} +
{\mathbf{v}} \cdot {\partial f \over \partial {\mathbf{r}}} -
{\partial \Phi \over \partial {\mathbf{r}}} \cdot
{\partial f \over \partial {\mathbf{v}}} = 0,$$ where $\Phi$ is the gravitational potential, is fully reversible [e.g. @vAvG77]; thus in some sense the original galaxies are still there, imposing a hidden symmetry on the dynamical state of a merging system. So we need not consider [*all*]{} possible distribution functions consistent with a given data cube; only a very small subset of these functions can possibly result from an encounter between two normal disk galaxies.
In practice, mergers are modeled by guessing initial conditions, numerically simulating the ensuing collision, and comparing the result to the morphology and kinematics of the system one wants to model. If the model fails to match the observations, go back and guess again until the results are satisfactory. Most of the guess-work focuses on selecting the disk orientations, typically specified by angles $(i_1,\omega_1)$ and $(i_2,\omega_2)$ for disks 1 and 2 [@TT72]; also needed are the eccentricity $e$ and pericentric separation $p$ of the initial orbit, as well as the galactic mass ratio $\mu$. So [*seven*]{} parameters are needed to specify the initial conditions for an encounter of two axisymmetric disk galaxies – not counting the parameters used to specify their internal structures[^2].
Once a simulation has been run, one must select another [*nine*]{} parameters when comparing the results to observational data: a time $t$ since pericenter, a viewing direction given, for example, by angles $(\theta_{\rm X},\theta_{\rm Y},\theta_{\rm Z})$, scale factors $\mathscr{L}$ and $\mathcal{V}$ for length and velocity[^3], and a center-of-mass position on the plane of the sky $(X_{\rm cm},Y_{\rm cm})$ and velocity $V_{\rm cm}$. All told, a minimum of [*sixteen*]{} parameters are needed to completely specify the initial conditions, time, point of view, and scale of a merger model. This plethora of parameters has long posed a challenge for systematic surveys of galactic collisions [@TT72; @FS82; @WS92; @HKBB93; @B98; @NB03]. The problem we address here is slightly different – instead of trying to survey the entire parameter space, we want to navigate toward a solution matching the morphology and kinematics of a given interacting system. Intuition and prior experience can guide this process by narrowing the range of parameter space explored. Nonetheless, given the size and complexity of this parameter space, it’s not surprising that many simulations must be run to attempt a match, or that detailed models of galactic collisions are not easy to produce.
“Identikit” Methodology {#sec:methodology}
=======================
While self-consistent simulations are useful to finalize dynamical models of tidally interacting galaxies, here we simulate galactic disks with test particles. Test particles have a long history [@PS61; @TT72; @CFWG80; @HQ87; @WS92] and nicely reproduce features such as bridges, tails, and shells which develop with little direct influence from self-gravity. To include orbital decay, which is crucial in modeling the more advanced stages of galaxy encounters and mergers, the test particles may be used to estimate the drag on the central masses [e.g., @TT72; @B84; @QG86]. However, orbit decay is largely driven by tidal interactions of galaxy [*halos*]{} [e.g., @T77; @W78; @B88], and it’s relatively easy to compute the self-consistent interaction of two halos using N-body simulations. Our initial approach was therefore to represent the mass of each galaxy with a spherical distribution of massive particles; in each of these spheres, we embedded [*multiple*]{} disks of test particles, and decided which disk to display [*after*]{} running the simulations[^4].
Building on this idea, and taking advantage of the faster processors now available, we have replaced these discrete collections of disks with spherical swarms of test particles moving on circular orbits – in effect, populating each galaxy model with [*all*]{} possible disks.
The modeling procedure begins by selecting mass models for the two galaxies, thereby fixing the mass ratio $\mu$. Our models include a bulge, a disk, and a halo; these components have cumulative mass profiles $m_{\rm b}(r)$, $m_{\rm d}(r)$, and $m_{\rm h}(r)$, respectively, where $m_c(r)$ is the total mass in component $c$ within radius $r$. For each galaxy, we compute the total mass profile $$m(r) = m_{\rm b}(r) + m_{\rm d}(r) + m_{\rm h}(r) \, ,
\label{eq:total-mass-prof}$$ calculate the corresponding isotropic distribution function using Eddington’s ([-@E16]) formula [e.g., @BT87 p. 236], and construct a spherical N-body realization of this profile using $N_{\rm
sphr}$ equal-mass particles. Details of this procedure are given in Appendix A.
Each of these spherical N-body realizations is then loaded with $N_{\rm test}$ test particles moving on circular orbits. We pick the orientation of the orbit of test particle $i$ by randomly drawing its normalized angular momentum $\hat{{\mathbf{s}}}_i$ from a uniform distribution on the unit sphere $\mathbf{S}^2$. The radial distribution of the test particles may be chosen at will. The simplest choice is to use a radial distribution following the cumulative profile of the disk, $m_{\rm d}(r)$; with this approach, the test particles representing a disk with a normalized spin vector $\hat{{\mathbf{s}}}_{\rm d}$ are those with $$1 - \hat{{\mathbf{s}}}_{\rm d} \cdot \hat{{\mathbf{s}}}_i \le \sigma \, ,
\label{eq:select_unbiased_disk}$$ where $\sigma \ll 1$ is a tolerance parameter proportional to the number of particles selected. In practice, however, this places many disk particles at small radii where they are largely immune to tides. The sampling at large radii can be improved by radially biasing the test particle distribution. We do this by multiplying the local disk particle density, $\rho_{\rm d}(r) = (4 \pi r^2)^{-1} d m_{\rm d}/d
r$, by a factor of $r^2$, and replacing (\[eq:select\_unbiased\_disk\]) with $$1 - \hat{{\mathbf{s}}}_{\rm d} \cdot \hat{{\mathbf{s}}}_i \le
\sigma \, / \, \mathrm{max}(q_i, r_{\rm min})^2 \, .
\label{eq:select_biased_disk}$$ Here $q_i$ is the [*initial*]{} orbital radius of particle $i$, and $r_{\rm min}$ is a parameter which keeps (\[eq:select\_biased\_disk\]) from diverging for small $q_i$. Particles selected using (\[eq:select\_biased\_disk\]) follow the original disk distribution down to radius $r_{\rm min}$, at smaller radii the disk is undersampled, but this has little effect if $r_{\rm min}$ is small.
Finally, two such configurations are placed on a relative orbit with a given pericentric separation $p$ and eccentricity $e$ and followed until they merge; we save particle positions and velocities every few time-steps, creating a data-base of several hundred frames tracing the system’s history from start to finish. This data can then be used to approximate [*any*]{} encounter with the chosen $\mu$, $p$, and $e$.
{width="85.00000%"}
Identikit software includes interactive routines allowing the user to select the disk orientations, viewing direction, scale factors, and centroid positions; the resulting test-particle coordinates are instantly projected on the $(X,Y)$, $(X,V)$, $(V,Y)$, and $(X,Z)$ planes. The user can also step forward or backward in time, switch between data-bases created using different values for $p$, $\mu$, or $e$, and vary the tolerance parameter $\sigma$. In addition, an observational data cube $F(X,Y,V)$ for a specific system to be matched may be projected onto the $(X,Y)$, $(X,V)$, and $(V,Y)$ planes, typically using grey-scale or contour images. A lap-top computer can easily store the necessary data and supply the modest processing power required when searching for a match. Fig. \[idkit\_match\] presents an example; the encounter and viewing parameters have been adjusted so the test particles (points) closely match the data cube (grey-scale).
Experimental Validation
=======================
The Identikit system is designed with the primary goal of analyzing observations. However, it’s not straightforward to perform empirical tests using observational data. For one thing, much of the available data is rather heterogeneous; data-sets with uniformly high resolution and signal-to-noise ratio are not easy to obtain. For another, we don’t know the true initial conditions and viewing parameters for more than a handful of the mergers that have been observed, so we don’t have any simple way to validate the results of our modeling. To determine if Identikit can actually reconstruct galactic encounters from the information contained in data-cubes, we tested it on an artificial data set of $36$ self-consistent disk galaxy merger simulations with random orientations, times since first passage, viewing directions, and scale factors.
Artificial Merger Data
----------------------
Our disk galaxy model has a spherical bulge [@H90a] containing $5$% of the mass, an exponential/isothermal disk [@dV59a; @dV59b; @F70; @vdKS81] containing another $15$%, and a spherical dark halo [@NFW96] containing the remaining $80$%. The density profiles for these components are $$\begin{array}{@{}l}
\rho_{\rm b}(r) \propto
r^{-1} (r + a_{\rm b})^{-3} \,, \\
\rho_{\rm d}(q,z) \propto
e^{- q / a_{\rm d}} \, \mathrm{sech}^2(z/z_{\rm d}) \,, \\
\rho_{\rm h}(r) \propto
r^{-1} (r + a_{\rm h})^{-2} \,,
\end{array}$$ where $a_{\rm b}$ is the scale length of the bulge, $q = \sqrt{x^2 +
y^2}$ is the cylindrical radius, $a_{\rm d}$ is the scale length of the disk, $z_{\rm d}$ is the scale height of the disk, and $a_{\rm h}$ is the scale length of the halo. Each galaxy was realized using a total of $N = 131072$ particles. The simulations used natural units with Newton’s constant $G = 1$. In these units, the galaxy model has total mass $m = 1.25$ and half-mass radius $r_{\rm med} \simeq 0.532$. The disk’s scale length $a_{\rm d} = 1/12$, and the median circular velocity of the disk material is $v_{\rm med} \simeq 1.23$; at a radius of $3 a_{\rm d}$ the orbital period is $t_{\rm orb} \simeq
1.23$.
We restricted our artificial data set to equal-mass ($\mu = 1$) encounters with parabolic initial orbits ($e = 1$); for each orbit, the pericentric separation $p$ was drawn from a uniform distribution in the range $[0.05,0.5] = [0.6,6] a_{\rm d}$. The instant when this idealized two-body orbit reaches pericenter defines $t = 0$; times $t
< 0$ are before pericenter, while times $t > 0$ are after pericenter. We adopt a coordinate system in which the orbital angular momentum vector is parallel to the $\hat{{\mathbf{z}}}$ axis. The normalized spin vector $\hat{{\mathbf{s}}}$ of each disk was chosen from a uniform distribution on the unit sphere $\mathbf{S}^2$; in practice, the inclination $i$ was chosen by drawing $\cos(i) = \hat{{\mathbf{s}}} \cdot
\hat{{\mathbf{z}}}$ from a uniform distribution in the range $[-1,1]$, and the argument $\omega$ was chosen from a uniform distribution in the range $[0^\circ,360^\circ]$. Further details on the galaxy models and the merger simulations are given in Appendix B.
For each of the $36$ merger simulations we chose a random time between first and second pericenter, rescaled the system by random factors in length and velocity, and “observed” it from a random direction. We first determined the relative orbit of each pair of galaxies, using the most tightly-bound $2048$ particles in each bulge to measure galactic positions. Let $t_1 \simeq 0$ and $t_2$ be times of first and second pericenter, respectively; the random time $t$ was drawn from a uniform distribution in the range $[t_1, t_2]$ and rounded down to the nearest available output time.
Next, we selected scale factors $\mathscr{L}$ and $\mathcal{V}$ for length and velocity, respectively. These were chosen so that the galaxy models obey a mass-radius-velocity relation of the form $M
\propto R^2 \propto V^4$ with a small amount of scatter [c.f. @TF77]. Let $\xi$ be drawn from a uniform distribution in the range $[-0.5,0.5]$, and $g_1$ and $g_2$ be drawn from a Gaussian distribution with zero mean and unit dispersion; then $$\mathscr{L} = 10^{\xi/2} \, 10^{0.05 g_1} \, ,
\qquad
\mathcal{V} = 10^{\xi/4} \, 10^{0.05 g_2} \, .$$
Finally, we chose a random viewing direction $\widehat{{\mathbf{Z}}}$ from a uniform distribution on the unit sphere $\mathbf{S}^2$. We drew a second vector $\widehat{{\mathbf{X}}}_0$ from the same distribution, and set $\widehat{{\mathbf{X}}} = \widehat{{\mathbf{X}}}_0 - \widehat{{\mathbf{Z}}}
(\widehat{{\mathbf{X}}}_0 \cdot \widehat{{\mathbf{Z}}})$ and $\widehat{{\mathbf{Y}}}
= \widehat{{\mathbf{Z}}} \times \widehat{{\mathbf{X}}}$. These vectors and scale factors were used to map the position ${\mathbf{r}}_i$ and velocity ${\mathbf{v}}_i$ of each particle $i$ to data-cube coordinates: $$X_i = \mathscr{L} \, \widehat{{\mathbf{X}}} \cdot {\mathbf{r}}_i \, , \qquad
Y_i = \mathscr{L} \, \widehat{{\mathbf{Y}}} \cdot {\mathbf{r}}_i \, , \qquad
V_i = \mathcal{V} \, \widehat{{\mathbf{Z}}} \cdot {\mathbf{v}}_i \, .$$
Particles from the disks of the two galaxies, transformed to $(X_i,
Y_i, V_i)$ coordinates, provide an N-body representation of a data cube $F_{\rm d}(X,Y,V)$ for the disk material. Such data is roughly comparable to the neutral hydrogen data-cubes $F_{\rm HI}(\alpha,
\delta, V_{\rm los})$ available for many interacting galaxies [e.g., @HvGRS01]. The simulated data has better resolution than most observational data-sets and is free of noise and interferometric artifacts; moreover, our simulations used collisionless particles instead of neutral gas. We could have run random mergers with gas to improve the correspondence between the simulations and real observational data, but the computing time required for a large suite of simulations with gas is non-trivial. Fortunately, collisionless simulations do a good job of reproducing the tidal features commonly detected in H[I]{} since the latter usually evolve ballistically once tidally extracted from their parent galaxies.
To present the simulated data in the form required for Identikit matching, we projected the disk particles for each of our $36$ mergers on the $(X,Y)$, $(X,V)$, and $(V,Y)$ planes; gridded particle distributions were lightly smoothed to produce grey-scale images. We also used tightly-bound particles from the bulge of each galaxy to determine its position and line-of-sight velocity; the results were plotted on top of the grey-scale images. Our images are thus analogous to H[I]{} maps supplemented with accurate nuclear coordinates and velocities.
The entire procedure outlined above, including both the generation of the simulations and the selection of viewing parameters, was performed by automated scripts without human intervention; we did not know the actual values of any parameters except $e$ and $\mu$. The resulting sample of merging galaxies, shown in Fig. \[random\_mergers\], possess a variety of morphologies; only a subset display the “double tails” characteristic of the best-known mergers [@T77].
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Identikit Matching {#sec:matching}
------------------
We prepared a series of eight Identikit simulations spanning a range of pericentric separations. Each simulation contained two identical ($\mu = 1$) configurations of $N_{\rm test} = 262144$ test particles and $N_{\rm sphr} = 81920$ massive particles; the mass model used to set up the massive particle distribution was a spherical version of the one used in the random mergers. These configurations were placed on parabolic ($e = 1$) relative orbits with pericentric separations $p
= 1/16$, $2/16$, $3/16$, $4/16$, $5/16$, $6/16$, $7/16$, and $8/16$, starting at $t = -2$ time units before first pericenter, and followed until $t = 8$, by which time even the widest passage had merged.
We used these simulations and the Identikit software to fit each of the random mergers by interactively matching the “observed” $(X,Y)$, $(X,V)$, and $(V,Y)$ projections with test particles. The modeling process usually began with rough guesses for the viewing direction, time since pericenter, and pericentric separation. A variety of clues guided these guesses. For example, short but pronounced tidal features point to a recent tidal encounter, while long but attenuated features suggest an older passage; in later stages, loops associated with tails show that material has started falling back. Likewise, if the galaxies display a large separation in projected velocity then the sight-line must be close to the orbital plane; conversely, a small difference in projected velocity implies either that the system is observed near apocenter or that the relative velocity vector is roughly perpendicular to the line of sight. Finally, other things being equal, closer passages generally yield stronger and more dramatic tidal features.
The next step was to adjust the orientations of the two disks, attempting to roughly match the morphology and kinematics of the system. This generally suggested further modifications to the viewing direction, separation, and time, as well as the scale factors and center-of-mass position and velocity. Further adjustment of all parameters continued until a satisfactory match was obtained or exhaustion set in. Our criteria for a satisfactory match were somewhat subjective[^5]; we placed a good deal of weight on matching tidal features (e.g., Fig. \[idkit\_match\]), while recognizing that test particles can’t accurately reproduce structures – such as tidally-induced spirals – which depend on self-gravity. Each match typically took a few hours, and the entire set of $36$ random mergers was matched in about one month; for comparison, a match to the NGC 7252 merger remnant [@HM95] took $74$ N-body runs over a three month time period, while a match to the NGC 4676 system [@B04] took $\sim 30$ runs over two months.
After changing the viewing direction, time since pericenter or orbital parameters, it’s usually necessary to reposition the centers of the models on top of the actual positions by adjusting the rotation about the viewing axis $\theta_{\rm Z}$, scale factor $\mathscr{L}$, and center of mass position $(X_{\rm cm}, Y_{\rm cm})$. We therefore implemented an option to “lock” the centers; when this option is invoked, $\theta_{\rm Z}$, $\mathscr{L}$, and $(X_{\rm cm}, Y_{\rm
cm})$ are recalculated on the fly, keeping the projected positions of the models invariant as other parameters are changed. Locking works quite well when the two galaxies are well-separated on the $(X,Y)$ plane; it’s less useful, and can be downright counter-productive, when the centers appear close together. In fine-tuning a nearly final match we sometimes found it useful to unlock the centers, trading off slight misalignments in central positions for improved matches to tidal features.
Fig. \[idkit\_match\], which shows our match to object 23 (see Fig. \[random\_mergers\]), illustrates many aspects of the matching process. From the start, it seemed likely that the viewing direction would be fairly close to the orbital plane, since the two galaxies have rather different systemic velocities. The galaxy on the lower right of the $(X,Y)$ projection appears nearly edge-on, as indicated by its morphology [*and*]{} its rather large velocity range. Since its tidal features lie more or less in the same plane as the disk itself, it seemed plausible that this galaxy has a relatively small inclination $i_1$ to the orbital plane, while its companion clearly has a higher inclination $i_2$ and appears more face-on from our viewpoint. The dual-valued velocities along the tail of the edge-on disk, which produce the “hook”-shaped feature seen in the $(X,V)$ and $(V,Y)$ projections, suggested that this tail is actually quite extended, but viewed so as to double back on itself.
For an initial match to this system, we tried a “middle-of-the-road” pericentric separation ($p = 0.25$); at a relatively early time ($t =
0.56$) we could roughly match the velocity difference and some aspects of the morphology and kinematics, including the spiral morphology of the face-on disk and the hooked tail in the $(V,Y)$ projection. However, other features of this initial match were less satisfactory. In the $(X,V)$ projection, the tail doubled back too soon, while in the $(V,Y)$ projection, the bridge did not span the velocity range between the galaxies, falling to the left of its ideal position. Moreover, the more face-on disk, while nicely rendered in the $(X,Y)$ projection, populated regions of phase space which the $(V,Y)$ projection showed to be empty. Trial and error revealed that wider passages and later times could repair most of these defects; the solution shown in Fig. \[idkit\_match\] uses a $p = 0.5$ passage viewed at $t = 1$. The tail in this match, while more extended than it was initially, is still a bit too short. Times $t > 1$ yield longer tails, but the velocity difference between the galaxies becomes too small, and bridge particles falling through the more face-on disk over-populate a relatively sparse region of phase-space. The adopted solution is therefore a compromise between several competing factors.
After comparing our Identikit models to the morphology and kinematics of all $36$ random mergers, we subjectively graded the solutions as “good” ($18$ cases), “fair” ($12$ cases), or “poor” ($6$ cases); these grades are indicated in Fig. \[random\_mergers\]. Good matches, like the one in Fig. \[idkit\_match\], strongly constrain the parameters. Fair matches generally appear plausible but allow more latitude in selecting parameter values; this group included several systems with twin edge-on tidal tails. Poor matches could be divided into two groups: systems with weak and diffuse tidal features, typically resulting from very wide encounters involving retrograde or highly inclined galaxies (objects 16 and 32), and systems with pronounced but confusing tidal features (objects 1, 6, 11, and 24).
Results: Parameters {#sec:parameters}
-------------------
![An abstract representation of the parameter space of galaxy interactions. The radial coordinate represents the initial orbit, the azimuthal coordinate represents the disk orientations, and the vertical coordinate represents the parameters chosen after a simulation is run. A conventional N-body simulation explores the parameter subspace represented by the dotted line, while a single Identikit simulation can explore entire cylindrical surface. \[interparam\]](figure03.eps){width="0.35\columnwidth"}
With the Identikit solutions in hand, we compared their parameter values to the true values used to generate the artificial merger data. Fig. \[interparam\] uses an abstract cylindrical coordinate system to portray the parameter space explored in these solutions. The radial coordinate represents the initial orbit of the two galaxies; the fits discussed here parameterize the orbit by the pericentric separation $p$, since the eccentricity $e$ and mass ratio $\mu$ were fixed beforehand. The azimuthal coordinate represents the four angles $(i_1, \omega_1)$ and $(i_2, \omega_2)$ required to specify the initial orientations of the two disks. Together, the radial and azimuthal coordinates of this abstract space completely specify the initial conditions for a galaxy interaction. The vertical coordinate represents the parameters selected [*after*]{} running a simulation: time since pericenter $t$, viewing angles $(\theta_{\rm X},\theta_{\rm
Y},\theta_{\rm Z})$, and scale factors $(\mathscr{L},
\mathcal{V})$; here the center-of-mass parameters are omitted since their values are not discussed below. A conventional N-body simulation starts at a point on the horizontal plane and explores the parameter subspace represented by the dotted line in this figure; a single Identikit simulation, in contrast, allows access to an entire cylindrical surface.
{width="94.00000%"}
to 0.042
{width="94.00000%"}
Fig. \[cmp\_incl\] presents a scatter plot comparing inclinations $i_{\rm fit}$ derived from the Identikit fits against the inclinations $i_{\rm true}$ used in the random merger sample. Here and in subsequent plots, color indicates the grade of each model; note that the disks are not graded individually, so both disks in a given model receive the same grade even if one fits better than the other. The good fits (shown in red) fall quite close to the diagonal line across the entire range of the plot. The fair fits (black) display more scatter but track the same relationship. In contrast, the poor fits (blue) have a very different distribution: almost all have inclinations $i_{\rm true} > 90^\circ$, and most fall quite far from the diagonal. This plot supports a couple of useful inferences. First, our subjective grades, based on the overall appearance of the Identikit models, correlate with $|i_{\rm fit} - i_{\rm true}|$; in other words, these grades mean something. Second, encounters involving disks with inclinations $i > 90^\circ$ are more difficult to model, presumably because the tidal features such encounters produce are less distinct and more ambiguous. Nonetheless, many high-inclination encounters were successfully modeled; other factors evidently influence the outcome of the modeling process.
Likewise, Fig. \[cmp\_omega\] compares fit and true values of the argument to pericenter, $\omega_{\rm fit}$ and $\omega_{\rm true}$. Since $\omega$ becomes indeterminate for inclinations near $i =
0^\circ$ or $i = 180^\circ$, we plot disks with $30^\circ \le i_{\rm
true} \le 150^\circ$ as filled circles, and disks outside this range as crosses. This plot shows that good fits yield arguments quite close to the true values, and the fair fits do nearly as well. At first sight it may seem that a few disks, represented by the one good and three fair points in the upper left and lower right of the plot, yield discrepant values of $\omega_{\rm fit}$, but this is an artifact of topology; $\omega$ is a periodic coordinate, and when opposing edges of the plot are identified, these apparent outliers are not so far from $\omega_{\rm fit} = \omega_{\rm true}$. The poor fits, in contrast, genuinely scatter throughout the plot. Most of the good and fair fits with $i_{\rm true} < 30^\circ$ or $150^\circ < i_{\rm true}$ still yield reasonable values for $\omega$, but these are a bit more scattered. For example, only one of the six good fits with $i_{\rm
true}$ in this range yields a $\omega_{\rm fit}$ within $10^\circ$ of $\omega_{\rm true}$; in contrast, just over half of the good fits with inclinations $30^\circ \le i_{\rm true} \le 150^\circ$ have $|\omega_{\rm fit} - \omega_{\rm true}| < 10^\circ$.
{width="94.00000%"}
to 0.042
{width="94.00000%"}
To examine our overall accuracy in determining initial disk orientations, we computed misalignments $\Delta_{\rm spin}$ between true and fitted spin vectors for all $72$ disks. Fig. \[cdf\_spin\] shows the cumulative distribution function of $\cos(\Delta_{\rm spin})
= \hat{{\mathbf{s}}}_{\rm fit} \cdot \hat{{\mathbf{s}}}_{\rm true}$; as in previous plots, color indicates the subjective grade of each fit. A perfect match yields $\cos(\Delta_{\rm spin}) = 1$, while if $\hat{{\mathbf{s}}}_{\rm fit}$ was uncorrelated with $\hat{{\mathbf{s}}}_{\rm
true}$ then $\cos(\Delta_{\rm spin})$ would be uniformly distributed in the range $[-1,1]$ and the plotted points would fall along a diagonal from lower left to upper right. The actual distribution is strongly peaked near $\cos(\Delta_{\rm spin}) = 1$, with the good fits showing the smallest misalignments, the poor fits showing the largest misalignments, and the fair fits falling in between. This is consistent with the previous figures, since $i$ and $\omega$ are just angular coordinates for $\hat{{\mathbf{s}}}$. For the entire sample, the median value is $\Delta_{\rm spin} = 18^\circ$, while for the $36$ disks in good fits the median is $\Delta_{\rm spin} = 12^\circ$.
In a similar fashion, Fig. \[cdf\_view\] shows the distribution of the misalignment in viewing direction, $\Delta_{\rm view}$, for all $36$ fits. Here $\cos(\Delta_{\rm view}) = \widehat{{\mathbf{Z}}}_{\rm
fit} \cdot \widehat{{\mathbf{Z}}}_{\rm true}$, perfect agreement again yields $\cos(\Delta_{\rm view}) = 1$, and perfect ignorance would distribute points along a diagonal from lower left to upper right. Viewing direction is quite well determined; for the entire sample the median $\Delta_{\rm view} = 13^\circ$, while for the $18$ good fits alone the median is only slightly smaller, $\Delta_{\rm view} =
12^\circ$. Note that an error circle with a radius of $12^\circ$ covers roughly $1$% of the solid angle of a sphere; these fits are [*much*]{} better than educated guesses!
{width="45.00000%"}to 0.1{width="45.00000%"}
Fig. \[cmp\_time\] compares the actual time $t_{\rm true}$ since pericenter against the time $t_{\rm fit}$ obtained from the Identikit fit. The scatter plot on the left shows that the fitted and true values are in good agreement, closely tracking each other throughout the entire range of times; there is no evidence of bias or systematic error, and the residuals appear to be random. The plot on the right shows the cumulative distribution of the fit/true ratio, $t_{\rm
fit}/t_{\rm true}$. The symmetric appearance of this curve provides further evidence that $t$ is accurately estimated by the Identikit models. Subjective fit quality appears to correlate with $t_{\rm
fit}/t_{\rm true}$; of the ten points at the two extremes of the distribution, only one comes from a good fit.
{width="45.00000%"}to 0.1{width="45.00000%"}
Fig. \[cmp\_rperi\] compares fitted and true values of the pericentric separation. Here the range of $p$ values is rather small, and the fact that $p_{\rm fit}$ can take on only eight discrete values is evident. There is a fair correlation between $p_{\rm fit}$ and $p_{\rm true}$, although the points show considerable scatter. The cumulative distribution of $p_{\rm fit}/p_{\rm true}$ plotted on the right is nonetheless fairly symmetric, and the median value of $p_{\rm
fit}/p_{\rm true}$ is very close to unity. Curiously, there’s not much sign that the grades assigned the models correlate with $p_{\rm
fit}/p_{\rm true}$; fair and poor fits appear interspersed with good ones throughout most of the distribution.
{width="45.00000%"}to 0.1{width="45.00000%"}
{width="45.00000%"}to 0.1{width="45.00000%"}
Identikit estimates of the length and velocity scale factors, $\mathscr{L}$ and $\mathcal{V}$, are compared with their true values in Figs. \[cmp\_lscale\] and \[cmp\_vscale\], respectively. The length scale $\mathscr{L}$ is well-determined; the plot on the left shows points scattered about the diagonal line, while the cumulative distribution on the right shows a narrow spread with a median value of $\mathscr{L}_{\rm fit}/\mathscr{L}_{\rm true}$ close to unity. In contrast, Identikit estimates of the velocity scale $\mathcal{V}$ show a small but very definite bias; the distribution of $\mathcal{V}_{\rm
fit}/\mathcal{V}_{\rm true}$ is narrow, but clearly offset from unity. Tests with isolated disks suggest that the absence of random motion in the Identikit test-particle disks is responsible; see Appendix C.
Table \[fit\_ratios\] lists statistics for the ratios of fitted to true values for $t$, $p$, $\mathscr{L}$, and $\mathcal{V}$; the physical parameters $T$ and $P$ will be discussed in the next section. Fit/true ratios for $t$ and $\mathscr{L}$ have fairly narrow distributions centered on unity, confirming that these parameters are determined accurately and without bias. The fit/true distribution for $p$, while somewhat broader, is also centered on unity. The fit/true distribution for the velocity scale factor $\mathcal{V}$ is quite narrow, but the median value is $\sim 10$% too high, showing again that a small bias is present in fitting $\mathcal{V}$.
Results: Residuals {#sec:residuals}
------------------
The errors determined by comparing the Identikit models with the actual mergers define six independent residuals: viewing direction ($\Delta_{\rm view}$), spin direction ($\Delta_{\rm spin}$), time since pericenter ($t_{\rm fit}/t_{\rm true}$), separation at pericenter ($p_{\rm fit}/p_{\rm true}$), length scale ($\mathscr{L}_{\rm fit}/\mathscr{L}_{\rm true}$), and velocity scale ($\mathcal{V}_{\rm fit}/\mathcal{V}_{\rm true}$). If each of these residuals is plotted against the others, a total of fifteen ($6 \times
5 / 2$) potential relationships can be examined. Most of these plots show no measurable correlation; only the four in Fig. \[residuals\] are significant (correlation coefficient $> 0.5$).
![Scatter-plots of statistically significant correlations between residuals in Identikit models. Color indicates quality of fit. The plot on the upper left contains twice as many points as the others because $\Delta_{\rm spin}$ is plotted for each disk. \[residuals\]](figure12.eps){width="\columnwidth"}
As shown in the upper left panel of Fig. \[residuals\], errors in disk orientation $\Delta_{\rm spin}$ are correlated with errors in viewing direction $\Delta_{\rm view}$. This correlation, which is largely driven by the poor fits, is not hard to explain. Once a viewing direction has been selected, the next step is usually to adjust the disk orientations; if clear guidance from tidal features is lacking, the best one can do is to match each disk’s position angle and apparent tilt with respect to the line of sight. A poor choice for the viewing direction will induce comparable errors in spin direction, as seen here.
The plot in the upper right panel shows that residuals in the time since pericenter $t$ correlate with residuals in pericentric separation $p$. This correlation also has a simple explanation; wider passages evolve and merge more slowly, so if $t_{\rm fit}$ is for some reason overestimated during the matching process then a larger $p_{\rm fit}$ can partly compensate for this error. As an extreme example, suppose $t_{\rm fit}$ was set so high that an encounter with the correct $p_{\rm fit}$ would already have merged by this time; by selecting a larger value for $p_{\rm fit}$, the merger can be delayed until some time after $t_{\rm fit}$, improving the match to the morphology.
A clear correlation between the residuals in $t$ and $\mathscr{L}$ of the form $\mathscr{L}_{\rm fit}/\mathscr{L}_{\rm true} \simeq (t_{\rm
fit}/t_{\rm true})^{-1}$ is shown in the lower right panel of Fig. \[residuals\]. Such a correlation arises naturally in cases where fits are strongly constrained by well-developed tidal tails. In a proper tail, material at the tip is moving fast enough to escape; at late times this material will have asymptotically constant velocity [@TT72], implying that a tail’s length grows in proportion to its age [@S77]. If $t_{\rm fit}$ is overestimated, tidal tails will be longer in direct proportion, and a smaller value of $\mathscr{L}_{\rm fit}$ will be needed to rescale them to their correct physical lengths. It’s likely that similar considerations also apply when other kinds of tidal features provide the primary constraints; this is consistent with the results presented here, which show that the good and fair fits fall close to $\mathscr{L}_{\rm
fit}/\mathscr{L}_{\rm true} \simeq (t_{\rm fit}/t_{\rm true})^{-1}$, while the poor ones are more scattered.
Finally, we conjecture that the correlation between residuals of $p$ and $\mathscr{L}$ shown in the lower left panel is induced by the $t_{\rm fit}/t_{\rm true}$–$\,p_{\rm fit}/p_{\rm true}$ and $t_{\rm
fit}/t_{\rm true}$–$\mathscr{L}_{\rm fit}/\mathscr{L}_{\rm true}$ correlations already described. Of the three, the $p_{\rm fit}/p_{\rm
true}$–$\mathscr{L}_{\rm fit}/\mathscr{L}_{\rm true}$ correlation shows the most scatter, and its outliers tend to match those in the $t_{\rm fit}/t_{\rm true}$–$\,p_{\rm fit}/p_{\rm true}$ correlation.
An interesting consequence of these correlations emerges when the dimensionless model parameters $t$ and $p$ are combined with the length and velocity scale factors $\mathscr{L}$ and $\mathcal{V}$. In matching an Identikit model to observational data, $X$ and $Y$ might be given in units of ${\rm kpc}$, and $V$ might be given in units of ${\rm km\,s^{-1}}$; the scale factors $\mathscr{L}$ and $\mathcal{V}$ would then have units of ${\rm kpc}$ and ${\rm km\,s^{-1}}$, respectively, and define a transformation from dimensionless model data to real physical values. The physical pericentric separation is $P = \mathscr{L} p$, while the time since pericenter is $T = (\mathscr{L}/\mathcal{V}) t$. A glance at the two lower plots in Fig. \[residuals\] suggests that these correlations may actually [*reduce*]{} errors in estimates of the physical parameters $T$ and $P$. Table \[fit\_ratios\] supports this; the uncertainty in $P$, as indicated by the ratio of $3^{\rm
rd}$ to $1^{\rm st}$ quartile values, is about half the uncertainty in $p$. For $T$ the improvement is not as striking; the ratio of $3^{\rm
rd}$ to $1^{\rm st}$ quartiles is slightly smaller than the ratio for $t$, but it appears that $\mathcal{V}$ introduces some additional scatter and a small bias.
Poor Matches {#sec:poor_match}
------------
For balance, it’s worth taking a closer look at some of the less successful matches. We judged six out of $36$ of our solutions to be poor fits to the “observed” data. As already noted, two of the systems with poor matches had very diffuse tidal features, while four had fairly strong features. Our difficulties in modeling the latter are somewhat surprising; why, given the information which must be present in strong tidal features, didn’t the models turn out better?
{width="30.00000%"}to 0.05{width="30.00000%"}to 0.05{width="30.00000%"}
Fig. \[poor\_match\] presents three versions of one of the more difficult systems (object 6 in Fig. \[random\_mergers\]). For ease of comparison, only the $(X,Y)$ and $(X,V)$ planes are shown; the $(V,Y)$ views are harder to interpret since the galaxies partly overlap. The left-hand version shows an Identikit model with the correct disk orientations, viewing direction, and velocity scale “dialed in” by hand. Only a finite set of pericentric separations are available in Identikit, so it was not possible to exactly match the pericentric separation $p_{\rm true} = 0.223$; on the whole, $p_{\rm fit} = 3/16$ gave a better match than $p_{\rm fit} = 1/4$ although the latter is numerically closer. This closer approach favored a somewhat different length scale ($\mathscr{L}_{\rm fit} =
0.644$ vs. $\mathscr{L}_{\rm true} = 0.590$) and a slightly earlier time ($t_{\rm fit} = 1.62$ vs. $t_{\rm true} = 1.69$); both of these adjustments are qualitatively consistent with the residual correlations discussed in the last section.
The “correct” solution on the left in Fig. \[poor\_match\] does a poor job of matching the morphology (top). It does better at matching the kinematics (bottom), although some fairly large regions of phase space are quite under-populated, and one of the spiral arms in the more face-on disk produces a badly-placed streak of particles across the $(X,V)$ projection. By our subjective criteria, this solution is a poor fit. Since the disk orientations and viewing direction [*exactly*]{} match those used to generate the grey-scale images, this mismatch must be largely due to the lack of disk self-gravity in the Identikit simulations. In particular, the spiral morphology of the more face-on disk is very poorly reproduced; with hindsight, we recognize that this spiral probably owes its form to self-gravity.
The middle and right parts of Fig. \[poor\_match\] present two of the solutions we found while trying to match this system. Essentially, each matches one galaxy while failing to match the other. The middle solution does a plausible job of matching the morphology [*and*]{} kinematics of the more edge-on disk, using a close passage ($p_{\rm
fit} = 1/16$) at an early time ($t_{\rm fit} = 1.12$). Although it does a very poor job with the more face-on disk, this solution is actually not too bad – the viewing angle is off by $47^\circ$, but the disks are within $30^\circ$ of their correct orientations. The right-hand solution approximates the morphology and kinematics of the more face-on disk, using a wider passage ($p_{\rm fit} = 3/16$) at the same early time. Ironically, although the decent-looking fit to the face-on disk and somewhat plausible match to the edge-on disk led us to adopt this solution as the best of a bad lot, it’s actually a [*much*]{} less accurate fit than the middle one – the viewing angle is off by $95^\circ$, while the face-on and edge-on disks are misaligned by $109^\circ$ and $94^\circ$, respectively! The splattering of particles between the galaxies in the $(X,V)$ projection, populating a region which should have been completely empty, was the critical flaw which saved us from assigning a “fair” grade to this solution.
The other three systems which we failed to fit despite their strong tidal features are a mixed lot. In object 24, much as in the example just discussed, the “correct” Identikit solution doesn’t match the morphology of either galaxy. In object 11, the correct solution matches one disk but fails to reproduce the morphology and kinematics of its partner. Finally, object 1 is reproduced quite accurately by the correct Identikit model. Our failure with object 1 was largely due to insufficient patience, while with object 24 the mismatch was so bad that the correct solution does not stand out when compared to other possible fits. Object 11 is intermediate; had we stumbled across the correct match to one disk we might have recognized the overall plausibility of this solution. In sum, human limitations and lack of self-gravity contribute about equally to the four poor matches of systems with strong tidal features.
Discussion
==========
Under somewhat idealized conditions, the Identikit methodology can recover the key parameters of a galactic encounter from a single data-cube. If the method can also be applied to real data, it will be a powerful tool for interpreting observations and reconstructing the dynamical histories of interacting galaxies. But before going further, some limitations should be discussed.
Limitations
-----------
First, as noted above, Identikit simulations are not completely self-consistent. Halos and bulges are treated self-consistently, but disks – of necessity – must be modeled with test particles. Disk structures requiring self-gravity, including bars and swing-amplified spirals, will not be reproduced. Consequently, these features can’t be used to match models to observations. Moreover, orbital decay of Identikit models is independent of disk orientation, whereas direct passages are expected to decay faster than retrograde ones [@W79; @B92]. For a pair of galaxies approaching their second passage, the estimated time $t_{\rm fit}$ since first pericenter may be off by as much as $\sim 20$%; in applications requiring an accurate estimate of $t_{\rm fit}$, Identikit models should be followed up with self-consistent simulations.
Second, while the test particles we used to model disks are collisionless, most kinematic tracers follow a specific phase of the interstellar medium – for example, H[I]{}, H$\alpha$, or CO. Collisionless particles can approximate gas dynamics, but only if the gas moves ballistically; streams of particles freely interpenetrate, whereas gas will be deflected if it encounters shocks. Examples include mass transfer via genuine bridges formed in low-inclination encounters [@TT72] and “splash bridges” due to hydrodynamic forces in interpenetrating encounters [@S97]. More subtle are the modest star–gas offsets seen in low-inclination encounters with extended gas disks [@M01]; it appears that the gas offset from the stars does follow ballistic trajectories, while some gas initially associated with the stars dissipates and falls back. Additionally, shocks may change the physical state of the gas; in particular, dense molecular gas is often associated with material which has undergone significant dissipation and hence [*cannot*]{} be modeled collisionlessly. But for most encounter geometries, gas in tidal features should be well-approximated by collisionless particles; exceptions can be recognized and allowed for in the fitting process. Stars, especially populations pre-dating the onset of an encounter, could be a useful complement to gas-phase tracers; absorption-line spectroscopy would require prohibitively large amounts of telescope time, but individual planetary nebulae are already providing kinematic data for nearby systems [e.g., @HFFD95; @DMFJC03].
Third, two parameters were not included in the present experiment: the orbital eccentricity $e$ and the mass ratio $\mu$. In principle it’s straightforward to include these parameters in the fitting process, although doing so increases the number of Identikit models needed. Both parameters have a priori constraints – orbits with $e \sim 1$ are favored theoretically (@TT72; but see @KB06), while $\mu$ can sometimes be estimated photometrically assuming constant $M/L$ ratios – so it seemed reasonable to exclude them from the initial experiments. Putting these parameters in play would probably increase the scatter in our fits; moreover, the three-way correlation between residuals of $t$, $p$, and $\mathscr{L}$ (§ \[sec:residuals\]) might expand to involve residuals of $e$ and $\mu$ as well. Within the context of the models examined here, there’s scant reason to expect much cross-talk between these parameters and others; in particular, disk orientations and viewing angles are robustly constrained by tidal features and should be well determined even if $e$ and $\mu$ are included in the fits.
Fourth, real galaxies have a range of rotation curve shapes, reflecting a diversity of mass profiles [e.g., @CvG91; @CGH06]; a single mass model is too limited. The fact that we used the same mass model for our artificial data [*and*]{} for the Identikit simulations no doubt helped to reduce the uncertainties in our solutions. There’s no reason why Identikit simulations cannot include a variety of mass models, although this will increase the number of choices to be made in modeling a galactic collision. On the other hand, if such simulations can discriminate between different mass models then they would provide a way to analyze the structure of disk galaxies.
Previous Studies
----------------
Models of interacting galaxies have a long history. @TT72 presented test-particle models of four systems: Arp 295, M 51, NGC 4676, and NGC 4038/9; the latter three have been revisited time and again by other workers. Kinematic information was initially scarce and of uneven quality, so early modeling attempts focused on reproducing the optical morphology. Better velocity data has given kinematics a more substantial role in more recent modeling efforts. At the same time, faster computers and N-body algorithms have enabled researchers to construct models incorporating self-gravity.
The methodology adopted by @HM95 in their model of the well-known merger remnant NGC 7252 includes several key practices found in other successful models:
1. Detailed velocity information, in the form of H[I]{} data [@HGvGS94], was available to constrain the model.
2. Astrophysical arguments were used to estimate several critical parameters – specifically, the mass ratio $\mu$, the initial orbital eccentricity $e$, and the pericentric separation $p$.
3. The model focused on reproducing the large-scale and morphology and kinematics of the tidal tails; these features evolved ballistically since first passage and therefore carry a memory of the initial encounter.
4. Test-particle models, with rigid galaxy potentials constrained to follow realistic merger trajectories, were used to narrow down the range of parameter space.
5. Fully self-consistent N-body models were used to refine the final model.
6. Simulation particles were plotted over orthogonal projections of the data cube to show that the final model reproduced [*both*]{} the morphology and the kinematics of NGC 7252.
Table \[dynamical\_models\] lists some interacting disk galaxies with dynamical models incorporating significant kinematical constraints. The progenitors of these systems span a range of mass ratios and morphological types. “S+S” systems involve two disk galaxies of roughly comparable mass, both generally displaying significant tidal features. Most of these pairs are observed between first and second passage; NGC 7252 is the only completed merger. As @SKBTEE05 note in modeling NGC 2207, earlier stages are generally easier to fit. In “S+d” encounters a disk galaxy is perturbed by a smaller companion, while in “E+S” systems the disk is disturbed by an elliptical of comparable mass. Finally, in “ring” galaxies a companion has plunged almost perpendicularly through a disk galaxy [@ASM96]; these systems are relatively straightforward to model since their geometry is fairly simple.
To varying degrees, the studies in Table \[dynamical\_models\] all followed the methodology used by @HM95. Most had access to detailed velocity information, usually obtained by H[I]{} interferometry or H$\alpha$ Fabry-Perot imaging, although a few models were based on long-slit spectroscopy. Adopted orbital eccentricities reflect a range of assumptions, not all equally plausible; for most systems, orbits with $e \sim 1$ seem more likely since $e < 1$ begs the question of what happened on the [*previous*]{} passage. Both test-particle and self-consistent techniques were used. Not all models were refined using fully self-consistent simulations; inasmuch as orbital decay is critical for many of these systems, the use of rigid potentials may be problematic in some cases.
As the third column of Table \[dynamical\_models\] shows, a wide range of criteria were used to define an acceptable match to the observations. Many studies still seemed more focused on morphology than kinematics, and less than half presented compelling quantitative comparisons between models and data. Matches labeled “kin2” used 2-D kinematical information, and presented the models and data in such a way that direct and unambiguous comparisons could easily be made – ideally, the model and data were overplotted, or at a minimum plotted to the same scale and orientation. Matches labeled “kin1” used 1-D data (e.g., long-slit spectra or mean velocities plotted as functions of a single coordinate), but again compared models and data directly. A few studies, labeled “gen” (for genetic), evaluated matches numerically; these will be discussed in § \[sec:genetic\]. Most of the remaining studies, while drawing on spatially-resolved kinematic information, presented essentially qualitative comparisons between models and data; these matches are labeled “qual” in the table. This designation is rather broad, ranging from studies which plotted models and data on different scales to studies which matched general kinematic trends or asserted, without providing quantitative evidence, that the model matched the data. Finally, a few studies which matched morphology only are designated “morph”; these are included when they served as precursors to more comprehensive modeling efforts.
The present study, while restricted to artificial data, closely parallels the approach of @HM95. We depart from them in treating the pericentric separation $p$ as a free parameter, and in not using fully self-consistent simulations to finalize the models. The latter, of course, is deliberate; one of our goals was to see if an approach combining test-particle disks with self-consistent halos can recover the encounter parameters of interacting disk galaxies. In practice, we envision using Identikit models to jump-start fully self-consistent simulations. We strongly concur that kinematic data provide an acid test which any dynamical model must pass [@TT72; @BR91], and that large-scale tidal features are the key to unlocking the dynamical history of galactic encounters. Finally, we emphasize that direct and unambiguous comparison between the simulations and the “observational” data was a necessary ingredient of our approach. Overplotting the particles on the data, as in Fig. \[idkit\_match\], is an effective way to present such comparisons.
### Genetic Algorithms {#sec:genetic}
To date, most attempts to model interacting galaxies have relied on expert judgement in selecting initial conditions and identifying good matches between simulations and observations. Recognizing the considerable labour involved, several groups have tried to automate the modeling process [@W98; @TK01; @GFP02]. The proposed algorithms have two essential components. First, they must replace the subjective comparison of the simulation particles ($\mathcal{P}$) and the observed data ($\mathcal{D}$) with an objective criterion $\mathcal{F}(\mathcal{P}, \mathcal{D})$ measuring goodness of fit. Second, they must perform an efficient search of a very large parameter space. In view of the number of parameters involved, a blind search is impractical; these groups have adopted strategies mimicking biological evolution, generally known as [*genetic algorithms*]{} [@H75].
Genetic algorithms create a population of $N_{\rm pop}$ individuals, each representing a possible solution to the problem at hand; in this case, an individual defines a set of initial conditions and viewing parameters. The evolutionary fitness of individual $i$ is determined by using its initial conditions and viewing parameters to produce a particle distribution $\mathcal{P}_i$ which is evaluated using $\mathcal{F}(\mathcal{P}_i, \mathcal{D})$. Once all $N_{\rm pop}$ individuals have been evaluated, the fittest among them are bred together to form a new generation, and the entire process is repeated. After $N_{\rm gen}$ generations, the population converges toward a nearly-optimal ensemble, with the fittest individual representing the best approximation to the desired solution.
Genetic algorithms for modeling interacting galaxies have been tested on artificial data [@W98; @TK01; @GFP02] and applied to real data for NGC 4449 [@TK01] and NGC 5194/95 [@WD01; @TS03]. Typical values of $N_{\rm pop} \simeq 10^2$ and $N_{\rm gen} \simeq 10^2$ imply that $\sim 10^4$ individuals must be evaluated to obtain a good match; with test-particle methods, this can be done in a few hours of CPU time. The Identikit methodology (§ \[sec:methodology\]) could be combined with a genetic algorithm, improving the treatment of orbital decay and substantially reducing the CPU time required to find a match.
However, the output of a genetic algorithm will be no better than the evaluation function $\mathcal{F}(\mathcal{P}, \mathcal{D})$ used to determine fitness. The simplest approach is to coarsely grid $\mathcal{P}$ on the $(X,Y)$ plane, and compare the result with an equally coarse gridding of $\mathcal{D}$. More recent implementations incorporate velocity information as well, and there’s no reason why gridding can’t be extended to 3-D $(X,Y,V)$ data. But the first two limitations of the Identikit method noted above also apply to existing evaluation functions. Features due to self-gravity in real systems can be discounted by an expert when attempting to fit a test-particle model, but may mislead an objective evaluation function, lowering the fitness of accurate solutions. And while H[I]{} is a good tracer of kinematics, its distribution in tidal features is often quite irregular; for example, a tail may appear as a series of clumps rather than a connected structure. An expert can recognize such tails as connected structures, but an automatic procedure may reject solutions which populate them with a smooth distribution of particles. To address these problems, recent genetic algorithm implementations include routines for masking or weighting the observational data; it remains to be seen if these techniques make genetic algorithms competitive with human experts.
Are Models Unique?
------------------
Can a dynamical model reproducing the morphology and kinematics of an interacting pair of galaxies be considered unique? Claims to this effect occasionally appear [e.g., @B88a; @B88b; @TK01]; skeptics, paraphrasing John von Neumann[^6], may be tempted to reply “with that many parameters I could fit an elephant”. The $16$ parameters introduced in § 1 are all physically motivated and necessary to describe a collision of two disk galaxies in 3-D; if a large number of parameters per se was really a flaw, models of spectral line formation in stellar atmospheres, requiring up to $\sim 90$ abundance parameters, would be on shaky ground indeed! Yet claims of uniqueness seem overconfident. A particular match may be unique [*within the universe of possibilities defined by a given model*]{}, implying that all the parameters appearing in the model can be determined within reasonable accuracy, and that no set of parameters outside this tolerance range yield as good a match. However, this is [*not*]{} the same thing as uniquely determining the dynamical state of a pair of colliding galaxies, which is specified by the distribution function $f({\mathbf{r}},{\mathbf{v}})$.
The problem of dynamically modeling isolated early-type galaxies, which has generated an extensive literature, illustrates some of the difficulties involved in determining $f({\mathbf{r}},{\mathbf{v}})$. Recent studies [e.g., @vdVdZvdB08; @vdBvdVVCdZ08] use Schwarzschild’s ([-@S79]) method to fit models with equilibrium distribution functions depending on three integrals of motion to stellar velocity data obtained from integral field spectroscopy. Such models are quite successful at describing the orbital structure of galaxies and diagnosing the presence of black holes and dark halos. But not many of these models are truly unique; the orientations and intrinsic shapes of axisymmetric models appear uncertain [@KCEMdZ05; @vdBvdVVCdZ08], while triaxial models pose additional difficulties [@vdVdZvdB08]. Determining $f({\mathbf{r}},{\mathbf{v}})$ for one galaxy is hard; doing so for a pair of galaxies seems harder still.
{width="45.00000%"}to 0.10{width="45.00000%"}
In the context of the model described here, it may not even be obvious that we can hope to constrain even the most basic parameters. The orientation of a single disk, unless it happens to be exactly edge-on, cannot be determined from a data cube $F(X,Y,V)$; for example, a disk tilted by $45^\circ$ to the line of sight produces [*exactly*]{} the same data cube as one tilted by $135^\circ$. (This ambiguity can be resolved by using dust lanes to determine which side of the disk is closer, although doing so goes beyond a strictly kinematic approach to galaxy modeling; see, e.g., the discussion in @SKBTEE05.) By extension, a pair of disks yield a four-fold degenerate solution, since each disk has two possible orientations. However, tidal interactions between disks break this degeneracy; as Fig. \[degeneracy\] shows, two disks which initially display identical $F(X,Y,V)$ distributions can be differentiated [*after*]{} an encounter. In this example, as in most of our Identikit solutions, the disk orientations $(i_1,\omega_1)$ and $(i_2,\omega_2)$ are strongly constrained; a little “wiggle room” exists around the actual values, but no radically different choice can reproduce the “observed” morphology and kinematics.
Taken together the results in § \[sec:parameters\] show that almost all of our $18$ “good” and $12$ “fair” fits accurately constrain the disk orientations, viewing direction, time since pericenter, pericentric separation, and scale factors. We did not examine errors in center-of-mass position and velocity, but the offsets in our fits were always very small, so presumably these parameters are also well-constrained. While some solutions are better than others, all $30$ of these fits appear to be unique; the estimated parameters are always in the neighborhood of the actual values, and often very close indeed.
The $6$ “poor” fits tell a different story. Object 16 (see Fig. \[random\_mergers\]), a wide retrograde encounter “observed” long after pericenter, has such attenuated tidal features that fitting this system is much like fitting a pair of disks [*before*]{} their first encounter; our fit is clearly degenerate. As § \[sec:poor\_match\] describes, object 6 also produced non-unique fits. Other poor solutions may also be degenerate, in view of the large misalignments in viewing direction and disk orientation they display (see Figs. \[cdf\_spin\] and \[cdf\_view\]).
The fact that we were able to produce well-constrained and reasonably accurate solutions for $30$ out of $36$ systems depends critically on the quality of the “observational” data we used. Data with lower resolution and added noise would yield a less favorable outcome; in particular, incomplete velocity information can seriously compromise a solution. Models of NGC 7252 illustrate this point; early attempts using very limited velocity data concluded that this object resulted from a merger of two [*retrograde*]{} disks [@BR91], but once high-quality H[I]{} data was available a very convincing solution involving a collision of two [*direct*]{} disks was soon found [@HM95]. Fortunately, instruments like the Expanded Very Large Array now coming on-line should make it far easier to obtain the detailed velocity information needed to accurately constrain dynamical models of interacting galaxies.
However, even a surfeit of data will not guarantee an unambiguous model in every case. One example may be the “Whirlpool” galaxy, NGC 5194/95; after three and a half decades of modeling based on ever-better H[I]{} data there is still no consensus regarding the number of passages required to account for NGC 5194’s optical morphology [*and*]{} extended H[I]{} tail [e.g. @DMFJC03 and references therein]. Models in which NGC 5195 makes only one passage can account for many of the observations but have trouble matching the tail velocities, while models in which this galaxy makes [*two*]{} passages match the velocities better but yield double tails [see @SL00 Fig. 4]. One possible solution to this puzzle is to assume that NGC 5194’s H[I]{} disk was warped even [*before*]{} its first and only encounter with NGC 5195; this could tilt the tail with respect to the inner disk in such a way as to match the observed velocities. Of course, pre-existing warps vastly complicate questions of uniqueness! In the specific case of NGC 5194/95, stronger evidence in favor of a single passage seems necessary to justify introducing such a warp; better data on [*stellar*]{} velocities may help [@DMFJC03].
Our limited knowledge of dark matter is a more fundamental barrier to the creation of truly unique models for interacting galaxies. Models of isolated galaxies in equilibrium can represent the dark matter as a potential imposed by hand. In contrast, models of interacting systems should really treat dark matter as a full participant in the dynamical equations[^7]. Given how little we know about dark matter, a good deal of guess-work will be part of any such model, and this is unlikely to change unless we can somehow measure the detailed distribution [*and*]{} kinematics of dark matter in individual galaxies. Nonetheless, models of interacting galaxies have already set limits on the radial structure of dark halos [@DMH96; @MDH98; @B99; @SW99], and further studies may yield additional insights into the nature, distribution, and dynamics of dark matter.
While it may never be possible to determine a unique distribution function $f({\mathbf{r}},{\mathbf{v}})$ for a specific pair of interacting galaxies, there are still compelling reasons to construct detailed models. A good model of a galactic collision can serve as a unifying hypothesis, providing a context to integrate and interpret a variety of observations; for example, a time-line of past encounters may help to understand interaction-induced star formation. Conversely, a bad model which contradicts other lines of evidence can be rejected. Viewed as a means to an end, dynamical modeling stands to teach us a good deal about galactic encounters and galaxies in general.
Extensions
----------
While the current Identikit system is essentially an interactive modeling tool, the basic approach offers other possibilities. For example, once the orbital parameters $p$, $e$, and $\mu$, time $t$, viewing angles $(\theta_{\rm X},\theta_{\rm Y},\theta_{\rm Z})$, and scale factors $\mathscr{L}$ and $\mathcal{V}$ have been fixed, it is possible to invert the mapping from initial disk orientation $(i,\omega)$ to the data cube $F(X,Y,Z)$, and ask which $(i,\omega)$ values allow a disk to populate a given point in the $(X,Y,V)$ space. As a rule, each $(X,Y,V)$ point corresponds to some range of $(i,\omega)$ values, but [*different*]{} points along one tidal structure should all be populated by the [*same*]{} disk, so this range can be constrained. This approach could effectively automate the process of finding disk orientations once other parameters have been selected; it could particularly useful in establishing confidence limits for key parameters once an initial match has been determined.
The Identikit technique has some interesting similarities to Schwarzschild’s ([-@S79]) method for building triaxial equilibrium systems. In a nutshell, Schwarzschild started with a stationary mass model, populated it with all possible orbits, and figured out how to add up the time-averaged density distributions of these orbits so as to recover the original mass model. We start with a time-dependent mass model generated by an encounter of two spherical systems; each sphere is initially populated with all possible circular orbits. But instead of finding a weighted sum of these orbits which reproduces the mass distribution, we try to select one co-planar family of orbits from each sphere to match the observed kinematics and morphology of interacting galaxies. Perhaps some variant of the algorithms used in Schwarzschild’s method to determine orbital weights could be applied to the problem of modeling interacting galaxies.
We hope to implement some of these extensions in Identikit 2.
J.E.B. thanks François Schweizer, Breanndán Nualláin, Shin Mineshige, and Piet Hut for valuable conversations and comments, and the Observatories of the Carnegie Institute of Washington, the California Institute of Technology, Kyoto University, and the Japan Society for the Promotion of Science for support and hospitality. We thank Curtis Struck for an open and constructive referee report. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
Identikit Model Construction
============================
To set up initial conditions for Identikit models we need to construct spherical equilibrium N-body models with mass profiles $m(r)$ given by (\[eq:total-mass-prof\]). We use Eddington’s ([-@E16]) formula [e.g., @BT87 p. 236] to compute the distribution function $f(E)$. @KMM04 used this approach to construct equilibrium halos with density profiles described by a three-parameter family of models [e.g., @Z96]. We go one step further by correcting for the finite resolution (i.e., “softening”) of the N-body force calculation.
We account for the effect of Plummer softening [@A63; @HB90] in the N-body simulations by introducing a quasi-empirical transformation of the total mass profile (Barnes, in preparation): $$\overline{m}(r) =
\left[
1 + (2/3)^{(\kappa/\alpha)} (\tilde{\epsilon}/r)^{\kappa}
\right]^{(\alpha/\kappa)} \, m(r) \, .
\label{eq:smooth-mass-profile}$$ where $\alpha$ is the logarithmic derivative of the density profile as $r \to 0$, the parameter $\tilde{\epsilon}$ is comparable to the softening length $\epsilon$, and the parameter $\kappa$ adjusts the shape of the transition near $r \sim \tilde{\epsilon}$. This smoothed mass profile is then used to compute the potential: $$\frac{d\Phi}{dr} = G \frac{\overline{m}(r)}{r^2} \, ,
\label{eq:potential-equation}$$ where $\Phi \to 0$ as $r \to \infty$. After expressing the original density profile $\rho(r) = (4 \pi r^2)^{-1} d m / d r$ as a function of this potential $\Phi$, we compute the distribution function: $$f(E) = \frac{1}{\sqrt{8} \pi^{2}} \, \frac{d}{dE}
\int_{E}^{0} d\Phi \, (\Phi - E)^{-1/2} \, \frac{d\rho}{d\Phi}
\label{eq:eddington-formula}$$
Once $f(E)$ has been calculated, generating an N-body realization with $N_{\rm sphr}$ massive particles is straightforward. For each particle $i$, let $\hat{{\mathbf{r}}}_i$ and $\hat{{\mathbf{v}}}_i$ be two vectors drawn from a uniform distribution on the unit sphere $\mathbf{S}^2$. Then the position of particle $i$ is ${\mathbf{r}}_i = r_i
\, \hat{{\mathbf{r}}}_i$, where $r_i$ is chosen by drawing a random number $x$ from a uniform distribution in the range $[0, m(\infty)]$ and solving $m(r_i) = x$, and the velocity is ${\mathbf{v}}_i = v_i \,
\hat{{\mathbf{v}}}_i$, where $v_i$ is chosen from the speed distribution $v^2 f(\frac{1}{2} v^2 + \Phi(r_i))$ using rejection sampling [@vN51]. The particle masses are $m_i = m(\infty) / N_{\rm
sphr}$.
The $N_{\rm test}$ test particles are initially placed in a single disk. If the test particle distribution follows $m_{\rm d}(r)$, the initial radius $q_i$ of particle $i$ may be selected by drawing a random number $x$ from a uniform distribution in the range $[0, m_{\rm
d}(\infty)]$ and solving $m_{\rm d}(q_i) = x$. However, we prefer to bias the distribution by a factor of $r^2$ to improve disk sampling at large $r$ (see § \[sec:methodology\]). Let $$\eta(r) =
\int_0^r d \chi \, \chi^2 \, m_{\rm d}^{\prime}(\chi) \, ,$$ where $m_{\rm d}^{\prime}(r) = d m_{\rm d} / d r$; then $q_i$ is selected by drawing $x$ from $[0,\eta(\infty)]$ and solving $\eta(q_i)
= x$. The orbital velocity $v_i$ of particle $i$ is calculated using the smoothed profile: $$v_i = \sqrt{G \, \overline{m}(q_i) / q_i} \, .
\label{eq:sphr-circ-vel}$$ Note that test particles are placed on [*exactly*]{} circular orbits, creating a perfectly “cold” disk. Finally, the position ${\mathbf{r}}_i$ and velocity ${\mathbf{v}}_i$ of particle $i$ are rotated to align ${\mathbf{r}}_i \times {\mathbf{v}}_i$ with a normalized angular momentum $\hat{{\mathbf{s}}}_i$ drawn from a uniform distribution on $\mathbf{S}^2$.
Random Merger Models
====================
Our galaxy construction procedure has some elements in common with @MD07. Like them, we compute isotropic distribution functions $f_{\rm b}(E)$ and $f_{\rm h}(E)$ for the bulge and halo, respectively, by approximating the disk’s gravitational field with its spherically averaged equivalent. Unlike them, we use the resulting bulge and halo “as is”, without first adiabatically imposing a flattened disk potential; the response of the bulge and halo to such an adiabatic transformation is so subtle that a good approximation to equilibrium is possible without it. This makes our procedure quite fast. What follows is a brief technical description of our procedure; a full discussion and numerical tests will be presented elsewhere (Barnes, in preparation).
The bulge follows a @H90a model out to a radius $b_{\rm b}$, and tapers at larger radii to avoid placing a small number of particles at extremely large distances: $$\rho_{\rm b}(r) =
\left\{
\begin{array}{ll}
\displaystyle
\frac{a_{\rm b} m_{\rm b}}{2 \pi} \,
\frac{1}{r (a_{\rm b} + r)^{3}} \, , &
r \le b_{\rm b} \\ [0.4cm]
\displaystyle
\rho_{\rm b}^{*} \, \left(\frac{b_{\rm b}}{r}\right)^2 \,
e^{-2 r / b_{\rm b}} \, , &
r > b_{\rm b} \\
\end{array}
\right.
\label{eq:bulge-model}$$ where $m_{\rm b}$ is the bulge mass, and $\rho_{\rm b}^{*}$ is fixed by requiring that $\rho_{\rm b}(r)$ be continuous at $r = b_{\rm b}$. For $b_{\rm b} \gg a_{\rm b}$, the slope $d \rho_{\rm b} / d r$ is also continuous at $r = b_{\rm b}$.
The halo follows a @NFW96 model out to a radius $b_{\rm h}$, and tapers at larger radii as proposed by @SW99: $$\rho_{\rm h}(r) =
\left\{
\begin{array}{ll}
\displaystyle
\frac{m_{\rm h}(a_{\rm h})}{4 \pi (\ln(2) - \frac{1}{2})} \,
\frac{1}{r (r + a_{\rm h})^2} \, , &
r \le b_{\rm h} \\ [0.4cm]
\displaystyle
\rho_{\rm h}^{*} \, \left(\frac{b_{\rm h}}{r}\right)^\beta \,
e^{- r / a_{\rm h}} \, , &
r > b_{\rm h} \\
\end{array}
\right.
\label{eq:halo-model}$$ where $m_{\rm h}(a_{\rm h})$ is the halo mass within radius $a_{\rm
h}$, and $\rho_{\rm h}^{*}$ and $\beta$ are fixed by requiring that $\rho_{\rm h}(r)$ and $d \rho_{\rm h} / d r$ are both continuous at $r
= b_{\rm h}$. The halo is tapered more abruptly than the bulge to tame the logarithmic divergence of the standard @NFW96 mass profile as $r \to \infty$.
The disk has an exponential radial profile [@dV59a; @dV59b; @F70] and a $\mathrm{sech}^2$ vertical profile [@vdKS81]: $$\rho_{\rm d}(q,\phi,z) =
\frac{m_{\rm d}}{4 \pi a_{\rm d}^2 z_{\rm d}} \,
e^{- q / a_{\rm d}} \,
\mathrm{sech}^2 \left( \frac{z}{z_{\rm d}} \right) \, ,
\label{eq:disk-model}$$ where $(q = \sqrt{x^2+y^2}, \phi, z)$ are cylindrical coordinates and $m_{\rm d}$ is the total mass of the disk.
For the bulge and halo, cumulative mass profiles are obtained by integrating (\[eq:bulge-model\]) and (\[eq:halo-model\]): $$m_{\rm b}(r) = \int_0^r d\chi \, 4 \pi \chi^2 \, \rho_{\rm b}(\chi) \, ,
\qquad {\rm and} \qquad
m_{\rm h}(r) = \int_0^r d\chi \, 4 \pi \chi^2 \, \rho_{\rm h}(\chi) \, .$$ We use the cumulative mass profile for an infinitely thin disk, $$m_{\rm d}(r) =
m_{\rm d} \, (1 - e^{- r / a_{\rm d}}) (1 + r / a_{\rm d}) \, ,$$ which is adequate for our purposes since $m_{\rm b}(r) \gg m_{\rm
d}(r)$ at small $r$. These functions are summed to get the total mass profile $m(r)$, the smoothed profile $\overline{m}(r)$ is computed using (\[eq:smooth-mass-profile\]), and the potential $\Phi(r)$ is computed using (\[eq:potential-equation\]). We then express $\rho_{\rm b}$ and $\rho_{\rm h}$ as functions of $\Phi$, insert these functions in (\[eq:eddington-formula\]) to obtain $f_{\rm b}(E)$ and $f_{\rm h}(E)$, and construct N-body realizations of the bulge and halo following the procedure described in Appendix A.
The disk is realized by sampling an approximate distribution function $$f_{\rm d}(q,\phi,z,v_{\rm q},v_\phi,v_{\rm z}) \, \propto \,
\rho_{\rm d}(q,\phi,z) \;
\mathcal{H}\!\left[\frac{v_{\rm q}}{\sigma_{\rm q}(q)}\right] \,
\mathcal{H}\!\left[\frac{v_\phi-\overline{v}(q)}{\sigma_\phi(q)}\right] \,
\mathcal{G}\!\left[\frac{v_{\rm z}}{\sigma_{\rm z}(q)} \right] \, ,
\label{eq:disk-dist-func}$$ where $v_{\rm q}$, $v_\phi$ and $v_{\rm z}$ are velocities in the radial, azimuthal, and vertical directions, respectively. The function $\overline{v}(q)$ is the mean rotation velocity, while $\sigma_{\rm q}(q)$, $\sigma_\phi(q)$, and $\sigma_{\rm z}(q)$ are dispersions in the radial, azimuthal, and vertical directions, respectively. The function $\mathcal{G}(x)$ is a Gaussian, while $\mathcal{H}(x)$ resembles $\mathcal{G}(x)$ but cuts off faster for large $|x|$: $$\mathcal{G}(x) \propto e^{- \frac{1}{2} x^2} \, ,
\qquad \qquad
\mathcal{H}(x) \propto e^{- \frac{1}{2} (x/c)^2 - \frac{1}{4} (x/c)^4} \, ,$$ where $c$ is fixed by requiring $\int dx \, x^2 \, \mathcal{H}(x) =
\int dx \, \mathcal{H}(x)$. This function is used instead of a Gaussian to avoid overpopulating the high-velocity tail of the distribution.
The local circular velocity $v_{\rm c}(q)$ is given by $$v_{\rm c}^2(q) =
G \, \frac{\overline{m}_{\rm s}(q)}{q} +
q \, \frac{d \Phi_{\rm d}}{d q} \, ,
\label{eq:circular-vel}$$ where $\overline{m}_{\rm s}(q)$ is the smoothed spheroid (bulge + halo) mass profile and $\Phi_{\rm d}$ is the potential due to the disk. To compute $\overline{m}_{\rm s}(r)$ we insert the spheroid profile $m_{\rm s}(r) = m_{\rm b}(r) + m_{\rm h}(r)$ in (\[eq:smooth-mass-profile\]). Our expression for $\Phi_{\rm d}$ explicitly takes “softening” into account: $$\frac{d \Phi_{\rm d}}{d q} = - \, \frac{G m_{\rm d}}{a_{\rm d}^3} \,
\int_0^\infty dk \,
\frac{k\,e^{-k\epsilon_{\rm d}}\,J_1(k q)}{(a_{\rm d}^{-2}+k^2)^{3/2}} \, ,
\label{eq:smoothed-disk-accel}$$ where $J_1(x)$ is the cylindrical Bessel function of order one, and setting $\epsilon_{\rm d} = \sqrt{\epsilon^2 + z_{\rm d}^2}$ allows – in an approximate way – for the finite thickness of the disk.
The vertical dispersion is given by the solution for an isothermal sheet [e.g., @BT87 p. 282]: $$\sigma_{\rm z}(q) = \sqrt{\pi G z_{\rm d} \Sigma(q)} \, ,
\label{eq:sigma-z}$$ where $\Sigma(q) = \int dz \, \rho_{\rm d}(q,z)$ is the surface density of the disk at cylindrical radius $q$. The radial dispersion is then determined by fixing the ratio $\sigma_{\rm q}/\sigma_{\rm
z}$: $$\sigma_{\rm q}(q) =
\mu(q) \, \sigma_{\rm z}(q) =
\left(1 + \frac{q}{q + q_\sigma}\right) \, \sigma_{\rm z}(q) \, ,
\label{eq:sigma-q}$$ where $q_\sigma$ is a scale parameter comparable to $a_{\rm d}$. The function $\mu(q)$ is chosen to make $\sigma_{\rm q}/\sigma_{\rm z}
\simeq 2$ in the body of the disk – roughly matching the solar neighborhood value [e.g., @DB98] – while letting $\sigma_{\rm
q}/\sigma_{\rm z} \to 1$ for $q \to 0$. The azimuthal dispersion is related to the radial dispersion [@BT87 p. 203]: $$\sigma_\phi(q) = \frac{\kappa(q)}{2 \Omega(q)} \, \sigma_{\rm q}(q) \, ,
\label{eq:sigma-phi}$$ where $\Omega(q) = v_{\rm c}(q) / q$ is the circular orbital frequency and $\kappa(q) = \sqrt{4 \Omega^2 + q d \Omega^2 / d q}$ is the epicyclic frequency.
Finally, the mean rotation velocity $\overline{v}(q)$ is determined using the axisymmetric Jeans equation [e.g., @BT87 p. 198]: $$\overline{v}^2(q) =
v_{\rm c}^2(q) +
\sigma_{\rm q}^2(q) \, \left(1 - \frac{2 q}{a_{\rm d}}\right) -
\sigma_\phi^2(q) +
\sigma_{\rm z}^2(q) \, q \, \frac{d \mu^2}{d q}
\label{eq:mean-velocity}$$
The parameter values needed to completely define the galaxy model are: $$\begin{array}{l@{\quad}l@{\quad}l}
m_{\rm b} = 0.0625 \, , &
a_{\rm b} = 0.02 \, , &
b_{\rm b} = 4.0 \, , \\
m_{\rm d} = 0.1875 \, , &
a_{\rm d} = 1/12 \, , &
z_{\rm d} = 0.0075 \, , \\
m_{\rm h}(a_{\rm h}) = 0.16 \, , &
a_{\rm h} = 0.25 \, , &
b_{\rm h} = 0.98015 \, , \\
\epsilon = 0.0075 \, , &
\tilde{\epsilon} = 0.0115 \, , &
\tilde{\epsilon}_{\rm s} = 0.0115 \, , \\
q_\sigma = 0.075 \, , &
\kappa = 1.975 \, , &
\kappa_{\rm s} = 2.025 \, .
\end{array}$$ A few remarks about these parameters are in order. First, tapering the bulge as in (\[eq:bulge-model\]) with $b_{\rm b} = 200 a_{\rm
b}$ reduces the total bulge mass by $\sim 0.5$%; to correct this, the value of $m_{\rm b}$ actually used in (\[eq:bulge-model\]) is adjusted upward accordingly. Second, the primary halo mass parameter is $m_{\rm h}(a_{\rm h})$; the halo taper radius $b_{\rm h}$ is adjusted to make the total halo mass $m_{\rm h}(\infty) = m_{\rm h} =
1.0$. Third, the softening parameter actually used in the N-body calculations is $\epsilon$; the values of $\tilde{\epsilon}$ and $\kappa$ listed here are chosen by computing $\Phi(r)$ using (\[eq:potential-equation\]) and comparing the result to an N-body calculation. Fourth, the parameters $\tilde{\epsilon}_{\rm s}$ and $\kappa_{\rm s}$ used to compute the smoothed spheroid mass profile $\overline{m}_{\rm s}(r)$ are likewise chosen by comparison with an N-body calculation.
Fig. \[galmod\] presents circular velocity profiles for the galaxy model adopted here. The left-hand panel shows profiles for the individual components, computed taking softening into account as described above. Also shown is the total circular velocity $v_{\rm
c}(q)$ given by (\[eq:circular-vel\]). The right-hand panel again shows $v_{\rm c}(q)$ and compares it with the mean rotation velocity $\overline{v}(q)$ given by (\[eq:mean-velocity\]) and the circular velocity for the equivalent spherical mass model given by (\[eq:sphr-circ-vel\]).
To check this model, we constructed a realization with $N_{\rm b} =
16384$ bulge particles, $N_{\rm d} = 49152$ disk particles, and $N_{\rm h} = 65536$ halo particles. This system was then evolved in isolation for $10$ time units, using a hierarchical N-body code[^8] with an accuracy parameter $\theta = 1$, quadrupole-moment corrections [@H87], a Plummer softening length $\epsilon = 0.0075$, and a leap-frog integrator with a time-step $\Delta t = 1/256$. During the first $0.25$ time units the ratio of kinetic to potential energy, $T/U$, fell from an initial value of $0.4980$ to $0.4945$; it then fluctuated around this value with an amplitude of $\sim 0.003$. This initial drop indicates that the model was not started in perfect equilibrium, but the implied rearrangement of mass is only a little larger than the $1/\sqrt{N}$ ($\simeq 0.0028$) fluctuations occurring in an N-body system with this $N$. Apart from transient spiral structure, this model showed no significant features until the disk begins to develop a bar at time $t \simeq 4$. It’s hard to completely suppress a weak bar instability in galaxy models with relatively massive disks like the one used here; however, this instability has little effect on the merger simulations since the galaxies interact with each other long before they would develop bars in isolation.
![Circular velocity profiles for the disk galaxy model. Left: circular velocities for the bulge (red), disk (grey), and halo (blue). The total circular velocity is also shown (black). Right: total circular velocity (black), mean rotation velocity (red), and circular velocity for the spherical mass model (blue). \[galmod\]](figure15.eps){width="0.5\columnwidth"}
The $36$ random merger simulations were run with the same number of particles per galaxy and N-body integration parameters used in the test just described. Energy was conserved to $\sim 0.05$% even in the most violent encounters. Particle positions and velocities were output every $\Delta t_{\rm out} = 1/32$ time units, providing a large data base which could be used to construct random samples like the one in Fig. \[random\_mergers\].
Velocity Scale Bias
===================
As noted at the end of § \[sec:parameters\], Identikit estimates of the velocity scale $\mathcal{V}$ are typically $\sim 10$% too high. The value of $\mathcal{V}$ is usually determined toward the end of the matching process; after values have been selected for most other parameters, we adjust $\mathcal{V}$ to obtain a good overall match between the particles and the grey-scale images in the $(X,V)$ and $(V,Y)$ planes. There are a number of factors which may influence the choice for $\mathcal{V}$, including the velocity difference between the galaxies and the characteristic velocities of tidal features. However, a key feature is the velocity widths of the galaxies; we generally try to adjust $\mathcal{V}$ so that the particles span the full range of velocities present in each galaxy.
![Identikit match to a single disk galaxy. The velocity scale factor $\mathcal{V}_{\rm fit}$ was adjusted by eye to match the width of the velocity distribution. \[fitvscale\]](figure16.ps){width="0.45\columnwidth"}
While matching the velocity widths of the individual galaxies [*seems*]{} reasonable, it appears to be the source of the bias in $\mathcal{V}$. Velocities in the test-particle Identikit disks are not identical to those in the self-gravitating disks used in the random mergers; the former are perfectly “cold”, while the latter have non-zero velocity dispersions. In addition, as the right-hand panel in Fig. \[galmod\] shows, the circular velocity profile of the Identikit model rises more rapidly, peaks at a smaller radius, and then falls slightly below either the circular ($v_{\rm c}(q)$) or mean ($\overline{v}(q)$) profiles in the self-gravitating model.
Fig. \[fitvscale\] shows an Identikit fit to a self-consistent disk galaxy model. This disk is tilted by $45^\circ$ to the line of sight; in fitting this model, the orientation and length scale factor were first set to their actual values. We then adjusted $\mathcal{V}_{\rm
fit}$ by eye, stretching the particle distribution to match the apparent velocity width of the grey-scale images in the $(X,V)$ and $(V,Y)$ projections. The value of $\mathcal{V}_{\rm fit}$ selected in this manner is $10.9$% too high, much as in the Identikit matches to interacting galaxies. In hindsight, we note that the $(X,V)$ projection shows a handful of points falling outside the rotation curve for large $|X|$; a fit giving priority to these points would have recovered a more accurate value for $\mathcal{V}$.
Compared to its Identikit analog, the larger velocity width of the self-consistent disk is a direct consequence of the random velocities imparted to individual disk particles. To test this, we constructed self-consistent disks with different dispersions and velocity scales, and selected best-fit values of $\mathcal{V}_{\rm fit}$ as in Fig. \[fitvscale\]. The ratio $\mathcal{V}_{\rm fit} /
\mathcal{V}_{\rm true}$ has a one-to-one relationship with the disk’s velocity dispersion; the larger the dispersion, the greater the overestimate of $\mathcal{V}$. It’s likely that by constructing Identikit models with random velocities comparable to those present in the self-gravitating disks we could largely remove this bias. However, “cold” disks may actually be more appropriate when fitting kinematics observed in cold gas tracers (e.g., H[I]{} or CO), since these have smaller velocity dispersions than most stellar components.
[lcrrr]{} time since pericenter & $t$ & $0.86$ & $1.02$ & $1.14$\
pericentric separation & $p$ & $0.79$ & $1.01$ & $1.25$\
length scale factor & $\mathscr{L}$ & $0.86$ & $1.01$ & $1.17$\
velocity scale factor & $\mathcal{V}$ & $1.05$ & $1.10$ & $1.15$\
\[0.25cm\] physical time & $T$ & $0.78$ & $0.95$ & $1.02$\
physical separation & $P$ & $0.87$ & $0.97$ & $1.13$\
[lrrccl]{} VV 784 & & 784 & ring & morph & @T78\
& & & & kin1 & @SMH93\
VV 347, Arp 119 & 119 & 347 & ring & qual & @HL01\
NGC 520 & 157 & 231 & S+S & kin1 & @SB91\
3C 48 & & & S+S & kin1 & @SEPZKS04\
NGC 672 / IC 1727 & & 338 & S+S & kin2 & @CFWG80\
NGC 1143/44 & 118 & 331 & E+S & qual & @LHG88\
IC 1908, AM 0313-545 & & & S+d & qual & @MBR93\
NGC 2207 / IC 2163 & & & S+S & qual & @ESKBE95\
& & & & kin2 & @SKBTEE05\
VV 785, AM 0644-741 & & 785 & ring & kin1 & @AW07\
NGC 2442 / AM 0738-692 & & & S+d & qual & @MB97\
II Hz 4 & & & ring & qual & @LT76\
NGC 2782 & 215 & & S+? & qual & @S94\
NGC 2992/93 & 245 & & S+S & kin2 & @DBSPWM00\
NGC 3031/77 & & & S+d & qual & @TD93\
NGC 3031/34/77 & & & S+S+d & qual & @BBCKB91\
& & & & qual & @Y99\
AM 1003-435 & & & S+S & qual & @GARD06\
NGC 3395/96 & 270 & 246 & S+S & kin1 & @CBAG99\
NGC 3448 / UGC 6016 & 205 & & S+d & qual & @NK86\
NGC 4038/39 & 244 & 245 & S+S & morph & @TT72\
& & & & kin2 & @vdH79\
& & & & qual & @MvdHB87\
& & & & qual & @B88\
& & & & qual & @MBR93\
& & & & kin2 & @H03\
NGC 4254/92? & & & S+S & qual & @DB08\
NGC 4435/38 & 120 & 188 & S0+S & morph & @CDCP88\
& & & & kin2 & @VBCS05\
NGC 4449 / DDO 125 & & & S+d & gen & @TK01\
NGC 4654/39? & & & S+S & kin2 & @V03\
NGC 4676 & 242 & 224 & S+S & morph & @TT72\
& & & & qual & @S74\
& & & & kin1 & @MBR93\
& & & & qual & @GS94\
& & & & kin1 & @SR98\
& & & & kin2 & @B04\
NGC 5194/95 & 085 & 001 & S+S0 & morph & @TT72\
& & & & kin2 & @T78\
& & & & qual & @H90b,\
& & & & kin2 & @SL00\
& & & & gen & @WD01\
& & & & gen & @TS03\
& & & & kin2 & @DMFJC03\
NGC 5216/18 & 104 & 033 & E+S & qual & @CAGS07\
NGC 5394/95 & 084 & 048 & S+S & qual & @KBEEKST99\
AM 2004-662 & & & E+d & kin1 & @DRDC00\
NGC 6872 / IC 4970 & & & S+S0 & qual & @MBR93\
& & & & qual & @HK07\
NGC 7252 & 226 & & S+S & qual & @BR91\
& & & & qual & @MBR93\
& & & & kin2 & @HM95\
& & & & qual & @MDH98\
NGC 7714/15 & 284 & 051 & ring & qual & @SW92\
& & & & kin2 & @SS03\
NGC 7752/53 & 086 & 005 & S+d & kin2 & @SL93\
[^1]: This software is available at [http://www.ifa.hawaii.edu/faculty/barnes/ \\linebreak
research/identikit/](http://www.ifa.hawaii.edu/faculty/barnes/ \linebreak
research/identikit/) .
[^2]: Selecting the correct internal structures is a separate problem, and one largely beyond the scope of this paper. Under some fairly general assumptions, the internal structure of an axisymmetric galaxy may be described by a distribution function $f =
f(E, J_\mathrm{z}, I_3)$ depending on the energy $E$, angular momentum about the symmetry axis $J_\mathrm{z}$, and a third integral of motion $I_3$. Formally speaking, an [*infinite*]{} number of parameters are needed to specify such a function.
[^3]: If the simulation is conducted in physical units then these parameters are not necessary – but additional parameters are required to describe the initial conditions, so the total parameter count is unchanged.
[^4]: In forensic investigations, “Identikit” is one of several systems used to construct portraits by selecting from a menu of facial features. Our approach is analogous.
[^5]: It’s not trivial to evaluate matches quantitatively; for more on this, see § \[sec:genetic\].
[^6]: See @D04 for one version: “with four parameters I can fit an elephant and with five I can make him wiggle his trunk”. John von Neumann could presumably fit a whole herd of elephants with the parameters used to describe a single galactic encounter!
[^7]: To be sure, not every study in Table \[dynamical\_models\] did this, but the speed of modern computers leaves little excuse for not treating the dark matter self-consistently in future work!
[^8]: See <http://www.ifa.hawaii.edu/faculty/barnes/treecode/treeguide.html> for a discussion of this code, which generalizes earlier modifications [@B90] of the original tree code [@BH86].
|
---
author:
- 'Yaohui Chen\*'
- 'Dongliang Mu\*'
- Jun Xu
- Zhichuang Sun
- Wenbo Shen
- Xinyu Xing
- Long Lu
- Bing Mao
bibliography:
- 'ref.bib'
title: '[[PTrix]{}]{}: Efficient Hardware-Assisted Fuzzing for COTS Binary'
---
[^1]
<ccs2012> <concept> <concept\_id>10002978.10003022.10003023</concept\_id> <concept\_desc>Security and privacy Software security engineering</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
[^1]: \*These two authors have contributed equally
|
---
abstract: 'We present 3-160 $\mu$m photometry obtained with the IRAC and MIPS instruments for the first five targets from the Spitzer Legacy Science Program “Formation and Evolution of Planetary Systems" and 4-35 $\mu$m spectro-photometry obtained with the IRS for two sources. We discuss in detail our observations of the debris disks surrounding HD 105 (G0V, 30 $\pm 10$ Myr) and HD 150706 (G3V, $\sim$ 700 $\pm 300$ Myr). For HD 105, possible interpretations include large bodies clearing the dust inside of 45 AU or a reservoir of gas capable of sculpting the dust distribution. The disk surrounding HD 150706 also exhibits evidence of a large inner hole in its dust distribution. Of the four survey targets without previously detected IR excess, spanning ages 30 Myr to 3 Gyr, the new detection of excess in just one system of intermediate age suggests a variety of initial conditions or divergent evolutionary paths for debris disk systems orbiting solar-type stars.'
author:
- 'M.R. Meyer, L.A. Hillenbrand, D.E. Backman, S.V.W. Beckwith, J. Bouwman, T.Y. Brooke, J.M. Carpenter, M. Cohen, U. Gorti, T. Henning, D.C. Hines, D. Hollenbach, J.S. Kim, J. Lunine, R. Malhotra, E.E. Mamajek, S. Metchev, A. Moro–Martin, P. Morris, J. Najita, D.L. Padgett, J. Rodmann, M.D. Silverstone, D.R. Soderblom, J.R. Stauffer, E.B. Stobie, S.E. Strom, D.M. Watson, S.J. Weidenschilling, S. Wolf, E. Young, C.W. Engelbracht, K.D. Gordon, K. Misselt, J. Morrison, J. Muzerolle, and K. Su.'
title: |
The Formation and Evolution of Planetary Systems:\
First Results from a Spitzer Legacy Science Program
---
\#1\#2\#3\#4\#5\#6\#7[ to ]{}
\#1\#2\#3\#4\#5\#6\#7
to\#2
------------------------------------------------------------------------
Introduction
============
A combination of optical, infrared, and millimeter observations has provided incontrovertible evidence over the past two decades that most stars are surrounded at birth by circumstellar accretion disks (e.g. Beckwith and Sargent, 1996). That at least some of these disks build planets has become clear from radial velocity and photometric studies revealing M sin$i$=0.2-15 M$_J$ planets orbiting nearby stars (e.g. Marcy et al. 2000). More indirect but still compelling evidence of planet formation comes from optical and infrared imaging of a few debris disks (e.g. Kalas, Liu, & Mathews 2004 and Weinberger et al. 1999). These solar system-sized dust disks are comprised of micron-sized grains produced as by-products of collisions between asteroid-like bodies with orbits expected to be dynamically stirred by massive planets (e.g. Lagrange et al. 2000).
Following the IRAS discovery of excess infrared emission associated with Vega (Auman et al. 1984), IRAS and ISO identified several dozen debris disks around fairly luminous main sequence stars. Neither observatory, however, had the sensitivity to detect [*solar-type stars*]{} down to photospheric levels at distances greater than a few pc, up to now preventing a complete census of solar-system-like debris disks over a wide ranges of stellar ages. The extant samples have made it difficult to infer the typical path of debris disk evolution (Habing et al. 1999; Meyer and Beckwith, 2000; Spangler et al. 2001). The unprecedented sensitivity of the Spitzer Space Telescope (Spitzer; Werner et al. 2004) will enable the Legacy Science Program [*The Formation and Evolution of Planetary Systems: Placing Our Solar System in Context*]{} to search for debris systems around 330 stars with spectral types F8V to K3V and ages ranging from 3 Myr to 3 Gyr. Observations and results described here are from early validation data taken with all three Spitzer instruments. Targets were selected from among those visible during the validation campaign of December 2003 to be representative of the overall FEPS sample, the primary control variable of which is stellar age. Observed were HD 105, HD 47875, HD 150706, HD 157664, and HD 161897 ranging in age from 30 Myr to 3 Gyr. We report here photometry for all of these stars, and low resolution spectro-photometry for two with observed IR excess, HD 105 and HD 150706.
Stellar properties for our validation sample are summarized here and reported in Table 1. HD 105 (G0V, Houk 1978; 40 $\pm$ 1 pc, Hipparcos) is a kinematic member of the Tuc-Hor moving group (Mamajek et al. 2004). It is a coronally and chromospherically active dwarf star with youth established through its Li I $\lambda$6707 equivalent width (e.g. Wichmann et al. 2003), its position in the HR diagram, and activity indicators such as X-ray emission (Cutispoto et al. 2002) and Ca II HK emission (Wright et al. 2004). We adopt an age of 30 $\pm 10$ Myr for HD 105. HD 150706 (G3V, Buscombe 1998; 27 $\pm$ 0.4 pc, Hipparcos) is an active main sequence dwarf with chromospheric Ca II HK emission (Wright et al. 2004) suggesting an age of 635-1380 Myr adopting the calibration of Donahue et al. (1996) and coronal X-ray emission (Voges et al 1999). The Li I $\lambda$6707 equivalent width (e.g. Soderblom & Mayor 1993) indicates an age consistent with possible kinematic membership in the UMa group (e.g. King et al. 2003) of $\sim$300-500 Myr. We adopt the weighted average of these two age estimates, 700 $\pm 300$ Myr for HD 150706. HD 47875 is a young active X-ray emitting star (Favata et al. 1995) with kinematics consistent with membership in the local association. We assign an age of 30–200 Myr to this source. HD 157664 is a galactic disk field star. Because there are no indications of youth and because volume–limited samples of sun–like stars are most likely 1–3 Gyr old (e.g. Rocha-Pinto et al. 2000), we tentatively assign this age to HD 157664. HD 161897 is similarly inactive and we assign a preliminary age based on its Ca II H & K emission (Wright et al. 2004) of 1–3 Gyr.
Spitzer Space Telescope Data
============================
Next we describe the data acquisition and reduction strategies for each instrument. The derived flux densities for all five sources are presented in Table 2.
IRAC (Fazio et al. 2004) observations in each of the four channels used the 32x32 pixel sub–array mode with an effective integration time of 0.01 sec per image (frame-time of 0.02 sec). The 64 images at each position in the four–point–random dither pattern provided a total integration time of 2.56 sec per channel. We began with the Basic Calibrated Data (BCD) products of the Spitzer Science Center (SSC) S9.1 data pipeline as described in the Spitzer Observer’s Manual v4.0 (hereafter SOM[^1]). Aperture photometry was performed using IDP3 (Schneider & Stobie 2002) v2.9. We used a 2-pixel radius aperture centered on the target and estimated background beyond an 8-pixel radius as the median of $\sim$820 pixels. Background flux was normalized to the area of the target aperture and subtracted from the summed target flux. The final source flux is the median of the 256 measures, corrected from a 2-pixel radius to the 10-pixel radius used for the IRAC instrumental absolute flux calibration. Measurement uncertainty was estimated as the standard error in the mean and added in quadrature to an absolute flux calibration uncertainty of 10%.
Low resolution ($\lambda/\Delta\lambda \ \simeq$ 70 – 120) spectra were obtained with the IRS (Houck et al. 2004) over the entire wavelength range available (5.2–38 $\mu$m) for all validation targets. We present here reduced spectral observations in Figure 1 for HD 105 and HD 150706 only. We used an IRS high–accuracy blue peak–up to acquire the source in the spectrograph slit. Integration times per exposure were 6 sec over the short-low wavelength range (5.2–14.5 $\mu$m), and either 6 sec (HD 150706) or 14 sec (HD 105) over the long-low wavelength range (14.0–38.0 $\mu$m). One cycle, resulting in spectra at two nod positions, was obtained in staring mode for averaging and estimating the noise. The BCDs resulting from the SSC pipeline S9.1 were reduced within the SMART software package (Higdon et al. 2004, in preparation). We used the [*droopres*]{} data products before stray–light and flat–field corrections were applied. Spectra were extracted assuming point source profiles with a fiducial width of 5-6 pixels in the center of the orders but allowing for variable width to account for increasing PSF size as a function of wavelength. Residual emission (mostly due to solar system zodiacal dust) was subtracted using adjacent pixels. The background–subtracted spectrum was divided by the spectrum of a photometric standard star ($\alpha$ Lac) and multiplied by an appropriately-binned template spectrum for this standard provided by the IRS instrument team. Random errors calculated from the difference between the two independent spectra were added in quadrature with an estimated 15 % uncertainty in absolute flux calibration to produce the spectra shown in Figure 1.
MIPS (Rieke et al. 2004) observations were obtained in all three bands using the small field photometry mode with 2 cycles of 3 sec Data Collection Events (DCEs) at 24 $\mu$m and 2 cycles of 10 sec DCEs at 160 $\mu$m, approaching the confusion limit (Dole et al. 2004). At 70$\mu$m, we observed HD 105, HD 150706 and HD 161897 for 2 cycles, HD 157664 for 4 cycles and HD 47875 for 7 cycles all with 10 sec DCEs. After initial processing by the SSC S9.1 pipeline to provide reconstructed pointing information, the MIPS data were further reduced using the MIPS Data Analysis Tool (DAT, v2.71) developed by the MIPS Instrument Team (Gordon et al. 2004). This includes the “enhancer“ portion of the DAT, which corrects for distortion in individual images and combines them onto a sub-sampled tangential plane mosaic. We present images of HD 105 and HD 150706 at 70 and 160 $\mu$m in Figure 2. Aperture photometry was performed in IDP3 with target apertures of 14.99”, 29.70“, and 47.8” at 24, 70, and 160 $\mu$m, respectively. We used background annuli of 29.97–42.46” for 24 $\mu$m, 39.6–79.2” for 70 $\mu$m, and 47.9–79.8” for 160 $\mu$m. The mean background per pixel was scaled to the appropriate aperture size and subtracted from the summed flux inside each aperture. Random uncertainties were determined from the ensemble of measurements for the 24 $\mu$m observations, and from the noise in the background for the mosaicked images at 70 and 160 $\mu$m. These error estimates were added in quadrature with uncertainties in the absolute calibration of 10, 20, and 40 % for 24, 70, and 160 $\mu$m respectively as measured for sources as faint at 60 mJy as 70 $\mu$m. Upper limits were derived for sources not detected based on photometry attempted at the source position estimated from the coordinates given in the image headers. Three times the estimated noise was used to determine an upper limit if the inferred SNR of the measured flux was $<$ 3$\sigma$.
Spectral Energy Distributions of Targets
========================================
We present SEDs for HD 105 and HD 150706 in Figure 1. We have adopted the recommended (SOM) central wavelengths and have not applied color–corrections which are still uncertain and much smaller than the quoted absolute calibration uncertainties. The photospheric emission component was modeled by fitting Kurucz atmospheres including convective overshoot to available $BV$ Johnson, $vby$ Stromgren, $B_TV_T$ Tycho, $H_p$ Hipparcos, $RI$ Cousins, and $J$, $H$, $K_s$ 2MASS photometry. Predicted magnitudes were computed as described in Cohen et al. (2003,and references therein) using the combined system response of filter, atmosphere (for ground-based observations), and detector. The best-fit Kurucz model was computed in a least squares sense with the effective temperature and normalization constant (i.e. radius) as free parameters, \[Fe/H\] fixed to solar metallicity, and surface gravity fixed to the value appropriate for the adopted stellar age and mass. Visual extinction was fixed to $A_V = 0^m$ for stars with distances less than 40 pc, assumed to be within the dust–free Local Bubble, but a free parameter for HD 47875 and HD 157664. The adopted stellar parameters are listed in Table 1.
Emission in excess of that expected from the stellar photospheres is found in two of our five validation targets, HD 105 and HD 150706. The remaining three validation targets have no significant excess emission and were not detected at 70 $\mu$m despite sensitivities comparable to the HD 105 and HD 150706 observations. We place limits on the ratio $L_{IR}/L_{*}$ for these three stars based on the observed upper limits (assuming a dust temperature of $\sim$ 40 K) in Table 1 and discuss the two excess sources in detail.
HD 105 was found to have an IR excess by Silverstone (2000) based on 60 and 90 $\mu$m ISO/ISOPHOT measurements. We confirm that with Spitzer measurements of excess at 70 and 160 $\mu$m but find no obvious excess at $\lambda$ $<$ 35 $\mu$m based on the IRS spectra or IRAC/MIPS photometry. The total flux in the excess calculated via trapezoidal integration from 24 to 1200 microns is $\sim 1 \times 10^{-14}$ W m$^{-2}$, corresponding to $L_{IR}/L_{*}$ = $\sim 3.9 \times 10^{-4}$. Assuming a temperature of $\sim$ 40 K, the estimated solid angle subtended by total effective particle cross-section is $\sim 2
\times 10^{-13}$ sr = 13 AU$^2$ for d = 40 pc. Based on the 24 $\mu$m measurement, we estimate there can be no more than $3 \times 10^{-3}$ as much radiating area at 100 K, and $<$ $4 \times 10^{-5}$ as much at 300 K, as there is at 40 K. HD 150706 has a newly discovered IR excess at 70 $\mu$m, but only an upper limit at 160 $\mu$m and no evidence for excess at $\lambda$ $<$ 35 $\mu$m. The total flux in the excess is $\sim 3 \times 10^{-15}$ W m$^{-2}$, corresponding to fractional IR luminosity $\sim 5.4
\times 10^{-5}$. Assuming a temperature $<$ 84 K, the total effective particle cross-section is $> 0.09$ AU$^{2}$ for d = 27 pc.
Disk Properties and Interpretation
==================================
Following arguments employed for Vega and the other main sequence debris disk archetypes discovered by IRAS (Backman and Paresce 1993), we assume the IR excess emission around HD 105 and HD 150706 is from grains orbiting, and in thermal equilibrium with radiation from, the central stars. Model inner and outer radii for disks containing the cold material around HD 105 and HD 150706 can be calculated with assumptions regarding grain composition, size distributions, and spatial distributions. The lack of distinct mineralogical features in the observed IRS spectra (which would constrain the dust properties) means there can be no unique model but rather a range of models that satisfy the observations. For HD 150706 the single data point for the IR excess translates into a limit that the material must lie farther from the star than $\sim$ 11 AU if it is in the form of “blackbody" grains larger than the longest wavelength of significant emission. Smaller grains would satisfy the same temperature constraint at larger distances from the star. The material around HD 105 is consistent with being distributed in a narrow ring with inner edge R$_{IN}$ at 42 $\pm$ 6 AU and an outer edge at R$_{OUT}$ - R$_{IN}$ $<$ 4 AU if “blackbody" grains are assumed. The ranges result from photometric uncertainty and are independent of the assumed grain surface density radial power law exponent within the range $\Sigma (r) \sim r^{[-2.0,0]}$. If material in the inner “hole" is assumed to have constant surface density with radius (as would be produced through the Poynting–Robertson (P–R) effect), the surface density in the zone at r $<$ R$_{IN}$ is less than $3 \times 10^{-2}$ of the model surface density at r $>$ R$_{IN}$. Another family of models containing intermediate-sized (graybody) grains with emissivity falling as $1/\lambda$ beyond $\lambda = 40 \mu$m possesses inner edges R$_{IN}$ ranging from about 50 to 70 AU and outer edges R$_{OUT}$ ranging from 250 to 1500 AU depending on the assumed radial power law exponent of the surface density distribution.
Given the above results for HD 105 and HD 150706 from simple models with strong (but reasonable) assumptions about the grain properties of the observed disks, we now explore ranges of disk models that are consistent with the data following Wolf & Hillenbrand (2003). For grain compositions we assumed “astronomical" silicate plus graphite in the ISM ratio and surface density distribution $\Sigma(r) \propto r^0$. The mass of the disk was adjusted to match the peak flux in the infrared excess. Parameters such as grain size distribution $n(a) \sim a^{-p}$ power–law exponent, minimum/maximum grain size, and the inner/outer edge of the disk, were varied to find the range of values consistent with the observed spectral energy distribution of HD 105. The models were relatively insensitive to the radial density distribution exponent. The wavelength at which the dust re-emission spectrum begins to depart significantly from the stellar photosphere was used to find the smallest grain size and smallest inner disk radius consistent with the data. These two parameters are degenerate resulting in single grain sizes in the range 0.3, 5, and 8 $\mu$m requiring inner gap sizes of 1000, 120, and 42 AU respectively. Adopting a minimum grain size of 5 $\mu$m and allowing for a grain size distribution up to 100 or 1000 $\mu$m produced lower $\chi^2$ fits and decreased the required inner radius from 120 to 45 AU (32 AU for a$_{MIN}$ $\sim$ 8 $\mu$m or larger). The upper grain size, if one exists, and the outer radius are not well constrained in the absence of sub-millimeter measurements. The mass in grains $<$1 mm for the above models is between 9$\times10^{-8}$ and 4$\times10^{-7}$ M$_\odot$. For HD 150706, we used the measured 70 $\mu$m flux and 160 $\mu$m upper limit and the methodology outlined above to model the disk. Compared to HD 105, we find a smaller minimum grain size (0.3 or 1 $\mu$m) and a narrower disk (inner radius $\sim$ 45 or $\sim$ 20 AU respectively with outer radius $<$ 100 AU). For the remaining three objects, we used the best–fit model for HD 150706, and scaled the results to the observed upper–limits on $L_{IR}/L_*$ to derive the dust mass upper limits given in Table 1.
In the cases of our two disk detections, assuming a grain density [^2] of 2.5 g/cm$^3$ we can calculate the P–R drag timescale. For HD 105 removal of 5 $\mu$m grains occurs $<$ 15 Myr at 45 AU compared to a stellar age of 30 Myr, suggesting that any such small grains are regenerated, perhaps through collisions of planetesimals. However, given the optical depth of the dust (15–300 AU$^2$, the radiating cross–sectional area), the timescale for dust grains to collide is $<$ 10$^6$ yrs suggesting that collisions as well as P–R drag are important in determining the actual size distribution of the dust as well as its radial surface density profile. For HD 150706 the P–R drag timescale for 1 $\mu$m grains at 20 AU is $<$ 1 Myr compared to an age of 700 $\pm$ 300 Myr suggesting regeneration of dust through collisions of planetesimals as in the case of HD 105. In both cases the lack of circumstellar material in the inner disk (expected from a model of P–R drag) suggests: 1) something is preventing dust at 20–45 30 AU from reaching the sublimation radius in the inner disk; and 2) a lack of significant numbers of colliding planetesimals inside of 20–45 AU. This suggests that the inner region is relatively clear of small bodies, consistent with some estimates for the timescale of terrestrial planet formation (e.g. Kenyon and Bromley, 2004). The presence of one or more large planets interior to $\sim$ 20–45 AU may explain the inner edge of the outer dust disk (e.g. Moro–Martin and Malhotra, 2003). The preceeding discussion presumed that the gas to dust mass ratio is $<$ 0.1, for which the dust dynamics are driven by interactions with the radiation field of the central star. Our high resolution IRS observations of the $\sim$ 30 Myr star HD 105, still under analysis, should be sensitive to small amounts of gas between 50–200 K. The presence of a remnant inner gas disk can influence dust migration and produce a ring morphology such as we infer for the dust (Takeuchi and Artymowicz, 2001).
HD 150706, with an age of 700 $\pm$ 300 Myr, is less likely to retain a gas–rich disk than HD 105 (e.g. Zuckerman et al. 1995). However, it does exhibit evidence for a previously undetected dust disk while HD 47875, HD 157664, and HD 161897, ranging in age from 30 Myr to 3 Gyr, do not at levels comparable to the HD 150706 detection (Table 1). The disk surrounding HD 150706 has a hole devoid of dust with inner radius of at least 20 AU, comparable to the disk surrounding HD 105. Perhaps the inner region of this disk is being kept clear by the presence of larger bodies as discussed above. While we cannot draw robust conclusions from these small samples, it is clear that other factors (range in primordial disk properties and/or variations in evolutionary histories?) are required to explain the growing body of debris disk observations besides a simple monotonic decrease in dust mass with age.
Acknowledgements
================
We would like to thank our colleagues at the Spitzer Science Center and members of the instrument teams, for their help in analyzing the Spitzer data. FEPS is pleased to acknowledge support through NASA contracts 1224768, 1224634, and 1224566 administered through JPL.
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[^1]: http://ssc.spitzer.caltech.edu/documents/som/
[^2]: This density is appropriate for solid silicate dust grains which we assume are the result of comminuted Kuiper Belt Object or cometary analogs whose bulk densities are lower.
|
---
abstract: 'In this paper, we consider $N$ non-intersecting Bessel paths starting at $x=a\geq 0$, and conditioned to end at the origin $x=0$. We derive the explicit formula of the distribution function for the maximum height. Depending on the starting point $a>0$ or $a=0$, the distribution functions are also given in terms of the Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively.'
author:
- 'Dan Dai$^{\ast}$ and Luming Yao$^{\dag}$'
title: 'The distribution function for the maximal height of $N$ non-intersecting Bessel paths'
---
*2010 Mathematics Subject Classification.* Primary 60J65, 33C47.\
*Keywords and phrases*: non-intersecting Bessel paths; maximum distribution; multiple orthogonal polynomials; Hankel determinant.
------------------------------------------------------------------------
width 65mm
$\ast$ Department of Mathematics, City University of Hong Kong, Hong Kong.\
Email: `dandai@cityu.edu.hk`
$\dag$ Department of Mathematics, City University of Hong Kong, Hong Kong.\
Email: `lumingyao2-c@my.cityu.edu.hk`
Introduction {#sec:intro}
============
Random walk models play an important role in various areas in physics, chemistry and computer sciences. For example, they are introduced to explain basic concepts in statistical physics [@reif] and stochastic algorithms [@motwani]. When a proper diffusion scaling limit is taken, the random walk models are reduced to Brownian motion models. The standard one-dimensional Brownian motion $\{W_t : t \ge 0\}$ initiated at $x$ is a stochastic process with the following properties (cf. [@borodin]):
- [$W_0=x$ almost surely;]{}
- [$W_t$ is continuous almost surely;]{}
- [for all $0=t_0<t_1<\cdots<t_n$, the increments $W_{t_i}-W_{t_{i-1}}$ are independent and normally distributed, with $\mathbf{E}(W_{t_i}-W_{t_{i-1}})=0$ and $\mathbf{E}(W_{t_i}-W_{t_{i-1}})^2=t_i-t_{i-1}$.]{}
There is a vast literature in the study of Brownian motions, among which people are interested in the maximal height of the outermost path in different non-intersecting Brownian motions. For example, let $0 \leq b_1^{(BE)} (t) < b_2^{(BE)} (t) < \cdots < b_N^{(BE)}(t)$ be $N$ non-intersecting Brownian excursions, i.e. non-intersecting Brownian motions with an absorbing wall located at $x=0$, conditioned to have the same starting point $x = 0$ and return to the origin at the end. The transition probability for a single Brownian motion with an absorbing wall at $x=0$, passing from $x$ to $y$ over the time interval $t$, is given by $$\label{absorb-prob}
p_{abs}(t, y|x)=\frac{1}{\sqrt{2\pi t}}\left(e^{-\frac{(x-y)^2}{2t}} - e^{-\frac{(x+y)^2}{2t}}\right).$$ Let $M$ be a positive constant. Then, the probability that the maximal height of the outermost path $\max\limits_{0<t<1}b_N^{(BE)}(t)$ is less than $M$ is given explicitly as follows: $$\label{BE-case}
\begin{split}
&\mathbb{P}(\max_{0<t<1}b_N^{(BE)}(t)<M) \\
& \qquad =\frac{2^{-N/2} \pi^{2N^2+N/2}}{M^{N(2N+1)}N! \prod_{k=0}^{N-1}(2k+1)!}\sum_{\mathbf x \in \mathbb{Z}^N}(\Delta(\mathbf {x^2}))^2 \left(\prod_{j=1}^N x_j^2 \right) \exp \biggl\{\frac{- \pi^2}{2M^2}\sum_{j=1}^N x_j^2 \biggr\},
\end{split}$$ where $\Delta(\mathbf x)$ is the Vandermonde determinant $$\label{vandermond}
\Delta(\mathbf x)=\prod_{1 \le j <k \le N}(x_k-x_j) \qquad \textrm{with} \quad \mathbf x=(x_1, x_2, \cdots, x_N).$$ The formula was first obtained by Schehr et al. [@gregory2008] through a path integral approach. Later, it was derived by using the Karlin-McGregor formula in Kobayashi et al. [@naoki2008], and by lattice paths in Feierl [@Feierl2012]. One may also consider $N$ non-intersecting Brownian motions $0 \leq b_1^{(R)} (t) < b_2^{(R)} (t) < \cdots < b_N^{(R)}(t)$ with a reflecting wall located at $x=0$, conditioned to have the same starting point $x = 0$ and return to the origin at the end. In this case, the transition probability is $$\label{reflecting-prob}
p_{ref}(t, y|x)=\frac{1}{\sqrt{2\pi t}}\left(e^{-\frac{(x+y)^2}{2t}}+e^{-\frac{(x-y)^2}{2t}}\right),$$ and the probability for the maximal height is given by $$\label{r}
\begin{split}
& \mathbb{P}(\max_{0<t<1}b_N^{(R)}(t)<M) \\
& \qquad =\frac{2^{-N/2} \pi^{2N^2-3N/2}}{M^{N(2N-1)}N! \prod_{k=0}^{N-1}(2k)!}\sum_{\mathbf x \in \{\mathbb{Z}-1/2\}^N}(\Delta(\mathbf {x^2}))^2\exp \biggl\{\frac{- \pi^2}{2M^2}\sum_{j=1}^N x_j^2 \biggr\};
\end{split}$$ see Liechty [@Karl2012 Eq. (1.18)]. It is worth mentioning that the above two probabilities and can be put in terms of Hankel determinants of discrete Gaussian orthogonal polynomial. Based on this observation, Liechty [@Karl2012] applied the Riemann-Hilbert approach to rigorously prove that, in a proper scaling limit as $N \to \infty$, the limiting distribution of the maximal height in both cases converges to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This justifies the formal derivations first given by Forrester, Majumdar and Schehr [@For:Maj:Sch2011].
In this paper, we focus on the distribution of the maximal height of the outermost path in $N$ non-intersecting Bessel paths. Recall that if $\{\mathbf{X}(t): t \ge 0\}$ is a $d$-dimensional Brownian motion, then the diffusion process $$\label{Rt-def}
R(t)=||\mathbf X(t)||_2=\sqrt{X_1(t)^2+\cdots+X_d(t)^2}, \qquad t \geq 0,$$ is called a $d$-dimensional Bessel process or a Bessel process of order $\alpha$ with parameter $\alpha=\frac{d}{2}-1$; see [@borodin; @karatzas; @revuz]. Bessel processes have important application in financial mathematics due to their close relation to financial models such as geometric Brownian motion and Cox-Ingersoll-Ross processes; see Göing-Jaeschke and Yor [@going] and references therein. When $d=1$, or equivalently $\alpha=-\frac{1}{2}$, the Bessel process $R(t)$ reduces to the Brownian motion with a reflecting wall located at the origin . While when $d=3$, or equivalently $\alpha=\frac{1}{2}$, the transition probability of $R(t)$ becomes $$\label{prob-alpha-1/2}
p(t, y|x)=\frac{y}{x}\frac{1}{\sqrt{2\pi t}}\left(e^{-\frac{(x-y)^2}{2t}}-e^{-\frac{(x+y)^2}{2t}}\right),$$ which is closely related to the Brownian motion with an absorbing wall located at the origin ; see Katori and Tanemura [@makoto2004; @makoto2007]. In recent years, there has been considerable interest in the study of non-intersecting (squared) Bessel paths; for example, see [@Del:Kui:Zha2012; @Katori2008; @Kuijlaars2009; @Kuijlaars2011].
Similar to the non-intersecting Brownian motions mentioned above, we consider a model of $N$ non-intersecting Bessel paths: $\{b_j(t)\}_{j=1}^N$. All paths start at $x=a \geq 0$, remain non-negative and are conditioned to end at time $t=1$ at $x=0$. That is, $$\label{bm-model}
\begin{split}
& b_1(0)=b_2(0)=\cdots=b_N(0)=a,\\
& b_1(1)=b_2(1)=\cdots=b_N(1)=0,\\
& 0 \le b_1(t)<b_2(t)< \cdots <b_N(t) \quad \text{for} \quad 0<t<1.
\end{split}$$ The transition probability for a single particle passes form $x$ to $y$ over the time interval $t$ is given by (cf. [@borodin; @ito1974]) $$\begin{aligned}
p(t,y|x)&=\frac{1}{t}\frac{y^{\alpha +1}}{x^{\alpha}}e^{-\frac{x^2+y^2}{2t}}I_{\alpha} \left( \frac{xy}{t} \right), \qquad x>0, \label{p1} \\
p(t,y|0)&=\frac{y^{2\alpha +1}}{2^{\alpha}\Gamma{(\alpha+1)}t^{\alpha+1}}e^{-\frac{y^2}{2t}} \label{p11}\end{aligned}$$ for $y \ge 0$, $t>0$ and $\alpha >-1$, where $I_{\alpha}$ is the modified Bessel function of the first kind $$\label{defI}
I_{\alpha}(z)=\sum^{\infty}_{k=0}\frac{(\frac{z}{2})^{2k+\alpha}}{k! \, \Gamma (k+\alpha +1)}.$$ One can see from that when $d=2(\alpha +1)$ is an integer, the Bessel process is the distance to the origin of a $d$-dimensional standard Brownian motion.
The main results of the paper are stated in the next section.
Statement of results {#sec:results}
====================
The maximal distribution
------------------------
Our first result is the explicit expression of the probability of the maximal height of the outermost path $\max\limits_{0<t<1}b_N(t)$.
\[the:theorem\] For $\alpha > -1$, let $$\label{bessel-zeros}
x_{1,\alpha} < x_{2,\alpha} < \cdots < x_{n,\alpha} < \cdots$$ be the zeros of the Bessel function $J_{\alpha}(x)$. For $ N \ge 1$, $M \ge a$ and the starting point $a>0$, we have $$\label{deter}
\begin{split}
& \mathbb{P}(\max_{0<t<1}b_N(t)<M) \\
& \qquad =c_N(\alpha) M^{-\frac{N(3N+2\alpha+1)}{2}} \det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}\frac{(-1)^{i-1}x_{n, \alpha}^{i+2j+\alpha-3}J_{\alpha}^{(i-1)}(\frac{a}{M}x_{n,\alpha})e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right],
\end{split}$$ where $c_N(\alpha)$ is a constant independent of $M$: $$\label{cn-def}
c_N(\alpha)=\frac{2^{\frac{N(3-N)}{2}} e^{\frac{a^2N}{2}}}{a^{\frac{N(N+2\alpha-1)}{2}}\prod_{j=1}^N \Gamma(j)}.$$ When the starting point is the origin, i.e., $a = 0$, we have $$\label{deter-a=0}
\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)= \widetilde{c}_N(\alpha) M^{-2N(N+\alpha)}\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2i+2j+2\alpha-4}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right],$$ where the constant $\widetilde{c}_N(\alpha)$ is given by $$\label{cn2-def}
\widetilde{c}_N(\alpha)=\frac{2^{2N- \alpha N-N^2}}{\prod_{j=1}^N \Gamma(j) \Gamma(\alpha +j)}.$$
Although the expressions and appear different, they indeed agree with each other. To see it, from properties of the Bessel function [@dlmf Eq. (10.6.2)], one gets the following expression for the higher order derivative of $J_{\alpha}(z)$: $$\label{kthde}
(-1)^n z^n J_{\alpha}^{(n)}(z) = \sum_{k=0}^n c_{n,k} z^k J_{\alpha+k}(z) \qquad \textrm{for } n \in \mathbb{N},$$ with the coefficient $c_{n,n}=1$. We replace $J_{\alpha}^{(i-1)}$ in by the right-hand side of the above formula, then apply row operations to eliminate $J_\nu$ for $\nu < \alpha+i-1$. This gives us $$\begin{split}
\mathbb{P}(\max_{0<t<1}b_N(t)<M) &=\frac{2^{\frac{N(3-N)}{2}} e^{\frac{a^2N}{2}}}{a^{\frac{N(N+2\alpha-1)}{2}}\prod_{j=1}^N \Gamma(j)} M^{-\frac{N(3N+2\alpha+1)}{2} }\\
&\qquad \times \det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}x_{n, \alpha}^{i+2j-3}\frac{J_{\alpha+i-1}(\frac{a}{M}x_{n,\alpha})x_{n, \alpha}^{\alpha}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right].
\end{split}$$ Since $J_\alpha(z) \sim \frac{z^\alpha}{2^\alpha \Gamma(\alpha +1)}$ as $z \to 0$, the determinant in the above formula behaves like $$\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}x_{n, \alpha}^{i+2j-3}\frac{J_{\alpha+i-1}(\frac{a}{M}x_{n,\alpha})x_{n, \alpha}^{\alpha}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right] \sim \det_{1 \le i, j \le N} \left[ \frac{(\frac{a}{2M})^{\alpha+i - 1}}{\Gamma(\alpha+j)}\sum_{n=1}^{\infty}x_{n, \alpha}^{2i+2j-4}\frac{x_{n, \alpha}^{2\alpha}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right]$$ as $a \to 0$. Then, follows from the above two formulae.
When the starting point $a$ is the origin and the number of Bessel paths $N$ is equal to 1, the distribution formula reduces to $$\mathbb{P}(\max_{0 < t < 1} b(t) <M)= \frac{2^{1- \alpha }}{\Gamma(\alpha +1)} M^{-2(\alpha+1)}\sum_{ n =1}^{\infty}\frac{x_{n, \alpha}^{2\alpha} e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n, \alpha})}.$$ This agrees with the results obtained by Pitman and Yor [@jim1999 Eq. (5)].
Relation to multiple orthogonal polynomials
-------------------------------------------
The maximum distributions in and can be put in terms of Hankel determinants of discrete Gaussian orthogonal polynomial; see Liechty [@Karl2012]. For the present model, we have similar results. Depending on the starting point $a>0$ or $a=0$, the distribution functions are given in terms of the Hankel determinants associated with *multiple discrete orthogonal polynomials* of type II or *discrete orthogonal polynomials*, respectively. For more properties of multiple orthogonal polynomials, one may refer to [@Arv:Cou:Van2003] and [@Ismail-book Ch. 23].
In the case where the starting point $a$ is positive, the corresponding two discrete weight functions for the multiple orthogonal polynomial are $$\begin{aligned}
\label{mop-weight}
w_1(x)=\frac{x^{\frac{\alpha}{2}}J_{\alpha}(\frac{a}{M}\sqrt{x})}{J_{\alpha+1}^2(\sqrt{x})}e^{-\frac{x}{2M^2}}, \qquad
w_2(x)&=\frac{x^{\frac{\alpha+1}{2}}J_{\alpha+1}(\frac{a}{M}\sqrt{x})}{J_{\alpha+1}^2(\sqrt{x})}e^{-\frac{x}{2M^2}}, \quad M > a,\end{aligned}$$ which are supported on the nodes $\{x_{n, \alpha}^2 \}_{n=1}^\infty$ with $x_{n, \alpha}$ being zeros of the Bessel function $J_\alpha(x)$ given in . The moments $m_k^{(j)}, j=1,2$, are given by $$\begin{aligned}
\label{mk-def}
m_k^{(1)}=\sum_{x \in \{x_{n, \alpha}^2, n \in \mathbb{N} \}} x^k w_1(x), \qquad
m_k^{(2)}=\sum_{x \in \{x_{n, \alpha}^2, n \in \mathbb{N} \}} x^k w_2(x).\end{aligned}$$ The Hankel determinant is $$\label{mop-hankel}
H_N:= \det\begin{pmatrix}
m_0^{(1)} & m_1^{(1)} & \dots & m_{N-1}^{(1)} \\
m_1^{(1)} & m_2^{(1)} & \dots & m_{N}^{(1)} \\
\vdots & \vdots & \ddots & \vdots \\
m_{n_1-1}^{(1)} & m_{n_1}^{(1)} & \dots & m_{N+n_1-2}^{(1)}\\
m_0^{(2)} & m_1^{(2)} & \dots & m_{N-1}^{(2)} \\
m_1^{(2)} & m_2^{(2)} & \dots & m_{N}^{(2)} \\
\vdots & \vdots & \ddots & \vdots \\
m_{n_2-1}^{(2)} & m_{n_2}^{(2)} & \dots & m_{N+n_2-2}^{(2)}
\end{pmatrix}$$ with $N=n_1+n_2$ and $$n_1=\begin{cases}
\frac{N}{2}, & \mbox{if } N \mbox{ is even,}\\
\frac{N+1}{2},& \mbox{if } N \mbox{ is odd,}
\end{cases}
\quad\text{and}\quad
n_2=\begin{cases}
\frac{N}{2}, & \mbox{if } N \mbox{ is even,}\\
\frac{N-1}{2},& \mbox{if } N \mbox{ is odd.}
\end{cases}$$ In the case where the starting point is located at the origin, the discrete weight function for the orthogonal polynomial is $$\widetilde{w}(x)=\frac{x^{\alpha}e^{-\frac{x}{2M^2}}}{J_{\alpha+1}^2(\sqrt{x})}, \qquad M>0,$$ which is also supported on the nodes $\{x_{n, \alpha}^2 \}_{n=1}^\infty$. The corresponding Hankel determinant is $$\label{op-hankel}
\widetilde{H}_N:= \det\begin{pmatrix}
\widetilde{m}_0 & \widetilde{m}_1 & \dots & \widetilde{m}_{N-1} \\
\widetilde{m}_1 & \widetilde{m}_2 & \dots & \widetilde{m}_{N} \\
\vdots & \vdots & \ddots & \vdots \\
\widetilde{m}_{N-1} & \widetilde{m}_{N} & \dots & \widetilde{m}_{2N-2}
\end{pmatrix},$$ where the moments $\widetilde{m}_k$ are given by $$\label{mk-def-2}
\widetilde{m}_k=\sum_{x \in \{x_{n, \alpha}^2, n \in \mathbb{N} \}} x^k \widetilde{w}(x).$$
Then, we have the following result.
\[thm-op\] Let the Hankel determinants $H_N$ and $\widetilde{H}_N$ be defined in and , respectively. For $a>0$, we have $$\label{hankel}
\mathbb{P}(\max_{0<t<1}b_N(t)<M)=c_N(\alpha)(-1)^{\frac{N(N-3)+2n_1^2}{4}} M^{-\frac{N(3N+2\alpha+1)}{2}} H_N,$$ where $c_N(\alpha)$ is given in . When $a=0$, we have $$\label{hankel-a=0}
\mathbb{P}(\max_{0<t<1}b_N(t)<M)=\widetilde{c}_N(\alpha) M^{-2N(N+\alpha)} \widetilde{H}_N,$$ where $\widetilde{c}_N(\alpha)$ is given in .
Special cases: $ \alpha = \pm \frac{1}{2}$
------------------------------------------
As mentioned in Section \[sec:intro\], the Bessel paths reduce to the Brownian motion with a reflecting wall at the origin when $\alpha=-\frac{1}{2}$. Indeed, recall that the Bessel functions reduce to the elementary functions when $\alpha = \pm \frac{1}{2}$: $$J_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}}\cos x, \quad \quad J_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}}\sin x \quad \textrm{and} \quad I_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}} \cosh x;$$ see [@dlmf Sec. 10.16]. Then, the transition probability becomes . When the starting point $a =0$, the model exactly reduces to the non-intersecting Brownian motion with a reflecting wall at the origin, conditioned to have the same starting point $x = 0$ and return to the origin at the end. Then, the probability for the maximum height given in also reduces to that of the Brownian motion with a reflecting wall given in . To see it, note that the zeroes in are $x_{n, \alpha}=(n-\frac{1}{2})\pi$, $n \in \mathbb{N}$ when $\alpha = -\frac{1}{2}$. Then, the determinant in becomes $$\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}x_{n, \alpha}^{2i+2j-4}\frac{x_{n, \alpha}^{2\alpha}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right]=\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}\frac{\pi^{2i+2j-3}}{2}(n-\frac{1}{2})^{2j+2i-4}e^{-\frac{(n-\frac{1}{2})^2 \pi ^2}{2M^2}} \right].$$ Moreover, as $\Gamma(j) \Gamma(-\frac{1}{2} +j)=2^{2-2j}\sqrt{\pi} \, \Gamma(2j-1)$, the constant in is $$\widetilde{c}_N(-\frac{1}{2})=\frac{2^{2N+1/2 N-N^2}}{\prod_{j=1}^N \Gamma(j) \Gamma(-\frac{1}{2} +j)}=\frac{2^{2N+1/2 N-N^2}}{\pi ^\frac{N}{2}2^{N-N^2}\prod_{j=0}^{N-1}(2j)!}.$$ With the above two formulae, the probability in reduces to $$\begin{aligned}
\mathbb{P}(\max_{0 < t < 1} b_N(t) <M) & =\frac{2^{N/2} \pi^{2N^2-3N/2}}{M^{N(2N-1)} \prod_{j=0}^{N-1}(2j)!}\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}(n-\frac{1}{2})^{2i+2j-4}e^{-\frac{(n-\frac{1}{2})^2 \pi ^2}{2M^2}} \right] \nonumber \\
& = \frac{2^{-N/2} \pi^{2N^2-3N/2}}{M^{N(2N-1)} \prod_{k=0}^{N-1}(2k)!}\det_{1 \le i, j \le N} \left[\sum_{x \in \{ \mathbb{Z}-1/2\}}x^{2i+2j-4}e^{-\frac{x^2 \pi ^2}{2M^2}} \right],\end{aligned}$$ which agrees with .
When $\alpha=\frac{1}{2}$, the Bessel paths are closely related to the Brownian motions with an absorbing wall at the origin; see the relation between their transition probabilities in . Although there is an additional factor $y/x$ in , the maximum distributions are exactly the same. Indeed, since the zeroes in are $x_{n,\alpha}=n\pi$, $n \in \mathbb{Z}$ when $\alpha = \frac{1}{2}$, we adopt similar computations as above and obtain from $$\label{alpha=1/2}
\mathbb{P}(\max_{0<t<1}b_N(t)<M) = \frac{2^{-N/2} \pi^{2N^2+N/2}}{M^{N(2N+1)} \prod_{k=0}^{N-1}(2k+1)!}\det_{1 \le i, j \le N} \left[\sum_{x \in \mathbb{Z} }x^{2i+2j-2}e^{-\frac{x^2 \pi ^2}{2M^2}} \right],$$ which agrees with .
The rest of the paper is organized as follows. In Section \[Sec:preliminary\], we apply the Karlin-McGregor formula and express the probability $\mathbb{P}(\max\limits_{0 < t < 1} b_N(t) <M)$ in terms of a ratio of two determinants. Properties for some general determinants are also discussed. In Section \[sec:derivation\], we derive the asymptotics for the determinants appearing in $\mathbb{P}(\max\limits_{0 < t < 1} b_N(t) <M)$. Then, our main results are proved with the asymptotic results. Finally, in Section \[sec:discussion\], we present some numerical computations and conclude with a summary.
Some preliminary work {#Sec:preliminary}
=====================
The Karlin-McGregor formula
---------------------------
Introduce the notations $$\label{vector-notations}
\begin{split}
& \mathbf{x}=(x_1, x_2, \cdots, x_N), \quad \mathbf{y}=(y_1, y_2, \cdots, y_N), \\
& \mathbf{x}^2=(x_1^2, x_2^2, \cdots, x_N^2), \quad |\mathbf {x}|=\sqrt{x_1^2+x_2^2+\cdots+x_N^2}.
\end{split}$$ and the constant vectors $$\mathbf{a}=(a, a, \cdots, a), \quad \mathbf{0}= (0, 0, \cdots, 0).$$ According to the Karlin-McGregor formula in the affine Weyl alcove of height $M$ (see [@makoto2007]) and the definition of $N$ non-intersecting Bessel processes, we obtain $$\label{p-relation-qqm}
\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)=\lim_{\mathbf{x} \to \mathbf{a}, \mathbf{y} \to \mathbf{0}}\frac{q^M(\mathbf{x},\mathbf{y})}{q(\mathbf{x},\mathbf{y})},$$ with $$\label{qm-def}
q^M(\mathbf{x},\mathbf{y})
=\det_{1 \le i, j \le N}[p^M(1,y_j|x_i)], \qquad q(\mathbf{x},\mathbf{y})=\det_{1 \le i, j \le N}[p(1,y_j|x_i)].$$ Here, $p(t,y|x)$ is the transition probability defined in and , and $p^M(t,y|x)$ is the transition probability of the particle passes from $x$ to $y$ over the time interval $t$ with an absorbing wall at the position $M$.
To find $p^M(t,y|x)$, let us consider the following diffusion equation $$\begin{cases}
\displaystyle u_t=\frac{1}{2} \left( u_{yy}+\frac{2 \alpha+1}{y}u_y \right), \\
\displaystyle\lim_{t \to 0}u(t,y)=\delta(y-x), \\
u(t,M)=u_{y}(t,0)=0;
\end{cases}$$ see similar equations in [@Abraham; @borodin]. Using the method of separation of variables, one gets the unique solution to the above equation $$u(t,y)=\sum^{\infty}_{n=1}\frac{1}{M^2J_{\alpha +1}^2(x_{n, \alpha})}\frac{1}{(xy)^{\alpha}}J_{\alpha}(\frac{x_{n,\alpha}}{M}x)J_{\alpha}(\frac{x_{n,\alpha}}{M}y)e^{-\frac{x^2_{n, \alpha}}{2M^2}t},$$ which is the transition density with respect to the speed measure $m_{\alpha}(dy)=2y^{2 \alpha +1}dy$. Therefore, we obtain $$\label{p2}
p^M(t,y|x)=\sum^{\infty}_{n=1}\frac{2}{M^2J_{\alpha +1}^2(x_{n, \alpha})}\frac{y^{\alpha +1}}{x^{\alpha}}J_{\alpha}(\frac{x_{n,\alpha}}{M}x)J_{\alpha}(\frac{x_{n,\alpha}}{M}y)e^{-\frac{x^2_{n, \alpha}}{2M^2}t}$$ where $J_{\alpha}(z)$ is the Bessel function of the first kind $$\label{besselj}
J_{\alpha}(z)=\sum^{\infty}_{k=0}(-1)^k \frac{(\frac{z}{2})^{2k+\alpha }}{k! \Gamma (k+\alpha +1)}$$ and $x_{n, \alpha}$’s are its zeros given in .
When $\alpha = -\frac{1}{2}$, the zeroes in are $x_{n,-\frac{1}{2}}=(n-\frac{1}{2})\pi$, $n \in \mathbb{N}$. Then, the transition probability becomes $$\label{pp2}
p^M(t, y|x)=\sum_{n=1}^{\infty} \frac{2}{M}\cos{\left(\frac{(n-\frac{1}{2})\pi}{M}x \right)}\cos{\left (\frac{(n-\frac{1}{2})\pi}{M}y \right)}e^{-\frac{(n-\frac{1}{2})^2\pi^2}{2M^2}t}.$$ By the Poisson summation formula, we have $$p^M(t, y|x)=\sum^{\infty}_{n=-\infty}\frac{(-1)^n}{\sqrt{2 \pi t}}\left[e^{-\frac{(y-x-2nM)^2}{2t}}+e^{-\frac{(y+x+2nM)^2}{2t}}\right].$$ It is easily seen from the above formula that $\lim\limits_{M \to \infty} p^M(t, y|x) = p(t, y|x)$. This, together with , gives us the desired result $\lim\limits_{M \to \infty}\mathbb{P}(\max\limits_{0 < t < 1} b_N(t) <M) = 1$.
Some properties for determinants
--------------------------------
We need some preliminary results for determinants.
\[lemma1\] Let $f$ and $g$ be functions with two independent variables, and $\mathbf{n}=(n_1, n_2, \cdots, n_N)$ be an $N$-dimensional vector. Then, we have $$\label{one-two-det}
\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)]=\frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)]\det_{1 \le i,j \le N}[g(y_i,n_j)].$$
From the Leibniz formula for the determinant, we have $$\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)] =
\sum_{\sigma} \left( \operatorname{sgn}\sigma\prod_{j=1}^N f(x_{\sigma (j)},n_j)g(y_j,n_j) \right),$$ where $\sigma$ is a permutations of the set $\{1, 2, \cdots, N\}$. Substituting the above formula into the left-hand side of , we get $$\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)] = \sum_{\mathbf{n} \in \mathbb{N}^N}\sum_{\sigma} \operatorname{sgn}\sigma\prod_{j=1}^N f(x_{\sigma (j)},n_j)g(y_j,n_j).$$ Let us change the index $n_j$ to $n_{\tau(j)}$, where $\tau $ is an arbitrary permutation of the set $\{1, 2, \cdots, N\}$. Since the summation is taken for all $n \in \mathbb{N}^N$, we have $$\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)] = \frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\sum_{\sigma}\sum_{\tau} \operatorname{sgn}\sigma\prod_{j=1}^N f(x_{\sigma (j)},n_{\tau(j)}) \prod_{j=1}^N g(y_j,n_{\tau(j)}).$$ Consider a new permutation $\rho$ defined as $ \rho: = \tau \circ \sigma^{-1}$. Denote $l = \sigma(j)$, then $\tau(j) = \tau (\sigma^{-1}(l)) = \rho(l)$. We change the index in the first product in the above formula and obtain $$\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)] =\frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\sum_{\tau}\sum_{\rho} \operatorname{sgn}\tau \operatorname{sgn}\rho \prod_{l=1}^N f(x_l,n_{\rho(l)}) \prod_{j=1}^N g(y_j,n_{\tau(j)}).$$ Separating the summations about $\tau$ and $\rho$, we get $$\begin{split}
\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)]
&=\frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\sum_{\rho} \operatorname{sgn}\rho\prod_{l=1}^N f(x_l,n_{\rho(l)})\sum_{\tau}\operatorname{sgn}\tau \prod_{j=1}^Ng(y_j,n_{\tau(j)}).
\end{split}$$ With the Leibniz formula for the determinants again, we obtain from the above formula.
This completes the proof of the lemma.
One may take matrix transpose in and rewrite the formula as $$\label{one-two-det-tran}
\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_j,n_i)g(y_i,n_i)]=\frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\det_{1 \le i,j \le N}[f(x_j,n_i)]\det_{1 \le i,j \le N}[g(y_j,n_i)].$$ Moreover, from the proof, it is easy to verify that Lemma \[lemma1\] still holds if an additional factor $\prod_{i=1}^{N} h(n_i)$ appears in the summation. More precisely, we have $$\label{one-two-det-2}
\begin{split}
&\sum_{\mathbf{n} \in \mathbb{N}^N}\left (\prod_{i=1}^{N} h(n_i)\right )\det_{1 \le i,j \le N}[f(x_i,n_j)g(y_j,n_j)]\\
& \qquad \qquad =\frac{1}{N!}\sum_{\mathbf{n} \in \mathbb{N}^N}\left (\prod_{i=1}^{N} h(n_i)\right )\det_{1 \le i,j \le N}[f(x_i,n_j)]\det_{1 \le i,j \le N}[g(y_i,n_j)].
\end{split}$$
To study the limits of determinants in , we will prove one more lemma below. First, let us consider the following Schur function $$\label{schur-def}
s_{\pmb{\mu}}(\mathbf{x})=\frac{\det \limits_{1 \le j,k \le N}[x_j^{{\mu}_k+N-k}]}{\det \limits_{1 \le j,k \le N}[x_j^{N-k}]}$$ with $\mathbf{x}$ being the vector defined in ; see [@macdonald]. Here $\pmb{\mu}=(\mu _1, \mu _2, \cdots, \mu _N)$ is a sequence of non-negative integers in decreasing order $\mu _1 \ge \mu _2 \ge \cdots \ge \mu _N$, which is called a partition. Obviously, both the numerator and denominator are Vandermonde determinants. For example, we have $$\det \limits_{1 \le j,k \le N}[x_j^{N-k}]=\prod_{1 \le j<k \le N}(x_j-x_k)=(-1)^{\frac{N(N-1)}{2}}\Delta (\mathbf x).$$ Note that $s_{\pmb{\mu}}(\mathbf{0})=0$ unless $\pmb{\mu} =\mathbf 0$. Moreover, we have the following property $$\label{schur-limit}
s_{\mathbf{0}}(\mathbf{x}) \equiv 1 \qquad \textrm{and} \qquad s_{\pmb{\mu}}(\mathbf{x}) = O(|\mathbf{x}|) \ \ \textrm{as } \mathbf{x} \to \mathbf{0} \ \ \textrm{if } \pmb{\mu} \ne \mathbf{0}.
$$ Then, we have the following result.
\[lemma2\] Let $f$ be a smooth function, then we have $$\label{fxy-limit}
\det_{1 \le i,j \le N}[f(x_i y_j)]=\left(\prod_{j=1}^N\frac{f^{(j-1)}(ay_{N+1-j})}{\Gamma(j)} \right)\Delta (\mathbf{x-a}) \Delta (\mathbf y)\{1+O(|\mathbf{x-a}|)\}$$ as $\mathbf{x} \to \mathbf{a}$.
We expand $f(x_i y_j)$ at $x_i=a$ into a Taylor series $$f(x_i y_j) = \sum_{k=0}^{\infty}\frac{f^{(k)}(ay_{j})}{k!}(x_i-a)^k y_j^k.$$ With and the above formula, we have $$\begin{split}
\det_{1 \le i,j \le N}[f(x_i y_j)]
&=\sum_{\mathbf k \in \mathbb{N}_0^N}\det_{1 \le i,j \le N} \left[\frac{f^{(k_j)}(ay_{j})}{k_j!}(x_i-a)^{k_j} y_j^{k_j} \right] \\
&=\frac{1}{N!}\sum_{\mathbf k \in \mathbb{N}_0^N}\prod_{j=1}^N\frac{f^{(k_j)}(ay_{j})}{k_j!}\det_{1 \le i,j \le N}[(x_i-a)^{k_j}]\det_{1 \le i,j \le N}[y_i^{k_j}],
\end{split}$$ where $\mathbb{N}_0=\{0, 1, 2, \cdots\}$. We order the index $\mathbf k = (k_1, k_2, \cdots, k_N)$ such that $\{ k_j \}_{j=1}^N$ is a decreasing sequence. This gives us $$\det_{1 \le i,j \le N}[f(x_i y_j)]
=\sum_{0 \le k_N < k_{N-1} <\cdots <k_1}\prod_{j=1}^N\frac{f^{(k_j)}(ay_{j})}{k_j!}\det_{1 \le i,j \le N}[(x_i-a)^{k_j}]\det_{1 \le i,j \le N}[y_i^{k_j}].$$ Changing the index from $\mathbf k$ to $\pmb\mu$ with $\mu_j=k_j-N+j$, we get from the definition of the Schur function in that $$\begin{split}
\det_{1 \le i,j \le N}[f(x_i y_j)] &=\sum_{\pmb{\mu}}\prod_{j=1}^N\frac{f^{(\mu_j+N-j)}(ay_{j})}{(\mu_j+N-j)!}s_{\pmb{\mu}}(\mathbf x-\mathbf a)\Delta(\mathbf{x-a})s_{\pmb{\mu}}(\mathbf y)\Delta(\mathbf y). \end{split}$$ Finally, using the asymptotics of the Schur function in , we obtain .
Derivation of the distribution function {#sec:derivation}
=======================================
In this section, we will prove our main theorem by studying some properties of $ q(\mathbf{x}, \mathbf{y})$ and $ q^M(\mathbf{x}, \mathbf{y})$ defined in .
Asymptotics of $q(\mathbf{x}, \mathbf{y})$ and $q^M(\mathbf{x}, \mathbf{y})$
----------------------------------------------------------------------------
From the transition probabilities $p(t,y|x)$ in and $p^M(t,y|x)$ in , we have $$\begin{aligned}
q(\mathbf{x}, \mathbf{y})
&=&\det_{1 \le i, j \le N} \left[\frac{y_j^{\alpha+1}}{x_i^{\alpha}}e^{-\frac{x_i^2+y_j^2}{2}}I_{\alpha}(x_i y_j) \right], \label{q-det-formula} \\
q^M(\mathbf{x}, \mathbf{y})&=&\det_{1 \le i, j \le N}\left[\sum^{\infty}_{n=1}\frac{2}{M^2J_{\alpha +1}^2(x_{n, \alpha})}\frac{y_j^{\alpha +1}}{x_i^{\alpha}}J_{\alpha}(\frac{x_{n,\alpha}}{M}x_i)J_{\alpha}(\frac{x_{n,\alpha}}{M}y_j)e^{-\frac{x^2_{n, \alpha}}{2M^2}}\right]. \label{qm-det-formula}\end{aligned}$$ To obtain the probability of the maximal height ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ in , we need the asymptotics of the above two functions as $\mathbf{x} \to \mathbf{a}$ and $\mathbf{y} \to \mathbf{0}$.
For $a>0, N \ge 1$, $M \ge a$, as $\mathbf{x} \to \mathbf{a}$ and $\mathbf{y} \to \mathbf{0}$, we have $$\label{q-limit-1}
\begin{split}
q(\mathbf{x}, \mathbf{y}) &
= \frac{a^{\frac{N(N-1)}{2}} e^{-\frac{a^2 N}{2} } }{2^{\frac{N(N-1+2\alpha)}{2}}} \left (\prod_{k=1}^N\frac{y_k^{2 \alpha +1} }{\Gamma(k)\Gamma(\alpha+k)} \right ) \\
& \qquad \qquad \times\Delta(\mathbf{x-a})\Delta(\mathbf{y}^2)\times \{1+O(|\mathbf{x-a}|)+O(|\mathbf{y}|)\}
\end{split}$$ and $$\label{qm-limit-1}
\begin{split}
q^M(\mathbf{x}, \mathbf{y})
&=\frac{1}{N!2^{N(N+\alpha -2)}M^{\frac{N(3N+2\alpha+1)}{2}}}\left(\prod_{k=1}^N \frac{y_k^{2 \alpha +1}}{x_k^{\alpha}(\Gamma (k))^2\Gamma (k+\alpha)}\right)\Delta(\mathbf {x-a})\Delta(\mathbf {y^2})\\
& \quad \times \left (\sum_{\mathbf n \in \mathbf N^N} \left (\prod_{s=1}^N \frac{(-1)^{s-1}x_{n_s, \alpha}^{ \alpha}J_{\alpha}^{(s-1)}(\frac{a}{M}x_{n_s,\alpha})e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})} \right ) \Delta (\mathbf{x_{n, \alpha}})\Delta(\mathbf{x_{n, \alpha}^2}) \right )\\
& \quad \times \{1+O(|\mathbf {x-a}|)+O(|\mathbf y|)\},
\end{split}$$ where $\Delta(\mathbf{x})$ is defined in and $$\mathbf{x_{n, \alpha}} = (x_{n_1, \alpha}, x_{n_2, \alpha}, \cdots , x_{n_N, \alpha}) .$$ When $a=0$, we have $$\label{q-limit-2}
q(\mathbf{x}, \mathbf{y})= \frac{1}{2^{N(N-1+\alpha)}} \left (\prod_{k=1}^N\frac{y_k^{2 \alpha +1} }{\Gamma(k)\Gamma(\alpha+k)} \right )\Delta(\mathbf{x}^2)\Delta(\mathbf{y}^2)\times \{1+O(|\mathbf{x}|)+O(|\mathbf{y}|)\}$$ and $$\label{qm-limit-2}
\begin{split}
q^M(\mathbf{x}, \mathbf{y})
&=\frac{1}{N! 2^{N(2N+2\alpha -3)} M^{2N(N+\alpha)}} \left(\prod_{k=1}^N \frac{y_k^{2 \alpha +1}}{(\Gamma (k))^2 \Gamma (k+\alpha)^2}\right)\Delta(\mathbf {x}^2)\Delta(\mathbf {y^2})\\
&\quad \times \left (\sum_{\mathbf n \in \mathbf N^N} \left (\prod_{s=1}^N \frac{x_{n_s, \alpha}^{2 \alpha}e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})} \right ) \left (\Delta(\mathbf{x_{n, \alpha}^2})\right )^2 \right ) \times \{1+O(|\mathbf {x}|)+O(|\mathbf y|)\}.
\end{split}$$
By the multilinearity of the determinant, we have from : $$\label{q-det-formula-2}
q(\mathbf{x}, \mathbf{y})
=\left(\prod_{k=1}^N\frac{y_k^{\alpha+1}}{x_k^{\alpha}}e^{-\frac{x_k^2+y_k^2}{2}} \right)\det_{1 \le i, j \le N}[I_{\alpha}(x_i y_j)].$$ With the definition of $I_{\alpha}(z)$ in , we get $$\begin{split}
\det_{1 \le i, j \le N}[I_{\alpha}(x_i y_j)] & =\det_{1 \le i, j \le N}\left [\sum_{k=0}^{\infty} \left (\frac{1}{2} \right )^{2k+\alpha}\frac{(x_i y_j)^{2k+\alpha}}{k! \, \Gamma (k+\alpha+1)}\right ] \\ &
= \frac{\prod \limits_{k=1}^N(x_k y_k)^{\alpha} }{2^{\alpha N} } \det_{1 \le i, j \le N}\left [\sum_{k=0}^{\infty} \left (\frac{1}{2} \right )^{2k}\frac{(x_i y_j)^{2k}}{k! \, \Gamma (k+\alpha+1)}\right ] .
\end{split}$$ Moreover, from Lemma \[lemma1\], we obtain the following expression $$\det_{1 \le i, j \le N}[I_{\alpha}(x_i y_j)]=
\frac{ \prod \limits_{k=1}^N(x_k y_k)^{\alpha} }{2^{\alpha N} N!} \sum_{\mathbf k \in \mathbb{N}_0^N}\frac{ 2^{ -\sum \limits_{j=1}^N 2 k_j}}{\prod\limits_{j=1}^N k_j! \, \Gamma(k_j+\alpha+1)} \det_{1 \le i,j \le N}[x_i^{2k_j}]\det_{1 \le i,j \le N}[y_i^{2k_j}] .$$ Since $\displaystyle \det_{1 \le i,j \le N}[x_i^{2k_j}]=0$ and $\displaystyle \det_{1 \le i,j \le N}[y_i^{2k_j}]=0$ for any $k_i = k_j$, we may order the index $\mathbf{k}=(k_1, k_2, \cdots, k_N)$ such that $k_1 > k_2 > \cdots >k_N$. Moreover, set $\mu _j= k_j-N+j$. Obviously, we have $\sum\limits_{j=1}^N k_j=\sum\limits_{j=1}^N \mu_j+N(N-1)/2$. Replacing the indexes $k_j$ by $\mu_j$ in the above formula, we obtain from the above formula and the definition of the Schur function in : $$\label{I-alpha-det}
\det_{1 \le i, j \le N}[I_{\alpha}(x_i y_j)]= \frac{ \prod \limits_{k=1}^N(x_k y_k)^{\alpha} }{2^{\alpha N} } \Delta(\mathbf{x}^2)\Delta(\mathbf{y}^2)\sum_{\pmb{\mu}}\frac{ 2^{ -\sum\limits_{j=1}^N 2\mu_j-N(N-1)}s_{\pmb{\mu}}(\mathbf x^2)s_{\pmb{\mu}}(\mathbf y^2)}{\prod \limits_{j=1}^N(\mu_{N-j+1}+j-1)! \, \Gamma(\mu_{N-j+1}+\alpha+j)}.$$ Recall that $s_{\pmb{\mu}}(\mathbf{0})=0$ unless $\pmb{\mu} =\mathbf 0$. As $\mathbf{y} \to \mathbf{0}$, the leading term in the above summation comes from the index $\pmb{\mu} = \mathbf{0}$. Moreover, since $s_{\pmb{\mu}}(\mathbf x^2)$ and $s_{\pmb{\mu}}(\mathbf y^2)$ always choose the same index $\pmb{\mu}$, we have $$\label{det-schur-limit}
\sum_{\pmb{\mu}}\frac{ 2^{-\sum\limits_{j=1}^N 2\mu_j-N(N-1)}s_{\pmb{\mu}}(\mathbf x^2)s_{\pmb{\mu}}(\mathbf y^2)}{\prod \limits_{j=1}^N(\mu_{N-j+1}+j-1)! \, \Gamma(\mu_{N-j+1}+\alpha+j)} = \frac{2^{- N(N-1)} }{\prod \limits_{j=1}^N(j-1)! \, \Gamma(\alpha+j)} \{1+O(|\mathbf{y}|) \},$$ as $\mathbf{y} \to \mathbf{0}$ uniformly for all $\mathbf{x}$. Since $$\Delta(\mathbf{x}^2) \sim (2a)^{\frac{N(N-1)}{2}} \Delta(\mathbf{x}-\mathbf{a}) \{ 1 + O(|\mathbf{x} - \mathbf{a}|)\}, \qquad \textrm{as } \mathbf{x} \to \mathbf{a} \quad \textrm{if} \quad a > 0,$$ we obtain for $a>0$ and for $a=0$, respectively, by using and the above three formulae.
Next, we consider $q^M(\mathbf{x}, \mathbf{y})$ in . Similar to the above computations, we obtain from the multilinearity of determinants and Lemma \[lemma1\], $$\begin{aligned}
&& q^M(\mathbf{x}, \mathbf{y}) = \frac{2^N \prod \limits_{k=1}^N \frac{y_k^{\alpha+1}}{x_k^{\alpha}} }{N! \, M^{2N}} \nonumber \\
&& \qquad \times \sum_{\mathbf n \in \mathbb{N}^N}\left(\prod_{s=1}^N \frac{e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})}\right) \det_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)\right]\det_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}y_i)\right] . \label{qm-det-formula-two}
\end{aligned}$$ Regarding the determinant $\displaystyle \det_{1 \le i, j \le N}[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)]$, we first expand $J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)$ into a Taylor series at $x_i = a$. Then, from Lemma \[lemma2\], we have $$\label{ja-det-a-asy}
\begin{split}
\det_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)\right]
&=\det_{1 \le i, j \le N}\left [\sum_{m=0}^{\infty} \frac{J_{\alpha}^{(m)}(\frac{a}{M}x_{n_j,\alpha})}{m!}(\frac{x_{n_j,\alpha}}{M})^m(x_i-a)^m\right]\\
&=\left(\prod_{j=1}^N\frac{J_{\alpha}^{(j-1)}(\frac{a}{M}x_{n_j,\alpha})}{(j-1)!M^{j-1}}\right)\Delta(\mathbf {x-a})\Delta(\mathbf{x_{n, \alpha}})\times\{1+O(|\mathbf{x-a}|)\},
\end{split}$$ as $\mathbf{x} \to \mathbf{a}$. If the starting point $a=0$, the expansion near $x = a$ is different as $J_\alpha(x)$ has an algebraic singularity at the origin; cf. . In this case, we adopt similar computations in deriving to obtain $$\begin{aligned}
&& \displaystyle \det_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)\right] =\det_{1 \le i, j \le N}\left[\sum_{m=0}^{\infty}(-1)^m\frac{(\frac{x_{n_j, \alpha}}{2M}x_i)^{2m+\alpha}}{m! \Gamma (m+\alpha+1)}\right] \nonumber \\
&& \displaystyle \qquad \qquad = \frac{\prod\limits_{k=1}^N (x_{n_k, \alpha}x_k)^{\alpha}}{(2M)^{\alpha N}}
\Delta(\mathbf{x}^2) \Delta (\mathbf{x^{\rm 2}_{n, \alpha}}) \nonumber \\
&& \qquad \qquad \qquad \times \sum_{\pmb{\mu}}\frac{ (-1)^{\sum\limits_{j=1}^N \mu_j+N(N-1)/2} (2M)^{ -\sum\limits_{j=1}^N 2\mu_j-N(N-1)}s_{\pmb{\mu}}(\mathbf x^2)s_{\pmb{\mu}}(\mathbf{x^{\rm 2}_{n, \alpha}} )}{\prod \limits_{j=1}^N(\mu_{N-j+1}+j-1)! \, \Gamma(\mu_{N-j+1}+\alpha+j)}.
\end{aligned}$$ Similar to , the above formula yields $$\label{ja-det-0-asy}
\det_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}x_i)\right] = \frac{\prod\limits_{k=1}^N (x_{n_k, \alpha}x_k)^{\alpha}}{(2M)^{N ( N -1 + \alpha ) }}
\Delta(\mathbf{x}^2) \Delta (\mathbf{x^{\rm 2}_{n, \alpha}}) \prod \limits_{j=1}^N \frac{ (-1)^{j-1} }{(j-1)! \, \Gamma(\alpha+j)} \{1+O(|\mathbf{x}|) \},$$ as $\mathbf{x} \to \mathbf{0}$. Of course, similar formula also holds for $\det\limits_{1 \le i, j \le N}\left[J_{\alpha}(\frac{x_{n_j,\alpha}}{M}y_i)\right]$ when $\mathbf{y} \to \mathbf{0}$.
Finally, with , and , we obtain the desired asymptotics in for $a>0$ and for $a =0$, respectively.
Proof of Theorem \[the:theorem\]
--------------------------------
The asymptotics for $q(\mathbf{x}, \mathbf{y})$ and $q^M(\mathbf{x}, \mathbf{y})$ in the previous are enough for us to derive the probability for the maximum height.
Recall the expression of ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ in terms of $q(\mathbf{x},\mathbf{y})$ and $q^M(\mathbf{x},\mathbf{y})$ in . When $a>0$, using the asymptotics obtained in and , we have $$\label{distribution}
\begin{split}
\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)
&= c_N(\alpha) M^{-\frac{N(3N+2\alpha+1)}{2}}\frac{1}{N!}\\
& \hspace{-1cm} \times \sum_{\mathbf n \in \mathbb{N}^N} \left [ \left( \prod_{s=1}^N \frac{(-1)^{s-1}x_{n_s, \alpha}^{ \alpha}J_{\alpha}^{(s-1)}(\frac{a}{M}x_{n_s,\alpha})e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})} \right) \Delta(\mathbf{x_{n, \alpha}})\Delta(\mathbf{x_{n, \alpha}^2})\right],
\end{split}$$ where $c_N(\alpha)$ is given in . Now, let us focus on the summation in the above formula and put it into a determinantal form. From the definition of Vandermonde determinants, we have $$\Delta(\mathbf{x_{n, \alpha}})\Delta(\mathbf{x_{n, \alpha}^2}) = \det_{1 \le i, j \le N}[x_{n_i, \alpha}^{j-1}]\det_{1 \le i, j \le N}[x_{n_i, \alpha}^{2(j-1)}].$$ Using , we get $$\begin{split}
& \sum_{\mathbf n \in \mathbb{N}^N} \left(\prod_{s=1}^N \frac{(-1)^{s-1}x_{n_s, \alpha}^{ \alpha}J_{\alpha}^{(s-1)}(\frac{a}{M}x_{n_s,\alpha})e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})} \right) \Delta(\mathbf{x_{n, \alpha}})\Delta(\mathbf{x_{n, \alpha}^2}) \\
& \qquad = N! \ \sum_{\mathbf n \in \mathbb{N}^N} \left(\prod_{s=1}^N \frac{(-1)^{s-1}x_{n_s, \alpha}^{ \alpha}J_{\alpha}^{(s-1)}(\frac{a}{M}x_{n_s,\alpha})e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})} \right) \det_{1 \le i, j \le N}\left[x_{n_i, \alpha}^{i+2j-3}\right].
\end{split}$$ Moving the preceding factors into the determinant and using the multilinearity of determinant, we obtain from the above formula.
When $a=0$, from and , the distribution formula can be written as $$\label{distribution-2}
\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)=\widetilde{c}_N (\alpha) M^{-2N(N+\alpha)}\frac{1}{N!}\sum_{\mathbf n \in \mathbb{N}^N} \left [\left (\prod_{s=1}^N \frac{x_{n_s, \alpha}^{2\alpha} e^{-\frac{x_{n_s, \alpha}^2}{2M^2}}}{J_{\alpha +1}^2(x_{n_s, \alpha})}\right ) (\Delta(\mathbf{x_{n, \alpha}^2}))^2 \right ],$$ where $\widetilde{c}_N (\alpha)$ is given in . With the aid of again, we obtain from above formula by similar computations.
This completes the proof of Theorem \[the:theorem\].
Proof of Theorem \[thm-op\]
---------------------------
Before proving Theorem \[thm-op\], we need one more property for the derivatives of the Bessel function $J_\alpha(z)$.
For $j \in \mathbb{N}$, we have the following differential relations $$\begin{aligned}
z^{2j-1}J_{\alpha}^{(2j-1)}(z)&=[(-1)^j z^{2j-1}+P_{2j-3}(z)]J_{\alpha+1}(z)+Q_{2j-2}(z)J_{\alpha}(z), \label{bessel-derivative-1} \\
z^{2j}J_{\alpha}^{(2j)}(z)&=\widetilde P_{2j-1}(z)J_{\alpha+1}(z)+[(-1)^j z^{2j}+\widetilde Q_{2j-2}(z)]J_{\alpha}(z), \label{bessel-derivative-2}
\end{aligned}$$ where $P_j(z)$, $Q_j(z)$, $\widetilde P_j(z)$ and $\widetilde Q_j(z)$ are polynomials of degree $j$. Moreover, $P_{2j-1}(z)$ and $\widetilde P_{2j-1}(z)$ are odd functions, while $Q_{2j}(z)$ and $\widetilde Q_{2j}(z)$ are even functions.
We prove this lemma by induction. It is well-known that the Bessel functions satisfy the following recurrence relations $$\begin{aligned}
\label{dr2}
zJ_{\alpha}'(z)=zJ_{\alpha-1}(z)-\alpha J_{\alpha}(z), \qquad
zJ_{\alpha}'(z)=-zJ_{\alpha+1}(z)+ \alpha J_{\alpha}(z);
\end{aligned}$$ see [@dlmf Eq. (10.6.2)]. From the above formulae, it is easy to see $$z^2J_{\alpha}''(z)=z J_{\alpha+1}(z)+(-z^2 + \alpha^2 - \alpha)J_{\alpha}(z).$$ The above formulae show that the lemma holds for $j=1$.
Assume the lemma holds for $j=k$. Then, we have $$\label{dr1}
z^{2k}J_{\alpha}^{(2k)}(z)=\widetilde P_{2k-1}(z)J_{\alpha+1}(z)+[(-1)^k z^{2k}+\widetilde Q_{2k-2}(z)]J_{\alpha}(z).$$ Taking derivative on both sides of the above formula and using again, we obtain $$\label{dr3}
z^{2k+1}J_{\alpha}^{(2k+1)}(z)=[(-1)^{k+1}z^{2k+1}+P_{2k-1}(z)]J_{\alpha+1}(z)+Q_{2k}(z)J_{\alpha}(z)$$ with $$\begin{aligned}
&P_{2k-1}(z)=-(2k+\alpha+1)\widetilde P_{2k-1}(z)+z\widetilde P'_{2k-1}(z)-z\widetilde Q_{2k-2}(z),\\
&Q_{2k}(z)=\alpha (-1)^{k}z^{2k}+z\widetilde P_{2k-1}(z)+(\alpha -2k)\widetilde Q_{2k-2}(z)+z\widetilde Q'_{2k-2}(z).
\end{aligned}$$ So, holds for $j = k+1$. Differentiating one more time, we also get for $j = k+1$ through similar computations.
This finish the proof of the lemma.
With the above preparation, we prove our second theorem below.
*Proof of Theorem \[thm-op\].* Let us focus on the determinant in . The entries in the first row are given by $$\label{hankl-row1}
\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2j+ \alpha-2}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})}J_{\alpha}(\frac{a}{M}x_{n,\alpha}) .$$ For the second row, by using , we replace $J_{\alpha}'(\frac{a}{M}x_{n,\alpha})$ with $J_{\alpha+1}(\frac{a}{M}x_{n,\alpha}) $ and $J_{\alpha}(\frac{a}{M}x_{n,\alpha})$. This gives us $$\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2j+ \alpha-2}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})}\left[x_{n, \alpha}J_{\alpha+1}(\frac{a}{M}x_{n,\alpha})-\frac{\alpha M}{a}J_{\alpha}(\frac{a}{M}x_{n,\alpha}) \right] .$$ Note that the coefficient of $J_\alpha$ term is independent of $n$. Applying a row operation, we eliminate the $J_\alpha$ term in the second row and obtain $$\label{hankl-row2}
\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2j+ \alpha-2}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})}x_{n, \alpha}J_{\alpha+1}(\frac{a}{M}x_{n,\alpha}).$$ Regarding the third row, we use to replace $x_{n,\alpha}^2 J_\alpha''(\frac{a}{M}x_{n,\alpha})$ and obtain $$\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2j+ \alpha-2}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})}\left[ d_1 x_{n, \alpha}J_{\alpha+1}(\frac{a}{M}x_{n,\alpha}) + \left( -x_{n, \alpha}^2 + d_0 \right) J_{\alpha}(\frac{a}{M}x_{n,\alpha})\right].$$ Note that $d_0$ and $d_1$ in the above formulae are two $n$-independent constants. Then, with the first two rows given in and , we again apply row operations to simplify the entries in the third row as $$\sum_{n=1}^{\infty}\frac{x_{n, \alpha}^{2j+ \alpha-2}e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \left[-x_{n, \alpha}^2J_{\alpha}(\frac{a}{M}x_{n,\alpha}) \right] .$$ One may repeat the above procedures by first replacing the $(-1)^{i-1} x_{n,\alpha}^{i-1} J_\alpha^{(i-1)}(\frac{a}{M}x_{n,\alpha})$ term with $J_{\alpha+1}$ and $J_{\alpha}$ functions, then apply row operations to simplify the entries. Here, we also need the properties that the polynomials $P_{2k-1}$, $\widetilde P_{2k-1}$ and $\widetilde Q_{2k}$, $Q_{2k}$ in and are even or odd functions. Therefore, the determinant in can be put into a form as follows: $$\label{moment}
\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}\frac{(-1)^{i-1}x_{n, \alpha}^{i+2j+\alpha-3}J_{\alpha}^{(i-1)}(\frac{a}{M}x_{n,\alpha})e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right] =\det_{1 \le i, j \le N} \left[\sum_{n=1}^{\infty}x_{n, \alpha}^{2j + \alpha-2}\frac{h_i(x_{n, \alpha}) e^{-\frac{x_{n, \alpha}^2}{2M^2}}}{J_{\alpha+1}^2(x_{n, \alpha})} \right],$$ where $$h_i(x)= \begin{cases}
(-1)^{\frac{i-1}{2}} x^{i-1} J_{\alpha}(\frac{a}{M}x),& \mbox{if } i \mbox{ is odd}, \\
(-1)^{\frac{i-2}{2}} x^{i-1} J_{\alpha+1}(\frac{a}{M}x),& \mbox{if } i \mbox{ is even}.
\end{cases}$$ Note that the entries in the right-hand side of are indeed the moments $m_{k}^{(i)}, \ i=1,2,$ defined in . Finally, we obtain by interchanging rows in , such that all the ones containing $J_{\alpha}(\frac{a}{M}x_{n,\alpha})$ are moved to the top block, while the ones with $J_{\alpha+1}(\frac{a}{M}x_{n,\alpha})$ are moved to the bottom block.
For the case $a=0$, it is straightforward to see that the entries in the determinant of are entries associated with the discrete orthogonal polynomials defined in . Then, the formula follows immediately.
This completes the proof of Theorem \[thm-op\].
Numerical simulations and conclusions {#sec:discussion}
=====================================
Numerical simulations
---------------------
Based on the explicit formulae in Theorem \[the:theorem\] and \[thm-op\], we compute ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ numerically. We set the number of Bessel paths $N$ to be $10$ and the order of Bessel functions $\alpha$ to be 1 as an illustration.
As expected, when the starting point $a$ is fixed, the probability ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ is positive and increase with respect to $M$. When $M$ is large, the probability tends to 1; see Figure \[fig1\] and Table \[table:M\]. When the upper constraint $M$ is fixed, the probability decreases with respect to $a$, and tends to 0 when $a$ is large; see Figure \[fig2\] and Table \[table:a\].
\[h\]
![The evolution of the probability ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ with respect to $a$ when $N=10, \alpha = 1$ and $M=5$.[]{data-label="fig2"}](M.pdf){width="3in"}
![The evolution of the probability ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ with respect to $a$ when $N=10, \alpha = 1$ and $M=5$.[]{data-label="fig2"}](a.pdf){width="3in"}
M $\displaystyle \mathbb{P}(\max_{0<t<1}b_N(t)<M)$
------ --------------------------------------------------
3.25 $2.3576\times 10^{-10}$
3.5 $9.82109\times 10^{-8}$
3.75 0.000233222
4 0.0183778
4.25 0.200657
4.5 0.604697
4.75 0.894626
5 0.984423
5.25 0.998446
5.5 0.999666
5.75 0.999961
6 0.999919
\[table:M\]
$a$ $\displaystyle \mathbb{P}(\max_{0<t<1}b_N(t)<M)$
------ --------------------------------------------------
1 0.984373
1.25 0.974007
1.5 0.954902
1.75 0.919556
2 0.856389
2.25 0.750728
2.5 0.592312
2.75 0.391808
3 0.194887
3.25 0.0617876
3.5 0.00968417
3.75 0.000500336
\[table:a\]
Conclusions
-----------
In summary, we derive the probability ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ for the maximum height of $N$ non-intersecting Bessel paths exactly. The probability can be put in terms of Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively, depending on the starting point $a>0$ or $a=0$. It is well-known that the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems is a powerful tool to study asymptotic problems related to (multiple) orthogonal polynomials. Therefore, this relation is important for us to derive the asymptotics of ${\displaystyle}\mathbb{P}(\max_{0 < t < 1} b_N(t) <M)$ as the number of paths $N \to \infty$. With suitable scaling, we expect the limiting distribution converges to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble due to its close relation to the Airy$_2$ process. We will leave the asymptotic study in a forthcoming publication.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11300115, CityU 11303016), and by grants from City University of Hong Kong (Project No. 7004864, 7005032).
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|
---
abstract: |
RESUMEN\
\
ABSTRACT\
author:
- '[**Carlos Echeverria$^{1,2}$[[^1]]{}, Wilmer Olivares-Rivas$^{2}$, and Kay Tucci$^{3}$**]{}'
title: ' SIMULACIÓN POR AUTÓMATAS CELULARES DE LAS CONFORMACIONES ESPACIALES DE POLIELECTROLITOS.'
---
.\
Introduction
============
Polyelectrolyte (PE) solutions are systems widely studied since they show properties that are of fundamental interest for applications in health science, food industry, water treatment, surface coatings, oil industry, among other fields. In fact, one of the problems found in genetic engineering in the appearance of conformational changes of the ADN molecule, which is a charged polyelectrolyte.[@YTY1].\
Here we study an infinite dilution polyelectrolyte solution, so that, the interaction among polyelectrolyte macromolecules are negligible. We model the polyelectrolyte as having dissociable functional groups that give rise to charged sites and counter-ions in aqueous solution. The long range interactions arising from these multiple charges are responsible for their macroscopic complex properties, which can not be explained by regular polymer theories. The spatial structures of these materials in solution have been studied extensively, particularly with a scaling theory[@DCR95; @DRO1] that are not appropriate for highly charged PE. The first simulations carried out for a single chain predicted the formation of groups of monomers, as the fraction of charged monomers increased. Such structures are known as pearl necklaces. The size of such pearls and the distance between then is determined by the balance between the electrostatic repulsion and steric effects.\
These pearl necklace structures have also been found in Molecular Dynamics (MD) simulations[@LH1; @MHK1; @MK1; @LH2; @LH3; @LHK1; @LHK2]. In this paper we are interested in the application of the much simpler Cellular Automata simulation to characterize the main features of a polyelectrolyte that could be responsible for such conformations. The complete simulation of this complex system requires the description of a model in terms of potential or forces. In the MD simulations of Limbach and Holm [@LH1], the monomers are connected along the chain by the finite extendible nonlinear elastic (FENE) bond represented by the potential energy.
$$U_{E}(r) = -\frac{1}{2}K_{ E}R_0^2\ln \left(1-\frac{r^2}{R_0^2} \right)$$
where $r$ is the distance between two bonded monomers, $K_{E}$, is the elastic bond constant, $\sigma$ is the monomer diameter, $k_B$ is Boltzmann’s constant, $T$ is the absolute temperature and the parameter $R_0$ represents the maximum extension of the bond between two neighbor monomers. Two charged sites i and j, with charges $eq_{i}$ and $eq_{j}$, a distance $r_{ij}$ apart, interact with the electrostatic Coulomb potential $$U_{C}(r_{ij}) = k_B T \frac{\lambda_Bq_iq_j}{r_{ij}}.$$ This potential is weighed by the Bjerrum length $\lambda_B=e^2/[4\pi \epsilon_o\epsilon_sk_BT]$, where $\epsilon_s$ and $\epsilon_o$ are the solvent permitivity and the vacuum permitivity respectively, $e$ is the electric charge unit. The parameter $\lambda_B$ is a measure of the strength of the electrostatic force as compared to the kinetic energy. The length ratio $\sigma/\lambda_B$ is a measure of the reduced temperature $ T/[e^2/(4\pi \epsilon_o\epsilon_s k_B\sigma )]$.
The short range and van der Waals interaction between any two particles or monomers is represented in the MD simulation by a typical truncated Lennard-Jones potential $$U_{LJ} (r_{ij}) = \left\lbrace
\begin{array}{lll}
4\epsilon \left[ \left( \frac{\sigma}{r_{ij}} \right)^{12} - \left(
\frac{\sigma}{r_{ij}} \right)^{6}
\right] + \epsilon_{cut} & r_{ij} < R_{c},\\
0 & r_{ij} > R_{c}
\end{array}\right.$$ where $\epsilon$ is the potential energy well depth and $\epsilon_{cut}$ is the cut off energy. This potential prevents the superposition of the bonding monomers. Counter-ions interact via a purely repulsive LJ interaction with $R_c= 2^{1/6}\sigma$.
Cellular Automata Model
=======================
Even though in the Cellular Automata simulation we do not use any form of potential energies or forces in an explicit manner, the [*rules*]{} for the movement of the different particles must be inspired on a model defined in terms of such potentials. We therefore establish our rules based on the essence of the previous three potentials.\
Polyelectrolyte Chain Construction Rules
----------------------------------------
The polymer is constructed by placing the monomers in a three dimensional cubic network of side $L$ and volume $L^{3}$. Each cell then has $26$ neighbors and represents a monomer with a monovalent charge $+1$, $-1$, or, for a neutral monomer, $0$, as depicted in Fig.(1) in two dimensions. Out of a total of $N$ monomers in the chain, it is assumed that a given fraction $f$ is charged. The polymer is then constructed by randomly binding consecutive sites in the network. Each monomer could be charged or uncharged, with a distribution chosen randomly. A key step on the construction of the polyion is the spatial location of the dissociated counter-ions. We place the counter-ions also randomly in free cells in the volume around the charged monomer within a distance $\lambda_B$, that is, in a volume $(2\lambda_B)^3$ centered on the charged site. The use of the Bjerrum parameter, which is related to the quality of the solvent, ensures the conservation of the total electroneutrality but gives a spatial distribution of counter-ions around the charged sites.\
So, each monomer $i$ of the system is represented in a $Nx4$ matrix where each element $M(m_x,m_y,m_z,q_i)$ indicates the $i^{th}$ polymer with charge $q_i$, that could be $-1, +1$ or $0$, at the positions $x,y,z$ given by the cell label $[m_x,m_y,m_z]$. The counter-ions with opposite charge are represented by a similar $fNx4$ matrix $M_C$.\
For simplicity we chose monomers with dissociable groups that give a site with a positive charge. We then set the following displacement rules for the different particles:\
Displacement Rules
------------------
[ ***Neutral Monomer Particle***]{}
1. Locate the unoccupied nearest neighbor sites. The new position where a move could be acceptable are those where it does not superimposed with any other particle in the system and where no bond is broken.
2. Count the amount of monomer particles around the current position and around every unoccupied neighbor, within a cube of volume $(2L)^3$ centered on it.
3. Move the test neutral monomer to the position that has the higher amount of monomers around it, including the current one if it were the case.
[***Positively Charged Monomer Particle*** ]{}
1. As before, locate the unoccupied acceptable nearest neighbor sites.
2. Count the amount of charge around the current position and around every unoccupied neighbor, within a cube of volume $(2\lambda_B)^3$ centered on it.
3. Move the test positive charged monomer to the position that has the lowest positive charge around it, including the current one if it were the case.
[***Negatively charged counter-ion*** ]{}
1. Move randomly to an unoccupied site within a cube of volume $(2\lambda_B)^3$ centered on the accepted new position of the corresponding positive monomer in the polymeric chain.
RESULTS AND DISCUSSION
=======================
We denote the position of monomer i with $r_i$ and the distance between two particles i and j with $r_ij$. The center of mass for the chain is then $R_s =
\frac{1}{N}\sum_{i=1}^N r_i$. and the center of mass coordinates are $x_i = r_i - R_s$. A parameter that is useful in the study of the spatial conformations of a polymer is the radius of gyration $R_G$, defined as $$R_g^2 = \frac{1}{N}\sum_{i=1}^N x_i^2$$
According with our construction and movement rules, we can vary the length of the chain $L$, or the number of monomers $N$, satisfying $L=N\sigma$ and the number of charged monomers, $N_c=fN$. We also take as an independent variable the parameter $\lambda_B$, which determines the number of cells, of size $\sigma$, where the range of the electrostatic attraction between a charged monomer and contra-ion extends. The charge distribution of the sequence of charged and uncharged monomers is determined randomly depending on the initial random seed.\
In Fig.(2) we show some of the equilibrium conformations obtained for a polyelectrolyte with $N=100$ monomers, with $\lambda_B=3\sigma$, for several charge fractions. The line represents the polyelectrolyte, the filled black bonded circles represent the charged monomers. The counter-ions in solution are represented by open non-bonded circles. For clarity, the neutral monomers are not shown. For a low charge fraction of $f=0.1$, the polyion presents an elongated string appearance. As the charge fraction is increased the polyion contracts and some groups of monomers tend to form clusters, so that, already for $f=0.5$, it shows the locally collapsed structures known as pearl necklace. These results are very similar to those obtained by the Molecular Dynamics simulations of Limbach and Holm [@DRO1; @LH1]\
It is important to notice that for a given total charge, determined by the fraction $f$, different distributions of the charged monomers give different conformations. To study this behavior, we have carried out simulations with several initial seeds for $N=100$ monomers, with a fixed fraction $f=0.5$ and a range parameter of $\lambda_B/\sigma = 3$. In Fig.(3), we show the temporal evolution of the radius of gyration $R_g$. In all cases the conformations change from the initial given $R_g$ value to a plateau value that correspond to the equilibrium structures. Fig.(3) clearly show that the plateau $R_g$ values, and thence, the final conformations depend on the charge distribution. In Fig.(4) we show snapshots of final structures corresponding with the different seeds of Fig.(3).\
As we can see from Fig.(4), the formation of the pearl necklace structures is independent of the charge sites distribution, for a given $f$ and $\lambda_B$. These clusters seem to be stabilized by the counter-ions as a consequence of the electroneutrality condition that we force to be satisfied. This is so because in our model the degree of freedom of the mobile counter-ions is much higher that that of the monomers tied to the chain. The strong repulsions that originate by the formation of clusters of neutral and positively charged monomers is compensated by the counter-ion cloud that forms around it.
In order to study the reproducibility of the configurations found, we carried out a large number of simulation runs, for fixed values of $N=150$, $f=0.5$, $\lambda_B/\sigma=3$ and $9$ and for the same initial charge distribution. In Fig.(5) we show the histograms for the frequency with which a given value of $R_g$ appears. We can see that the distribution of the $R_g$ is very closely a gaussian with a reasonably low dispersion of less than a $10\%$ about the mean value.\
In Fig.(5B) we can see that similar distribution is obtained when the range parameter $\lambda_B/\sigma$ increases to a value of $9$. We have further tested the effect of the parameter $\lambda_B$ by generating structures for different values of it. In Fig.(6) we show some equilibrium conformations for $\lambda_B/\sigma$ equal to A) 3,B) 6,C) 9 and D) 12. Here we use a large charge fraction $f=0.5$ and $N=100$ and we used the same initial charge distributions in all cases. As we can observe as Bjerrum length increases the final polyelectrolyte structures become more compact. The number or pearls or conglomerates is higher for the lower values of $\lambda_B$, a result similar to that obtained by MD simulations of Limbach y Holm [@LH1]-[@LHK2]
Conclusions
===========
With the simple technique described here, we were able to reproduce the complex structure of model polyelectrolytes that fare very well with those predicted by the more sophisticated Molecular Dynamic and Monte Carlo simulations[[@DRO1; @LH1]]{}. We even predict situations with single conglomerates and with pearl necklace type conglomerates. We thus show the potentiality of the Celular Automata in the simulation of the trends in the formation of the various types of spatial conformations of polyelectrolytes. We remark on the importance of the charge distribution once the fractional charge is fixed.
Acnowledgement
==============
This work was supported in part by the grant 04-005-01 from the Decanato de Investigación of Universidad Nacional Experimental del Táchira, in part by the grant G-9700741 of FONACYT, and in part by the grant C-1279-0402-B from Consejo de Desarrollo Cientíco Humanstico y Tecnológico of Universidad de Los Andes.\
Numerical calculations were carried out at the computer center CeCalcULA.\
[25]{} A.V. Dobrynin, R.H. Colby and M. Rubinstein,[*Macromolecules* ]{},[**28**]{},1859,(1995). A. V. Dobrynin; M. Rubinstein; S. P. Obukhok,[*Macromolecules*]{},[**29**]{},2974,(1996). Y. Yasmasaki, Y. Teramoto, and K. Yoshikawa,[*Biophys. J.*]{},[**[80]{}**]{},2974,(2001). H. J. Limbach y C. Holm,[*J. Phys. Chem. B*]{},[**107**]{},8041-8055,(2003). U. Micka; C. Holm; K. Kremer,[*Langmuir*]{},[**15**]{},4033,(1999). U. Micka; K. Kremer,[*Europhys. Lett.*]{},[**49**]{},189-195,(2000). H. J. Limbach y C. Holm, [*J. Chem. Phys.*]{},[**114**]{},9674-9682,(2002). H. J. Limbach y C. Holm,[*Comput. Phys. Commun.*]{},[**147**]{},321-324,(2002). H. J. Limbach y C. Holm; K. Kremer, [*J. Phys: Condens. Matter*]{},[**15**]{},S205-S211,(2003). H. J. Limbach y C. Holm; K. Kremer, [*Europhys. Lett.*]{},[**60**]{},566-572,(2002).
[^1]: Corresponding author: e-mail: cecheve@ula.ve
|
---
abstract: |
Let $\xi$ be an algebraic number and let $\alpha,\beta\in \mathbb
Q[\xi]$. An explicit formula for the coordinates of the product $\alpha\beta$ is given in terms of the coordinates of $\alpha$ and $\beta$ and the companion matrix of the minimal polynomial of $\xi$. The formula as well as its proof extend to fairly general simple integral extensions.
---
$$\text{\bf\Large A Closed Formula for the Product}$$ $$\text{\bf\Large in Simple Integral Extensions}$$
[\
Av Corrientes 3985 6A, (1194) Buenos Aires, Argentina\
email: nguersenz@fibertel.com.ar]{}
and
[\
Department of Mathematics and Statistics, University of Regina, Saskatchewan, Canada\
email: fernando.szechtman@gmail.com]{}
[**Keywords:**]{} Integral extension, companion matrix.
[**AMS Classification:**]{} 15A24
Let $\xi$ be an algebraic number of degree $n$ and minimal polynomial $f\in \mathbb Q[X]$, say $f=X^{n}+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$. Given $\alpha\in \mathbb
Q(\xi)$ let $[\alpha]$ stand for the coordinates of $\alpha$ relative to the basis $1, \xi, \dots , \xi^{n-1}$. For $g,h$ in $\mathbb Q[X]$ let $\beta=g(\xi),\gamma=h(\xi)\in\mathbb Q(\xi)$. Obviously $[\beta+\gamma]=[\beta]+[\gamma]$. On the other hand $[\beta\gamma]$ is usually obtained either by reducing high powers of $\xi$ to smaller ones using the relation $\xi^n=
-a_{n-1}\xi^{n-1} - \cdots - a_1\xi -a_0$, or by finding the remainder of dividing $gh$ by $f$. We wonder if there is also a closed formula for $[\beta\gamma]$ in terms of $[\beta]$ and $[\gamma]$ that replaces these procedures. Surprisingly, such a formula exists, and it is so natural that it works in greater generality, as described below.
[**Theorem 1.**]{} [*Let $R$ be a ring with identity. Assume that $\xi$ belongs to an overing of $R$, that $\xi$ commutes with all elements of $R$ and that $f=X^{n}+a_{n-1}X^{n-1}+\cdots+a_1X+a_0\in R[X]$ is the minimal polynomial of $\xi$ over $R$, i.e., that $\xi$ is a root of $f$ but not a root of any polynomial in $R[X]$ of degree less than $n$. Let $\alpha,\beta\in R[\xi]$ have coordinates $[\alpha],[\beta]$ relative to the basis $1,\xi,\dots,\xi^{n-1}$. Then $$\label{form1}
[\alpha\beta]=(I\; C\;\dots\; C^{n-1})([\alpha]\otimes[\beta]),$$ where $C\in M_n(R)$ is the companion matrix of $f$, $$C=\left(\begin{array}{ccccc}
0 & 0 & \cdots & 0 & -a_0 \\
1 & 0 & \cdots & 0 & -a_1 \\
0 & 1 & \cdots & 0 & -a_2 \\
\vdots & \vdots & \cdots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & -a_{n-1} \\
\end{array}\right),$$ and for $x,y\in R^n$ their left Kronecker product $x\otimes y\in
R^{n^2}$ is given by $$x\otimes y=\left(\begin{array}{c}
xy_1 \\
\vdots \\
xy_n \\
\end{array}\right).$$*]{} [*Proof.*]{} Observe first of all that the coefficients of $f$ belong to the center of $R$. Indeed, let $y\in R$. Then $0=yf(\xi)-f(\xi)y=(ya_0-a_0y)+(ya_1-a_1y)\xi+\cdots+(ya_{n-1}-a_{n-1}y)\xi^{n-1}$. Thus $R[C]$ is a ring with $C$ in its center. Clearly $f(C)=0$ and $I, C, \dots , C^{n-1}$ is a basis of the $R$-module $R[C]$. Let $[A]$ be the coordinates of $A\in R[C]$ in this basis. The map $R[C]\to R[\xi]$ given by $p(C)\mapsto p(\xi)$, $p\in R[X]$, is a ring isomorphism that preserves coordinates, so it suffices to prove the result in $R[C]$. To this end, note that if $A\in R[C]$ $$\label{uno} A = ([A] \;C[A] \;\dots \;C^{n-1}[A]).$$ Indeed, let $e_1,\dots,e_n$ be the canonical basis of $R^n$. We have $A = y_0I + y_1C + \cdots + y_{n-1}C^{n-1}$ with $y_j\in R$, so $Ae_1 = y_0e_1 + y_1e_2 + \cdots + y_{n-1}e_n = [A]$. But $Ae_j
= AC^{j-1}e_1=~C^{j-1}Ae_1$, so $Ae_j=C^{j-1}[A]$, for all $2\le
j\le n$. Thus the matrices in question have the same columns.
We know from (\[uno\]) that $[AB]$ is the first column of $AB$, i.e. $[AB] = A[B]$, also by (\[uno\]). Let 0 stand for the zero column vector of length $n$. Applying (\[uno\]) once more gives $$\begin{aligned}
\quad\quad\quad [AB] & = ([A] \;C[A]\; \dots \; C^{n-1}[A])[B]\\
& = (I\; C \dots C^{n-1})\!\left(\begin{array}{cccc}
[A] & 0 & \cdots & 0 \\
0 & [A] & \cdots & 0 \\
\vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & [A] \\
\end{array}\right)\![B]\\
& = (I\; C\;\dots\; C^{n-1})
([A]\otimes[B]).\qquad\qquad \qquad\blacksquare\end{aligned}$$
As a typical example, let $R=\mathbb R$ and $\xi=i$, so that $f(X)=X^2+1$. Then $$[(b_0+b_1 \xi)(c_0+c_1 \xi)]=\left(\begin{array}{ccccc}
1 & 0 & | & 0 & -1 \\
0 & 1 & | & 1 & 0 \\
\end{array}\right)\left(\begin{array}{c}
b_0c_0 \\
b_1c_0 \\
b_0c_1 \\
b_1c_1 \\
\end{array}\right)=\left(\begin{array}{c}
b_0c_0-b_1c_1 \\
b_1c_0+b_0c_1 \\
\end{array}\right),$$ as expected from the usual definition of the product in $\mathbb
R^2$ that produces $\mathbb C$. This can be generalized by defining an operation on the column space $R^n$ that makes it a ring isomorphic to $R[\xi]$ via the map $R[\xi]\mapsto R^n$ given by $\alpha\mapsto [\alpha]$. We just define $$\label{g8} x\cdot y=(I\; C\;\cdots \; C^{n-1})(x\otimes y),\quad
x,y\in R^n.$$
Suppose next $A\in M_n(R)$ has coefficients in the center of $R$. Then its characteristic polynomial, say $f$, is monic of degree $n$ with all its coefficients in the center of $R$ and satisfies $f(A)=0$. However, $A$ need not have a minimal polynomial, and even if it has one it may be harder to determine than $f$. For cases such as these we have the result below, where $R_n[X]=\{p\in R[X]: \,p=0 \text{ or } \deg \,p <n\}$ and $[p]$ denotes the column vector in $R^n$ formed by the coefficients of $p\in R_n[X]$.
[**Theorem 2.**]{} [*Let $R$ be a ring with identity. Assume that $\xi$ belongs to an overing of $R$, that $\xi$ commutes with all elements of $R$, and that $f(\xi)=0$ for some monic polynomial $f\in R[X]$ of degree $n$ with all its coefficients in the center of $R$. Let $C$ denote the companion matrix of $f$. Then for all $g, h\in R_n[X]$ we have $$g(\xi)h(\xi) = (g\odot h)(\xi),$$ where $g\odot h$ denotes the polynomial in $R_n[X]$ whose coefficients are given by $$[g\odot h]= (I\; C\;\cdots\;
C^{n-1})([g]\otimes[h]).$$*]{} [*Proof.*]{} This follows from Theorem 1 via the epimorphism $R[C]\mapsto R[\xi]$.$\quad\qquad \qquad\qquad
\blacksquare$
As an example, let $\xi =A \in M_3(R)$ have coefficients in the center of $R$ and characteristic polynomial $f=X^3-1$. Let $g=b_0+b_1X+b_2X^2$, $h=c_0+c_1X+c_2X^2$ be in $R_3[X]$. Then $g(A)h(A)= (g\odot h)(A)$, where $$\begin{aligned}
\label{example}
[g\odot h] &=\left(\begin{array}{ccccccccccc}
1 & 0 & 0 &|& 0 & 0 & 1& |&0&1&0\\
0 & 1 & 0 &|& 1 & 0 & 0 &|&0&0&1\\
0 & 0 & 1&| & 0 & 1 & 0&|&1&0&0\end{array}\right)
\!\!\left(\begin{array}{c}
b_0c_0 \\
\vdots \\
b_1c_2\\
b_2c_2\\
\end{array}\right)= \left(\begin{array}{c}
b_0c_0+b_2c_1 +b_1c_2\\
b_1c_0 +b_0c_1+b_2c_2 \\
b_2c_0+ b_1c_1+ b_0c_2\end{array}\right).\end{aligned}$$ This identity can be easily generalized for $f=X^n-1$ using the formula $[g\odot h] = g(C)[h]$, which is particularly useful when $g(C)$ has a specified form, as in this example, related to the well known [*circulant*]{} matrices.
|
=1
[\
]{}
[**Abstract**]{}
We present a thorough analysis of the electromagnetic response of strongly coupled neutral plasmas described by the gauge/gravity correspondence. The coupling of the external electromagnetic field with the tower of quasi-normal modes of the plasmas supports the presence of various electromagnetic modes with different properties. Among them we underline the existence of negative refraction with low dissipation for a transverse non-hydrodynamical mode. Previous hydrodynamical approaches have shown the ubiquitous character of negative refraction in charged plasmas and the absence thereof in neutral plasmas. Our results here extend the analysis for neutral plasmas beyond the hydrodynamical regime. As an application of these new insights we briefly discuss the case of the quark gluon plasma in the temperature dominated regime.
Introduction
============
The study of the linear response of a system to a small external perturbation is an essential tool to gain insight on the collective behavior of its constituents and to provide information about the transport properties at thermal equilibrium[^1]. Media having electrically movable components that can easily respond to an external electromagnetic field are ubiquitous in physics: high-energy physics plasmas and condensed matter superconductors are instances of this sort. For such media usually the electromagnetic field interacts with thermal and mechanical modes; hence the electromagnetic energy finds various channels of propagation and dissipation through the medium. The response of the system to an external electromagnetic field is then in general non-local and the medium is said to be spatially dispersive (see for instance [@Landau]). This rich dynamics generates an intriguing phenomenology which can feature exotic electromagnetic effects such as negative refraction (NR) [@SpatNeg] and additional light-waves (ALW) [@ALW] as we will shortly review.
Exotic electromagnetic phenomena have recently attracted an intense theoretical and experimental (and even technological) interest especially in relation to artificial systems called meta-materials [@Metamat]. Inspired by the amazing developments obtained on meta-materials, in [@Noi1; @Noi2; @Noi3] it was first discovered that charged fluids which admit a hydrodynamical description present exotic electromagnetic properties as a general feature. The negative refraction phenomenon (in a certain range of frequencies) and the presence of additional light-waves are shown respectively in [@Noi1; @Noi2] and [@Noi3]. The former implies that the electromagnetic energy flux and the phase velocity of an electromagnetic wave through the medium propagate in opposite directions [@SpatNeg; @Metamat]; the latter means that, even if isotropic, a medium supports multiple electromagnetic waves with the same frequency but different wave-vectors [@ALW]. Following these results, various authors investigated these topics in a wide variety of different setups, see for example [@OtherNegRef]. In particular [@DJD] provides an analysis of the actual signature of the abovementioned phenomena in condensed matter systems and it proposes suitable experiments to measure such effects.
Even though much of the previously stated theoretical progress about NR and ALW was obtained within the framework of strongly coupled plasmas, it turned out that the qualitative behavior of the discovered phenomena has a more general valence and it constitutes a universal property of electrically charged hydrodynamical systems [@Noi2]. The main purpose of the present paper is to go beyond the regime described by hydrodynamics in systems characterized by strong coupling physics. In particular, we will show that the coupling of the external electromagnetic field with the tower of quasi-normal modes of strongly coupled plasmas leads to the presence of various electromagnetic modes which present different properties. For instance, it is important to stress that we found negative refraction with low dissipation for a transverse non-hydrodynamical mode.
Gaining insight on the strongly coupled dynamics is crucial in order to describe some very interesting phases of matter which have recently been realized both in high energy physics experiments (e.g. the quark gluon plasma (QGP), see for example [@CasalderreySolana:2011us]) and in condensed matter systems (e.g. high-$T_c$ superconductors, see for example [@Zaanen:2010yk] for a recent review[^2]). Moreover, the existence of strong interactions among the various resonances in some of the artificially engineered meta-materials, the so called stereo-meta-materials [@Stereo], provides yet another interesting phenomenological stimulus to investigate the electromagnetic behavior and response of strongly coupled systems.
The idea of performing an analysis beyond the hydrodynamical regime aims at studying the electromagnetic wave modes that are not controlled by the hydrodynamical universal behavior accounted by the equations of fluid dynamics[^3]. Particular attention is then paid to observe whether the exotic phenomena arising already in hydrodynamics are present also beyond the long wavelength and long time approximations and what are their properties in this regime.
It is well known that the dynamics at strong coupling is usually very difficult to study and, outside the regime constrained by hydrodynamics, it is quite difficult to provide universal predictions. However, some interesting theoretical progress has been recently obtained thanks to techniques coming from string theory. These new methods exploit a conjectured duality between a strongly coupled quantum field theory and a weakly coupled gravity theory; such duality is usually referred to as gauge/gravity correspondence [@AdSCFT]. The gauge/gravity duality (and similar correspondences) allows us to quantitatively study the correlation functions of specific models and, at the same time, to understand better many interesting and generic qualitative behaviors associated to various (universality) classes of strongly coupled systems. Insights from the gauge/gravity correspondence appeared to be relevant both to high energy physics and condensed matter (see for instance [@KovOrov] to have two paradigmatic examples) especially in relation to the characterization of the phase structure and transport properties. Within the gauge/gravity correspondence framework, quantitative computations of expectation values and correlation functions of strongly coupled quantum systems at finite temperature are reduced to classical gravity calculations in specific black hole backgrounds (which are actually finite temperature gravitational systems). The thermodynamical observables of the strongly coupled system in equilibrium are mapped to classical properties of the dual gravity model. A semi-classical study of the equations of motion for the fields of the dual gravity model unveils both the equilibrium thermodynamics and the linear response (i.e. the slightly out of equilibrium physics). The latter being obtained considering linearized equations of motion (equipped with appropriate boundary conditions) describing small fluctuations around the gravitational black hole background.
To our aim, the benefit of the gauge/gravity framework is twofold: first of all it allows us to study correlation functions for any value of the frequency and wave-vector in a well defined and complete setup (essential to go beyond hydrodynamics); secondly, it provides interesting hints on generic responses of strongly coupled systems beyond the hydrodynamical regime.
A possible phenomenological application of our results is to QGP physics. The neutral plasma that we study could indeed mimic the QGP plasma in a regime dominated by the temperature where the finite chemical potential is neglected. In particular, we stress that the electrodynamical, non-hydrodynamical modes described in this paper have smaller wavelengths when compared to the hydrodynamical modes studied in [@Noi2]; hence they probe smaller distances which could be relevant for actual QGP samples produced in experiments.
The paper is organized as follows. In Section \[scene\] we briefly review the linear response theory needed to study the electromagnetic properties of a medium. Section \[results\] contains our main results obtained by studying the first transverse electromagnetic waves through the strongly coupled plasma. We show in particular the generic presence of ALW and we demonstrate the existence of a propagating electromagnetic mode with negative refraction and very small dissipation. In Section \[mod\] we describe the actual calculations venturing into technical details of both the gauge/gravity setup and the peculiarities of the necessary renormalization procedure. Section \[nume\] is dedicated to the description of our numerical and semi-analytical methods and the checks that our results have undergone. In Section \[QGP\] we discuss very briefly some phenomenological applications of our study to the QGP. In Section \[long\] we show the results of an analogous analysis regarding the longitudinal light-wave modes. Eventually, in Section \[con\] we conclude stressing the significance of the results and possible future prospects. The appendices contain a characterization of the quasi-normal modes of the plasma (both transverse and longitudinal) and the details about the computation regarding the longitudinal light-wave sector.
Setting the scene {#scene}
=================
The electromagnetic response of a system in local equilibrium to an external electromagnetic perturbation is described by the 2-point retarded correlation function of the electromagnetic current [@Dress]. If, as in the case we are considering, the system has spatial dispersion, the response function depends on the space positions where the currents are evaluated. In the particular case of a homogeneous and isotropic system, the correlation functions depend only on the distances and they can be further decomposed into a transverse and a longitudinal part with respect to the wave-vector $q$ of the external perturbation. In this case, as shown for example in [@SpatNeg; @Landau], it is possible to describe the macroscopic electromagnetic properties of the system with three fields: $D$, $E$ and $B$. Moreover, the linear response of the medium (in Fourier space) is accounted for by a single tensorial function depending both on the frequency $\omega$ and the wave-vector $q$, the so-called permittivity tensor $\epsilon_{ij}( \omega, q)$, $$\label{DEeps}
D_i =\epsilon_{ij}( \omega, q)E_j \ .$$ As a consequence of the isotropic assumption, the permittivity tensor is expressed in terms of only two scalar functions $$\label{DEeps}
\epsilon_{ij}( \omega, q) =\epsilon_{T}( \omega, q) \left( \delta_{ij} - \frac{q_i q_j}{q^2} \right) + \epsilon_{L}( \omega, q) \frac{q_i q_j}{q^2}\ ,$$ describing the transverse and longitudinal response with respect to the spatial momentum $q$ of the external perturbation.
The 2-point retarded correlation functions, denoted as ${\cal G}_T(\omega,q)$ and ${\cal G}_L(\omega,q)$ for the transverse and the longitudinal part respectively, characterize completely the permittivity tensor: $$\label{eps_tra}
\epsilon_{T,L}(\omega,q) = 1 - 4 \pi e^2 \, \frac{ {\cal G}_{T,L}(\omega,q)}{\omega^2}\ ,$$ where $e$ is the electromagnetic coupling constant. The knowledge of ${\cal G}_{T,L}(\omega,q)$ allows us to study the propagation and dissipation of the electromagnetic waves inside the medium at the linear level. Indeed, the Maxwell equations for the electromagnetic field provide the dispersion relation between $q$ and $\omega$ both for the transverse and the longitudinal channel[^4]. The Maxwell equations for the transverse and longitudinal part of the electric field $E$ imply the dispersion equations: $$\label{eps_traK}
\epsilon_T(\omega,q)\ = \frac{q^2}{\omega^2}\ , \qquad \epsilon_L(\omega,q)\ =0 \ ,$$ which admit solutions $q(\omega)$ corresponding to possible electromagnetic waves through the medium[^5]. The information about the propagation and dissipation of an electromagnetic mode through the system is encoded in the refractive index which is defined as $n^2(\omega)=q^2(\omega)/\omega^2$ considered on the dispersion relation of the mode under study.
Albeit we study both the transverse and the longitudinal electromagnetic sectors, we focus mainly on the detailed analysis of the transverse sector (whose properties are more interesting to us). An analogue detailed treatment of the longitudinal sector is reported in Appendix \[lon\_det\].
Results
=======
In this Section we describe the main results of our analysis. We focus on the refractive indexes of the first electromagnetic transverse modes inside a strongly coupled plasma composed of electrically charged constituents but globally neutral. The plasma we study is the finite temperature state of a relativistic and conformally invariant particle physics theory composed of fermions and bosons[^6]. The choice of this particular kind of plasma is due to the existence of very powerful theoretical tools (provided by the gauge/gravity correspondence) to study its strongly coupled dynamics. Indeed the present analysis can be read as a first example to investigate the potentiality of the gauge/gravity correspondence to explore exotic electromagnetic phenomena of strongly coupled, finite temperature systems beyond the hydrodynamic regime.
The plasma is assumed to be homogeneous and isotropic[^7]. Previous studies have shown that plasmas described with the gauge/gravity correspondence have in general an infinite set of discrete quasi-normal (complex valued) self-frequencies [@Quasi; @Amado:2007yr; @Amado:2008ji], that appear as poles in the two-point retarded correlation function of the currents. In this paper we are interested to couple these quasi-normal modes of the plasma to an external electromagnetic field and study the possible electromagnetic modes supported by the system. It is important to observe that, even if the quasi-normal modes of the medium appear as poles of the correlation functions, due to high dissipation, the quasi-particle approximation is usually not valid. This implies that at any frequency the response function can not be approximated by a single resonance. Indeed, the self-frequencies of the system are in general complex and the amount of dissipation and propagation of the quasi-normal modes is usually comparable[^8].
![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three transverse electromagnetic modes of the neutral plasma under study.[]{data-label="q"}](Req.pdf "fig:"){width="78mm"} ![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three transverse electromagnetic modes of the neutral plasma under study.[]{data-label="q"}](Imq.pdf "fig:"){width="78mm"}
![Real (left) and imaginary (right) parts of the refraction index $n$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three transverse electromagnetic modes of the neutral plasma under study.[]{data-label="n"}](Ren.pdf "fig:"){width="78mm"} ![Real (left) and imaginary (right) parts of the refraction index $n$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three transverse electromagnetic modes of the neutral plasma under study.[]{data-label="n"}](Imn.pdf "fig:"){width="78mm"}
![Absolute value of the ratio of the dissipation/propagation ratio for the first three electromagnetic modes of the plasma (left). Rescaled plot of the negatively refractive mode (right).[]{data-label="prop"}](prop.pdf "fig:"){width="78mm"} ![Absolute value of the ratio of the dissipation/propagation ratio for the first three electromagnetic modes of the plasma (left). Rescaled plot of the negatively refractive mode (right).[]{data-label="prop"}](propzoom.pdf "fig:"){width="78mm"}
The propagation and dissipation of an electromagnetic wave is described by the corresponding refractive index: the real part of the index accounts for the propagation of the wave while the imaginary part describes its attenuation. We focus here on the transverse modes of the electromagnetic field. The analysis of the longitudinal channel is contained in Section \[long\] and Appendix \[lon\_det\]. A study of the first three transverse modes (i.e. those associated to the lowest wave-numbers) reveals the rich structure of the electromagnetic response of the medium. We have analyzed the real and imaginary parts of their refractive index and of the wave-vector $q$ as functions of the frequency, and, in order to characterize the propagation/dissipation rate of the aforementioned modes, we have studied the absolute value of the ratio $\text{Im}(n)/\text{Re}(n)$[^9].
The results are presented in Figures \[q\], \[n\] and \[prop\]. For reasons that will become clear in the next Sections we plot the results using the rescaled wave-vector ${\mathfrak q}$ and the rescaled frequency ${\mathfrak w}$, obtained by dividing the wave-vector number and the frequency by the temperature of the system $$\begin{aligned}
\label{Twq}
\mathfrak{w} = \frac{\omega}{2 \pi T}\ ,
\ \ \ \
\mathfrak{q} = \frac{q}{2 \pi T}\ .\end{aligned}$$ A quite interesting picture arises from our analysis of the electrodynamic response of the plasma. The system can indeed support various electromagnetic modes, some of them evanescent and other propagating, with different peculiar characteristics. The wave-vector of one of the modes vanishes for vanishing frequency while those corresponding to the other two modes reach a finite value at zero frequency (one purely real and the other purely imaginary). The presence of a finite value of $\mathfrak{q}$ at null frequency (regardless of its real, imaginary or complex character) is due to the coupling of the electromagnetic waves to non-hydrodynamical quasi-normal modes of the plasma. The mode presenting an imaginary $\mathfrak{q}$ at zero frequency is very dissipative in the low-frequency region. On the contrary, the one presenting a real $\mathfrak{q}$ at zero frequency propagates with very small damping. It is very interesting to observe that the real part of the refractive index of this last propagating electromagnetic mode is negative implying that it is a negatively refractive mode, i.e. its energy flux and phase velocity are directed in opposite directions[^10].
In some previous studies [@Noi1; @Noi2] it has been proven that for charged plasmas in the hydrodynamic regime it always exists a mode with negative refraction for small enough frequency, while there is no negative refraction for globally uncharged plasmas. In the present paper we extended the investigation beyond the hydrodynamical modes and we proved that globally neutral, strongly coupled plasmas can support modes with negative refraction too. Moreover, in [@Noi3], the presence of multiple electromagnetic modes was shown to be generic for the hydrodynamical regime. Here we extended such analysis to entail also non-hydrodynamical modes and we showed that ALW still appear to be present. In this extended context it is natural to relate the presence of ALW to the coupling of the electromagnetic radiation to the infinite tower of quasi-normal modes.
Note that we studied the modes at frequencies $\mathfrak{w} \lesssim 1$. In this regime a neat hierarchy between the modes arises from their dissipation over propagation ratio. From Figure \[prop\] one can observe that the negative refraction mode dominates the signal propagation. In line with the collected numerical data one expects that, increasing the frequency, more electromagnetic waves propagate significantly. To have a faithful picture of the plasma response (in terms of propagating waves) at higher frequency it is therefore necessary to include higher modes into the analysis[^11]. The negative refraction phenomenon of the plasma at hand is then likely to be significant in the low-frequency regime. Of course a careful study of the boundary conditions would be required to actually understand which modes are excited in a specific experimental setup. In this paper we concentrate on the bulk properties of the medium and we leave this problem for future investigation.
As a final comment, let us mention that in this Section we presented the results without making reference to the specific value of the temperature or other physical parameters of the system. Let us anticipate that the qualitative behavior of the system is robust and valid in a wide range of parameters. A precise treatment of the quantitative predictions of our model requires a careful analysis of the regularization/renormalization procedure which we postpone to later sections.
Stringy setup and gauge/gravity computation {#mod}
===========================================
In this Section we provide some details about the actual class of systems we consider, the computations we performed and the associated dual gravitational setup. The material collected in the first two Subsections is quite standard and describes how to perform computations in the gauge/gravity correspondence at finite temperature. Nevertheless, we decided to report it in some detail since some subtleties of the computation are relevant for our analysis concerning the electromagnetic modes. Experts in gauge/gravity correspondence could however speed through the discussion below and go to Subsection \[lw\].
As we have previously stated we consider the electromagnetic response of strongly coupled neutral plasmas. The gauge/gravity correspondence allows us to describe the quantum dynamics at strong coupling of a gauge field theory defined on 4-dimensional Minkowski space-time in terms of a classical gravitational model living in a 5-dimensional Anti-de Sitter space-time ($AdS_5$). Furthermore, the gauge/gravity correspondence maps the finite temperature phase of a quantum field theory into a finite temperature gravity configuration: namely a black hole solution which becomes asymptotically $AdS_5$ at large distances ($AdS$ black hole).
We do not restrict ourselves to any particular model, but we study instead the generic behavior of the universal sector of a strongly coupled theory containing the energy momentum tensor $T_{ \mu \nu}$ and the $U(1)$ conserved current $J_{ \mu}$.
In the context of the gauge/gravity correspondence, these two minimal ingredients are mapped respectively to a 5-dimensional metric $g_{mn}$ and a 5-dimensional vector field $A_{m}$, whose dynamics is encoded in the 5-dimensional Maxwell-Einstein action[^12] $$\label{action}
S = - \frac{N^2}{32 \pi^2} \int d^5x \sqrt{-g} \left( R - 2\Lambda \right)
+ \frac{N^2}{16 \pi^2} \int d^5x\; \sqrt{-g}\;\frac{1}{4} F_{mn} F^{ mn} \ ,$$ where $\Lambda$ is the cosmological constant and $N$ is, in the gravity model perspective, a normalization constant. The gravitational constant has been related to the $AdS_5$ radius of curvature $L$ and the string constant $\alpha'$ which have been both put to 1.
The ground state at finite temperature $T$ and zero charge density of a strongly coupled quantum field theory is dually described by the neutral $AdS_5$ black hole solution of the action specified by the following bulk metric written in the Poincaré patch $$\label{bg}
ds^2 = \frac{\pi^2 T^2}{u} \left[-f(u) dt^2 + dx_i dx^i\right] + \frac{1}{4 u^2 f(u)} du^2\ ,$$ where $f(u) = 1-u^2$. The horizon of the black hole is located at $u=1$, while the $AdS_5$ conformal boundary corresponds to $u=0$.
In the gauge/gravity framework, the study of the fluctuations of the 5-dimensional photon $A_{m}$ on the background specified by provides information on the retarded 2-point correlation functions of the $U(1)$ current $J_{ \mu}$ of the dual $(3+1)$-dimensional quantum field theory [@Son:2002sd]. As already stated, the knowledge of the retarded 2-point correlation function of the currents is the required element to discuss the electromagnetic properties of such strongly coupled plasmas.
Vector transverse fluctuations
------------------------------
We focus on the fluctuations of the 5-dimensional bulk photon $A_m$. We fix partially the 5-dimensional gauge invariance imposing $A_u = 0$; note that this $5$-dimensional “radial” gauge condition still leaves the usual $4$-dimensional gauge freedom for the remaining components of the vector potential $A_\mu$. In other words, we study the fluctuations of the bulk gauge field along the $4$-dimensional space-time directions parametrized with $\mu=t,x,y,z$ where the dual theory is actually defined. Moreover, we consider the fluctuations which are transverse to the direction of propagation, namely the fluctuations that are perpendicular to the spatial wave-vector $\vec{q}$.
We decompose $A_\mu$ in $4$-dimensional Fourier modes: $$\label{FAi}
A_\mu(u,t,\vec{x}) = \int \frac{d\omega d^3q}{(2\pi)^4} e^{ -i (\omega t - \vec{ q } \cdot \vec{x} )} A_\mu (u,\omega,\vec{q})
\ .$$ Since the system is invariant with respect to spatial rotations, we can in general take the spatial momentum of the fluctuations to be along the $z$ direction so that the $4$-momentum is $k^\mu = (\omega,0,0,q)$. The transverse fluctuations are therefore those along the $x$ and $y$ directions and we introduce the label $\alpha = x,y$ to refer to this transverse space. Note that, as we have cylindrical symmetry around the $z$ direction, we can focus on the gauge field fluctuations along $x$ without spoiling the generality of the treatment. Specifically, we adopt the following ansatz for the Fourier components $$\label{ans}
A_\alpha(u,\omega,q)\, dx^\alpha = \phi(u,\omega,q)\, dx\ .$$ Plugging this ansatz into the Maxwell equation derived from the action and considered on the background (\[bg\]), we obtain $$\label{transverse}
\phi'' + \frac{f'(u)}{f(u)} \phi' + \frac{\mathfrak{w}^2-\mathfrak{q}^2 f(u)}{u f(u)^2} \phi = 0 \ ,$$ which, by cylindrical symmetry, is valid for the fluctuations along a generic transverse direction. The $u$, $q$ and $\omega$ dependence of $\phi$ is understood, the prime indicates the derivative with respect to the radial coordinate $u$; we also adopted the dimensionless frequency $\mathfrak{w} = \omega/(2\pi T)$ and momentum $\mathfrak{q}=q/(2\pi T)$ already introduced in , where $T$ is the temperature of the dual field theory[^13] or, equivalently, of the $AdS$ black hole in (\[bg\]).
A near-horizon analysis of Equation (\[transverse\]) yields the following asymptotic behavior for the field $\phi$ $$\label{IR_exp}
\phi = (1-u)^{\pm i \mathfrak{w}/2} \left[ a_{\pm} + b_{\pm} (1-u) + c_{\pm} (1-u)^2 + ...\right]\ .$$ The generic solution to the differential equation is given by the superposition of an in-going and an out-going solutions. We select the in-going solution imposing the condition $a_{+}=0$ in accordance with the idea that nothing can be emitted by the black hole horizon at the classical level. This choice corresponds to compute retarded correlators in the dual field theory. As we are confronted with a second order differential equation we need to impose one further boundary condition. Relying on the linearity of Equation we choose $a_{-}=1$; as we will shortly explain, we are eventually interested in ratios like $\phi'/\phi$ which are completely insensitive to this specific choice. As we will describe in detail in Section \[nume\], once equipped with these horizon boundary conditions, we are able to solve the differential problem propagating the solution from the horizon.
We now turn the attention to the near-boundary region corresponding to small $u$. This asymptotic UV analysis is necessary to extract from our model the physical quantities we are interested in. A near-boundary term-wise study of the equation of motion near the $AdS$ boundary at $u=0$ shows that the asymptotic expansion of the field $\phi$ is there $$\label{UV_phi}
\phi = \phi_0 + u\, \phi_1 + u \ln(u)\, \tilde{\phi}_1
+ u^2 \phi_2 + u^2\ln(u)\, \tilde{\phi}_2 + ...\; ,$$ where the $\phi_i$ and $\tilde{\phi}_i$ coefficients are independent of the $u$ coordinate, and satisfy[^14] $$\begin{aligned}
\label{rel}
(\mathfrak{w}^2 - \mathfrak{q}^2) \phi_0 + \tilde{\phi}_1 &=& 0\ , \\
2 \tilde{\phi}_2 + (\mathfrak{w}^2 - \mathfrak{q}^2) \tilde{\phi}_1 &=& 0\ , \\
2 \phi_2 + 3 \tilde{\phi}_2 + (\mathfrak{w}^2 - \mathfrak{q}^2) \phi_1 &=& 0\ .\end{aligned}$$ Note that, since the equation of motion is a second order differential equation, we can express all the coefficients of this term-wise solution as functions of the first two coefficients $\phi_0$ and $\phi_1$.
Correlation functions and holographic renormalization {#Green}
-----------------------------------------------------
In this Section we follow the gauge/gravity prescription to compute the retarded two-point correlator of the transverse currents of the strongly coupled plasma. Indeed, once we found the solution to Equation (\[transverse\]) with the boundary condition explained in the previous section, we can obtain the retarded correlator of the current [@Son:2002sd]. More specifically, as it is usual in quantum field theory, one can derive the correlators through functional differentiation with respect to the sources; from the gravity model standpoint, the procedure corresponds to functionally differentiate the on-shell bulk action with respect to the boundary value of the fluctuating fields.
The on-shell gauge field action for the transverse vector field (described by the solution $\phi$ of the previous Section) becomes $$\label{os_lag_tra}
\frac{(N T)^2}{16} \int \frac{d\omega d^3q}{(2\pi)^4}\ f(u) (\phi')^2\ ,$$ which, after integration by parts and having considered the equation of motion, leads to the boundary term $$\label{bou_ter}
\frac{(N T)^2}{16} \left[ f(u) \phi \phi' \right]' \ .$$ This latter is to be integrated on the boundary manifold. Interestingly, the on-shell action reduces to boundary terms and only the contribution from the $AdS$ conformal boundary at $u=0$ is not vanishing. The contribution from the horizon at $u=1$ actually vanishes because $f(u)$ is there null. As a consequence, we just focus on the contribution to the primitive of at asymptotically small $u$; we expand the field $\phi$ and its derivative as in discarding the terms that vanish at $u=0$ and we obtain $$\label{asy_bou}
\frac{(N T)^2}{16}\ \int \frac{d\omega d^3 q}{(2\pi)^4}\ \phi_0 \left[ \phi_1 - {\phi}_0 (\mathfrak{w}^2 - \mathfrak{q}^2) (1 +\ln(u)) \right]\ .$$ It is important to observe that there exists a logarithmically diverging term. This signals the need to renormalize the model. The occurrence of a large-volume divergence in the gravity theory living on the asymptotically $AdS$ geometry corresponds (through gauge/gravity correspondence) to the UV divergences of the dual quantum field theory.
Let us describe the renormalization procedure [@RenAdSCFT] of the model at hand. At first we regularize the on-shell action considering a small $u=\epsilon$ cutoff, $$S_{\text{reg}} =\frac{(N T)^2}{16} \int_{u=\epsilon} \frac{d\omega d^3 q}{(2\pi)^4}\ \phi_0 \left[ \phi_1 - \phi_0 (\mathfrak{w}^2 - \mathfrak{q}^2) (1 + \ln(\epsilon))\right]\ .$$ We then add an appropriate boundary counter-term $$\label{ct}
S_{\text{c.t.}} = - \frac{N^2}{16 ( 2 \pi )^2} \int_{u=\epsilon} \frac{d\omega d^3q}{(2\pi)^4}\
\sqrt{-\gamma}\ \frac{1}{2} (\ln(\epsilon) + \tilde{c}) F_{ij} F^{ij}\ ,$$ where $\gamma$ represents the determinant of the metric induced by the bulk metric on the $u=\epsilon$ shell, the indexes $i,j$ run only over the 4 boundary directions and $\tilde{c}$ is a real numerical constant (we will later comment on this constant). Notice that the overall factor is chosen to express $S_{\text{c.t.}}$ in terms of the gothic variables $\mathfrak{w}$ and $\mathfrak{q}$. The renormalized action is obtained from $$S_{\text{ren}} = \lim_{\epsilon \rightarrow 0} \left[ S_{\text{reg}} + S_{\text{c.t.}}\right]\ ,$$ and, explicitly, we have $$\label{onshelRen}
S_{\text{ren}} = \frac{(N T)^2}{16} \int \frac{d\omega d^3 q}{(2\pi)^4}\
\phi_0 \left[ \phi_1 - {\phi}_0 (\mathfrak{w}^2 - \mathfrak{q}^2) \frac{c}{2} \right]\ ,$$ where we have introduced $c = 2(1 - \tilde{c})$ for later convenience.
We remind the reader that $\phi_0$ represents the boundary value of the bulk fluctuation field which, through the gauge/gravity dictionary, is mapped to a source of the dual boundary theory. It is therefore with respect to $\phi_0$ that we are interested in taking derivatives of the on-shell bulk action. Specifically, the renormalized 2-point correlation function for the transverse current is obtained taking the second order derivative of the renormalized on-shell action (\[onshelRen\]) with respect to $\phi_0$. In Fourier space we obtain $$\label{dd}
\begin{split}
\frac{\delta^2 S_{\text{ren}}}{(\delta \phi_0)^2}
&= \frac{(N T)^2}{16}\ \left[ \frac{\delta\phi_1}{\delta \phi_0}
- c(\mathfrak{w}^2 - \mathfrak{q}^2) \right]\\
&\sim \frac{(N T)^2}{16}\ \left[ \frac{\phi_1}{\phi_0}
- c(\mathfrak{w}^2 - \mathfrak{q}^2) \right]\ .
\end{split}$$ In the last passage we exploited the linearity assumption and the fact that for zero source $\phi_0$ the fluctuation profile becomes trivial[^15].
From we can define the renormalized retarded correlator for the transverse current: $$\label{grin}
G^{(c)}(\mathfrak{w},\mathfrak{q}) = - \frac{(N T)^2}{16}\ \left[ \frac{\phi_1}{\phi_0}
- c(\mathfrak{w}^2 - \mathfrak{q}^2) \right]\ ;$$ the label $(c)$ is a reminder of the dependence of the correlation function on the coefficient $c$ in front of the contact term. The correlation function is the fundamental quantity that allows us to compute the electromagnetic response of the strongly coupled plasma according to equations (\[eps\_tra\]) and , as explained in Section \[scene\]. It is easy to check that the function defined in equation satisfies the usual properties for a response function of a causal quantum field theory: namely it has poles in the negative imaginary part of the complex plane and its imaginary part is negatively defined when $\omega$ and $q$ are real.
### Contact term {#ct}
As explained above, the computation of the current-current correlator requires the renormalization of the on-shell action. In particular, the renormalization demands to consider a counter-term to cancel the logarithmic divergence in . However, as usual in quantum field theory, the choice of the counter-term is not unique, and only its diverging part is specified by the renormalization procedure; the finite part should instead be fixed by some experimental measure. Such ambiguity is accounted by the real constant $c$ in the definition of the correlator in (\[grin\]), that is actually not fixed by any consistency requirement related to symmetries and Ward identities. The arbitrariness of the finite part of the counter-term introduces a term proportional to $\mathfrak{w}^2-\mathfrak{q}^2$ in the correlator (\[grin\]) which is usually referred to as a *contact term*[^16]. Therefore, the constant $c$ in the correlator should be in principle fixed with an experimental measure of an observable containing it. Once the value of $c$ is fixed, the model does provide quantitative results depending on $N$ (a quantity which, roughly speaking, is associated to the number of degrees of freedom of the system) and the temperature $T$ (which represents a physical scale of the model considered on the black hole solution). As we will see, it is particularly important to underline that the qualitative behavior of the electromagnetic modes in the medium, and more specifically of the refractive index, is a robust feature with respect to the actual value of $c$.
Electromagnetic modes {#lw}
---------------------
As explained in Section \[scene\], the possible electromagnetic transverse modes supported by the plasma are obtained solving the dispersion equation[^17] involving $\mathfrak{q}$ and $\mathfrak{w}$: $$\label{lwe}
\frac{\mathfrak{q}^2}{\mathfrak{w}^2} = 1 - \frac{4\pi e^2}{(2\pi T)^2 \mathfrak{w}^2}\; G^{(c)}(\mathfrak{w},\mathfrak{q})\ ,$$ where $e$ is the electromagnetic coupling constant as previously introduced in (\[eps\_tra\])[^18]. Substituting the explicit expression of the correlation function obtained in , the wave equation becomes $$\label{lwe2}
\frac{\mathfrak{q}^2}{\mathfrak{w}^2} = 1 + \frac{1}{\mathfrak{w}^2} \
\frac{e^2 N^2}{16\pi}\ \left[ \frac{\phi_1}{\phi_0}
- c(\mathfrak{w}^2 - \mathfrak{q}^2) \right]\\ .$$ Note that, since our system is conformally invariant at zero temperature, as long as we work with the dimensionless quantities $\mathfrak{w}$ and $\mathfrak{q}$, we have no explicit dependence on $T$ in the wave equation. $T$ actually provides only a scale with respect to which we measure the actual physical quantities. Furthermore it could be interesting to notice that in the presence of a finite value of $c$ can be reabsorbed by an overall rescaling of the correlation function; more precisely $$\label{lwe3}
G^{(c)}(\mathfrak{w},\mathfrak{q}) \leftrightarrow \frac{G^{(c=0)}(\mathfrak{w},\mathfrak{q})}{1 - \frac{c\, e^2 N^2}{16 \pi}} \ .$$ The rescaling suggests that, as far as the study of the wave equation is concerned, it is possible to trade the contact term with a normalization factor in front of the correlator, this latter being related to $e N$. We could therefore work without specifying the value of $e N$ and reducing everything to a contact term to be fixed against a physical measure performed on the electromagnetic modes. One could for example think to fix $c$ with the requirement that the value of $\mathfrak{q}(\mathfrak{w}=0)$ for a specified mode coincides with that measured in an experiment.
To check the soundness of our results we did a scan of the electromagnetic modes over a broad range of values of $c$ and $e N$. The results show that the qualitative behaviour presented in Section \[results\] is the same for a very broad range of parameters inside the validity of the numerical computation. For convenience in the paper we decided to plot the results for the specific values: $e N=3$ and $c=5.5$.
Semi-numerical analysis and checks {#nume}
==================================
In this Section we would like to explain briefly how we actually performed the computations and characterized the various electromagnetic modes supported by the strongly coupled plasma. We refer again to Figures \[q\], \[n\] and \[prop\] in Section \[results\]. The aim is to solve the wave equation (\[lwe\]) and find the dispersion relations $\mathfrak{q}_A=\mathfrak{q}_A(\mathfrak{w})$ (the label $A$ distinguishes the different modes) connecting the complex wave-vector $\mathfrak{q}$ and the real frequency $\mathfrak{w}$.
At first we solve the equation for the transverse vector fluctuations ; once a solution is obtained, we can read the near-boundary coefficients and plug them into (\[grin\]) to find the correlator. To solve the differential equation we use two different methods whose results are later compared and cross-checked. The first method consists in integrating numerically the equation of motion as explained in [@Hartnoll:2009sz]. The second method is semi-analytical and consists in expanding the function $\phi(u)$ near the horizon (as in ) and solving Equation order by order along the lines of [@Kovtun:2005ev]; this infrared solution is then matched with an analogous term-wise solution computed at the boundary, see . It turns out that, while the numerical method yields more precise answers, the semi-analytical method is instead more agile.
Once a solution to is obtained, it is then possible to follow the gauge/gravity recipe and define a correlation function as described in Subsection \[Green\]. For very small values of $\mathfrak{w}$ and $\mathfrak{q}$ we check that the correlation function agrees with the analytical result of [@Policastro:2002se]. The correlator is then inserted in the electromagnetic wave equation (\[grin\]) to search for the dispersion relations $\mathfrak{q}_A=\mathfrak{q}_A(\mathfrak{w})$ of the various transverse modes. As the correlator is a complicated rational function of $\mathfrak{q}$ and $\mathfrak{w}$, the electromagnetic wave equation has in general many solutions. The set of possible solutions $\mathfrak{q}_A(\mathfrak{w})$ can be studied relying on a simplifying approximation: We build the correlator using the previously introduced semi-analytical matched solution of ; we then write the correlator as the ratio of two polynomials in $\mathfrak{w}$ and $\mathfrak{q}$; we expand both the numerator and the denominator up to a suitable order in $\mathfrak{q}$ and $\mathfrak{w}$ around a chosen point in the $(\mathfrak{w},\mathfrak{q})$ space (this statements will be shortly made more precise). The results we obtained with the aforementioned approximations are afterwards checked against the correlation function computed from the full numerical solution.
Let us be more specific on the computational procedure. We find solutions to the electromagnetic wave equation expressed as complex functions $\mathfrak{q}_A(\mathfrak{w})$ where the frequency is a real quantity. We test the soundness of the solutions as follows: we first repeat the procedure with an increased order in $\mathfrak{w}$ and $\mathfrak{q}$ up to which we expand the numerator and the denominator of the correlation function; we than check that the solutions found previously (i.e. with shallower expansions) still solve the equation within the numerical tolerance. As an additional test we check that the solutions fall in the $(\mathfrak{w},\mathfrak{q})$ region where the approximated correlation function adhere well to the correlation function obtained in the fully numerical approach. Eventually and more stringently we also check that the solutions obtained with the approximate method solve the electromagnetic wave equation written in terms of the fully numerical correlation function within the numerical tolerance.
We start looking for reliable electromagnetic wave solutions in the small $\mathfrak{w}$ and $\mathfrak{q}$ region ($\mathfrak{w},\mathfrak{q}\lesssim 1$); we then follow the modes iterating the approximate procedure and expanding around a generic point of their dispersion relation $\mathfrak{q}(\mathfrak{w})$. This allows us to analyze the modes at higher values of the frequency.
Phenomenology: wavelengths and the QGP {#QGP}
======================================
The analysis presented so far should be able to capture the electromagnetic response of a generic globally neutral strongly coupled plasma described by the gauge/gravity correspondence. A natural candidate for phenomenological application of the model at hand is the QGP studied in high energy physics experiments such as RHIC or LHC, where the plasma is dominated by the temperature, while the charge density is comparatively smaller. A similar phenomenological investigation has been already performed in [@Noi2] for the hydrodynamical modes of strongly coupled and charged plasmas. However in [@Noi2] it was observed that, even if the fluid dynamic analysis demonstrates the generic presence of negative refraction, the wavelength of the electromagnetic mode is larger than the dimension of the typical QGP sample produced in actual experiments.
In this Section, without pretending to provide any specific experimental or phenomenological prediction, we briefly investigate the wavelengths of the modes studied in the paper. In particular we show that the wavelengths of some non-hydrodynamical modes are smaller than the those of the hydrodynamical modes considered in [@Noi2] and that they are actually comparable with the typical dimension of the sample (namely few fm). This result supports the possibility that the coupling of the electromagnetic field with the non-hydrodynamical quasi-normal modes could in principle have phenomenological and experimental relevance and it calls for further investigation.
![ Wavelengths of the transverse electromagnetic modes. This values has been obtained considering $T=200$ MeV. We set $e N=3$ and $c=5.5$.[]{data-label="Olambda"}](Olambda.pdf){width="78mm"}
In Figure \[Olambda\] we report the wavelengths of our electromagnetic transverse modes. The wavelength of a mode is given by $\lambda = 2\pi/ \text{Re}[q] = 1/ T \text{Re}[\mathfrak{q}]$ in accordance with . To associate a physical order of magnitude to the wavelength of the analyzed modes we need to specify the temperature of interest. In relation to the QGP a typical value for the temperature is $200$ MeV for experiments like RIHC or LHC[^19]. Re-introducing the dimensionful physical constants, we find the following estimate for the wavelength $$\lambda = \frac{\hbar c}{T\; \text{Re}[\mathfrak{q}]} \sim 1\, \text{fm}\ ,$$ for frequency of the order of $10^{24}$ Hz, as reported in Figure \[Olambda\]. In particular, note that the negative refracting mode of Figure \[Olambda\] (solid line) presents a wavelength of order $1$ fm for the full range of frequency.
The size of the QGP samples in current experiments is actually of the order of few fm’s [@CasalderreySolana:2011us; @vanHees:2011vb; @Aamodt:2011mr], therefore (higher) non-hydrodynamical modes could actually probe the plasma and perhaps leave there their footprint. We leave a deeper exploration of the higher modes and their possible effects on the QGP for future work.
Longitudinal channel {#long}
====================
![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal electromagnetic modes.[]{data-label="longlight"}](Req_long.pdf "fig:"){width="78mm"} ![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal electromagnetic modes.[]{data-label="longlight"}](Imq_long.pdf "fig:"){width="78mm"}
In this Section we want to analyze briefly the longitudinal electromagnetic modes supported by the strongly coupled plasma and we proceed with a similar spirit as with the transverse modes. As we have previously recalled (see Section \[scene\]), in the presence of spatial dispersion the response function of a medium depends also on the wave-number vector $q$. This fact implies that also the longitudinal component of the electric field can propagate when the Maxwell equation for the longitudinal modes of the electric field (in Fourier space) $$\label{eps_lon}
\epsilon_{L}(\omega,q) = 1 - 4 \pi e^2\, \frac{ {\cal G}_{L}(\omega,q)}{\omega^2}\ =0$$ is satisfied. ${\cal G}_{L}(\omega,q)$ is the retarded correlator of the longitudinal current and, as in the transverse modes case, it is a rational function whose poles correspond to the longitudinal quasi-normal modes of the plasma. Equation (\[eps\_lon\]) provides a set of dispersion relations $q_A=q_A(\omega)$ between the complex longitudinal wave-vectors $q$ and the real frequency $\omega$ of the mode. In this Section we simply give an account of some of the results whose computational details can be found in Appendix \[lon\_det\]. In Figure \[longlight\] and \[long\_ref\] we plot the real and imaginary parts of the wave-vector[^20] $\mathfrak{q}$, the real part of the longitudinal refractive index $n=\mathfrak{q}/\mathfrak{w}$ and the ratio between the imaginary and the real parts of $n$ for the first three electromagnetic modes supported by the strongly coupled plasma. These Figures should be compared with Figures \[q\], \[n\] and \[prop\] in Section \[results\] which instead refer to the transverse sector.
It is very interesting to observe that also the longitudinal channel supports various electromagnetic modes. However, we did not find any negative refractive longitudinal mode. All the modes have finite momentum at vanishing frequency. Nevertheless for two of them $\mathfrak{q}(\mathfrak{w}=0)$ is purely imaginary, while for the remaining mode is purely real. The modes having an imaginary momentum at vanishing frequency are highly dissipative in the IR regime even though the increase of the real part of $\mathfrak{q}$ with the frequency indicates that they could become more propagating at higher frequencies, as it can be seen in Figure \[long\_ref\]. The mode with purely real momentum at $\mathfrak{w}=0$ is highly propagating already in the low-frequency region and it keeps this characteristic in the whole range of frequency that we considered. Notice that the sign of the real part of the momentum shows that all these modes have positive refracting index.
![Real part of the refraction index $n$ and of the ratio $\text{Im}(n)/\text{Re}(n)$ as functions of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal electromagnetic modes.[]{data-label="long_ref"}](Ren_long.pdf "fig:"){width="78mm"} ![Real part of the refraction index $n$ and of the ratio $\text{Im}(n)/\text{Re}(n)$ as functions of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal electromagnetic modes.[]{data-label="long_ref"}](prop_long.pdf "fig:"){width="78mm"}
Conclusion {#con}
==========
In this paper we studied the electromagnetic linear response of strongly coupled neutral plasmas described by the gauge/gravity correspondence and characterized the electromagnetic modes with the lowest wave-vectors. The salient features of the present analysis are the presence of multiple electromagnetic waves with different refractive indexes and a propagating negative refracting mode with very small dissipation. Our study has been performed without adopting any hydrodynamical approximation; hence we extended some previous hydrodynamical studies of strongly coupled plasmas beyond the regime of small frequencies and wave-vectors.
Our simple model highlights the potential richness of the electrodynamic response of strongly coupled plasmas and this calls for further investigations. In particular, on a phenomenological level, we showed that the characteristic wavelengths of the electromagnetic modes in the plasma could be comparable to the typical size of the QGP samples produced in high-energy physics experiments. This fact provides some arguments supporting the possible relevance of the presented exotic phenomena for actual physical systems.
Our results suggest many future lines of investigation. Similar systematic analyses of the electromagnetic properties can be indeed performed using different kinds of gravitational backgrounds featuring appealing characteristics such as finite charge density, spontaneous symmetry breaking, non-relativistic or non-isotropic setups and the presence of magnetic fields. These extensions could find application in various physical systems such as the QGP but also in condensed matter and astrophysics.
We know that strong spatial dispersion is a crucial ingredient in producing the exotic electromagnetic response we are concerned with, however it would be desirable to understand whether some more precise connection between the modes of the plasma and its associated electromagnetic modes can be clarified. In particular it would be nice to have a direct understanding of the presence or not of negative refraction for the various electromagnetic modes based on the QNM structure.
An interesting and ambitious aim would be to provide a systematic and complete analysis of all the non-hydrodynamical modes.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is our great pleasure to acknowledge Alberto Mariotti for relevant discussions in the first phases of the project and Antonio Amariti for very useful suggestions and for precious comments on the draft. Moreover we would like to thanks Carlo Maria Becchi, Riccardo Argurio, Giovanni Villadoro, Diego Redigolo, Flavio Porri, Matteo Bertolini, Andrea Amoretti, Nicola Maggiore, Nicodemo Magnoli, Alessandro Braggio, Antonello Scardicchio, Sebastiano Pilati, Grazia Luparello, Luciano Ramello, Shira Chapman for interesting discussions and for having shared with us their valuable insight. D.F would like to acknowledge the kind hospitality of the LPTHE, where part of this research has been implemented.
D.F. is “Chargé de recherches” of the Fonds de la Recherche Scientifique F.R.S.-FNRS (Belgium), and his research is supported by the F.R.S.-FNRS. The work of D.F and A.M. is partially supported by IISN - Belgium (conventions 4.4511.06 and 4.4514.08), by the Communauté Française de Belgique“ through the ARC program and by the ERC through the SyDuGraM” Advanced Grant.
Details about the longitudinal channel {#lon_det}
======================================
In this appendix we would like to provide some details on the actual computation done for the longitudinal electromagnetic modes. The procedure is similar to that explained in the main text for the transverse modes. However there exist some differences to be taken into account. Using the rotation invariance of the system we fix the four momentum as $k=(\omega,0,0,q)$. The longitudinal current then corresponds to the $z$ direction of the field $A$ in the dual gravity setup: $A_z$. However in this case $A_z$ mixes with the time component $A_t$ [@Kovtun:2005ev]. We consider the following ansatz $$A_m dx^m = A_t e^{-i(\omega t - q z)} dt + A_z e^{-i(\omega t - q z)} dz\ .$$ and define the gauge invariant combination: $$\label{psi_def}
\psi = \mathfrak{q} A_t + \mathfrak{w} A_z\ ,$$ which represents the electric field in the $z$ direction. The $A_t$ and $A_z$ equations then leads to a single equation for $\psi$, namely $$\psi''(u) + \frac{\mathfrak{w}^2 f'(u)}{f(u)\left[\mathfrak{w}^2 - \mathfrak{q}^2 f(u)\right]} \psi'(u)
+ \frac{\mathfrak{w}^2 - \mathfrak{q}^2 f(u)}{u f(u)^2} \psi(u) = 0\ .$$ From a near-boundary study we find the asymptotic behavior of the field $$\psi(u) = \psi_0 + u \psi_1 + u \ln(u) \tilde{\psi}_1 + u^2 \psi_2 + u^2 \ln(u) \tilde{\psi}_2 + ...\ ,$$ which is identical to that of $\phi$ written in . Recall indeed that in the limit of vanishing $q$ the longitudinal and the transverse sector coincide. Moving to finite $q$ does not affect the general form of the UV asymptotic expansions. The same UV asymptotic behavior for the longitudinal and traverse modes implies that the holographic renormalization procedure is analogous in both channels.We can indeed use the same counter-term $c(\mathfrak{w}^2 - \mathfrak{q}^2)$ for both polarizations.
The on-shell action for the combination (\[psi\_def\]) is: $$\label{os_lag_lon}
\frac{(N T)^2}{16}
\int \frac{d\omega d^3 q}{(2 \pi)^4}
\frac{f(u)}{\mathfrak{q}^2 f(u) - \mathfrak{w}^2}\ \psi' \psi\ ;$$ This is the longitudinal version of Equation . Hence the $A'_z A_z$ term of the on-shell action is $$\frac{(N T)^2}{16}
\int \frac{d\omega d^3 q}{(2 \pi)^4}
\frac{\mathfrak{w}^2 f(u)}{\mathfrak{q}^2 f(u) - \mathfrak{w}^2}\ A_z' A_z\ .$$ We understand that, in a similar way as for the transverse channels, the $zz$ current-current correlator related to the second functional differentiation of the on-shell action with respect to $A_z$ can be expressed as follows $$G_{zz}^{(c)}(\mathfrak{w},\mathfrak{q}) = - \frac{(N T)^2}{16} \frac{\mathfrak{w}^2}{\mathfrak{q}^2-\mathfrak{w}^2}
\left[\frac{\psi_1}{\psi_0} - c (\mathfrak{w}^2-\mathfrak{q}^2)\right]\ .$$ These last passages are done in line with [@Kovtun:2005ev] to which we refer for further details. Note that the $tt$ and $tz$ correlators are obtained in a similar way leading to $$G_{tt}^{(c)}(\mathfrak{w},\mathfrak{q}) = - \frac{(N T)^2}{16} \frac{\mathfrak{q}^2}{\mathfrak{q}^2-\mathfrak{w}^2}
\left[\frac{\psi_1}{\psi_0} - c (\mathfrak{w}^2-\mathfrak{q}^2)\right]$$ and $$G_{tz}^{(c)}(\mathfrak{w},\mathfrak{q}) = - \frac{(N T)^2}{16} \frac{\mathfrak{w} \mathfrak{q}}{\mathfrak{q}^2-\mathfrak{w}^2}
\left[\frac{\psi_1}{\psi_0} - c (\mathfrak{w}^2-\mathfrak{q}^2)\right]\ .$$
It is important to notice that the set of $zz$, $tt$ and $tz$ correlators satisfy the Ward identity $$k^\mu G^{(c)}_{\mu\nu} (\mathfrak{w},\mathfrak{q}) = 0\ .$$ This fact is related to the structure of the frequency and momentum dependent factors in front of the expressions of the correlators. As a consequence, the contact term proportional to $c$ is not fixed by the Ward identities or, in other words, it is not constrained by symmetry requirements. All the arguments about the contact terms that we have described in relation to the transverse sector can be repeated for the longitudinal sector.
Quasi-normal modes
==================
In order to characterize better the strongly coupled plasma under study in the main text, we report here an analysis of its internal modes. The retarded correlation function accounting for the electromagnetic response of the plasma presents poles at specific values for the momentum and frequency of the external perturbation. Such poles correspond in the dual gravitational picture to quasi-normal modes of the black hole solution (see for instance [@Kokkotas:1999bd; @Kovtun:2005ev; @Nollert:1999ji; @Horowitz:1999jd; @Son:2002sd; @Amado:2007yr; @Amado:2008ji; @Starinets:2002br; @Nunez:2003eq]). To find the dispersion relations between $\mathfrak{q}$ and $\mathfrak{w}$ of the above-mentioned quasi-normal modes it is enough to look at the zeros of the inverse of the correlation function. These solutions can be represented (for instance) as complex functions $\mathfrak{q}(\mathfrak{w})$ of the real frequency[^21]. In this Appendix we perform a characterization of the first quasi-normal modes of the plasma both in the longitudinal and transverse sectors.
The results for the transverse sector are summarized in Figure \[trans\_mech\]. All the modes we found are qualitatively similar: they all show a finite imaginary part and null real part for $\mathfrak{q}$ as the frequency goes to zero.
![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first transverse quasi-normal modes. The numerical analysis suggest the presence of an infinite tower of analogous modes.[]{data-label="trans_mech"}](Req_transv_mech.pdf "fig:"){width="78mm"} ![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first transverse quasi-normal modes. The numerical analysis suggest the presence of an infinite tower of analogous modes.[]{data-label="trans_mech"}](Imq_transv_mech.pdf "fig:"){width="78mm"}
![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal quasi-normal modes.[]{data-label="long_mech"}](Req_long_mech.pdf "fig:"){width="78mm"} ![Real (left) and imaginary (right) parts of the rescaled wave-vector ${\mathfrak q}$ as a function of the rescaled frequency ${\mathfrak w}$ for the first three longitudinal quasi-normal modes.[]{data-label="long_mech"}](Imq_long_mech.pdf "fig:"){width="78mm"}
Such circumstance corresponds to having a highly dissipative (and non-propagating) set of modes at low frequency. At higher values of the frequency, however, the real part of $\mathfrak{q}$ increases until it becomes significantly bigger than its imaginary part. The modes are then propagating for higher values of the dimensionless frequency $\mathfrak{w}$.
The longitudinal sector features different kinds of modes (see Figure \[long\_mech\]): we have a diffusive mode whose complex momentum vanishes as the frequency goes to zero. This is the hydrodynamical longitudinal mode discussed[^22] in [@Policastro:2002se]. The remaining modes are closely analogous to those we found in the transverse sector. They have a purely imaginary momentum $\mathfrak{q}$ as the frequency vanishes. The numerical investigation leads us to imagine that we have an infinite tower of such modes which again are strongly dissipating at low frequency.
As a final comment we want to underline that whenever possible we checked our results with [@Amado:2008ji].
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[^1]: The topic of linear response is of course extremely wide, for the fundamental material of use here we refer to standard text-books and in particular to [@FosterKM].
[^2]: To have a recent and brief introduction on the gauge/gravity approach to describe strongly coupled superconductors see for instance [@Musso:2014efa].
[^3]: A correlated question which is very interesting and treatable within the framework of the gauge/gravity correspondence is the relation among thermalization and hydrodynamization. To have a recent account on this topic we indicate [@Janik:2014kfa; @Stricker:2014cma] and references therein.
[^4]: In a spatially dispersive medium also the longitudinal electromagnetic waves do propagate.
[^5]: A description of the electromagnetic waves in terms of functions $\omega(q)$ is possible as well; see for instance [@Ginzburg]
[^6]: Further details of the actual theory and setup will be provided in Section \[mod\].
[^7]: These are assumptions that make our analysis easier. Non-homogeneous or non-isotropic cases, charged plasmas, superconducting phases and non-relativistic theories are very interesting to investigate too and are left for future work.
[^8]: Note that only in this paragraph we refer to the framework in which $\omega$ is complex and $q$ is real.
[^9]: Since $n=q/\omega$, studying the ratio $\text{Im}(n)/\text{Re}(n)$ is equivalent to studying $\text{Im}(q)/\text{Re}(q)$; recall indeed that $\omega$ is here a real quantity. Therefore, the ratio $\text{Im}(n)/\text{Re}(n)$ can be read as the comparison between the wavelength and the characteristic attenuation length of the signal.
[^10]: The system we are studying is a passive medium which can only dissipate and not source energy. This implies that the sign of the imaginary part of the refractive index agrees with the direction of propagation of the electromagnetic energy flux in the medium [@mccall]. Intuitively, in a passive medium, a wave propagates in the direction in which it is damped.
[^11]: Our methods allow also a study at higher frequency though a heavier numerical effort is required. For instance, the negatively refracting mode has been studied up to $\mathfrak{w}\sim 2.5$, see the right plot of Figure \[prop\].
[^12]: Our notations are coherent with [@Hartnoll:2009sz; @Argurio:2013uba].
[^13]: Equations (\[Twq\]) provide a meaningful definition of the concept of small or large frequency and wave-vector. Recall that the microscopic theory that we describe is conformally invariant at zero temperature; hence the temperature is the only scale against which we can meaningfully compare the magnitude of the frequency and wave-vector.
[^14]: The background specified in is asymptotically $AdS_5$ and a similar study of the near-boundary behavior of the transverse vector fluctuations on pure $AdS_5$ leads to the same relations among the first UV coefficients.
[^15]: More precisely, the differential problem , being linear, is invariant under a rescaling of the field $\phi(u)$. The solution then depends actually on only one parameter; said otherwise, $\phi_1$ is proportional to $\phi_0$.
[^16]: It is indeed a polynomial function of $\mathfrak{w}$ and $\mathfrak{q}$ that corresponds to space-time delta-functions or derivatives thereof. These contributions are clearly associated to the behavior of the correlator at coincident points.
[^17]: This is actually the wave equation for the transverse component of the electric field inside the medium.
[^18]: In the gauge/gravity correspondence setting, $J$ is the conserved current of a global $U(1)$ symmetry of the quantum field theory. In order to obtain the electromagnetic response of the medium, the standard procedure is to introduce in the QFT action the interaction term $e J \mathcal{A}$ where $\mathcal{A}$ is the electromagnetic field considered as *external* and $e$ is the associated electromagnetic coupling constant considered to be perturbatively small. The retarded correlator of the global current provides, at leading order in $e$, the exact result for the retarded correlator of the local current.
[^19]: Due to our numerical procedure, with the choice of temperature $T=200$ MeV, the regime of validity of our results corresponds to a frequency interval roughly between $10^{14}$ Hz and $10^{24}$ Hz.
[^20]: As for the transverse modes, we use the rescaled quantities ${\mathfrak q}= q / 2 \pi T$, ${\mathfrak w }= \omega / 2 \pi T$.
[^21]: It is important to underline that these dispersion relations between $\mathfrak{q}$ and $\mathfrak{w}$ are the ones associated to the [*quasi-normal modes*]{} of the plasma and they are different from the dispersion relations for the [*electromagnetic modes*]{} supported by the plasma and discussed in the main text. Indeed the first ones come from solving the equations: $(G_{T,L}(\omega,q))^{-1}=0$, while the second ones solve equations which look approximately as $q^2 = \omega^2 - G_{T}(\omega,q)$ and $\omega^2 - G_{L}(\omega,q)=0$.
[^22]: We also checked explicitly that at low frequency this longitudinal hydrodynamical mode is well approximated by the diffusive pole structure with diffusion constant $D = 2\pi T$ (referred to dimensionful $\omega$ and $k$).
|
---
abstract: 'Novák conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,{\lambda})$-designs with $1 \leq {\lambda}\leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and ${\lambda}=1$ and also when ${\lambda}\leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.'
author:
- Tao Feng
- Daniel Horsley
- Xiaomiao Wang
title: 'Novák’s conjecture on cyclic Steiner triple systems and its generalization'
---
Introduction
============
Let $V$ be a set of $v$ [*points*]{}, and ${\mathcal{B}}$ be a collection of $k$-subsets of $V$ called [*blocks*]{}. A pair $(V, {\mathcal{B}})$ is called a [*$(v,k,\lambda)$-design*]{} if every pair of distinct elements of $V$ is contained in precisely $\lambda$ blocks of ${\mathcal{B}}$. A $(v,3,1)$-design is called a [*Steiner triple system*]{} of order $v$ and is written as an STS$(v)$.
An [*automorphism*]{} of a $(v,k,\lambda)$-design $(V, {\mathcal{B}})$ is a permutation on $V$ leaving ${\mathcal{B}}$ invariant. A $(v,k,\lambda)$-design is said to be [*cyclic*]{} if it admits an automorphism consisting of a cycle of length $v$. Without loss of generality we identify $V$ with ${\mathbb Z}_v$, the additive group of integers modulo $v$. The blocks of a cyclic $(v,k,\lambda)$-design can be partitioned into orbits under ${\mathbb Z}_{v}$. We can choose any fixed block from each orbit and then call these [*base blocks*]{}. If the cardinality of an orbit is equal to $v$, the orbit is [*full*]{}. Otherwise, it is [*short*]{}. It follows from the orbit-stabilizer theorem that the cardinality of any orbit is a divisor of $v$ and is at least $v/k$. If $\gcd(v,k)=1$, then all orbits of a cyclic $(v,k,\lambda)$-design are full (see [@k Lemma 1]). It is known that a cyclic STS$(v)$ exists if and only if $v\equiv 1,3\pmod{6}$ and $v\neq 9$ (see [@triplesystem Theorem 7.3]).
A useful tool for generating cyclic designs is the concept of a difference family. A [*$(v,k,{\lambda})$-cyclic difference family*]{} is a family ${\mathcal{F}}$ of $k$-subsets (called [*base blocks*]{}) of ${\mathbb Z}_{v}$ such that the multiset $\Delta {\mathcal{F}}:=\{x-y:x,y\in F, x\not=y,F\in {\mathcal{F}}\}$ contains every element of ${\mathbb Z}_{v}\setminus \{0\}$ exactly $\lambda$ times. Such a family is denoted as a $(v,k,\lambda)$-CDF. It consists of $\lambda(v-1)/(k(k-1))$ base blocks. A $(v,k,\lambda)$-CDF ${\mathcal{F}}$ can generate a cyclic $(v,k,\lambda)$-design with block-multiset $\dev {\mathcal{F}}:=\{F+t:F\in {\mathcal{F}},t\in {\mathbb{Z}}_v\}$ (see [@s Theorem 3.46]). Furthermore, when $\gcd(v,k)=1$, ${\mathcal{F}}$ is a $(v,k,\lambda)$-CDF if and only if $\dev {\mathcal{F}}$ is a cyclic $(v,k,\lambda)$-design (see [@bjl Proposition VII.1.5]).
A $(v,k,\lambda)$-CDF is said to be *disjoint* and written as a $(v,k,\lambda)$-DDF when its base blocks are mutually disjoint. Novák [@novak] conjectured in 1974 that for any cyclic STS$(v)$ with $v\equiv 1\pmod{6}$, it is always possible to find a set of $(v-1)/6$ disjoint base blocks which come from different block orbits to form a $(v,3,1)$-DDF (see also [@abel Remark 16.22] or [@triplesystem Work point 22.5.2]).
\[weak conjecture\] Every cyclic STS$(v)$ with $v\equiv 1\pmod{6}$ is generated by a $(v,3,1)$-DDF.
Conjecture \[weak conjecture\] is widely believed to be true but not much progress has been made on it. So far it is only known that Conjecture \[weak conjecture\] holds for all $v\equiv 1\pmod{6}$ and $v\leq 61$ (see [@triplesystem Theorem 22.3]). On the other hand, Dinitz and Rodney [@dinitzUtil] proved that a $(v,3,1)$-DDF exists for any $v\equiv 1 \pmod{6}$ by taking a suitable $(v,3,1)$-CDF and then replacing each of its base blocks $B_i$ by a suitable translate $B_i+t_i$. For more information on $(v,3,1)$-DDFs with $v\equiv 3 \pmod{6}$, interested readers are referred to [@buratti36; @dinitz36].
Recently, using the Combinatorial Nullstellensatz, Karasev and Petrov [@karasev] proved the following result.
\[pairwisedisjoint\] Let $\mathbb{F}$ be an arbitrary field, and let $m$ and $d$ be positive integers such that $(md)!/(d!)^m\neq0$ in $\mathbb{F}$. Let $X_1,\ldots, X_m$ and $T_1,\ldots, T_m$ be subsets of $\mathbb{F}$ such that $$\forall i < j\ \ |X_i-X_j|\leq2d,\ \ \ \forall i\ \ |T_i|\geq (m-1)d+1,$$ where $X_i-X_j:=\{x-y:x\in X_i,y\in X_j\}$. Then there exists a system of representatives $t_i\in T_i$ such that the sets $X_1+t_1,\ldots, X_m+t_m$ are pairwise disjoint.
We now apply Lemma \[pairwisedisjoint\] to show that Conjecture \[weak conjecture\] is true whenever $v$ is a prime.
\[resultprime\] Let $k\geq 2$ and let $p$ be a prime. Every cyclic $(p,k,1)$-design is generated by a $(p,k,1)$-DDF.
We may assume $p>k$ because otherwise the result is trivial. Since $\gcd(p,k)=1$, a cyclic $(p,k,1)$-design has $m=(p-1)/(k(k-1))$ full orbits and no short orbits. Let $B_1,\ldots, B_m$ be base blocks of a cyclic $(p,k,1)$-design and let $d=\lceil k^2/2\rceil$. Then $|B_i-B_j|\leq2d$ for any $1\leq i<j\leq m$. Let $T_1=\cdots=T_m=\mathbb Z_p$. Then $|T_i|=p\geq(m-1)d+1$ for $k\geq2$. Since $md<p$ when $k\geq2$, $(md)!/(d!)^m\not\equiv 0 \pmod{p}$. Therefore, by Lemma \[pairwisedisjoint\], there exists a system of representatives $t_i\in T_i$ such that $B_1+t_1,\ldots, B_m+t_m$ are pairwise disjoint. So $B_1+t_1,\ldots, B_m+t_m$ form a $(p,k,1)$-DDF.
Theorem \[resultprime\] motivates us to present the following conjecture on cyclic $(v,k,1)$-designs, which also allows for designs with short orbits.
\[conj:any k\] For any cyclic $(v,k,1)$-design, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint.
The existence of $(v,k,1)$-DDFs is in general quite a hard problem. Conjecture \[conj:any k\], if true, would reduce the existence of $(v,k,1)$-DDFs to the existence of $(v,k,1)$-CDFs. The following results on CDFs are known in the literature.
\[lem:456-wilson\]
- For any prime $p\equiv 1\pmod{12}$, there exists a $(p, 4, 1)$-CDF.
- For any prime $p\equiv 1\pmod{20}$, there exists a $(p, 5, 1)$-CDF.
- For any prime $p\equiv 1\pmod{30}$ and $p\neq 61$, there exists a $(p, 6, 1)$-CDF.
- Let $p\equiv 1\pmod{k(k-1)}$ be a prime. Then a $(p,k,1)$-CDF exists if $p>\binom{k}{2}^{k(k-1)}$.
As a corollary of Theorem \[resultprime\] together with Lemma \[lem:456-wilson\], we obtain the following existence results on DDFs.
Let $p\equiv 1\pmod{k(k-1)}$ be a prime.
- There exists a $(p,k,1)$-DDF for each $k\in\{4,5,6\}$ and $(k,p)\neq (6,61)$.
- There exists a $(p,k,1)$-DDF whenever $p>\binom{k}{2}^{k(k-1)}$.
We remark that by using Weil’s theorem on estimates of character sums, Wu, Yang and Huang [@wuActa] also established the existence of a $(p,4,1)$-DDF for any prime $p\equiv 1\pmod{12}$. We also observe that the main result of [@EhaGloJoo] shows that, for fixed $k$ and large $v$, one can find a family $\mathcal{F}$ of $(1-o(1))\frac{v-1}{k(k-1)}$ pairwise disjoint base blocks of size $k$ such that $\Delta \mathcal{F}$ contains each difference at most once. This is accomplished by letting $H$ be the disjoint union of $(1-o(1))\frac{v-1}{k(k-1)}$ copies of $K_k$ and applying [@EhaGloJoo Theorem 1.2] to find a rainbow copy of $H$ in the complete graph on ${\mathbb{Z}}_v$ with edges coloured according to their differences.
In this paper, we shall provide a proof of Conjecture \[conj:any k\] when $v$ is sufficiently large compared to $k$. In fact, we consider a more general setting. We shall examine cyclic $(v,k,\lambda)$-designs with $k \geq 2\lambda+1$. As the main result of this paper, we prove Theorem \[thm:main theorem\] below. In fact we prove a stronger statement which sometimes guarantees the existence of a family of mutually disjoint blocks containing many blocks from each orbit (see Theorem \[thm:asympt\]).
\[thm:main theorem\] Let $k$ and $\lambda$ be fixed positive integers such that $k\geq 2{\lambda}+1$. There exists an integer $v_0$ such that, for any cyclic $(v,k,\lambda)$-design with $v\geq v_0$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint.
Combining Theorems \[resultprime\] and \[thm:main theorem\] yields the following corollary.
Let $k \geq 3$ be a fixed integer. With the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-DDF.
Preliminaries
=============
For any positive integer $c$, let $[c]:=\{1,\ldots,c\}$. We will make use of the following simple lemma which shows that, for large $v$ and fixed $k$ and ${\lambda}$, a cyclic $(v,k,{\lambda})$-design has few short orbits.
\[L:fewShort\] Let $k \geq 2$ and $\lambda \geq 1$ be fixed integers. If $(V,{\mathcal{B}})$ is a cyclic $(v,k,{\lambda})$-design with $h$ short orbits and $m$ full orbits, then
- $h \leq 2{\lambda}\sqrt{k}$; and
- $\frac{{\lambda}(v-1)}{k(k-1)}-2{\lambda}\sqrt{k} \leq m \leq \frac{{\lambda}(v-1)}{k(k-1)} \leq m+h \leq \frac{{\lambda}(v-1)}{k(k-1)}+2{\lambda}\sqrt{k}$.
Let the point set of $(V,{\mathcal{B}})$ be ${\mathbb{Z}}_v$ and let ${\mathcal{B}}_1,\ldots,{\mathcal{B}}_{h}$ be the short orbits of $(V,{\mathcal{B}})$. Let $i \in [h]$. Recall that by the orbit-stabilizer theorem we have $|{\mathcal{B}}_i|=\ell_i$ where $\frac{v}{k} \leq \ell_i<v$ and $\ell_i \mid v$. Let $B_i$ be a base block from ${\mathcal{B}}_i$ such that $B_i$ contains the point $0$. Since $|{\mathcal{B}}_i|=\ell_i$, $B_i+\ell_i=B_i$. It follows that $B_i$ contains all multiples of $\ell_i$. Write $S_i:=\{0,\ell_i,2\ell_i,\ldots,(\frac{v}{\ell_i}-1)\ell_i\}$. Then $S_i\subseteq B_i$. Furthermore, for any $a\in B_i$, $a+S_i\subseteq B_i$, and so $B_i$ is a disjoint union of some cosets of $S_i$ in $\mathbb Z_v$, which implies $|S_i|\mid |B_i|$. That is, $\frac{v}{\ell_i}\mid k$. Also, because exactly ${\lambda}$ blocks in ${\mathcal{B}}$ contain the pair $\{0,\ell_i\}$, we have that at most $\lambda$ of the orbits ${\mathcal{B}}_1,\ldots,{\mathcal{B}}_h$ have cardinality $\ell_i$.
Thus, $h \leq \lambda \sigma_0(k)$ where $\sigma_0(k)$ denotes the number of divisors of $k$. We know that $\sigma_0(k) \leq 2\sqrt{k}$ for any positive integer $k$ by using the fact that $d\mid k$ if and only if $\frac{k}{d}\mid k$, and so (i) follows. Then (ii) follows from (i) by routine calculation after observing that $mv+\sum_{i=1}^h\ell_i=|{\mathcal{B}}|=\frac{{\lambda}v(v-1)}{k(k-1)}$.
An [*$r$-uniform hypergraph*]{} $G$ is a pair $(V, E)$ where $V$ is a vertex set and $E$ is a set of $r$-subsets of $V$ called edges. The [*degree*]{} $\deg_G(x)$ of a vertex $x\in V$ is the number of edges of $G$ containing $x$. For distinct vertices $x$ and $y$ of $G$, the [*codegree*]{} ${\mathop{\mathrm{codeg}}\nolimits}_G(x, y)$ is the number of edges of $G$ containing both $x$ and $y$. We write $\delta_G:=\min\limits_{x\in V} \deg_{G}(x)$, $\Delta_G:=\max\limits_{x\in V} \deg_{G}(x)$ and $\Delta^c_G:=\max\limits_{x,y\in V,x\neq y} {\mathop{\mathrm{codeg}}\nolimits}_G(x,y)$.
A [*proper edge-colouring*]{} of a hypergraph $G=(V,E)$ with $c$ colours is a function $f: E\longrightarrow [c]$ such that no two edges that share a vertex get the same colour. The following powerful result of Pippenger and Spencer [@PipSpe] (based on the Rödl nibble) shows that every almost regular $r$-uniform hypergraph $G$ with small maximum codegree can be edge-coloured with close to $\Delta_G$ colours.
\[thm:pip\_sp\] Let $r\geq 2$ be an integer. For each real number $\eta>0$, there exists a real number $\eta^*>0$ and an integer $n_0$ such that if $G$ is an $r$-uniform hypergraph on $n\geq n_0$ vertices satisfying $\delta_G\geq (1-\eta^*)\Delta_G$ and $\Delta^c_G \leq \eta^* \Delta_G$, then $G$ has a proper edge-colouring with $(1+\eta)\Delta_G$ colours.
Proof of Theorem \[thm:main theorem\]
=====================================
A [*partial parallel class*]{} of a $(v,k,\lambda)$-design is a set of pairwise disjoint blocks. Let $(V,{\mathcal{B}})$ be a cyclic $(v,k,\lambda)$-design with orbits ${\mathcal{B}}_1,\ldots,{\mathcal{B}}_t$, let ${\mathcal{P}}$ be a partial parallel class of $(V,{\mathcal{B}})$ and let $s=\lfloor\frac{k-1}{\lambda}\rfloor$. For any nonnegative integer $a$ we define $T_a({\mathcal{P}})=\{i \in [t]: |{\mathcal{P}}\cap {\mathcal{B}}_i|=a\}$ to be the set of indices of orbits of $(V,{\mathcal{B}})$ that contain exactly $a$ blocks in ${\mathcal{P}}$, and we define ${\tau}_a({\mathcal{P}})=|T_a({\mathcal{P}})|$. Also, we say that a block $B \in {\mathcal{B}}$ is *${\mathcal{P}}$-good* if, for each $i \in [t]$, $B$ intersects at most one block in ${\mathcal{P}}\cap {\mathcal{B}}_i$ and, for each $i \in T_0({\mathcal{P}}) \cup \cdots \cup T_{s-1}({\mathcal{P}})$, $B$ intersects no block in ${\mathcal{P}}\cap {\mathcal{B}}_i$. Blocks in ${\mathcal{B}}$ that are not ${\mathcal{P}}$-good are *${\mathcal{P}}$-bad*. Finally we define, if $s \geq 2$, $$d({\mathcal{P}})=\medop\sum_{a=0}^{s-2}(s-1-a){\tau}_a({\mathcal{P}}).$$ The definitions of ${\mathcal{P}}$-good and $d({\mathcal{P}})$ are implicitly dependent on the value of $s=\lfloor\frac{k-1}{\lambda}\rfloor$.
Our strategy is to first, in Lemma \[L:applyPipSpe\] below, apply Lemma \[thm:pip\_sp\] to an auxiliary hypergraph in order to obtain a partial parallel class in the design that contains $s$ blocks from almost every orbit. We then, in Lemma \[L:greedy\], prove that if each orbit of a cyclic design contains sufficiently many good blocks relative to some partial parallel class, then this class can be modified so that it contains $s$ blocks from almost every orbit and $s-1$ blocks from each remaining orbit. Finally, to prove Theorem \[thm:asympt\], we show that Lemma \[L:greedy\] can successfully be applied to a partial parallel class obtained by making some modifications to a class given by Lemma \[L:applyPipSpe\].
\[L:applyPipSpe\] Let $k$ and ${\lambda}$ be positive integers and let $s=\lfloor\frac{k-1}{{\lambda}}\rfloor$. For each real number ${\epsilon}^*>0$, there exists an integer $v_0$ such that, for each integer $v \geq v^*_0$, any cyclic $(v,k,{\lambda})$-design with $m$ full orbits has a partial parallel class ${\mathcal{P}}$ that contains $s$ blocks from each of $(1-{\epsilon}^*)m$ full orbits and no blocks from any other orbit.
Let $(V,{\mathcal{B}})$ be a cyclic $(v,k,{\lambda})$-design with full orbits ${\mathcal{B}}_1,\ldots,{\mathcal{B}}_m$. Observe that, by Lemma \[L:fewShort\](ii), $\frac{{\lambda}(v-1)}{k(k-1)}-2{\lambda}\sqrt{k} \leq m \leq \frac{{\lambda}(v-1)}{k(k-1)}$. Hence, supposing $v$ is sufficiently large, we have $\frac{{\lambda}(v-1)}{k^2} < m \leq \frac{{\lambda}(v-1)}{k(k-1)}$. Let $w=\lfloor\frac{v-1}{k}\rfloor$ and choose integers $s_1,\ldots,s_m \in \{s,s+1\}$ such that $s_1+\cdots+s_m =w$. Such integers exist because $sm \leq \frac{v-1}{k}$ using $s \leq \frac{k-1}{{\lambda}}$ and $m \leq \frac{{\lambda}(v-1)}{k(k-1)}$, and because $(s+1)m > \frac{v-1}{k}$ using $s+1 \geq \frac{k}{{\lambda}}$ and $m > \frac{{\lambda}(v-1)}{k^2}$. Let $W=\{u_{i,j}: i \in [m],j \in [s_i]\}$ be a set of $w$ vertices disjoint from $V$. We form a $(k+1)$-uniform hypergraph $G$ with vertex set $V \cup W$ and edge set $$\{B \cup \{u_{i,j}\}: B \in {\mathcal{B}}_i, i \in [m], j \in [s_i]\}.$$
Observe that, for each $x \in V$, we have $\deg_G(x)=ks_1+\cdots+ks_m=kw$ because $x$ is in $k$ blocks in each full orbit, and hence we have $v-k \leq \deg_G(x) \leq v-1$. Also, $\deg_G(x)=v$ for each $x \in W$ because each full orbit contains $v$ blocks. Furthermore ${\mathop{\mathrm{codeg}}\nolimits}_G(x,y) \leq \lambda(s+1) \leq k+{\lambda}-1$ for all distinct $x,y \in V$ because $(V,{\mathcal{B}})$ is a design of index $\lambda$, ${\mathop{\mathrm{codeg}}\nolimits}_G(x,y)=0$ for all distinct $x,y \in W$, and ${\mathop{\mathrm{codeg}}\nolimits}_G(x,y) = k$ for all $x \in V$ and $y \in W$ because $k$ blocks from any full orbit contain a given vertex in $V$. So $G$ has $v+w$ vertices, $vw$ edges, $\delta_G \geq v-k$, $\Delta_G \leq v$, and $\Delta^c_G \leq k+{\lambda}-1$. Thus Lemma \[thm:pip\_sp\] implies that for any real number ${\epsilon}^* > 0$, supposing $v$ is sufficiently large, $G$ has a proper edge-colouring with $(1+\frac{{\epsilon}^*}{s+1})v$ colours.
Let $\mathcal{C}$ be a largest colour class of this colouring. Then $\mathcal{C}$ is a set of disjoint edges of $G$ and, because $G$ has $vw$ edges, $|\mathcal{C}| \geq \frac{(s+1)w}{s+1+{\epsilon}^*}>w-{\epsilon}^*\frac{w}{s+1}>w-{\epsilon}^* m$ where the last inequality follows because $w < (s+1)m$. Let $$M=\bigl\{i \in [m]: |\{j \in [s_i]: \text{$u_{i,j}$ is in an edge in $\mathcal{C}$}\}| \geq s\bigr\}.$$ Observe that $|M| > (1-{\epsilon}^*)m$ because each edge of $G$ contains exactly one vertex in $W$ and hence there are less than ${\epsilon}^* m$ vertices in $W$ that are not in an edge of $\mathcal{C}$. Let $\mathcal{C}'$ be the set of edges in $\mathcal{C}$ that contain a vertex in $\{u_{i,j}:i \in M, j \in [s_i]\}$ and let ${\mathcal{P}}=\{E \cap V: E \in \mathcal{C}'\}$. Then, by the definitions of $G$ and $\mathcal{C}'$, ${\mathcal{P}}$ is a partial parallel class in $(V,{\mathcal{B}})$ that contains at least $s$ blocks from ${\mathcal{B}}_i$ for each $i \in M$ and no other blocks. So, by deleting some blocks from ${\mathcal{P}}$ if necessary, we can obtain a partial parallel class with the desired properties.
\[L:greedy\] Let $k$ and ${\lambda}$ be positive integers such that $k \geq 2{\lambda}+1$ and let $s=\lfloor\frac{k-1}{{\lambda}}\rfloor$. Let $(V,{\mathcal{B}})$ be a cyclic -design with orbits ${\mathcal{B}}_1,\ldots,{\mathcal{B}}_t$, and let ${\mathcal{P}}'$ be a partial parallel class that contains at most $s$ blocks from each orbit. If, for each $i \in T_0({\mathcal{P}}') \cup \cdots \cup T_{s-2}({\mathcal{P}}')$, ${\mathcal{B}}_i$ contains more than $k^2(ks-k+1)(d({\mathcal{P}}')-1)$ ${\mathcal{P}}'$-good blocks, then there is a partial parallel class ${\mathcal{P}}''$ of $(V,{\mathcal{B}})$ such that ${\tau}_{s-1}({\mathcal{P}}'') \leq (k+1)d({\mathcal{P}}')+{\tau}_{s-1}({\mathcal{P}}')$ and ${\tau}_s({\mathcal{P}}'') = t-{\tau}_{s-1}({\mathcal{P}}'')$.
Note that $s \geq 2$ by our hypotheses. We prove the result by induction on the quantity $d({\mathcal{P}}')$. If $d({\mathcal{P}}')=0$ then ${\tau}_0({\mathcal{P}}')=\cdots={\tau}_{s-2}({\mathcal{P}}')=0$ and we can take ${\mathcal{P}}''={\mathcal{P}}'$ to complete the proof. So suppose that $d({\mathcal{P}}')=\ell$ for some positive integer $\ell$ and that the result holds for $d({\mathcal{P}}') < \ell$. Let $j \in T_0({\mathcal{P}}') \cup \cdots \cup T_{s-2}({\mathcal{P}}')$ and let $B_j$ be a ${\mathcal{P}}'$-good block in ${\mathcal{B}}_j$ (such a block exists by the hypotheses of the lemma because $d({\mathcal{P}}') \geq 1$). Let ${\mathcal{Q}}$ be the set of blocks in ${\mathcal{P}}'$ which intersect $B_j$ and let ${\mathcal{P}}^*=({\mathcal{P}}' \cup \{B_j\}) \setminus {\mathcal{Q}}$. Note that ${\mathcal{P}}^*$ is a partial parallel class of $(V,{\mathcal{B}})$ and that $|{\mathcal{Q}}| \leq |B_j|=k$. Also, because $B_j$ was ${\mathcal{P}}'$-good, ${\tau}_1({\mathcal{Q}})=|{\mathcal{Q}}|$ and $T_1({\mathcal{Q}}) \subseteq T_s({\mathcal{P}}')$.
Observe that $|{\mathcal{P}}^* \cap {\mathcal{B}}_j|=|{\mathcal{P}}' \cap {\mathcal{B}}_j|+1$, $|{\mathcal{P}}^* \cap {\mathcal{B}}_i|=s-1$ for each $i \in T_1({\mathcal{Q}})$, and ${\mathcal{P}}^* \cap {\mathcal{B}}_i={\mathcal{P}}' \cap {\mathcal{B}}_i$ for all $i \in [t] \setminus (T_1({\mathcal{Q}}) \cup \{j\})$. Thus $d({\mathcal{P}}^*)=d({\mathcal{P}}')-1$ and ${\tau}_{s-1}({\mathcal{P}}^*) \leq {\tau}_{s-1}({\mathcal{P}}')+k+1$. Any block in ${\mathcal{B}}$ that was ${\mathcal{P}}'$-good but is ${\mathcal{P}}^*$-bad must intersect one of the at most $ks-k+1$ blocks in $\{B_j\} \cup \bigcup_{i\in T_1({\mathcal{Q}})}({\mathcal{P}}^*\cap{\mathcal{B}}_i)$. For each $i \in T_0({\mathcal{P}}^*) \cup \cdots \cup T_{s-2}({\mathcal{P}}^*)$, at most $k^2$ blocks in ${\mathcal{B}}_i$ intersect each of these blocks and so, because more than $k^2(ks-k+1)(d({\mathcal{P}}')-1)$ blocks in ${\mathcal{B}}_i$ were ${\mathcal{P}}'$-good, more than $$k^2(ks-k+1)(d({\mathcal{P}}')-1)-k^2(ks-k+1)=k^2(ks-k+1)(d({\mathcal{P}}')-2)=k^2(ks-k+1)(d({\mathcal{P}}^*)-1)$$ blocks in ${\mathcal{B}}_i$ are ${\mathcal{P}}^*$-good. Thus we can apply our inductive hypothesis to ${\mathcal{P}}^*$ to establish the existence of a partial parallel class ${\mathcal{P}}''$ of $(V,{\mathcal{B}})$ such that ${\tau}_{s-1}({\mathcal{P}}'') \leq (k+1)d({\mathcal{P}}^*)+{\tau}_{s-1}({\mathcal{P}}^*)$ and ${\tau}_s({\mathcal{P}}'') =t-{\tau}_{s-1}({\mathcal{P}}'')$. The proof is now complete by observing that $${\tau}_{s-1}({\mathcal{P}}'') \leq (k+1)d({\mathcal{P}}^*)+{\tau}_{s-1}({\mathcal{P}}^*) \leq (k+1)(d({\mathcal{P}}')-1)+{\tau}_{s-1}({\mathcal{P}}') + k+1=(k+1)d({\mathcal{P}}')+{\tau}_{s-1}({\mathcal{P}}').$$
\[thm:asympt\] Let $k$ and ${\lambda}$ be fixed positive integers such that $k \geq 2{\lambda}+1$ and let $s=\lfloor\frac{k-1}{{\lambda}}\rfloor$. For each real number $\epsilon>0$, there is an integer $v_0$ such that, for each integer $v \geq v_0$, any cyclic $(v,k,{\lambda})$-design with $t$ orbits has a partial parallel class that contains $s-1$ blocks from each of at most ${\epsilon}t$ orbits and contains $s$ blocks from each other orbit.
Note that $s \geq 2$ by our hypotheses. We may assume that ${\epsilon}< \frac{1}{4k^2}$. Let ${\epsilon}^*=\frac{{\epsilon}}{2(k+1)s}$. Let $(V,{\mathcal{B}})$ be a cyclic $(v,k,{\lambda})$-design with orbits ${\mathcal{B}}_1, \ldots, {\mathcal{B}}_t$ and suppose that $m$ of these orbits are full. Throughout this proof, we will tacitly assume $v$ is sufficiently large whenever necessary and will use asymptotic notation with respect to this regime. Note that $t=\frac{{\lambda}(v-1)}{k(k-1)}+O(1)$ by Lemma \[L:fewShort\](ii) and hence $t=\Theta(v)$. By Lemma \[L:applyPipSpe\] there is a partial parallel class ${\mathcal{P}}$ of $(V,{\mathcal{B}})$ such that $T_0({\mathcal{P}})$ contains at most ${\epsilon}^*m \leq \epsilon^* t$ indices of full orbits and every other index of a full orbit is in $T_s({\mathcal{P}})$. Let $$R = \{i \in [t]: \text{${\mathcal{B}}_i$ contains at least $\tfrac{1}{2}st$ ${\mathcal{P}}$-bad blocks}\}.$$
A block in ${\mathcal{B}}$ is ${\mathcal{P}}$-bad if and only if it intersects at least two blocks in ${\mathcal{P}}\cap {\mathcal{B}}_i$ for some $i \in T_s({\mathcal{P}})$. At most $k^2 {\lambda}\binom{s}{2}$ blocks of ${\mathcal{B}}$ intersect at least two blocks in ${\mathcal{P}}\cap {\mathcal{B}}_i$ for each $i \in T_s({\mathcal{P}})$, and so it follows that at most $k^2 {\lambda}\binom{s}{2}{\tau}_s({\mathcal{P}})\leq k^2 {\lambda}\binom{s}{2}t$ blocks in ${\mathcal{B}}$ are ${\mathcal{P}}$-bad. Thus, by the definition of $R$, we have $|R| \leq k^2s{\lambda}$.
We can greedily choose a partial parallel class ${\mathcal{R}}$ in $(V,{\mathcal{B}})$ such that $|{\mathcal{R}}\cap {\mathcal{B}}_i|=s$ for each $i \in R$ and ${\mathcal{R}}\cap {\mathcal{B}}_i=\emptyset$ for each $i \in [t] \setminus R$. To see this, suppose that $x<s|R|\leq k^2s^2{\lambda}$ blocks of the class have already been chosen and note that, for each $i \in R$, at most $k^2$ of blocks in ${\mathcal{B}}_i$ intersect each already chosen block and $$|{\mathcal{B}}_i| \geq \tfrac{v}{k} \gg k^4s^2{\lambda}>k^2x.$$ Thus we can indeed choose a suitable ${\mathcal{R}}$ greedily.
Now let $$Q = \{i \in [t] \setminus R: \text{some block in ${\mathcal{P}}\cap {\mathcal{B}}_i$ intersects some block in ${\mathcal{R}}$}\}.$$ Observe that $ks|R| \leq k^3s^2{\lambda}$ vertices in $V$ are in a block in ${\mathcal{R}}$ and hence $|Q| \leq k^3s^2{\lambda}$.
Let ${\mathcal{P}}'={\mathcal{R}}\cup \bigcup_{i \in [t] \setminus Q}({\mathcal{P}}\cap {\mathcal{B}}_i)$ and note that ${\mathcal{P}}'$ is a partial parallel class in $(V,{\mathcal{B}})$. So $T_s({\mathcal{P}}')=(T_s({\mathcal{P}}) \cup R) \setminus Q$ and $T_0({\mathcal{P}}')=[t] \setminus T_s({\mathcal{P}}')$. Thus ${\tau}_1({\mathcal{P}}')=\cdots={\tau}_{s-1}({\mathcal{P}}') = 0$ and ${\tau}_0({\mathcal{P}}') \leq {\tau}_0({\mathcal{P}})+|Q|$. Furthermore, $T_0({\mathcal{P}})$ contains at most $\epsilon^* t$ indices of full orbits and, by Lemma \[L:fewShort\](i), at most $2{\lambda}\sqrt{k}$ indices of short orbits. From this it follows that $$\label{E:dPbound}
d({\mathcal{P}}') = (s-1){\tau}_0({\mathcal{P}}') \leq (s-1)({\tau}_0({\mathcal{P}})+|Q|) <\mfrac{{\epsilon}t}{2(k+1)}+O(1) \ll \mfrac{{\epsilon}t}{k+1}.$$
Any block in ${\mathcal{B}}$ that was ${\mathcal{P}}$-good but is ${\mathcal{P}}'$-bad must intersect two of the $s$ blocks in ${\mathcal{P}}' \cap {\mathcal{B}}_i$ for some $i \in R$. For each $i \in R$, at most $k^2 {\lambda}\binom{s}{2}$ blocks in ${\mathcal{B}}$ intersect two of the blocks in ${\mathcal{P}}' \cap {\mathcal{B}}_i$. So at most $k^2 {\lambda}\binom{s}{2}|R|\leq k^4s{\lambda}^2\binom{s}{2}$ blocks in ${\mathcal{B}}$ were ${\mathcal{P}}$-good but are ${\mathcal{P}}'$-bad. Thus, for each $i \in T_0({\mathcal{P}}')$, because $i \notin R$ and hence less than $\frac{1}{2}st$ blocks in ${\mathcal{B}}_i$ were ${\mathcal{P}}$-bad, the number of ${\mathcal{P}}'$-bad blocks in ${\mathcal{B}}_i$ is less than $\tfrac{1}{2}st + k^4s{\lambda}^2\binom{s}{2}$. Now $t \leq \frac{{\lambda}(v-1)}{k(k-1)}+2{\lambda}\sqrt{k}$ by Lemma \[L:fewShort\](ii) and hence $st \leq \frac{v}{k}+O(1)$. So, since $|{\mathcal{B}}_i|\geq \frac{v}{k}$, more than $\frac{v}{k}-\tfrac{1}{2}st - k^4s{\lambda}^2\binom{s}{2} \geq \frac{1}{2}st-O(1)$ blocks in ${\mathcal{B}}_i$ are ${\mathcal{P}}'$-good. Thus ${\mathcal{P}}'$ satisfies the conditions of Lemma \[L:greedy\] because $$k^2(ks-k+1)(d({\mathcal{P}}')-1) < {\epsilon}k^2 s t < \tfrac{1}{4}st \ll \tfrac{1}{2}st-O(1)$$ where the first inequality follows by because $ks-k+1<s(k+1)$ and the second follows because ${\epsilon}< \frac{1}{4k^2}$. Thus, by applying Lemma \[L:greedy\] to ${\mathcal{P}}'$, there is a partial parallel class ${\mathcal{P}}''$ of $(V,{\mathcal{B}})$ such that ${\tau}_s({\mathcal{P}}'')=t-{\tau}_{s-1}({\mathcal{P}}'')$ and $${\tau}_{s-1}({\mathcal{P}}'') \leq (k+1)d({\mathcal{P}}') < {\epsilon}t$$ where the last inequality follows by .
Note that in the special case where ${\lambda}$ divides $k-1$, the partial parallel class given by Theorem \[thm:asympt\] uses all but at most $\epsilon kt+1$ points of the design.
[**Proof of Theorem \[thm:main theorem\]**]{} This follows directly from Theorem \[thm:asympt\], noting that $s \geq 2$ because $k \geq 2{\lambda}+1$.
Concluding remarks
==================
A $(v,k,\lambda)$-DDF necessarily has $1\leq \lambda \leq k-1$ apart from the trivial case of a $(k,k,k)$-DDF (see [@burattik-1]). Theorem \[thm:main theorem\] requires $1 \leq {\lambda}\leq (k-1)/2$. It is natural to ask whether it is possible to relax this condition. We make the following conjecture.
\[con:any k any lambda\] Let $k$ and ${\lambda}$ be fixed positive integers such that $k \geq \lambda+1$. There exists an integer $v_0$ such that, for any cyclic $(v,k,\lambda)$-design with $v\geq v_0$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint.
Compared with Conjecture \[conj:any k\], Conjecture \[con:any k any lambda\] is stated for sufficiently large $v$. This is from the observation that the union of $\lambda$ copies of a $(k(k-1)+1,k,1)$-CDF forms a $(k(k-1)+1,k,\lambda)$-CDF which yields a cyclic $(k(k-1)+1,k,\lambda)$-design without short orbits. Note that a $(k(k-1)+1,k,1)$-CDF is often called a [*cyclic difference set*]{} (see [@baumert]) and it generates a symmetric design, any two blocks of which must intersect in one point. Thus the resulting cyclic $(k(k-1)+1,k,\lambda)$-design cannot be generated by a DDF.
Finally we remark that Novák made a stronger conjecture on cyclic STS$(v)$ than Conjecture \[weak conjecture\] in 1974. A $(v,3,1)$-DDF for $v\equiv 1\pmod{6}$ is called *symmetric* if its base blocks can be chosen in such a way that for any nonzero $x$ of $\mathbb Z_v$, at most one of $x$ and its complement $v-x$ occurs in the base blocks and no base block contains zero.
\[strong conjecture\] Every cyclic STS$(v)$ with $v\equiv 1 \pmod{6}$ is generated by a symmetric $(v,3,1)$-DDF.
So far it is only known that Conjecture \[strong conjecture\] holds for all $v\equiv 1\pmod{6}$ and $v\leq 61$ (see [@triplesystem Theorem 22.3]).
Acknowledgments {#acknowledgments .unnumbered}
===============
Research for this paper was carried out while the first and third authors were visiting Monash University. They express their sincere thanks to the School of Mathematics at Monash University for their kind hospitality.
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abstract: 'We construct a new family of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$, whose Albanese map is a quadruple cover of an abelian surface with polarization of type $(1, \,3)$. We also show that this family provides an irreducible component of the moduli space of surfaces with $p_g=q=2$ and $K^2=6$. Finally, we prove that such a component is generically smooth of dimension $4$ and that it contains the $2$-dimensional family of product-quotient examples previously constructed by the first author. The main tools we use are the Fourier-Mukai transform and the Schrödinger representation of the finite Heisenberg group $\mathscr{H}_3$.'
author:
- Matteo Penegini and Francesco Polizzi
title: 'A new family of surfaces with $p_g=q=2$ and $K^2=6$ whose Albanese map has degree $4$'
---
Introduction {#sec.intro}
============
In recent years, both the geographical problem and the fine classification for *irregular* algebraic surfaces (i.e. surfaces with irregularity $q>0$) have attracted the attention of several authors; in particular minimal surfaces of general type with $\chi({\mathscr{O}})=1$, i.e. $p_g=q$, were deeply investigated.
In this case some well-known results imply $p_g \leq 4$ . While surfaces with $p_g=q=4$ and $p_g=q=3$ have been completely described, see [@D82], [@CCML98], [@HP02], [@Pi02], the classification of those with $p_g=q=2$ is still missing (see [@PP10] and [@PP11] for a recent account on this topic). As the title suggests, in the present paper we consider some new surfaces $\widehat{S}$ with $p_g=q=2$ and $K^2=6$ whose Albanese map $\hat{\alpha} \colon {\widehat{S}}{\longrightarrow}{\widehat{A}}:=\textrm{Alb}({\widehat{S}})$ is a quadruple cover of an abelian surface ${\widehat{A}}$.
Our construction presents some analogies with the one presented in [@CH06] and [@PP10] for the case $p_g=q=2$ and $K^2=5$. Indeed, in both situations the Tschirnhausen bundle $\mathscr{E}^{\vee}$ associated with the Albanese cover is of the form $\Phi^{\mathscr{P}}(\mathscr{L}^{-1})$, where $\mathscr{L}$ is a polarization on $A$ (the dual abelian variety of ${\widehat{A}}$) and $\Phi^{\mathscr{P}}$ denotes the Fourier-Mukai transform. More precisely, in the case studied in [@CH06] the surfaces are triple covers, $\mathscr{E}^{\vee}$ has rank $2$ and $\mathscr{L}$ is a polarization of type $(1, \,
2)$; in our setting, instead, the cover has degree $4$, the bundle $\mathscr{E}^{\vee}$ has rank $3$ and $\mathscr{L}$ is a polarization of type $(1, \, 3)$.
The results of the paper can be summarized as follows, see Theorem \[theo.main\].
There exists a $4$-dimensional family $\mathcal{M}_{\Phi}$ of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ such that, for the general element $\widehat{S} \in \mathcal{M}_{\Phi}$, the canonical class $K_{{\widehat{S}}}$ is ample and the Albanese map $\hat{\alpha}\colon \widehat{S} \longrightarrow \widehat{A}$ is a finite cover of degree $4$.
The Tschirnhausen bundle ${\mathscr{E}}^{\vee}$ associated with $\hat{\alpha}$ is isomorphic to $\Phi^{\mathscr{P}}(\mathscr{L}^{-1})$, where $\mathscr{L}$ is a polarization of type $(1, \, 3)$ on $A$.
The family $\mathcal{M}_{\Phi}$ provides an irreducible component of the moduli space $\mathcal{M}_{2, \, 2, \,6}^{\emph{can}}$ of canonical models of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$. Such a component is generically smooth and contains the $2$-dimensional family of product-quotient surfaces constructed by the first author in *[@Pe11]*.
As far as we know, the family $\mathcal{M}_{\Phi}$ gives the only examples currently known of minimal surfaces with $p_g=q=2$, $K^2=6$ and Albanese map of degree $4$.
Our Main Theorem is obtained by extending the construction given in [@CH06] to the much more complicated case of quadruple covers. More precisely, in order to build a quadruple cover $\hat{\alpha} \colon \widehat{S} {\longrightarrow}\widehat{A}$ with Tschirnhausen bundle ${\mathscr{E}}^{\vee}$, we first build a quadruple cover $\alpha \colon S {\longrightarrow}A$ with Tschirnhausen bundle $\phi_{{\mathscr{L}}^{-1}}^*{\mathscr{E}}^{\vee}$ (here $\phi_{{\mathscr{L}}} \colon A {\longrightarrow}\widehat{A}$ denotes the group homomorphism sending $x \in A$ to $t_x^* \mathscr{L}^{-1} \otimes \mathscr{L} \in \widehat{A}$) and then, by using the Schrödinger representation of the finite Heisenberg group $\mathscr{H}_3$ on $H^0(A, \, {\mathscr{L}})$, we identify those covers of this type that descend to a quadruple cover $\hat{\alpha} \colon \widehat{S} {\longrightarrow}\widehat{A}$. For the general surface ${\widehat{S}}$, the branch divisor $\widehat{B} \subset {\widehat{A}}$ of $\alpha \colon {\widehat{S}}{\longrightarrow}{\widehat{A}}$ is a curve in the linear system $|\widehat{\mathscr{L}}^{\otimes 2}|$, where $\widehat{\mathscr{L}}$ is a polarization of type $(1, \,3)$ on ${\widehat{A}}$, with six ordinary cusps and no other singularities; such a curve is $\mathscr{H}_3$-equivariant and can be associated with the dual of a member of the Hesse pencil of plane cubics in $\mathbb{P}^2$.
Let us explain now the way in which this paper is organized.
In Section \[sec.prel\] we set up notation and terminology and we present some preliminaries. In particular we recall the theory of quadruple covers developed by Casnati-Ekedahl [@CE96] and Hahn-Miranda [@HM99] and we briefly describe the geometry of the Hesse pencil.
Sections \[sec.constr\], \[sec.moduli\], \[sec.simple\] are devoted to the proof of the Main Theorem.
In Section \[sec.constr\] we present our construction, we prove that a general surface $\widehat{S}$ in our family is smooth and we compute its invariants (Propositions \[prop.sing.covers\] and \[prop.num.invariants.cover\]).
In Section \[sec.moduli\] we examine the subset of the moduli space corresponding to our surfaces, showing that it is generically smooth, of dimension $4$ and that it closure $\mathcal{M}_{\Phi}$ provides an irreducible component of $\mathcal{M}_{2, \, 2, \,
6}^{\textrm{can}}$ (Proposition \[prop.irred.comp\]). We also observe that the general surface in $\mathcal{M}_{\Phi}$ admits no pencil over a curve of strictly positive genus (Proposition \[rem-no-irr\]).
In Section \[sec.simple\] we prove that the moduli space $\mathcal{M}_{2, \, 2, \,
6}^{\textrm{can}}$ has a $3$-dimensional singular locus (Corollary \[cor.sing.loc.M\]). Moreover, we show that the irreducible component $\mathcal{M}_{\Phi}$ contains the $2$-dimensional family of product-quotient examples constructed by the first author in [@Pe11] (Proposition \[wT-in-Mphi\]).
Finally, in Section \[sec.open.prob\] we present some open problems.
The paper also contains two appendices. In the Appendix $1$ we show the following technical result needed in the proof of the Main Theorem: for a general choice of the pair $(A, \,
\mathscr{L})$, the three distinguished divisors coming from the Schrödinger representation of the Heisenberg group $\mathscr{H}_3$ on $H^0(A, \, \mathscr{L})$ are smooth and intersect transversally. Appendix $2$ contains the computer algebra script we used to compute the equation of the branch curve $B$ of $\alpha \colon S {\longrightarrow}A$; it is written in the Computer Algebra System `MAGMA`, see [@Magma].
**Acknowledgments** Both authors were partially supported by the DFG Forschergruppe 790 *Classification of algebraic surfaces and compact complex manifolds*.
F. Polizzi was partially supported by Progetto MIUR di Rilevante Interesse Nazionale *Geometria delle Variet$\grave{a}$ Algebriche e loro Spazi di Moduli*. He thanks the Mathematisches Institut, Universit$\ddot{\textrm{a}}$t Bayreuth for the invitation and the hospitality in the period October-November 2011.
The authors are indebted with F. Catanese for sharing with them some of his ideas on this subject. They are also grateful to I. Bauer, W. Liu, E. Sernesi, L. van Geemen for useful discussions and suggestions.
**Notation and conventions.** We work over the field $\mathbb{C}$ of complex numbers.
If $A$ is an abelian variety, we call ${\widehat{A}}:= \textrm{Pic}^0(A)$ its dual abelian variety.
If $\mathscr{L}$ is a line bundle on $A$ we denote by $\phi_{\mathscr{L}}$ the morphism $\phi_{{\mathscr{L}}} \colon A
\longrightarrow {\widehat{A}}$ given by $x \mapsto t^*_x {\mathscr{L}}\otimes
{\mathscr{L}}^{-1}$. If ${\mathscr{L}}$ is non-degenerate then $\phi_{{\mathscr{L}}}$ is an isogeny, and its kernel is denoted by $K({\mathscr{L}})$. In this case the *index* of ${\mathscr{L}}$ is the unique integer $i({\mathscr{L}})$ such that $H^j(A, \, {\mathscr{L}})=0$ unless $j=i({\mathscr{L}})$.
Throughout the paper we use italic letters for line bundles and capital letters for the corresponding Cartier divisors, so we write for instance $\mathscr{L}={\mathscr{O}}_S(L)$. The corresponding complete linear system is denoted either by $|\mathscr{L}|$ or by $|L|$.
If $L$ is an ample divisor on $A$, then it defines a positive line bundle $\mathscr{L}={\mathscr{O}}_A(L)$, whose first Chern class is a polarization on $A$. By abuse of notation we consider both the line bundle $\mathscr{L}$ and the divisor $L$ as polarizations.
By “*surface*” we mean a projective, non-singular surface $S$, and for such a surface $\omega_S={\mathscr{O}}_S(K_S)$ denotes the canonical class, $p_g(S)=h^0(S, \, \omega_S)$ is the *geometric genus*, $q(S)=h^1(S, \, \omega_S)$ is the *irregularity* and $\chi(\mathscr{O}_S)=1-q(S)+p_g(S)$ is the *Euler-Poincaré characteristic*.
Preliminaries {#sec.prel}
=============
Quadruple cover of algebraic varieties
--------------------------------------
The two papers [@CE96] and [@HM99] deal with the quadruple covers of algebraic varieties; the former only considers the Gorenstein case, whereas the latter develops the theory in full generality. The main results are the following.
*[@CE96 Theorems 1.6 and 4.4]*\[theo.casnati\] Let $Y$ be a smooth algebraic variety. Any quadruple Gorenstein cover $f\colon X \longrightarrow Y$ is determined by a locally free $\mathscr{O}_{Y}-$module ${\mathscr{E}}$ of rank $3$, a locally free $\mathscr{O}_{Y}-$module ${\mathscr{F}}$ of rank $2$ with $\bigwedge^2 {\mathscr{F}}\cong \bigwedge^3 {\mathscr{E}}^{\vee}$ and a general section $\eta \in H^0(Y,
\, S^2{\mathscr{E}}^{\vee} \otimes {\mathscr{F}}^{\vee} )$.
*[@HM99 Theorem 1.2]*\[theo.mir\] Let $Y$ be a smooth algebraic variety. Any quadruple cover $f\colon
X \longrightarrow Y$ is determined by a locally free $\mathscr{O}_Y$-module $\mathscr{E}$ of rank $3$ and a totally decomposable section $\eta \in H^0(Y, \, \bigwedge^2 S^2
\mathscr{E}^{\vee} \otimes \bigwedge^3 \mathscr{E})$.
In order to make the notation consistent, in Theorem \[theo.casnati\] we called ${\mathscr{E}}^{\vee}$ the sheaf which is called ${\mathscr{E}}$ in [@CE96]. In Theorem \[theo.mir\], *totally decomposable* means that, for all $y \in Y$, the image of $$\eta|_y \colon (\bigwedge^3 {\mathscr{E}}^{\vee})_y \longrightarrow
(\bigwedge^2 S^2 \mathscr{E}^{\vee})_y$$ is totally decomposable in $(\bigwedge^2 S^2 \mathscr{E}^{\vee})_y$, i.e. of the form $\xi_1 \wedge \xi_2$ with $\xi_i \in (S^2
\mathscr{E}^{\vee})_y$.
The vector bundle ${\mathscr{E}}^{\vee}$ is called the *Tschirnhausen bundle* of the cover. We have $f_{*}\mathscr{O}_X= \mathscr{O}_Y
\oplus {\mathscr{E}}$, so $$\label{eq.inv.hacca}
h^i(X, \, \mathscr{O}_X)=h^i(Y, \, \mathscr{O}_Y)+h^i(Y, \, {\mathscr{E}}).$$
By [@CE96 Proposition 5.1] there is an exact sequence $$\label{suc.effe}
0 \longrightarrow {\mathscr{F}}\stackrel{\varphi}{\longrightarrow} S^2
{\mathscr{E}}^{\vee} \longrightarrow f_*\omega^{2}_{X | Y} \longrightarrow 0$$ and the associated Eagon-Northcott complex tensored with $\bigwedge^2 {\mathscr{F}}^{\vee}$ yields $$\label{suc.igor.notcot}
0 \longrightarrow S^2{\mathscr{F}}\otimes \bigwedge^2 {\mathscr{F}}^{\vee}
\longrightarrow S^2 {\mathscr{E}}^{\vee} \otimes {\mathscr{F}}^{\vee} \longrightarrow
\bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}}.$$
The induced map in cohomology $$H^0(Y, \, S^2 {\mathscr{E}}^{\vee} \otimes {\mathscr{F}}^{\vee}) \longrightarrow H^0(Y,
\, \bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}})$$ provides the bridge between Theorem \[theo.casnati\] and Theorem \[theo.mir\]. In fact a straightforward computation shows that it sends the element $\varphi \in \textrm{Hom}({\mathscr{F}}, \, S^2 {\mathscr{E}}^{\vee}) \cong H^0(Y,
\, S^2 {\mathscr{E}}^{\vee} \otimes {\mathscr{F}}^{\vee})$ to the totally decomposable element $2 \varphi \wedge \varphi \in \textrm{Hom}(\bigwedge^2 {\mathscr{F}},
\, \bigwedge^2S^2 {\mathscr{E}}^{\vee}) \cong H^0(Y, \, \bigwedge^2 S^2
{\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}})$.
When $X$ and $Y$ are smooth surfaces one has the following useful formulae.
*[@CE96 Proposition 5.3]*\[prop.invariants\] Let $X$ and $Y$ be smooth, connected, projective surfaces, $f\colon
X \longrightarrow Y$ a cover of degree $4$ and $R \subset X$ the ramification divisor of $f$. Then:
- $\chi (\mathscr{O}_X)=
4\chi(\mathscr{O}_Y) +
\frac{1}{2}c_1({{\mathscr{E}}^{\vee}})K_Y+\frac{1}{2}c^2_1({\mathscr{E}}^{\vee})-c_2({\mathscr{E}}^{\vee})$;
- $K^2_X= 4K^2_Y +
4c_1({{\mathscr{E}}^{\vee}})K_Y+2c^2_1({\mathscr{E}}^{\vee})-4c_2({\mathscr{E}}^{\vee})+c_2({\mathscr{F}})$;
- $p_a(R)= 1 +
c_1({{\mathscr{E}}^{\vee}})K_Y+2c^2_1({\mathscr{E}}^{\vee})-4c_2({\mathscr{E}}^{\vee})+c_2({\mathscr{F}})$.
Fourier-Mukai transforms of (W)IT-sheaves {#sub.fourier}
-----------------------------------------
Let $A$ be an abelian variety of dimension $g$ and ${\widehat{A}}:=
\textrm{Pic}^0(A)$ its dual abelian variety. We say that a coherent sheaf $\mathscr{F}$ on $A$ is
- a *IT-sheaf of index i* (or, equivalently, that ${{\mathscr{F}}}$ *satisfies IT of index i*) if $$H^j(A, \, \mathscr{F} \otimes \mathscr{Q})=0 \quad \textrm{for all }
\mathscr{Q} \in \textrm{Pic}^0(A) \quad \textrm{and } j\neq i;$$
- a *WIT-sheaf of index i* (or, equivalently, that ${{\mathscr{F}}}$ *satisfies WIT of index i*) if $$R^j \pi_{{\widehat{A}}\, *}(\mathscr{P} \otimes
\pi^*_{A}\mathscr{F})=0 \quad \textrm{for all} \quad j \neq i,$$ where $\mathscr{P}$ is the normalized Poincaré bundle on $A \times
{\widehat{A}}$.
If ${\mathscr{F}}$ is a WIT-sheaf of index $i$, the coherent sheaf $$\Phi^{\mathscr{P}}(\mathscr{F}):={R}^i\pi_{{\widehat{A}}\, *}(\mathscr{P}
\otimes \pi^*_{A}\mathscr{F})$$ is called the *Fourier-Mukai transform* of $\mathscr{F}$.
For simplicity of notation, given any WIT-sheaf $\mathscr{G}$ of index $i$ on $\widehat{A}$ we use the same symbol $\Phi^{\mathscr{P}}$ for its Fourier-Mukai transform $$\Phi^{\mathscr{P}}(\mathscr{G}):={R}^i\pi_{A \,
*}(\mathscr{P} \otimes \pi^*_{{\widehat{A}}}\mathscr{G}).$$
By the Base Change Theorem (see [@Mum70 Chapter II]) it follows that ${\mathscr{F}}$ satisfies IT of index $i$ if and only it satisfies WIT of index $i$ and $\Phi^{\mathscr{P}}(\mathscr{F})$ is locally free. In particular, any non-degenerate line bundle ${\mathscr{L}}$ of index $i$ on $A$ is a IT-sheaf of index $i$ and its Fourier-Mukai transform $\Phi^{\mathscr{P}}(\mathscr{L})$ is a vector bundle of rank $h^i(A,
\, {\mathscr{L}})$ on $\widehat{A}$.
*[@Mu81] [@BL04 Theorem 14.2.2]* \[prop.mukai\] Let ${\mathscr{F}}$ be a WIT-sheaf of index $i$ on $A$. Then $\Phi^{\mathscr{P}}(\mathscr{F})$ is a WIT-sheaf of index $g-i$ on $\widehat{A}$ and $$\Phi^{\mathscr{P}} \circ \Phi^{\mathscr{P}} ({\mathscr{F}}) = (-1)_{A}^*{\mathscr{F}}.$$
\[rem.FM\] In general the Fourier-Mukai transform induces an equivalence of categories between the two derived categories $D(A)$ and $D(\widehat{A})$, such that for all $\mathscr{F} \in D(A)$ and $\mathscr{G} \in D({\widehat{A}})$ one has $$\Phi^{\mathscr{P}} \circ \Phi^{\mathscr{P}}(\mathscr{F}) =
(-1)_{A}^*(\mathscr{F}) [-g] \quad \textrm{and} \quad
\Phi^{\mathscr{P}} \circ \Phi^{\mathscr{P}}(\mathscr{G}) =
(-1)_{\widehat{A}}^*(\mathscr{G}) [-g],$$ where $[-g]$ means “shift the complex $g$ places to the right". When ${\mathscr{F}}$ is a $WIT$-sheaf the complex $\Phi^{\mathscr{P}}(\mathscr{F})$ can be identified with a coherent sheaf, since it is different from zero at most at one place.
\[cor.menouno\] Let ${\mathscr{F}}$ be a WIT-sheaf on $A$. Then $$\Phi^{\mathscr{P}} ((-1)_A^* {\mathscr{F}}) =
(-1)_{\widehat{A}}^*\Phi^{\mathscr{P}} ({\mathscr{F}}).$$
Set $\mathscr{F}':=\Phi^{\mathscr{P}} ((-1)_A^* {\mathscr{F}})$; then by Proposition \[prop.mukai\] $${\mathscr{F}}=\Phi^{\mathscr{P}} \circ \Phi^{\mathscr{P}}((-1)_A^*
{\mathscr{F}})=\Phi^{\mathscr{P}}(\mathscr{F}'),$$ hence $$\Phi^{\mathscr{P}}({\mathscr{F}})=\Phi^{\mathscr{P}} \circ
\Phi^{\mathscr{P}}(\mathscr{F}')=(-1)_{\widehat{A}}^*
\mathscr{F'}=(-1)_{\widehat{A}}^* \Phi^{\mathscr{P}} ((-1)_A^* {\mathscr{F}}).$$
The Hesse pencil and the family of its dual curves {#sub.hal}
--------------------------------------------------
In the sequel we will use some classical facts about the Hesse pencil of cubic curves, that are summarized here for the reader’s convenience. We follow the treatment given in [@AD09].
The *Hesse pencil* is the one-dimensional linear system of plane cubic curves defined by $$E_{t_0, \, t_1}\colon t_0(x^3+y^3+z^3)+t_1xyz=0, \quad [t_0 : \,
t_1] \in \mathbb{P}^1.$$ Its nine base points are the inflection points of any smooth curve in the pencil. There are four singular members in the Hesse pencil and each one is the union of three lines: $$\begin{array}{ll}
E_{0,\, 1} \colon & xyz=0, \\
E_{1, \, -3} \colon & (x+y+z)(x+\omega y+ \omega^2 z)(x+ \omega^2 y + \omega z)=0, \\
E_{1,\, -3\omega}\colon & (x+\omega y + z)(x+ \omega^2 y + \omega^2 z)(x+y+ \omega z)=0, \\
E_{1, \, -3\omega^2} \colon & (x+\omega^2 y+ z)(x+ \omega y + \omega
z)(x+y+\omega^2 z)=0.
\end{array}$$ We call the singular members the *triangles*.
The dual curve $\mathbf{B}_{m_0, \, 3m_1}$ of a smooth member $E_{m_0,3m_1}$ of the Hesse pencil is a plane curve of degree six with nine cusps, whose equation in the dual plane $(\mathbb{P}^{2})^{\vee}$ is $$\label{eq.dual.Hesse}
\begin{split}
& m^4_0(X^6+Y^6+Z^6)-m_0(2m^2_0+32m^3_1)(X^3Y^3+X^3Z^3+Y^3Z^3) \\
-24 & m^2_0m^2_1XYZ(X^3+Y^3+Z^3)-(24m^3_0m_1+48m^4_1)X^2Y^2Z^2=0.
\end{split}$$ Notice that the dual of a triangle becomes a triangle taken with multeplicity two. For any pair $(m_0, \, m_1)$ there is a unique cubic $\mathbf{C}_{m_0,\, 3m_1}$ passing through the nine cusps of $\mathbf{B}_{m_0,\, 3m_1}$. The general element of the pencil generated by $\mathbf{B}_{m_0, \, 3m_1}$ and $2 \mathbf{C}_{m_0,\,
3m_1}$ is an irreducible curve of degree six with nine nodes at the nine cusps of $\mathbf{B}_{m_0,\, 3m_1}$. Such a pencil is called the *Halphen pencil* associated with $\mathbf{B}_{m_0,\, 3m_1}$, see [@H1882].
The construction {#sec.constr}
================
The aim of this section and the next one is to prove the main result of the paper, namely the following
\[theo.main\] There exists a $4$-dimensional family $\mathcal{M}_{\Phi}$ of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ such that, for the general element $\widehat{S} \in \mathcal{M}_{\Phi}$, the canonical class $K_{\widehat{S}}$ is ample and the Albanese map $\hat{\alpha}\colon \widehat{S} \longrightarrow \widehat{A}$ is a finite cover of degree $4$.
The Tschirnhausen bundle ${\mathscr{E}}^{\vee}$ associated with $\hat{\alpha}$ is isomorphic to $\Phi^{\mathscr{P}}(\mathscr{L}^{-1})$, where $\mathscr{L}$ is a polarization of type $(1,\,3)$ on $A$.
The family $\mathcal{M}_{\Phi}$ provides an irreducible component of the moduli space $\mathcal{M}_{2, \, 2, \,6}^{\emph{can}}$ of canonical models of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$. Such a component is generically smooth.
We first outline the main idea of our construction, which is inspired by the one used in [@CH06] in the simpler case where the Albanese map has degree $3$. In order to build a quadruple cover $\hat{\alpha} \colon \widehat{S} {\longrightarrow}\widehat{A}$ with Tschirnhausen bundle ${\mathscr{E}}^{\vee}$, we first build a quadruple cover $\alpha \colon S {\longrightarrow}A$ with Tschirnhausen bundle $\phi_{{\mathscr{L}}^{-1}}^*{\mathscr{E}}^{\vee}$ and then we identify those covers of this type that descend to a quadruple cover $\hat{\alpha} \colon
\widehat{S} {\longrightarrow}\widehat{A}$ (Propositions \[prop.dis.sec\] and \[prop.tot.decomp\]). Furthermore we prove that for a general such a cover the surface $\widehat{S}$ is smooth and we compute its invariants (Propositions \[prop.sing.covers\] and \[prop.num.invariants.cover\]). Finally, in Section \[sec.moduli\] we examine the subset $\mathcal{M}_{\Phi}$ of the moduli space corresponding to our surfaces.
We start by considering a $(1, \,3)$-polarized abelian surface $(A,
\, {\mathscr{L}})$. For all $\mathscr{Q} \in \textrm{Pic}^0(A)$ we have $$h^0(A, \, {\mathscr{L}}\otimes \mathscr{Q} )=3, \quad h^1(A, \, {\mathscr{L}}\otimes
\mathscr{Q})=0, \quad h^2(A, \, {\mathscr{L}}\otimes \mathscr{Q})=0,$$ so by Serre duality the line bundle ${\mathscr{L}}^{-1}$ satisfies IT of index $2$. Its Fourier-Mukai transform $\mathscr{E}^{\vee}:=
\Phi^{\mathscr{P}}({\mathscr{L}}^{-1})$ is a rank $3$ vector bundle on ${\widehat{A}}$ which satisfies WIT of index $0$, see Proposition \[prop.mukai\]. The isogeny $$\phi:=\phi_{{\mathscr{L}}^{-1}}\colon A \longrightarrow {\widehat{A}}, \quad
\phi(x)=t_x^*{\mathscr{L}}^{-1} \otimes {\mathscr{L}}$$ has kernel $K({\mathscr{L}}^{-1})=K({\mathscr{L}}) \cong ({\mathbb{Z}}/3 {\mathbb{Z}})^2$ and by [@Mu81 Proposition 3.11] we have $$\label{eq.mukai}
\phi^* {\mathscr{E}}^{\vee} = {\mathscr{L}}\oplus {\mathscr{L}}\oplus {\mathscr{L}}.$$ Since $\Phi^{\mathscr{P}}({\mathscr{E}}^{\vee})= \Phi^{\mathscr{P}} \circ
\Phi^{\mathscr{P}}({\mathscr{L}}^{-1})=(-1)_A^* {\mathscr{L}}^{-1}$ is locally free, it follows that ${\mathscr{E}}^{\vee}$ is actually a IT-sheaf of index $0$. By we have $$\label{eq-chern.phi.E}
c_1(\phi^*{\mathscr{E}}^{\vee})=3L, \quad c_2(\phi^*{\mathscr{E}}^{\vee})=18,$$ hence $$\label{eq-chern.E}
c_1({\mathscr{E}}^{\vee})=\widehat{L}, \quad c_2({\mathscr{E}}^{\vee})=2,$$ where $\widehat{L}$ is a polarization of type $(1, \, 3)$ on $\widehat{A}$. Therefore Hirzebruch-Riemann-Roch yields $\chi(\widehat{A}, \, {\mathscr{E}}^{\vee})=1$, which in turn implies $$\label{eq.cohom.EE^vee}
h^0(\widehat{A}, \, {\mathscr{E}}^{\vee})=1, \quad h^1(\widehat{A}, \,
{\mathscr{E}}^{\vee})=0, \quad h^2(\widehat{A}, \, {\mathscr{E}}^{\vee})=0.$$ Now we want to construct a quadruple cover $\widehat{\alpha} \colon
\widehat{S} \longrightarrow \widehat{A}$ with Tschirnhausen bundle ${\mathscr{E}}^{\vee}$. By [@HM99] it suffices to find those totally decomposable elements in $$H^0 \big(A, \, \bigwedge^2 S^2
(\phi^*{\mathscr{E}}^{\vee}) \otimes \bigwedge^3 (\phi^* {\mathscr{E}}) \big) \cong
H^0(A, \, {\mathscr{L}})^{\oplus 15}$$ that are $K(\mathscr{L}^{-1})$-invariant and therefore descend to elements in $H^0 \big(\widehat{A}, \, \bigwedge^2 S^2 {\mathscr{E}}^{\vee}
\otimes \bigwedge^3 {\mathscr{E}}\big)$.
Let us consider the Heisenberg group $$\mathscr{H}_3:=\{(k, \, t, \, l) \mid k \in {\mathbb{C}}^* , \, t \in
{\mathbb{Z}}/3{\mathbb{Z}}, \, l \in \widehat{{\mathbb{Z}}/3{\mathbb{Z}}} \},$$ whose group law is $$(k, \, t, \, l)\cdot (k', \, t', \, l')=(kk'l'(t), \, t+t', \,
l+l').$$
By [@BL04 Chapter 6] there exists a canonical representation, known as the Schrödinger representation, of $\mathscr{H}_3$ on $H^0(A, \, {\mathscr{L}})$, where the latter space is identified with the vector space $V:={\mathbb{C}}({\mathbb{Z}}/3{\mathbb{Z}})$ of all complex-valued functions on the finite group ${\mathbb{Z}}/3{\mathbb{Z}}$. Such an action is given by $$(k, \, t, \, l)f(x)=kl(x)f(t+x).$$ Let $\{X, \, Y, \, Z\}$ be the basis of $H^0(A, \, {\mathscr{L}})$ corresponding to the characteristic functions of $0, \, 1, \, 2$ in $V$.
\[prop.X.Y.Z-art\] For a general choice of the pair $(A, \, \mathscr{L})$, the three effective divisors defined by $X$, $Y$, $Z \in H^0(A, \,
\mathscr{L})$ are smooth and intersect transversally.
See Appendix $1$, in particular Proposition \[prop.X.Y.Z\].
\[prop.dis.sec\] The $K(\mathscr{L}^{-1})$-invariant subspace of $H^0 \big(A, \,
\bigwedge^2 S^2 (\phi^*{\mathscr{E}}^{\vee}) \otimes \bigwedge^3 (\phi^* {\mathscr{E}})
\big)$ can be identified with $$\label{eq.dis.sec}
\begin{split}
\phi^*H^0 \big(\widehat{A}, \, \bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}}\big) = & \{(aZ, \, bY, \, cY, \, dX, \, -cZ, \, eX, \, -bX, \\
& -eZ, \, -dY, \, -dZ, \, eY, \, -aX, \, aY, \, cX, \, -bZ) \mid \\ & a, \, b, \, c, \, d, \, e \in {\mathbb{C}}\} \subset H^0(A, \, {\mathscr{L}})^{\oplus 15}.
\end{split}$$
With respect to the basis $\{X, \, Y, \, Z\}$, the Schrödinger representation can be written as $$\begin{array}{lll}
(1, \, 0, \, 0) \mapsto \left( \begin{matrix}
1&0&0\\
0&1&0\\
0&0&1
\end{matrix}\right), &
(1, \, 0, \, 1) \mapsto \left( \begin{matrix}
0&1&0\\
0&0&1\\
1&0&0
\end{matrix}\right), &
(1, \, 0, \, 2) \mapsto \left( \begin{matrix}
0&0&1\\
1&0&0\\
0&1&0
\end{matrix}\right),
\end{array}$$
$$\begin{array}{lll}
(1, \, 1, \, 0) \mapsto \left( \begin{matrix}
1&0&0\\
0&\omega &0\\
0&0& \omega^2
\end{matrix}\right), &
(1, \, 1, \, 1) \mapsto \left( \begin{matrix}
0&1&0\\
0&0&\omega \\
\omega^2 &0&0
\end{matrix}\right), &
(1, \, 1, \, 2) \mapsto \left( \begin{matrix}
0&0&1\\
\omega &0&0\\
0& \omega^2 &0
\end{matrix}\right),
\end{array}$$
$$\begin{array}{lll}
(1, \, 2, \, 0) \mapsto \left( \begin{matrix}
1&0&0\\
0&\omega^2 &0\\
0&0& \omega
\end{matrix}\right), &
(1, \, 2, \, 1) \mapsto \left( \begin{matrix}
0&1&0\\
0&0&\omega^2 \\
\omega &0&0
\end{matrix}\right), &
(1, \, 2, \, 2) \mapsto \left( \begin{matrix}
0&0&1\\
\omega^2 &0&0\\
0& \omega &0
\end{matrix}\right),
\end{array}$$
where $\omega:=e^{2\pi i/3}$. There is an induced representation of $\mathscr{H}_3$ on the $45$-dimensional vector space $\bigwedge^2
S^2 V^{\vee} \otimes \bigwedge^3 V \otimes V$, and a long but straightforward computation shows that this gives in turn a representation of $K(\mathscr{L}^{-1}) \cong ({\mathbb{Z}}/3{\mathbb{Z}})^2$. Denoting by $\{\widehat{X}$, $\widehat{Y}$, $\widehat{Z}\}$ the dual basis of $\{X, Y, Z\}$, one checks that the space of $K(\mathscr{L}^{-1})$-invariants vectors in $\bigwedge^2 S^2
V^{\vee} \otimes \bigwedge^3 V \otimes V$ has dimension $5$ and is generated by $$\begin{split}
v_1 = -(\widehat{X} \widehat{Z} \wedge \widehat{Z}^2) \otimes (X \wedge Y \wedge Z )\otimes X & + (\widehat{Y}^2 \wedge \widehat{Y} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes Y \\ & +(\widehat{X}^2 \wedge \widehat{X} \widehat{Y}) \otimes (X \wedge Y \wedge Z )\otimes Z, \\
v_2 = -(\widehat{X} \widehat{Y} \wedge \widehat{Y}^2) \otimes (X \wedge Y \wedge Z )\otimes X & + (\widehat{X}^2 \wedge \widehat{X} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes Y \\ & +(\widehat{Z}^2 \wedge \widehat{Y} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes Z, \\
v_3 = (\widehat{Y}^2 \wedge \widehat{Z}^2) \otimes (X \wedge Y \wedge Z )\otimes X & + (\widehat{X}^2 \wedge \widehat{Y}^2) \otimes (X \wedge Y \wedge Z )\otimes Y \\ & -(\widehat{X}^2 \wedge \widehat{Z}^2) \otimes (X \wedge Y \wedge Z )\otimes Z, \\
v_4 = (\widehat{X}^2 \wedge \widehat{Y} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes X & - (\widehat{X} \widehat{Y} \wedge \widehat{Z}^2) \otimes (X \wedge Y \wedge Z )\otimes Y \\ & -(\widehat{X} \widehat{Z} \wedge \widehat{Y}^2) \otimes (X \wedge Y \wedge Z )\otimes Z, \\
v_5 = (\widehat{X} \widehat{Y} \wedge \widehat{X} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes X & + (\widehat{X} \widehat{Z} \wedge \widehat{Y} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes Y \\ & -(\widehat{X} \widehat{Y} \wedge \widehat{Y} \widehat{Z}) \otimes (X \wedge Y \wedge Z )\otimes Z. \\
\end{split}$$ Therefore any $K(\mathscr{L}^{-1})$-invariant vector can be written as $av_1+bv_2+cv_3+dv_4+ev_5$ for some $a$, $b$, $c$, $d$, $e \in
\mathbb{C}$. Now the claim follows by choosing for $\bigwedge^2 S^2
V^{\vee}$ the basis $$\begin{array}{ccccc}
\{ \widehat{X}^2 \wedge \widehat{X}\widehat{Y}, & \widehat{X}^2 \wedge \widehat{X}\widehat{Z}, & \widehat{X}^2 \wedge \widehat{Y}^2, & \widehat{X}^2 \wedge \widehat{Y}\widehat{Z}, & \widehat{X}^2 \wedge \widehat{Z}^2, \\
\widehat{X}\widehat{Y} \wedge \widehat{X}\widehat{Z}, & \widehat{X}\widehat{Y} \wedge \widehat{Y}^2, & \widehat{X}\widehat{Y} \wedge \widehat{Y}\widehat{Z}, & \widehat{X}\widehat{Y} \wedge \widehat{Z}^2, & \widehat{X}\widehat{Z} \wedge \widehat{Y}^2, \\
\widehat{X}\widehat{Z} \wedge \widehat{Y}\widehat{Z}, &
\widehat{X}\widehat{Z} \wedge \widehat{Z}^2, & \widehat{Y}^2 \wedge
\widehat{Y}\widehat{Z}, & \widehat{Y}^2 \wedge \widehat{Z}^2, &
\widehat{Y}\widehat{Z} \wedge \widehat{Z}^2 \}.
\end{array}$$
\[prop.tot.decomp\] A section $\eta \in \phi^*H^0 \big(\widehat{A}, \, \bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}}\big)$ corresponding to a vector as in is totally decomposible if and only if
- $b=-a$, $d=0$ and $e=-a^2/c$, or
- $a=b=d=e=0$, or
- $a=b=c=d=0$.
In other words, the totally decomposable sections $($up to a multiplicative constant$)$ are in one-to-one correspondence with the points of the smooth conic of equations $$a^2+ce=0, \quad b+a=0, \quad d=0$$ in the projective space $\mathbb{P}^4$ with homogeneous coordinates $[a:b:c:d:e]$.
Proposition \[prop.dis.sec\] allows one to identify the building data $c_{ij}$ in [@HM99 p. 7] as follows: $$\label{eq.cij}
\begin{array}{lllll}
c_{12}=aZ, & c_{13}=bY, & c_{14}=cY, & c_{15}=dX, & c_{16}=-cZ, \\
c_{23}=eX, & c_{24}=-bX, & c_{25}=-eZ, & c_{26}=-dY, & c_{34}=-dZ, \\
c_{35}=eY, & c_{36}=-aX, & c_{45}=aY, & c_{46}=cX, & c_{56}=-bZ.
\end{array}$$ By [@HM99 Theorem 3.1] the corresponding section $\eta\in
\phi^*H^0 \big(\widehat{A}, \, \bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes
\bigwedge^3 {\mathscr{E}}\big)$ is totally decomposable if and only if the $c_{ij}$ satisfy the Plücker relations: $$\label{eq.plu}
\begin{array}{ccc}
c_{12}c_{34}-c_{13}c_{24}+c_{14}c_{23}=0, & c_{12}c_{35}-c_{13}c_{25}+c_{15}c_{23}=0, & c_{12}c_{36}-c_{13}c_{26}+c_{16}c_{23}=0, \\
c_{12}c_{45}-c_{14}c_{25}+c_{15}c_{24}=0, & c_{12}c_{46}-c_{14}c_{26}+c_{16}c_{24}=0, & c_{12}c_{56}-c_{15}c_{26}+c_{16}c_{25}=0, \\
c_{13}c_{45}-c_{14}c_{35}+c_{15}c_{34}=0, & c_{13}c_{46}-c_{14}c_{36}+c_{16}c_{34}=0, & c_{13}c_{56}-c_{15}c_{36}+c_{16}c_{35}=0, \\
c_{14}c_{56}-c_{15}c_{46}+c_{16}c_{45}=0, & c_{23}c_{45}-c_{24}c_{35}+c_{25}c_{34}=0, & c_{23}c_{46}-c_{24}c_{36}+c_{26}c_{34}=0, \\
c_{23}c_{56}-c_{25}c_{36}+c_{26}c_{35}=0, & c_{24}c_{56}-c_{25}c_{46}+c_{26}c_{45}=0, & c_{34}c_{56}-c_{35}c_{46}+c_{36}c_{45}=0. \\
\end{array}$$ Substituting in the values given in we get the result.
Let us take now a point $[a: c] \in \mathbb{P}^1$. By Theorem \[theo.mir\] and Proposition \[prop.tot.decomp\] there exists a quadruple cover $\alpha \colon S \longrightarrow A$ induced by the totally decomposable section $\eta\in \phi^*H^0 \big(\widehat{A}, \,
\bigwedge^2 S^2 {\mathscr{E}}^{\vee} \otimes \bigwedge^3 {\mathscr{E}}\big)$ represented by the point $[a: -a: c: 0: -a^2/c] \in \mathbb{P}^4$; note that cases $\boldsymbol{(ii)}$ and $\boldsymbol{(iii)}$ in Proposition \[prop.tot.decomp\] correspond to $[a:c]=[0:1]$ and $[a:c]=[1:0]$, respectively. By construction the cover $\alpha$ is $K(\mathscr{L}^{-1})$-equivariant, so it induces a quadruple cover $\hat{\alpha} \colon \widehat{S} \longrightarrow \widehat{A}$ that fits into a commutative diagram $$\label{dia.projs}
\begin{xy}
\xymatrix{S \ar[d]_{\alpha} \ar[rr]^{\psi} & & \widehat{S} \ar[d]^{\widehat{\alpha}} \\
A \ar[rr]^{\phi_{\mathscr{L}^{-1}}} & & \widehat{A}, \\
}
\end{xy}$$ where $\psi$ is an étale $(\mathbb{Z}/3 \mathbb{Z})^2$-cover.
We denote by $B$ and $R$ the branch divisor and the ramification divisor of $\alpha \colon S {\longrightarrow}A$ and by $\widehat{B}$ and $\widehat{R}$ those of $\hat{\alpha} \colon {\widehat{S}}{\longrightarrow}{\widehat{A}}$.
Following [@HM99], over an affine open subset $U \subset A$ the surface $S$ is given by $$\textrm{Spec} \, \frac{\mathscr{O}_{U}[u, \,v, \, w]}{(F_1, \ldots ,
\, F_6)},$$ with $$\label{eq.qua.cover}
\begin{split}
F_1 &=u^2 -(a_1u+a_2v+a_3w+b_1), \\
F_2 &=uv -(a_4u+a_5v+a_6w+b_2), \\
F_3 &=uw -(a_7u+a_8v+a_9w+b_3),\\
F_4 &=v^2 -(a_{10}u+a_{11}v+a_{12}w+b_4), \\
F_5 &=vw - (a_{13}u+a_{14}v+a_{15}w+b_5), \\
F_6 &=w^2 -(a_{16}u+a_{17}v+a_{18}w+b_6).\\
\end{split}$$ Here $$\begin{array}{lll}
a_{1}=\frac{1}{2}c_{23}, & a_{2}=-c_{13}, & a_{3}=c_{12}, \\ a_{4}=\frac{1}{4}c_{25}-\frac{1}{2}c_{34}, & a_{5}=-\frac{1}{2}c_{15}-\frac{1}{4}c_{23}, & a_{6}=c_{14}, \\
a_{7}=\frac{1}{2}c_{26}-\frac{1}{4}c_{35}, & a_{8}=-c_{16}, &
a_{9}=\frac{1}{2}c_{15}-\frac{1}{4}c_{23}, \\ a_{10}=c_{45},
& a_{11}=-\frac{1}{2}c_{25}, & a_{12}=c_{24}, \\
a_{13}=c_{46}, & a_{14}=-\frac{1}{2}c_{26}-\frac{1}{4}c_{35}, &
a_{15}=\frac{1}{4}c_{25}+\frac{1}{2}c_{34}, \\ a_{16}=c_{56}, &
a_{17}=-c_{36}, & a_{18}=\frac{1}{2}c_{35}
\end{array}$$ and $$\begin{array}{l}
b_1 =-a_1a_5+a_2a_4-a_2a_{11}-a_3a_{14}+a_5^2+a_6a_8, \\
b_2 =a_2a_{10}+a_3a_{13}-a_4a_5-a_6a_7, \\
b_3 =a_2a_{13}+a_3a_{16}-a_4a_8-a_7a_9, \\
b_4 =-a_1a_{10}+a_4^2-a_4a_{11}+a_5a_{10}+a_6a_{13}-a_7a_{12}, \\
b_5 =-a_5a_{13}+a_8a_{10}+a_{12}a_{17}-a_{14}a_{15}, \\
b_6 =-a_1a_{16}-a_4a_{17}+a_7^2-a_7a_{18}+a_8a_{13}+a_9a_{16}, \\
\end{array}$$ where the $c_{ij}$ are given in .
Using the `MAGMA` script in the Appendix $1$, one checks that the branch locus $B$ of $\alpha\colon S \longrightarrow A$ is the element in $|6L|$ of equation $$\label{eq.branch.S}
\begin{split}
a^8c^4(X^6+Y^6+Z^6)& + a^2c\bigg(-\frac{4}{27}a^9-\frac{2}{9}a^6c^3-\frac{64}{9}a^3c^6 +\frac{256}{27}c^9\bigg)(X^3Y^3+X^3Z^3+Y^3Z^3)\\
& + a^4c^2\bigg(-\frac{2}{3}a^6+\frac{16}{3}a^3c^3-\frac{32}{3}c^6\bigg)XYZ(X^3+Y^3+Z^3) \\
& +\bigg(-\frac{1}{27}a^{12}-\frac{92}{27}a^9c^3+\frac{112}{9}a^6c^6+\frac{256}{27}a^3c^9-\frac{256}{27}c^{12}\bigg)X^2Y^2Z^2=0.
\end{split}$$
We can also see as the equation of a sextic curve in the dual projective plane $(\mathbb{P}^{2})^{\vee}$ with homogeneous coordinates $[X:Y:Z]$; we shall denote such a curve by $\mathbf{B}$. Varying the point $[a:c] \in \mathbb{P}^1$, the curves $\mathbf{B}$ form a (non-linear) pencil in $\mathbb{P}^{2
\vee}$, which turns out to be the pencil of the dual curves of members of the Hesse pencil. In fact, $\mathbf{B}$ is precisely the dual of the curve $E_{m_0,\, 3m_1}$ with $m_0=a^2c$ and $m_1=a^3/2-2c^3$, see Subsection \[sub.hal\].
\[rem.cuspidal.sextic\] The $\mathscr{H}_3$-equivariant sextic $\mathbf{B}$ also appears, for some special choices of the pair $(A, \, \mathscr{L})$, as a component of the branch locus of the $6$-fold cover $\varphi_{|\mathscr{L}|} \colon A {\longrightarrow}\mathbb{P}H^0(A, \,
\mathscr{L})^{\vee}$. See for instance *[@BL94]* and *[@Cas99]*.
The group $K(\mathscr{L}^{-1})$ acts on $\mathbf{B}$ and this induces an action on $B$. The quotient of $B$ by this action is a curve $\widehat{B} \in |2\widehat{L}|$, where $\widehat{L}$ is the polarization of type $(1, \, 3)$ on ${\widehat{A}}$ appearing in ; the curve $\widehat{B}$ is precisely the branch locus of the quadruple cover $\hat{\alpha} \colon \widehat{S}
\longrightarrow \widehat{A}$.
Now set $$\begin{split}
T_1 &:=\{ [1:1], \, [\omega:1], \, [\omega^2:1], \, [1:0] \}, \\
T_2 &:=\{ [-2:1], \, [-2\omega:1], \, [-2\omega^2:1], \, [0:1] \}.
\end{split}$$
If $[a : c] \notin T_1 \cup T_2$ then $\mathbf{B}$ has nine ordinary cusps as only singularities; since $X, \,Y, \,Z \in H^0(A, \,
\mathscr{L})$ and $L^2=6$, by Proposition \[prop.X.Y.Z-art\] it follows that, for a general choice of the pair $(A, \,
\mathscr{L})$, the curve $B$ has $54$ ordinary cusps as only singularities. In this case a computation using `MAGMA` shows that $S$ is smooth; since $\psi \colon S \longrightarrow
\widehat{S}$ is étale, the surface $\widehat{S}$ is also smooth. Moreover, the nine cusps of $\mathbf{B}$ belong to a single $K({\mathscr{L}}^{-1})$-orbit and the stabilizer of each cusp is the identity. Therefore the $54$ cusps of $B$ fall in precisely six orbits and consequently $\widehat{B}$ is an irreducible curve with six cusps as only singularities.
If $[a : c] \in T_1 \cup T_2$ then $\mathbf{B}=2\mathbf{B}'$, where $\mathbf{B}'$ is a triangle. Consequently, $B=2B'$, where $B'$ has $18$ ordinary double points as only singularities. In this case the `MAGMA` script shows that $S$ is smooth if and only if $[a : c]
\notin T_1$; if instead $[a : c] \in T_1$ then the singular locus of $S$ coincides with the preimage of $B$. Moreover, the group $K({\mathscr{L}}^{-1})$ acts on the three sides of the triangle $\mathbf{B}'$; each side has stabilizer isomorphic to ${\mathbb{Z}}/ 3 {\mathbb{Z}}$ and there is only one orbit; consequently, $\widehat{B}=2\widehat{B}'$, where $\widehat{B}'$ is an irreducible curve with two ordinary double points as only singularities.
In any case, the preimage of a general point in the branch divisor $\widehat{B}$ consist of *three* distinct points. This shows that the quadruple cover $\alpha \colon \widehat{S} {\longrightarrow}\widehat{A}$ cannot factor through a double cover, hence it must be the Albanese map of $\widehat{S}$.
Summing up, we have proven
\[prop.sing.covers\] Let $(A, \, \mathscr{L})$ be a general $(1,\, 3)$-polarized abelian surface. Then the following holds:
- the surface $\widehat{S}$ is smooth precisely when $[a:c] \notin T_1$, whereas if $[a:c] \in
T_1$ it has a $1$-dimensional singular locus;
- if $\widehat{S}$ is smooth and $[a:c]
\notin T_2$ then the branch locus $\widehat{B}$ of $\hat{\alpha}\colon \widehat{S} \longrightarrow \widehat{A}$ is an irreducible curve in $|2 \widehat{L}|$, where $\widehat{L}$ is a polarization of type $(1, \, 3)$ on ${\widehat{A}}$, with six ordinary cusps as only singularities;
- if $\widehat{S}$ is smooth and $[a:c] \in T_2$ then $\widehat{B}=2\widehat{B}'$, where $\widehat{B}'$ is an irreducible curve in $|\widehat{L}|$ with two ordinary double points as only singularities;
- $\hat{\alpha} \colon \widehat{S} {\longrightarrow}\widehat{A}$ is the Albanese map of $\widehat{S}$.
We can now compute the invariants of $\widehat{S}$.
\[prop.num.invariants.cover\] If $[a:c] \notin T_1$ and $\widehat{A}$ is general, then $\widehat{S}$ is a minimal surface of general type with $p_g=q=2$ and $K^2=6$. The canonical class $K_{\widehat{S}}$ is ample and the general element of $|K_{\widehat{S}}|$ is smooth and irreducible.
Using and we obtain $p_g(\widehat{S})=q(\widehat{S})=2$. By Hurwitz formula we get $$K_{\widehat{S}}=\hat{\alpha}^*K_{\widehat{A}}+\widehat{R}=
\widehat{R},$$ so $\widehat{R} \in |K_{\widehat{S}}| $. If $\widehat{S}$ is general, the restriction $\hat{\alpha} \colon \widehat{R}
\longrightarrow \widehat{B}$ is the normalization map and by Proposition \[prop.sing.covers\] we infer that $\widehat{R}$ is a smooth curve of genus $7$. Thus the genus formula gives $$2K^2_{\widehat{S}}=\widehat{R}(K_{\widehat{S}}+\widehat{R})=2g(\widehat{R})-2=12,$$ hence $K^2_{\widehat{S}}=6$. Moreover, by Bertini’s theorem the general element of $|K_{\widehat{S}}|$ is smooth and irreducible, because $\widehat{R}$ is so. Finally, since $\hat{\alpha} \colon
{\widehat{S}}{\longrightarrow}{\widehat{A}}$ is a finite map onto an abelian variety, the surface ${\widehat{S}}$ contains no rational curves. In particular ${\widehat{S}}$ is a minimal model and $K_{\widehat{S}}$ is ample.
\[rem-cusps-non-gen-pos\] The six cusps of the branch curve $\widehat{B}$ are not in general position: in fact, by the results of Subsection *\[sub.hal\]* it follows that there exists a unique element $\widehat{C} \in |\widehat{L}|$ containing them. The general element of the pencil generated by $\widehat{B}$ and $2
\widehat{C}$ is an irreducible curve in $|2 \widehat{L}|$ with six nodes at the six cusps of $\widehat{B}$ and no further singularities. Hence the cuspidal curve $\widehat{B} \in |2
\widehat{L}|$ can be obtained as a limit of nodal curves belonging to the same linear system.
In the sequel we denote by $\mathscr{A}_{1,\,3}$ the moduli space of $(1,3)$-polarized abelian surfaces. It is a quasi-projective variety of dimension $3$, see [@BL04 Chapter 8]. The following result will be used in the next section.
\[prop.R.nonhyp\] For a general choice of $\widehat{A} \in \mathscr{A}_{1,3}$ and $[a:c] \in \mathbb{P}^1$ the curve $\widehat{R}$ is not hyperelliptic. In particular, for any line bundle $\mathcal{N}$ on $\widehat{R}$ with $\deg (\mathcal{N}) =6$ we have $h^0(\widehat{R},
\, \mathcal{N}) \leq 3$.
This follows from a rigidity result of hyperelliptic curves on general abelian varieties, see [@Pi89]. More precisely, one can show that the only hyperelliptic deformations of a (possibly singular) hyperelliptic curve on a fixed simple abelian variety are just the translations. This implies that a linear system on a general abelian surface contains at most a finite number of hyperelliptic curves. In our situation, we have an equisingular, $1$-dimensional family $\{\widehat{B} \}$ of cuspidal curves in the linear system $|2 \widehat{L}|$, parametrized by the points $[a:c]
\in \mathbb{P}^1$. Then, if ${\widehat{A}}$ is simple, the general curve $\widehat{B}$ is non-hyperelliptic, that is its normalization ${\widehat{R}}$ is non-hyperelliptic.
If $\mathcal{N}$ is any line bundle on ${\widehat{R}}$ with $\deg(\mathcal{N})=6$, Riemann-Roch yields $h^0({\widehat{R}}, \,
\mathcal{N})-h^1({\widehat{R}}, \, \mathcal{N})=0$; in particular, if $\mathcal{N}$ is non-special we obtain $h^0({\widehat{R}}, \, \mathcal{N})=0$. If instead $\mathcal{N}$ is special, since ${\widehat{R}}$ is non-hyperelliptic Clifford’s theorem implies $$h^0({\widehat{R}}, \, \mathcal{N}) -1 < \frac{1}{2} \deg(\mathcal{N}),$$ that is $h^0({\widehat{R}}, \, \mathcal{N}) \leq 3$.
The moduli space {#sec.moduli}
================
We denote by $\mathcal{M}_{\Phi}^0$ the family of canonical models $\widehat{X}$ of minimal surfaces of general type with $p_g=q=2$, $K^2 =6$ such that:
- the Albanese map $\hat{\alpha} \colon
\widehat{X} {\longrightarrow}\widehat{A}$ is a finite, quadruple cover;
- the Tschirnhausen bundle ${\mathscr{E}}^{\vee}$ associated with $\hat{\alpha} \colon \widehat{X} {\longrightarrow}\widehat{A}$ is of the form ${\mathscr{E}}^{\vee}=\Phi^{\mathscr{P}}(\mathscr{L}^{-1})$, where $\mathscr{L}$ is a polarization of type $(1,\,3)$ on $A$.
Notice that $\mathcal{M}_{\Phi}^0$ coincides with the family of canonical models of surfaces $\widehat{S}$ constructed in Section \[sec.constr\], and that such a family depends on four parameters (three parameters from ${\widehat{A}}\in \mathscr{A}_{1,\,3}$ and one parameter from $[a:c] \in \mathbb{P}^1$). More precisely, there is a generically finite, dominant map $$\mathbf{P}^0 {\longrightarrow}\mathcal{M}_{\Phi}^0,$$ where $\mathbf{P}^0$ is a Zariski-dense subset of a $\mathbb{P}^1$-bundle over $\mathscr{A}_{1,\,3}$. Then $\mathcal{M}_{\Phi}^0$ is irreducible and $\dim \mathcal{M}_{\Phi}^0=4$.
\[prop.E.stable\] If ${\widehat{S}}\in \mathcal{M}_{\Phi}^0$ then the Tschirnhausen bundle ${\mathscr{E}}^{\vee}$ is stable with respect to the polarization $\widehat{L}$. In particular it is simple, that is $H^0({\widehat{A}}, \,
{\mathscr{E}}\otimes {\mathscr{E}}^{\vee})= \mathbb{C}$.
Using we obtain $$\label{eq.LdL}
\phi^* ({\mathscr{E}}\otimes {\mathscr{E}}^{\vee})= \bigoplus_{i=1}^9 (\mathscr{L}^{-1}
\otimes \mathscr{L}),$$ hence $$\label{eq.LdL.global.sec}
H^0(A, \,\phi^* ({\mathscr{E}}\otimes {\mathscr{E}}^{\vee}))= \bigoplus_{i=1}^9 H^0(A,
\,\mathscr{L}^{-1} \otimes \mathscr{L}).$$ There is a natural action of $K(\mathscr{L}^{-1}) \cong (\mathbb{Z}/
3 \mathbb{Z})^2$ on the vector space $H^0(A, \,\phi^* ({\mathscr{E}}\otimes
{\mathscr{E}}^{\vee}))$ and the direct summands in the right-hand side of correspond precisely to the eigenspaces associated with the $9$ irreducible characters. In particular, the invariant eigenspace is isomorphic to $H^0(A, \, \mathscr{L}^{-1}
\otimes \mathscr{L})$, hence $$H^0({\widehat{A}}, \, {\mathscr{E}}\otimes {\mathscr{E}}^{\vee}) \cong H^0(A, \,\mathscr{L}^{-1}
\otimes \mathscr{L})=\mathbb{C}$$ and this shows that ${\mathscr{E}}^{\vee}$ is simple. Then, by and [@BL04 Exercise 2 page 476], it follows that ${\mathscr{E}}^{\vee}$ is semi-homogeneous. Finally, any simple, semi-homogeneous vector bundle on an abelian variety is stable with respect to any polarization, see [@BL04 Exercise 1 page 476].
We denote by $\mathcal{M}_{\Phi}$ the closure of $\mathcal{M}_{\Phi}^0$ in the moduli space $\mathcal{M}_{2,\,6,\,6}^{\emph{can}}$ of canonical models of minimal surfaces of general type with $p_g=q=2$, $K^2=6$.
By the previous considerations, $\mathcal{M}_{\Phi}$ is irreducible and $\dim \mathcal{M}_{\Phi}=4$. Now we want to prove that $\mathcal{M}_{\Phi}$ provides an irreducible component of the moduli space $\mathcal{M}_{2,\, 2,
\, 6}^{\textrm{can}}$ and that such a component is generically smooth. In order to do this, we must prove that for the general surface $\widehat{S} \in \mathcal{M}_{\Phi}$ one has $h^1(\widehat{S}, \, T_{\widehat{S}})= \dim \mathcal{M}_{\Phi}=4$.
By [@Se06 p. 262] we have an exact sequence $$\label{seq.deform}
0 \longrightarrow T_{\widehat{S}} \stackrel{d
\hat{\alpha}}{\longrightarrow} \hat{\alpha}^*T_{\widehat{A}}
\longrightarrow \mathcal{N}_{\hat{\alpha}} \longrightarrow 0,$$ where $\mathcal{N}_{\hat{\alpha}}$ is a coherent sheaf supported on $\widehat{R}$ and called the normal sheaf of $\hat{\alpha}$; we denote by $\varphi= \hat{\alpha}|_{\widehat{R}} \colon \widehat{R}
{\longrightarrow}\widehat{B} \subset \widehat{A}$ the normalization map of $\widehat{B}$.
Let $\Delta \subset \widehat{R}$ be the divisor formed by all points $p$ such that $\varphi(p)$ is a cusp of $\widehat{B}$, each counted with multiplicity $1$, and let $i \colon \widehat{R} {\longrightarrow}\widehat{S}$ be the inclusion of $\widehat{R}$ in $\widehat{S}$; then $\hat{\alpha} \circ i=\varphi$. By [@CF11 Section 7] there are two commutative diagrams $$\label{dia.1}
\xymatrix{
& & & 0 \ar[d]& \\
& 0 \ar[d] & & \mathscr{O}_{\Delta} \ar[d] & \\
0 \ar[r] & T_{{\widehat{R}}} \ar[r]^{d \varphi} \ar[d] & \varphi^*T_{{\widehat{A}}}
\ar[r] \ar@{=}[d] & \mathcal{N}_{\varphi}
\ar[r] \ar[d]& 0 \\
& i^*T_{{\widehat{S}}} \ar[r]^{i^*(d \hat{\alpha})} & \varphi^*T_{{\widehat{A}}} \ar[r] & i^* \mathcal{N}_{\hat{\alpha}} \ar[r] \ar[d] &
0 \\
& & & 0 & }$$ and $$\label{dia.2}
\xymatrix{
& & & 0 \ar[d]& \\
& 0 \ar[d] & & \mathscr{O}_{\Delta} \ar[d] & \\
0 \ar[r] & T_{{\widehat{R}}} \ar[r]^{d \varphi} \ar[d] & \varphi^*T_{{\widehat{A}}}
\ar[r] \ar@{=}[d] & \mathcal{N}_{\varphi}
\ar[r] \ar[d]& 0 \\
& T_{{\widehat{R}}} \otimes {\mathscr{O}}_{{\widehat{R}}}(\Delta) \ar[r]^{d \varphi} & \varphi^*T_{{\widehat{A}}} \ar[r] & \mathcal{N}'_{\varphi} \ar[r] \ar[d] &
0, \\
& & & 0 & }$$ where $\mathcal{N}'_{\varphi}$ is a line bundle on ${\widehat{R}}$ satisfying $$\label{eq.deg.Nphi}
\begin{split}
\deg (\mathcal{N}'_{\varphi})=\deg(\mathcal{N}_{\varphi})-\deg
(\Delta) & =\deg (\varphi^* T_{{\widehat{A}}}) - \deg (T_{{\widehat{R}}}) - \deg
(\Delta)\\ & = \deg({\mathscr{O}}_{{\widehat{R}}}^{\oplus 2}) + \deg (K_{{\widehat{R}}})- \deg
(\Delta)=6.
\end{split}$$ The fact that $\mathcal{N}_{\hat{\alpha}}$ is supported on $\widehat{R}$, together with and , imply that $$\label{eq.hi(N)}
h^i({\widehat{S}}, \, \mathcal{N}_{\hat{\alpha}})=h^i({\widehat{R}}, \,
i^*\mathcal{N}_{\hat{\alpha}})=h^i({\widehat{R}}, \,
\mathcal{N}'_{\varphi}), \quad i=0,\,1,\,2,$$ so for a general choice of the abelian variety ${\widehat{A}}\in
\mathscr{A}_{1,3}$ and of the point $[a:c] \in \mathbb{P}^1$ one has $$\label{eq.dimNp}
h^0({\widehat{S}}, \, \mathcal{N}_{\hat{\alpha}})=h^0({\widehat{R}}, \,
\mathcal{N}'_{\varphi}) \leq 3,$$ see Proposition \[prop.R.nonhyp\]. Now we can prove the desired result.
\[prop.irred.comp\] If $\widehat{S}$ is a general element of $\mathcal{M}_{\Phi}$ then $h^1(\widehat{S}, \, T_{\widehat{S}})=4$. Hence $\mathcal{M}_{\Phi}$ provides an irreducible component of the moduli space $\mathcal{M}_{2,\, 2, \,6}^{\emph{can}}$ of canonical models of minimal surfaces with $p_g=q=2$ and $K^2=6$. Such a component is generically smooth, of dimension $4$.
Since $\dim \mathcal{M}_{\Phi} =4$, it is sufficient to show that for a general choice of $\widehat{S}$ one has $$\label{h1N leq 4}
h^1({\widehat{S}}, \, T_{{\widehat{S}}}) \leq 4.$$
The surface $\widehat{S}$ is of general type, so we have $H^0(\widehat{S}, \, T_{\widehat{S}})=0$ and yields the following exact sequence in cohomology $$\label{seq.deform.coh}
0 \longrightarrow H^0(\widehat{S}, \, \hat{\alpha}^*T_{\widehat{A}})
\longrightarrow H^0(\widehat{S}, \, \mathcal{N}_{\hat{\alpha}})
\longrightarrow H^1(\widehat{S}, \, T_{\widehat{S}})
\stackrel{\varepsilon}{\longrightarrow} H^1(\widehat{S}, \,
\hat{\alpha}^*T_{\widehat{A}}).$$ By the same argument used in [@PP11 Section 3] one proves that the image of $\varepsilon \colon H^0(\widehat{S}, \,
T_{\widehat{S}}) \longrightarrow H^1(\widehat{S}, \,
\hat{\alpha}^*T_{\widehat{A}})$ has dimension $3$ (this is essentially a consequence of the fact that when one deforms $S$ the complex torus ${\widehat{A}}$ remains algebraic). Then follows from and .
\[rem-no-irr\] The general element $\widehat{S}$ of $\mathcal{M}_{\Phi}$ admits no pencil $p \colon \widehat{S} {\longrightarrow}T$ over a curve $T$ with $g(T) \geq
1$.
Assume that ${\widehat{A}}$ a simple abelian surface. Then the set $$V^1(\widehat{S}):=\{ \mathscr{Q} \in \textrm{Pic}^0(\widehat{S}) \;
| \; h^1(\widehat{S}, \, \mathscr{Q}^{\vee}) >0 \}$$ cannot contain any component of positive dimension, so $\widehat{S}$ does not admit any pencil over a curve $T$ with $g(T) \geq 2$, see [@HP02 Theorem 2.6]. If instead $g(T)=1$, the universal property of the Albanese map yields a surjective morphism $\widehat{A} {\longrightarrow}T$, contradicting the fact that ${\widehat{A}}$ is simple.
Quadruple covers with simple Tschirnhausen bundle {#sec.simple}
=================================================
\[prop.simple\] Let $(\widehat{A}, \, \widehat{\mathscr{L}})$ be a $(1,3)$-polarized abelian surface and ${\mathscr{E}}^{\vee}$ be a *simple* rank $3$ vector bundle on $\widehat{A}$ with $$h^0(\widehat{A}, \, {\mathscr{E}}^{\vee})=1, \quad h^1(\widehat{A}, \,
{\mathscr{E}}^{\vee})=0, \quad h^2(\widehat{A}, \, {\mathscr{E}}^{\vee})=0,$$ $$c_1({\mathscr{E}}^{\vee})= \widehat{\mathscr{L}}, \quad c_2({\mathscr{E}}^{\vee})=2.$$ Then there exists a polarization $\mathscr{L}$ of type $(1,3)$ on $A$ such that ${\mathscr{E}}^{\vee}=\Phi^{\mathscr{P}}(\mathscr{L}^{-1})$.
Since ${\mathscr{E}}^{\vee}$ is simple and $2c^2_1({\mathscr{E}}^{\vee})-6c_2({\mathscr{E}}^{\vee})=0$, by [@O71 Corollary p. 249] and [@BL04 Ex. 2 p. 476] there exist an abelian surface $Z$, a line bundle $\mathscr{G}$ on $Z$ and an isogeny $g\colon Z \longrightarrow \widehat{A}$ such that $g^*{\mathscr{E}}^{\vee}=\mathscr{G}^{\oplus 3}$. Hence we obtain $$3c_1(\mathscr{G})=c_1(g^*{\mathscr{E}}^{\vee})=\deg(g)\cdot
\widehat{\mathscr{L}},$$ that is $\mathscr{G}$ is ample. For any $\mathscr{Q} \in
\textrm{Pic}^0(\widehat{A})$ we have $g^*\mathscr{Q} \in
\textrm{Pic}^0(Z)$ and $g^*({\mathscr{E}}^{\vee} \otimes
\mathscr{Q})=(\mathscr{G} \otimes g^*\mathscr{Q})^{\oplus 3}$, so the ampleness of $\mathscr{G}$ implies $$H^i(Z, \, g^*({\mathscr{E}}^{\vee} \otimes \mathscr{Q}))= H^i(Z, \,
\mathscr{G} \otimes g^*\mathscr{Q})^{\oplus 3}=0, \quad i=1, \, 2$$ for all $\mathscr{Q} \in \textrm{Pic}^0(\widehat{A})$. Since $g$ is a finite map, we get $$H^i(\widehat{A}, \, {\mathscr{E}}^{\vee} \otimes \mathscr{Q})\cong
g^*H^i(\widehat{A}, \, {\mathscr{E}}^{\vee} \otimes \mathscr{Q}) \subseteq
H^i(Z, \, g^*({\mathscr{E}}^{\vee} \otimes \mathscr{Q}))=0, \quad i=1, \, 2$$ that is ${\mathscr{E}}^{\vee}$ satisfies IT of index $0$. Since $\textrm{rank}({\mathscr{E}}^{\vee})=3$ and $h^0(\widehat{A}, \,
{\mathscr{E}}^{\vee})=1$, the Fourier-Mukai transform $\Phi^{\mathscr{P}}({\mathscr{E}}^{\vee})$ is a line bundle of type $(1, \,
3)$ on $A$ that we denote by $\mathscr{M}^{-1}$.
Therefore we have $$(-1)_{\widehat{A}}^*{\mathscr{E}}^{\vee} = \Phi^{\mathscr{P}} \circ
\Phi^{\mathscr{P}}({\mathscr{E}}^{\vee}) =
\Phi^{\mathscr{P}}(\mathscr{M}^{-1}),$$ so ${\mathscr{E}}^{\vee}=\Phi^{\mathscr{P}}((-1)_A^*\mathscr{M}^{-1})$ by Corollary \[cor.menouno\]. Setting $\mathscr{L}^{-1}:=(-1)_A^*\mathscr{M}^{-1}$ we are done.
The following corollary can be seen as a converse of Proposition \[prop.E.stable\].
\[cor.simple\] Let $\widehat{S}$ be a surface of general type with $p_g=q=2$, $K_{{\widehat{S}}}^2=6$ such that the Albanese map of its canonical model $\widehat{X}$ is a finite, quadruple cover $\hat{\alpha} \colon
\widehat{X} {\longrightarrow}{\widehat{A}}$, where ${\widehat{A}}$ is a $(1, \, 3)$-polarized abelian surface. Assume in addition that the Tschirnhausen bundle $\mathscr{E}^{\vee}$ associated with $\hat{\alpha}$ is simple. Then $\widehat{X}$ belongs to $\mathcal{M}_{\Phi}$.
Two remarkable sub-families of $\mathcal{M}_{\Phi}$ {#sec.two-sub}
===================================================
A $3$-dimensional family of surfaces contained in the singular locus of $\mathcal{M}_{2, \, 2, \,6}^{\textrm{can}}$
--------------------------------------------------------------------------------------------------------------------
If we choose $[a:c]=[0:1]$ in the construction of Section \[sec.constr\], by we obtain $$\label{eq.cij.bidouble}
c_{12}=c_{13}=c_{15}=c_{23}=c_{24}=c_{25}=c_{26}=c_{34}=c_{35}=c_{36}=c_{45}=c_{56}=0.$$ Then implies that the local equations of the surface $S$ are $$\label{eq.bidouble}
\begin{array}{ll}
u^2=YZ \\
uv=Yw \\
uw=Zv \\
v^2=XY \\
vw=Xu \\
w^2=XZ
\end{array}$$ and this shows that the Albanese map $\alpha \colon S {\longrightarrow}A$ is a *bidouble cover*, i.e. a Galois cover with Galois group $(\mathbb{Z}/2 {\mathbb{Z}})^2$. Conversely, relations are precisely the conditions ensuring that $\alpha \colon S {\longrightarrow}A$ is a bidouble cover, see [@HM99 p. 25-27].
We use the theory of bidouble covers developed in [@Ca84]. The cover $\alpha \colon S {\longrightarrow}A$ is branched over the three divisors $D_X$, $D_Y$, $D_Z \in |L|$ corresponding to the distinguished sections $X$, $Y$, $Z \in H^0(A, \, {\mathscr{L}})$; for simplicity of notation we write $D_1$, $D_2$, $D_3$ instead of $D_X$, $D_Y$, $D_Z$, respectively. Then the building data of $\alpha \colon S {\longrightarrow}A$ consist of three line bundles $\mathscr{L}_1$, $\mathscr{L}_2$, $\mathscr{L}_3$ on $A$, with $\mathscr{L}_i=\mathscr{O}_A(L_i)$, such that $$\begin{array}{ll}
2L_1 \cong D_2 + D_3 \\
2L_2 \cong D_1+ D_3 \\
2L_3 \cong D_1+ D_2
\end{array}$$ and $\alpha_* \mathscr{O}_S = \mathscr{O}_A \oplus
\mathscr{L}_1^{-1} \oplus \mathscr{L}_2^{-1} \oplus
\mathscr{L}_3^{-1}$. On the other hand, by we obtain $\alpha_* \mathscr{O}_S = \mathscr{O}_A \oplus \phi^* \
\mathscr{E}=\mathscr{O}_A \oplus \mathscr{L}^{-1} \oplus
\mathscr{L}^{-1} \oplus \mathscr{L}^{-1}$; therefore [@At56] implies ${\mathscr{L}}_1={\mathscr{L}}_2={\mathscr{L}}_3= {\mathscr{L}}$.
By [@Ca84 p. 497] there is an exact sequence $$\label{seq.obstructed}
\begin{split}
0 & {\longrightarrow}H^0(S, \, \alpha^* T_A) {\longrightarrow}\bigoplus_{i=1}^3
H^0(\mathscr{O}_{D_i}(D_i) \oplus \mathscr{O}_{D_i})
\stackrel{\partial}{{\longrightarrow}} H^1(S, \, T_S) \\ &
\stackrel{\varepsilon}{{\longrightarrow}} H^1(S, \,\alpha^* T_A) {\longrightarrow}\cdots,
\end{split}$$ whose meaning is the following: the image of $\partial$ are the first-order deformations coming from the so called “natural deformations" (see [@Ca84 Definition 2.8 p. 494]) and such first order deformations are trivial if they arise from automorphism of $A$. Moreover, if one considers the map $$\partial' \colon \bigoplus_{i=1}^3
H^0(\mathscr{O}_A(D_i) \oplus \mathscr{O}_A) {\longrightarrow}H^1(S, \, T_S),$$ obtained as the composition of the direct sum of the restriction maps with $\partial$, then for any $\bigoplus _i(a_i \oplus
\delta_i) \in \bigoplus_i H^0(\mathscr{O}_A(D_i) \oplus
\mathscr{O}_A)$ the element $\partial'(\bigoplus_i(a_i \oplus
\delta_i)) \in H^1(S, \, T_S)$ is the Kodaira-Spencer class of the corresponding natural deformation of $S$. More explicitly, taking $$a_i = \alpha_i X + \beta_i Y + \gamma_i Z, \quad \alpha_i, \,
\beta_i, \, \gamma_i \in \mathbb{C}$$ the local equations of the associated natural deformation of the bidouble cover are $$\label{eq.bidouble.deformed}
\begin{array}{ll}
u^2=(Y+ \alpha_3 X + \beta_3 Y + \gamma_3 Z+\delta_3 w)(Z+ \alpha_2 X + \beta_2 Y + \gamma_2 Z+\delta_2 v) \\
uv= (Y + \alpha_3 X + \beta_3 Y + \gamma_3 Z)w+ \delta_3 w^2 \\
uw= (Z+ \alpha_2 X + \beta_2 Y + \gamma_2 Z)v+ \delta_2 v^2 \\
v^2=(X+ \alpha_1 X + \beta_1 Y + \gamma_1 Z+\delta_1 u )(Y+ \alpha_3 X + \beta_3 Y + \gamma_3 Z+\delta_3 w) \\
vw= (X+\alpha_1 X + \beta_1 Y + \gamma_1 Z)u+ \delta_1 u^2 \\
w^2=(X+\alpha_1 X + \beta_1 Y + \gamma_1 Z + \delta_1 u )(Z + \alpha_2 X + \beta_2 Y + \gamma_2
Z + \delta_2 v).
\end{array}$$
By using the restriction exact sequence $$0 {\longrightarrow}H^0(A, \, \mathscr{O}_A) {\longrightarrow}H^0(A, \, \mathscr{O}_A(D_i))
\stackrel{\rho_i}{{\longrightarrow}} H^0(D_i, \, \mathscr{O}_{D_i}(D_i)) {\longrightarrow}H^1(A, \, \mathscr{O}_A) {\longrightarrow}0$$ we obtain $$\label{eq.HDi}
H^0(D_i, \, \mathscr{O}_{D_i}(D_i))= \textrm{im} \, \rho_i \oplus
H^1(A, \, \mathscr{O}_A),$$ where $$\label{eq.im.rho}
\begin{split}
\textrm{im} \, \rho_1 & = \textrm{span}(Y/X, \, Z/X) \\
\textrm{im} \, \rho_2 & = \textrm{span}(X/Y, \, Z/Y) \\
\textrm{im} \, \rho_3 & = \textrm{span}(X/Z, \, Y/Z). \\
\end{split}$$
There is an action of $(\mathbb{Z}/ 3 \mathbb{Z})^2=\langle r, \, s
\, | \, r^3=s^3=[r,s]=1 \rangle$ on $S$ given by $$\label{eq.action}
\begin{split}
r(u, \, v, \, w, \, X, \, Y, \, Z) & :=(u, \, \omega v, \, \omega^2
w, \, X, \, \omega^2 Y, \, \omega Z), \\
s(u, \, v, \, w, \, X, \, Y, \, Z) & := (v, \, w, \, u, \, Z, \, X,
\, Y)
\end{split}$$ and the corresponding quotient map is precisely $\psi \colon S {\longrightarrow}\widehat{S}$; notice that the Albanese map $\hat{\alpha} \colon
\widehat{S} {\longrightarrow}\widehat{A}$ is *not* a Galois cover. We shall denote by $\chi_0$ the trivial character of $(\mathbb{Z}/ 3
\mathbb{Z})^2$ and by $\chi$ any non-trivial character; moreover, if $V$ is a vector space with a $(\mathbb{Z}/ 3
\mathbb{Z})^2$-representation, we indicate by $V^{\chi_0}$ the invariant eigenspace and by $V^{\chi}$ the eigenspace relative to $\chi$. For instance, since $q(A)=q(\widehat{A})$ and $q(S)=q(\widehat{S})$, it follows $$\label{eq.invariant-sub-H1}
\begin{split}
H^i(A, \, \mathscr{O}_A)^{\chi_0}&=H^i(A, \, \mathscr{O}_A), \quad
i=0, \, 1,\\
H^i(A,\, \mathscr{O}_A)^{\chi}&=0 \quad \chi \neq \chi_0, \quad
i=0, \, 1,
\end{split}$$ $$\label{eq.invariant-sub-H1*}
\begin{split}
H^i(S, \, \mathscr{O}_S)^{\chi_0}&=H^i(S, \, \mathscr{O}_S), \quad
i=0, \, 1,\\
H^i(S,\, \mathscr{O}_S)^{\chi}&=0 \quad \chi \neq \chi_0, \quad
i=0, \, 1.
\end{split}$$ Since $\alpha^*T_A = \mathscr{O}_S^{\oplus 2}$, we also have $$\label{eq.invariant-sub-H0}
\begin{split}
H^i(S, \, \alpha^*T_A)^{\chi_0}&=H^i(S, \, \alpha^*T_A), \quad
i=0, \, 1, \\
H^i(S, \, \alpha^*T_A)^{\chi}&=0 \quad \chi \neq \chi_0, \quad i=0,
\, 1.
\end{split}$$
\[prop.im.delta\] The action ** induces an action of $(\mathbb{Z}/ 3 \mathbb{Z})^2$ on $\emph{im}\, \partial$ such that $$\dim (\emph{im} \, \partial)^{\chi_0} =5.$$
Using , and , by a tedious but straightforward computation one proves that induces an action of $(\mathbb{Z}/ 3 \mathbb{Z})^2$ on the vector space $\bigoplus_i
H^0(\mathscr{O}_{D_i}(D_i) \oplus \mathscr{O}_{D_i}) \cong
\mathbb{C}^{15}$, such that $$\dim \bigg( \bigoplus_i H^0(\mathscr{O}_{D_i}(D_i) \oplus
\mathscr{O}_{D_i}) \bigg)^{\chi_0}=7$$ whereas the eight remaining eigenspaces are all $1$-dimensional. By the $2$-dimensional subspace $\ker
\partial = H^0(S, \, \alpha^* T_A)$ is contained in the invariant eigenspace, so the claim follows.
\[prop.M.non.normal\] Assume that $\alpha \colon S {\longrightarrow}A$ is a bidouble cover, that is $[a:c]=[0:1]$. Then $h^1({\widehat{S}}, \, T_{{\widehat{S}}})=8$.
By we have $$H^1(S, \, T_S)=\textrm{im} \, \partial \oplus \textrm{im} \,
\varepsilon$$ and, as remarked in the proof of Proposition \[prop.irred.comp\], we can prove that $ \textrm{im} \,
\varepsilon \subset H^1(S, \, \alpha^* T_A)$ has dimension $3$. Using Lemma \[prop.im.delta\] and the fact that $H^1(S, \,
\alpha^* T_A)$ is contained in the invariant eigenspace, see , it follows that the dimension of $H^1(S, \, T_S)^{\chi_0}$ equals $5+3=8$. Finally, the $(\mathbb{Z}/3 \mathbb{Z})^2$-cover $\psi \colon S {\longrightarrow}{\widehat{S}}$ is unramified, so by [@Pa91 Proposition 4.1] we infer $H^1(S, \,
T_S)^{\chi_0}=H^1({\widehat{S}}, \, T_{{\widehat{S}}})$ and we are done.
Since ${\widehat{S}}$ is contained in the $4$-dimensional, irreducible component $\mathcal{M}_{\Phi}$ of $\mathcal{M}_{2, \, 2,
\,6}^{\textrm{can}}$, Proposition \[prop.M.non.normal\] implies that ${\widehat{S}}$ is a singular point of the moduli space. So we obtain
\[cor.sing.loc.M\] The moduli space $\mathcal{M}_{2, \,
2, \,6}^{\emph{can}}$ contains a $3$-dimensional singular locus.
A $2$-dimensional family of product-quotient surfaces
-----------------------------------------------------
In the paper [@Pe11 Theorem 4.15], the first author constructed a $2$-dimensional family of product-quotient surfaces (having precisely two ordinary double points as singularities) with $p_g=q=2$, $K^2=6$ and whose Albanese map is a generically finite quadruple cover. We will now recall the construction and show that this family is actually contained in $\mathcal{M}_{\Phi}$.
Let us denote by $\mathfrak{A}_4$ the alternating group on four symbols and by $V_4$ its Klein subgroup, namely $$V_4=\langle id, \, (12)(34), \, (13)(24), \, (14)(23) \rangle
\cong (\mathbb{Z}/2 \mathbb{Z})^2.$$ $V_4$ is normal in $\mathfrak{A}_4$ and the quotient $H:=\mathfrak{A}_4 / V_4$ is a cyclic group of order $3$. By using Riemann’s existence theorem is possible to construct two smooth curves $C_1$, $C_2$ of genus $4$ endowed with an action of $\mathfrak{A}_4$ such that the only non-trivial stabilizers are the elements of $V_4$. Then
- $E_i':=C_i/\mathfrak{A}_4$ is an elliptic curve;
- the $\mathfrak{A}_4$-cover $f_i \colon C_i {\longrightarrow}E_i'$ is branched at exactly one point of $E_i'$, with branching order $2$.
It follows that the product-quotient surface $$\widehat{X}:=(C_1 \times C_2)/\mathfrak{A}_4,$$ where $\mathfrak{A}_4$ acts diagonally, has two rational double points of type $\frac{1}{2}(1, \, 1)$ and no other singularities. It is straightforward to check that the desingularization $\widehat{S}$ of $\widehat{X}$ is a minimal surface of general type with $p_g=q=2$, $K_{{\widehat{S}}}^2=6$ and that $\widehat{X}$ is the canonical model of $\widehat{S}$.
The $\mathfrak{A}_4$-cover $f_i \colon C_i {\longrightarrow}E_i'$ factors through the bidouble cover $g_i \colon C_i {\longrightarrow}E_i$, where $E_i:=C_i/V_4$. Note that $E_i$ is again an elliptic curve, so there is an isogeny $E_i {\longrightarrow}E_i'$, which is a triple Galois cover with Galois group $H$. Consequently, we have an isogeny $$p \colon E_1 \times E_2 {\longrightarrow}{\widehat{A}}:=(E_1 \times E_2)/H,$$ where the group $H$ acts diagonally, and a commutative diagram $$\label{dia.example.pq}
\begin{xy}
\xymatrix{C_1 \times C_2 \ar[d]_{g_1 \times g_2} \ar[rr]^{\tilde{p}} & & (C_1 \times C_2 )/ \mathfrak{A}_4 =\widehat{X} \ar[d]^{\widehat{\alpha}} \\
E_1 \times E_2 \ar[rr]^{p} & & (E_1 \times E_2)/H = {\widehat{A}}. \\
}
\end{xy}$$ In this way one constructs a two-dimensional family $\mathcal{M}_{PQ}$ of product-quotient surfaces $\widehat{X}$ which are canonical models of surfaces of general type with $p_g=q=2$ and $K^2=6$. The morphism $\hat{\alpha} \colon \widehat{X} {\longrightarrow}\widehat{A}$ is the Albanese map of $\widehat{X}$; it is a finite, non-Galois quadruple cover.
\[wT-in-Mphi\] The family $\mathcal{M}_{PQ}$ is contained in $\mathcal{M}_{\Phi}$.
We must show that $\widehat{X}$ belongs to $\mathcal{M}_{\Phi}$. From the construction it follows that the Tschirnhausen bundle of $\hat{\alpha} \colon \widehat{X} {\longrightarrow}\widehat{A}$ is of the form $$\mathscr{E}^{\vee}= p_* \mathscr{D},$$ where $\mathscr{D}$ is a principal polarization of product type on $E_1 \times E_2$. Since $K(\mathscr{D})=0$ one clearly has $t^*_x
\mathscr{D} \neq \mathscr{D}$ for all $x \in \ker p \setminus
\{0\}$, so by [@O71 Theorem 2, p. 248] the canonical injection $\mathscr{O}_{\widehat{A}} \longrightarrow {\mathscr{E}}\otimes
{\mathscr{E}}^{\vee}$ induces isomorphisms $$h^i(\widehat{A}, \, \mathscr{O}_{\widehat{A}}) \longrightarrow
h^i(\widehat{A}, \, {\mathscr{E}}\otimes {\mathscr{E}}^{\vee}), \quad i=0, \, 1.$$ In particular $H^0(\widehat{A}, \, {\mathscr{E}}\otimes {\mathscr{E}}^{\vee} ) \cong
\mathbb{C}$, that is ${\mathscr{E}}^{\vee}$ is simple. Then the claim follows from Corollary \[cor.simple\].
The branch locus of the Albanese map $\hat{\alpha} \colon
\widehat{X} {\longrightarrow}{\widehat{A}}$ is a curve $\widehat{B}=2(\widehat{B}_1+\widehat{B}_2)$, where $\widehat{B}_1$ and $\widehat{B}_2$ are the images via $p \colon
E_1 \times E_2 {\longrightarrow}(E_1 \times E_2)/H$ of two elliptic curves belonging to the two natural fibrations of $E_1 \times E_2$; then $\widehat{B}_1 \widehat{B}_2=3$. Notice that $\widehat{B}$ is not of the form described in Proposition *\[prop.sing.covers\]*; the reason is that the $(1,
\,3)$-polarized abelian surface ${\widehat{A}}=(E_1 \times E_2)/H$ is not general, since it is an étale $\mathbb{Z}/3
\mathbb{Z}$-quotient of a product of elliptic curves.
Open problems {#sec.open.prob}
=============
- Is $\mathcal{M}_{\Phi}$ a *connected* component of $\mathcal{M}_{2, \, 2,\, 6}^{\textrm{can}}$? This is equivalent to ask whether it is *open* therein; in other words, given a smooth family $\mathscr{X} {\longrightarrow}\Delta$ over a small disk such that $X_0 \in \mathcal{M}_{\Phi}$, is $X_t \in
\mathcal{M}_{\Phi}$ for $t$ small enough?
Notice that any surface which is deformation equivalent to a surface in $\mathcal{M}_{\Phi}$ must have Albanese map of degree $4$, since the Albanese degree of a surface with $q=2$ and maximal Albanese dimension is a topological invariant, see [@Ca91] and [@Ca11 Section 5]. This leads naturally to the next question:
- What are the possible degrees for the Albanese map of a minimal surface with $p_g=q=2$ and $K^2=6$?
And, more generally:
- What are the irreducible/connected components of $\mathcal{M}_{2, \, 2,\, 6}^{\textrm{can}}$?
Appendix 1. The divisors corresponding to $X$, $Y$, $Z \in H^0(A, \, \mathscr{L})$ {#appendix-1.-the-divisors-corresponding-to-x-y-z-in-h0a-mathscrl .unnumbered}
==================================================================================
Let $D_X$, $D_Y$, $D_Z$ be the three divisors on $A$ which correspond to the distinguished sections $X$, $Y$, $Z \in H^0(A,
\, \mathscr{L})$ defined by the Schrödinger representation of the Heisenberg group $\mathscr{H}_3$, see Section \[sec.constr\]. In this Appendix we show that, for a general choice of the pair $(A, \, \mathscr{L})$, the curves $D_X$, $D_Y$, $D_Z$ are smooth and intersect transversally. We believe that this fact is well-known to the experts, however we give a proof for lack of a suitable reference.
Let us start with a couple of auxiliary results.
\[lem.triple\] Let $(B, \, \mathscr{M})$ be a principally polarized abelian surface and $K_1 \cong \mathbb{Z}/3 \mathbb{Z}$ a subgroup of the group $B[3]$ of points of order $3$ on $B$. Then there exist a $(1, \,
3)$-polarized abelian surface $(A, \, \mathscr{L})$ and an isogeny $f \colon A {\longrightarrow}B$ of degree $3$ such that:
- $\mathscr{L}=f^* \mathscr{M}$;
- $K(\mathscr{L})= \ker f \oplus f^{-1}(K_1)$.
In a suitable basis, the period matrix for $B$ is $$\left(
\begin{array}{cccc}
z_{11} & z_{12} & 1 & 0 \\
z_{21} & z_{22} & 0 & 1 \\
\end{array}
\right),$$ where $Z:=\left(
\begin{array}{cc}
z_{11} & z_{12} \\
z_{21} & z_{22} \\
\end{array}
\right)$ satisfies ${}^tZ=Z$ and $\textrm{Im}\,Z
>0$. Then $B=\mathbb{C}^2/ \Lambda'$, with $$\Lambda'=\lambda_1 \mathbb{Z} \oplus \lambda_2 \mathbb{Z} \oplus
\mu_1 \mathbb{Z} \oplus \mu_2 \mathbb{Z}$$ and $$\lambda_1:=\left(
\begin{array}{c}
z_{11} \\
z_{21} \\
\end{array}
\right), \quad
\lambda_2:=\left(
\begin{array}{c}
z_{12} \\
z_{22} \\
\end{array}
\right), \quad
\mu_1:=\left(
\begin{array}{c}
1 \\
0 \\
\end{array}
\right), \quad
\mu_2:=\left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right).$$ Up to a translation, we may assume $K_1= \langle
\overline{\lambda_2/3} \rangle$, where the symbol “$\overline{\phantom{1}}$" denotes the image in the complex torus. The lattice $$\Lambda:=\lambda_1 \mathbb{Z} \oplus \lambda_2 \mathbb{Z} \oplus
\mu_1 \mathbb{Z} \oplus 3\mu_2 \mathbb{Z}$$ verifies $[\Lambda' : \Lambda]=3$, hence setting $A:=\mathbb{C}^2 /
\Lambda$ the identity map $\mathbb{C}^2 {\longrightarrow}\mathbb{C}^2$ induces a degree $3$ isogeny $f \colon A {\longrightarrow}B$ such that $\ker f = \langle
\bar{\mu}_2 \rangle$.
Now the polarization $\mathscr{L}:=f^* \mathscr{M}$ is of type $(1,
\, 3)$ and satisfies $$K(\mathscr{L})= \langle \bar{\mu}_2, \, \overline{\lambda_2 /3}
\rangle=\ker f \oplus f^{-1}(K_1).$$
\[lem.translates\] Let $(A, \, \mathscr{L})$ and $(B, \, \mathscr{M})$ be as in Lemma *\[lem.triple\]*. Then, up to a simultaneous translation, the three divisors $D_X, \, D_Y, \, D_Z$ corresponding to $X$, $Y$, $Z
\in H^0(A, \, \mathscr{L})$ are given by $$f^*M, \quad f^*t_x^*M, \quad f^*t_{2x}^*M,$$ where $x \in B$ is a generator of $K_1$ and $M$ is the unique effective divisor in the linear system $|\mathscr{M}|$.
Up to translation, we may assume that $\mathscr{M}$ is a line bundle of characteristic zero with respect to the decomposition $\Lambda=\Lambda_1 \oplus \Lambda_2$, where $\Lambda_1=\langle
\bar{\lambda}_1, \, \bar{\lambda}_2 \rangle$ and $\Lambda_2=\langle \bar{\mu}_1, \, 3\bar{\mu}_2 \rangle.$ Hence, by the isogeny theorem for finite theta functions (see [@BL04 page 168]), it follows $D_X=f^*M$.
On the other hand, if $y \in A$ is any point such that $f(y)=x$, that is $y \in f^{-1}(K_1) \subset K(\mathscr{L})$, one has $$f^*t_x^*M \in |t_y^*f^* \mathscr{M}|=|t_y^* \mathscr{L}|=
|\mathscr{L}|$$ and the same holds for $f^*t_{2x}^*M$. In other words, $f^*t_x^*M$ and $f^*t_{2x}^*M$ are effective divisors corresponding to sections of $H^0(A, \, \mathscr{L})$, and straightforward computations as in [@BL04 Chapter 6] show that they are $D_Y$ and $D_Z$, respectively.
Now we can prove the desired result.
\[prop.X.Y.Z\] For a general choice of the pair $(A, \, \mathscr{L})$, the three divisors $D_X$, $D_Y$, $D_Z \in |\mathscr{L}|$ are smooth and intersect transversally.
It is sufficient to exhibit an example in which each claim is satisfied.
First, take $B=J(C)$, the Jacobian of a smooth curve $C$ of genus $2$, and let $\mathscr{M}$ be the natural principal polarization. Choose as $K_1$ any cyclic subset of $B[3]$ and construct $(A, \,
\mathscr{L})$ as in Lemma \[lem.triple\]. The unique effective divisor $M \in |\mathscr{M}|$ is smooth, so the same is true for its translates $t_x^*M$ and $t_{2x}^*M$. By Lemma \[lem.translates\] it follows that $D_X$, $D_Y$, $D_Z$ are smooth, too. Hence they are smooth for a general choice of the pair $(A, \, \mathscr{L})$.
Next, take $(B, \, \mathscr{M})=(E_1 \times E_2, \,
p_1^*\mathscr{M}_1 \otimes p_2^*\mathscr{M}_2)$, where $E_i$ is an elliptic curve, $\mathscr{M}_i$ is a divisor of degree $1$ on $E_i$ and $p_i \colon E_1 \times E_2 {\longrightarrow}E_i$ is the projection. Now choose a $3$-torsion point on $E_1 \times E_2$ of the form $x=(x_1, \, x_2)$, where $x_i$ is a non-trivial $3$-torsion point on $E_i$, take $K_1=\langle x \rangle$ and construct $(A, \,
\mathscr{L})$ as in Lemma \[lem.triple\]. If $M$ is the unique effective divisor in $|\mathscr{M}|$ then the three divisors $M$, $t_x^*M$, $t_{2x}^*M$ are pairwise without common components and intersect transversally, hence by Lemma \[lem.translates\] the same is true for $D_X$, $D_Y$, $D_Z$. Therefore $D_X$, $D_Y$, $D_Z$ intersect transversally for a general choice of the pair $(A, \, \mathscr{L})$.
This completes the proof.
\[rem.D\] One can show that $(-1)_A \colon A {\longrightarrow}A$ acts on $D_X$ as an involution, whereas $D_Y$ and $D_Z$ are exchanged by $(-1)_A$. See *[@BL04 Exercise 13 p. 177]*.
Appendix 2. The MAGMA script used to calculate the equation of the branch divisor of $\alpha: S {\longrightarrow}A$ {#appendix-2.-the-magma-script-used-to-calculate-the-equation-of-the-branch-divisor-of-alpha-s-longrightarrowa .unnumbered}
===================================================================================================================
QuadrupleCover:=function(Q,a,g)
//
// Given the field Q and the two parameters a and g of Proposition 2.4, the function QuadrupleCover
// gives:
// (1) the local equations of S "CoverAff5";
// (2) the local equation of the ramification divisor "RamCurve";
// (3) the local equation of the branch locus "AffBranch". Moreover
// (4) it checks whether these objects are singular or not, and it gives, if
// possible, a description of their singularities.
//
R<Y,Z,u,v,w>:=PolynomialRing(Q,5);
// Here Y,Z are sections in H^0(A,L)
A5 <Y,Z,u,v,w>:= AffineSpace(Q,5);
/////////////////////////////////////////////////////////////////////////////
// The local model of the cover S
/////////////////////////////////////////////////////////////////////////////
CoverAff5:=function(A5,R,a,g,T)
//
// Given a 5-dimensional affine space A5, a polynomial ring in the same
// coordinates R and three integers a,g,T (T is a translation), the function "CoverAff5"
// returns an affine scheme in A5 which is the local model of S with local equations in coordinates
// Y,Z,u,v,w.
//
b := -a; d := 0; if g eq 0 then e := 1; else e := -a^2/g; end if;
////////////////////////////////////////////////////////////////////////////
c12:=a*(Z-T); c13:=b*(Y-T); c14:=g*(Y-T); c15:=d; c16:=-g*(Z-T);
c23:=e; c24:=-b; c25:=-e*(Z-T); c26:=-d*(Y-T); c34:=-d*(Z-T);
c35:=e*(Y-T); c36:=-a; c45:=a*(Y-T); c46:=g; c56:=-b*(Z-T);
/////////////////////////////////////////////////////////////////////////////
a1:=1/2*c23; a2:=-c13; a3:=c12; a4:=1/4*c25-1/2*c34;
a5:=-1/2*c15-1/4*c23; a6:=c14; a7:=1/2*c26-1/4*c35; a8:=-c16;
a9:=1/2*c15-1/4*c23; a10:=c45; a11:=-1/2*c25; a12:=c24; a13:=c46;
a14:=-1/2*c26-1/4*c35; a15:=1/4*c25+1/2*c34; a16:=c56; a17:=-c36;
a18:=1/2*c35;
/////////////////////////////////////////////////////////////////////////////
b1:=-a1*a5+a2*a4-a2*a11-a3*a14+(a5)^2+a6*a8;
b2:=a2*a10+a3*a13-a4*a5-a6*a7; b3:=a2*a13+a3*a16-a4*a8-a7*a9;
b4:=-a1*a10+(a4)^2-a4*a11+a5*a10+a6*a13-a7*a12;
b5:=-a5*a13+a8*a10+a12*a17-a14*a15;
b6:=-a1*a16-a4*a17+(a7)^2-a7*a18+a8*a13+a9*a16;
/////////////////////////////////////////////////////////////////////////////
f1:=u^2-(a1*u+a2*v+a3*w+b1); f2:=u*v-(a4*u+a5*v+a6*w+b2);
f3:=u*w-(a7*u+a8*v+a9*w+b3); f4:=v^2-(a10*u+a11*v+a12*w+b4);
f5:=v*w-(a13*u+a14*v+a15*w+b5); f6:=w^2-(a16*u+a17*v+a18*w+b6);
/////////////////////////////////////////////////////////////////////////////
I:=ideal<R|[f1,f2,f3,f4,f5,f6]>;
/////////////////////////////////////////////////////////////////////////////
return Scheme(A5,Generators(I)); end function;
/////////////////////////////////////////////////////////////////////////////
// The ramification curve
/////////////////////////////////////////////////////////////////////////////
RamCurveA5:=function(Cover,A5,R)
//
// Given a local model of the "Cover", the affine space A5 in which the
// cover sits and a polynomial ring in the same coordinates,
// the function "RamCurveA5" returns an affine scheme in A5 which is
// the ramification curve of \ALPHA.
//
f1:= Generators(Ideal(Cover))[1]; f2:= Generators(Ideal(Cover))[2];
f3:= Generators(Ideal(Cover))[3]; f4:= Generators(Ideal(Cover))[4];
f5:= Generators(Ideal(Cover))[5]; f6:= Generators(Ideal(Cover))[6];
/////////////////////////////////////////////////////////////////////////////
I:=ideal<R|[f1,f2,f3,f4,f5,f6]>;
/////////////////////////////////////////////////////////////////////////////
L:=[Derivative(f1,u),Derivative(f1,v),Derivative(f1,w),
Derivative(f2,u),Derivative(f2,v),Derivative(f2,w),
Derivative(f3,u),Derivative(f3,v),Derivative(f3,w),
Derivative(f4,u),Derivative(f4,v),Derivative(f4,w),
Derivative(f5,u),Derivative(f5,v),Derivative(f5,w),
Derivative(f6,u),Derivative(f6,v),Derivative(f6,w)];
/////////////////////////////////////////////////////////////////////////////
J:=Matrix(R,6,3,L); Ms:=Minors(J,3); J3:=ideal<R | Ms>;
RJ3:=Radical(J3); H:=RJ3+I; U:=Generators(H);
/////////////////////////////////////////////////////////////////////////////
return Scheme(A5,U); end function;
/////////////////////////////////////////////////////////////////////////////
// MAIN ROUTINE
/////////////////////////////////////////////////////////////////////////////
a:=a; g:=g; T:=0;
/////////////////////////////////////////////////////////////////////////////
printf "\n --------BEGINNING------- \n ";
printf "\n a=%o,\n g=%o,\n Q=%o \n", a,g,Q;
/////////////////////////////////////////////////////////////////////////////
CoverA5:=CoverAff5(A5,R,a,g,T); printf "\n The cover is a ";
CoverA5; printf "\n The affine dimension of the cover is ";
Dimension(CoverA5); printf "The cover is singular: ";
IsSingular(CoverA5); if IsSingular(CoverA5) then
SingSchCoverA5:=SingularSubscheme(CoverA5); printf"The dimension of
the singular locus is: "; Dimension(SingSchCoverA5); if
Dimension(SingSchCoverA5) eq 0 then
SingPCoverA5:=SingularPoints(CoverA5); printf"The rational singular
points are: "; SingPCoverA5; else printf"The cover is not normal!";
end if; else RamCurveA5:=RamCurveA5(CoverA5,A5,R); printf "\n The
ramification divisor is a "; RamCurveA5; printf "The affine
dimension of the ramification divisor is "; Dimension(RamCurveA5);
printf"The ramification divisor is singular: ";
IsSingular(RamCurveA5); printf"\n \n
---------------------------- \n \n";
/////////////////////////////////////////////////////////////////////////////
// The branch curve
/////////////////////////////////////////////////////////////////////////////
A2<q,l>:=AffineSpace(Q,2); f := map< A5 -> A2 | [Y,Z] >;
AffBranch:=f(RamCurveA5); printf"The branch divisor is singular:
"; IsSingular(AffBranch); if IsSingular(AffBranch) then
SingSchBranch:=SingularSubscheme(AffBranch); printf"The dimension
of its singular locus is "; Dimension(SingSchBranch); end if; if
Dimension(SingSchBranch) eq 0 then
SingPBranch:=SingularPoints(AffBranch); printf"The rational
singular points of the branch are "; SingPBranch; printf" The
branch is a "; AffBranch; printf" \n \n
---------------------------- \n \n";
/////////////////////////////////////////////////////////////////////////////
end if; end if; printf" \n ------------THE END---------- \n"; return
0; end function;
/////////////////////////////////////////////////////////////////////////////
[9]{} M. Artebani, I. Dolgachev, *The Hesse pencil of plane cubic curves*. Enseign. Math. (2), VOL. **55**, (2009), 235–273. M. F. Atiyah, *On the Krull-Schmidt theorem with application to sheaves*. Bulletin de la S. M. F., tome **84** (1956), 307-317.
C. Birkenhake, H. Lange, *A family of abelian surfaces and curves of genus $4$*, Manuscripta Math. **85** (1994), 393-407.
C. Birkenhake, H. Lange, *Complex abelian varieties*. Grundlehren der Mathematischen Wissenschaften, Vol **302**, Second edition, Springer-Verlag, Berlin, 2004. F. Catanese, *On the moduli spaces of surfaces of general type*. J. Differential Geometry **19** (1984), 483-515.
F. Catanese, *Moduli and classification of irregular Kaehler manifolds and algebraic varieties with Albanese general type fibrations*. Invent. Math. **104** (1991), 263–289.
F. Catanese, *A superficial working guide to deformations and moduli*, e-print arXiv:1106.1368, to appear in in the Handbook of Moduli, a volume in honour of David Mumford, to be published by International Press.
G. Casnati, *The cover associated to a $(1,\,3)$-polarized bielliptic abelian surface and its branch locus*. Proceedings of the Edinburgh Mathematical Society **42** (1999), 375-392.
F. Catanese, C. Ciliberto, M. Mendes Lopes, *On the classification of irregular surfaces of general type with nonbirational bicanonical map*. Trans. Amer. Math. Soc. **350**, no. 1 (1998), 275–308. G. Casnati, T. Ekedahl, *Covers of algebraic varieties I. A general structure theorem, covers of degree $3$, $4$ and Enriques surfaces*. J. Algebraic Geom. **5** (1996), 439–460. J. Chen, C. Hacon, *A surface of general type with $p_g=q=2$ and $K^2=5$*. Pacific. J. Math. **223** (2006), 219–228.
C. Ciliberto, F. Flamini: *On the branch curve of a general projection of a surfaces to a plane*. Trans. Amer. Math. Soc. **363** (2011), 3457–3471.
O. Debarre, *[In$\acute{e}$galit$\acute{e}$s num$\acute{e}$riques pour les surfaces de type g$\acute{e}$n$\acute{e}$ral]{}*. Bull. Soc. Math. France **110** (1982), 319–346.
G.H. Halphen, *Sur les courbes planes du sixième degré à neuf points double*. Bull. Soc. Math. France , **10**, (1882),162–172. R. Hartshorne, *Algebraic Geometry*. Graduate Texts in Mathematics **52**, Springer 1977. C. Hacon, R. Pardini, *Surfaces with $p_g=q=3$*. Trans. Amer. Math. Soc. **354** (2002), 2631–2638.
D.W. Hahn, R. Miranda, *Quadruple covers of algebraic varieties*. J. Algebraic Geom. **8**, 1999. http://magma.maths.usyd.edu.au/magma/
S. Mukai, *Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves*. Nagoya Math. J. **81** (1981), 153–175. D. Mumford, *Abelian Varieties*. Oxford University Press (1970).
T. Oda, *Vector bundles on Abelian surfaces*. Invent. Math. **13** (1971), 247–260.
R. Pardini, *Abelian covers of algebraic varieties*. J. Reine Angew. Math. **417** (1991), 191–213. M. Penegini, *The classification of isotrivial fibred surfaces with $p_g=q=2$*. Collect. Math. **62**, No. **3**, (2011), 239–274. M. Penegini, F. Polizzi, *On surfaces with $p_g=q=2$, $K^2=5$ and Albanese map of degree $3$*. e-print arXiv:1011.4388 (2010). M. Penegini, F. Polizzi, *On surfaces with $p_g=q=2$, $K^2=6$ and Albanese map of degree $2$*. e-print arXiv:1105.4983 (2011), to appear in Canad. J. Math.
G. P. Pirola, *Curves on generic Kummer varieties*. Duke Math. J. **59** (1989), 701-708.
G.P. Pirola, *Surfaces with $p_g=q=3$*. Manuscripta Math. **108** no. 2 (2002), 163–170. E. Sernesi, *Deformations of Algebraic Schemes*. Grundlehren der Mathematischen Wissenschaften, Vol **334**, Springer-Verlag, Berlin, 2006.
Matteo Penegini, Dipartimento di Matematica *“Federigo Enriques”*, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy\
*E-mail address:* `matteo.penegini@unimi.it`\
\
Francesco Polizzi, Dipartimento di Matematica, Università della Calabria, Cubo 30B, 87036\
Arcavacata di Rende (Cosenza), Italy\
*E-mail address:* `polizzi@mat.unical.it`
|
---
abstract: |
We tackle the problem of accurate optical tracking of distant man-made probes, on Halo orbit around the Earth-Sun libration point L2 and beyond, along interplanetary transfers. The improved performance of on-target tracking, especially when observing with small-class telescopes is assessed providing a general estimate of the expected S/N ratio in spacecraft detection. The on-going [Gaia]{} mission is taken as a pilot case for our analysis, reporting on fresh literature and original optical photometry and astrometric results.
The probe has been located, along its projected nominal path, with quite high precision, within $0.13_{\pm 0.09}$ arcsec, or $0.9_{\pm0.6}$ km. Spacecraft color appears to be red, with $(V-R_c) = 1.1_{\pm0.2}$ and a bolometric correction to the $R_c$ band of $(Bol-R_c) = -1.1_{\pm0.2}$. The apparent magnitude, $R_c = 20.8_{\pm0.2}$, is much fainter than originally expected. These features lead to suggest a lower limit for the Bond albedo $\alpha = 0.11_{\pm 0.05}$ and confirm that incident Sun light is strongly reddened by [Gaia]{} through its on-board MLI blankets covering the solar shield.
Relying on the [Gaia]{} figures, we found that VLT-class telescopes could yet be able to probe distant spacecraft heading Mars, up to 30 million km away, while a broader optical coverage of the forthcoming missions to Venus and Mars could be envisaged, providing to deal with space vehicles of minimum effective area ${\cal A} \ge 10^6$ cm$^2$. In addition to L2 surveys, 2m-class telescopes could also effectively flank standard radar-ranging techniques in deep-space probe tracking along Earth’s gravity-assist maneuvers for interplanetary missions.
address: 'INAF - Osservatorio Astronomico di Bologna, Via Ranzani 1 40127 Bologna Italy'
author:
- Alberto Buzzoni
- Giuseppe Altavilla
- Silvia Galleti
title: |
Optical tracking of deep-space spacecraft in Halo L2 orbits and beyond:\
the Gaia mission as a pilot case
---
Space vehicles; Techniques: high angular resolution; Astrometry and celestial mechanics; Techniques: photometric
=0.5 cm
{width="0.58\hsize"}
Introduction
============
The exploitation of the Sun-Earth Lagrangian points, especially L1 and L2, along the Sun-Earth direction [@farquhar73; @nariai75; @rawal91; @farquhar02] has been an extraordinary challenge for space exploration in the recent years. For their particular position, some $1.5~10^6$ km away, on opposite sides of Earth and therefore well beyond the Moon, both locations are ideal lookouts for astrophysical observatories aimed at studying the Sun (L1) and the deep Universe (L2), far from any anthropic contamination. The L2 point, in particular, has been hosting a number of important astrophysical missions, starting with the [Wmap]{}, [Planck]{} and [Herschel]{} probes, and currently continuing with the [Gaia]{} mission, aimed at performing an exhaustive census of the Milky Way stellar population [@debruijne12; @cacciari15]. Following [Gaia]{}, other major space facilities are planned to be located in a so-called Halo L2 orbit in the forthcoming years. These include the James-Webb Space Telescope [[Jwst]{}; @gardner06], the [Euclid]{} cosmological probe [@laureijs10], and the [Athena]{} X-ray observatory [@barret13].
Optical ground tracking is yet of recognized importance for any L2 mission. For the co-rotating orbit to be maintained within its nominal figures, in fact, we need to carefully check spacecraft during its course along a complex Lissajous trajectory, as seen from Earth [e.g. @bray67; @zagouras85; @liu07; @kolemen12; @dutt11; @qiao14]. This is also of special interest for any space observatory (like [Gaia]{}, or the next [Jwst]{}) as its absolute inertial position is required with exquisite precision to allow, for instance, a confident measure of astronomical parallaxes of distant stars with the on-board instruments. In this regard, radar-ranging techniques may actually provide a better measurement of spacecraft distance and radial velocity [@imbriale03], but telescope observations, from their side, take advantage of a superior angular resolution, providing in principle more accurate astrometry and finer proper motion estimates [@altmann11].
Compared to the observation of near-Earth satellites, however, optical tracking of distant probes, in L2 and on route to even farther interplanetary distances, has to deal with much fainter target magnitudes, a drawback that urges a substantial improvement in terms of telescope skills and especially of observing techniques to effectively assess our deep-space situational awareness [e.g. @mooney06; @ruprecht14; @woods14].
In this contribution we want therefore to briefly assess some technical issues (Sec. 2) dealing with accurate ground tracking of deep-space probes at optical wavelength, taking fresh observations of the [Gaia]{} spacecraft (Sec. 3) as a pilot case for tuning up our theoretical analysis. The relevant photometric figures for [Gaia]{} will constrain the required telescope performance, for the optical observations to consistently complement standard radar-tracking techniques as in the forthcoming space missions to Mars and other planets of the solar system (Sec. 4). Our conclusions will be briefly stressed in Sec. 5.
Apparent magnitude of distant spacecraft
========================================
Depending on its physical properties, a satellite under solar illumination reflects a fraction $\alpha$ (the so-called Bond albedo) of the incident flux. The remaining fraction of the input energy is retained and heats the body up to an equilibrium temperature that leads to a balance between the absorbed and re-emitted flux. At Earth’s heliocentric distance, this temperature cannot exceed 120$^\circ$C [e.g. @gilmore02], so that thermal emission of spacecraft in the terrestrial neighborhood (and beyond) is only relevant at mid/far-infrared wavelength.
If a probe offers a cross-section $s^2$ to Sun’s light, being $s$ its reference scale-size, and if we assume the illuminated body to reflect isotropically, then the apparent bolometric magnitude of a spacecraft placed at a distance $d$ from Earth’s surface (at Sun’s opposition)[^1] simply scales as the ratio of the incident solar flux at Earth and at the spacecraft distance, so that $$m_{\rm bol} - m^{\rm bol}_\odot = -2.5 \log \left[\left(\frac{D_\odot}{D_\odot+d}\right)^2
\left(\frac{s^2}{4 \pi d^2}\right)\right],
\label{eq:1}$$ where $D_\odot = 1.49\,10^{13}$ cm is the astronomical unit (AU) and $m^{\rm bol}_\odot = -26.85$ is the apparent bolometric magnitude of the Sun, as seen from Earth [e.g. @karttunen96]. With the relevant substitutions, and expressing the spacecraft distance $u = d/D_\odot$ in AU, eq. (\[eq:1\]) takes the form: $$m_{\rm bol} = -5 \log s +5 \log [u\,(1+u)] +k,
\label{eq:2}$$ where the numerical constant is $$k = 2.5 \log (4\pi D_\odot^2) +m^{\rm bol}_\odot = 41.77,
\label{eq:2b}$$ providing the satellite scale-size $s$ is set in cm.
In Fig. \[f01\] we report an illustrative summary of the expected bolometric magnitude for distant man-made probes of different characteristic size, compared with a few small planetary bodies. Just as a guideline, the limiting magnitude as for observing Sun-type stars, reached by mid-class (2m aperture) and new-generation telescopes (i.e. the ESO 8m VLT and the forthcoming 40m E-ELT) is also marked on the plot.
As a fraction $(1-\alpha)$ of the incident solar flux is “diverted” into the infrared, to convert the bolometric figures to other broad-band [*optical*]{} magnitudes, say for instance the Johnson-Cousins $R_c$ band, we have to dim the re-processed optical flux such as $$m_R = m_{\rm bol} -BC_R' -2.5\,\log \alpha,
\label{eq:3}$$ being $BC_R' = (m_{\rm bol} - m_R)$ the bolometric correction to the band, if satellite were a perfect acromatic reflector (i.e. for $\alpha \to 1$). This value has to be estimated on the basis of satellite’s physical and geometrical properties. If we lack this information, then either the solar value ($BC_R' = +0.3$) could be taken as a first approximation or, if a color is known for our target, the $BC_R'$ correction for a more appropriate star could be chosen, as from standard calibrations in the literature [e.g. @johnson66; @bessell79; @buzzoni10].
![The surface-brightness dimming for trailing satellites, according to eq. (\[eq:dimming\]). Different exposure times are assumed, as labelled. Each strip has a lower and upper envelope for a 2“ and 0.5” FWHM seeing figure, respectively. The reference angular speed for satellites in LEO, MEO, and GEO [@veis63] is reported together with the mean sky motion for other relevant solar bodies. In addition, we also display the mean angular velocity of spacecraft [Gaia]{}, along its Halo L2 orbit. []{data-label="f02"}](fig02.ps){width="0.98\hsize"}
Motion dimming
--------------
For fixed exposure time, any moving satellite tends to appear more elusive than fixed stars of the same magnitude on CCD images. In fact, while a star is supposed to concentrate most of its photons across a spot of area $a = \pi (\textsc{Fwhm}/2)^2$, whose FWHM depends on the telescope point-spread-function, a target moving with angular speed $\zeta$, along a time $t_{\rm exp}$, would spread its light across a larger area $a' = \zeta\,t_{\rm exp}\,\textsc{Fwhm} + \pi (\textsc{Fwhm}/2)^2$. If both the star and the moving target have the same magnitude, the latter would actually be harder to catch, because of a lower mean surface brightness and a correspondingly poorer S/N ratio. The surface-brightness dimming can be quantified in $$\Delta \mu_{\rm sat} = 2.5 \log (a'/a) = 2.5 \log \left(1 + {4\,t_{\rm exp}\,\zeta \over {\pi \textsc{Fwhm}}}\right).
\label{eq:dimming}$$ Note that the effect does not depend on target magnitude. Rather, and quite interestingly, it becomes more severe with increasing exposure time and with improving seeing (that is for a better FWHM value), as shown in Fig. \[f02\].
![The magnitude limit reached by a 8m telescope, with a $\textsc{Dqe} = 0.8$, at a 5 and 100 S/N detection level in 100 (upper envelope) and 1000 sec (lower envelope of the curves) exposure time, as from eq. (\[eq:sn\]). We assume to observe in the $R_c$ band under two extreme seeing conditions, namely with a FWHM of 0.5 and 2.0 arcsec, and with a dark sky ($\mu_{\rm sky}^R = 20.5$ mag arcsec$^{-2}$). The indicative proper motion of some reference objects is reported, as from Fig. \[f02\]. []{data-label="f03"}](fig03.ps){width="\hsize"}
According to telescope diameter $\cal{D}$ and the exposure time $t_{\rm exp}$, the S/N ratio of a trailing object can eventually be computed as $$\left({S\over N}\right)_{\rm trail} = {\cal R}\,{10^{-0.4\,\left[m_{\rm sat}-\left({{\mu_{\rm sky}-\Delta\mu_{\rm sat}}\over
2}\right)\right]}},
\label{eq:sn}$$ with $${\cal R} = \frac{\cal{D}}{\textsc{Fwhm}} \left(n_o\,{\textsc{Dqe}}\,t_{\rm exp}\right)^{1/2}.
\label{eq:snb}$$ In previous equations, $\mu_{\rm sky}$ is sky surface brightness, $n_o$ and [Dqe]{} are the reference photon number for the magnitude zero point and the detector quantum efficiency, respectively, according to the photometric band of our observations. By definition, $n_o = (f_o\,\lambda_o\,\Delta_\lambda)/(h\,c)$, being $f_o$ the zero-mag reference flux, $\lambda_o$ and $\Delta_\lambda$ the effective wavelength and width of the photometric band, respectively, $h$ the Planck constant and $c$ the speed of light, as usual. For the $R_c$ band, $n_o = 1.05~10^{6}$ photons cm$^{-2}$ s$^{-1}$ [@buzzoni05]. Notice that, if $\zeta \to 0$, then the $\Delta \mu_{\rm sat}$ term vanishes and eq. (\[eq:sn\]) approaches the S/N ratio for a fixed star.
As an illustrative case, in Fig. \[f03\] we report the limiting magnitude in the $R_c$ band, that can be achieved for a trailing target at a S/N = 5 and 100 detection level with a 8m telescope ($\textsc{Dqe} = 0.8$) in a 100 and 1000 sec exposure time. We assume to observe from a good astronomical site [$\mu_{\rm sky}^R = 20.5$ mag arcsec$^{-2}$, e.g. @patat03], under two extreme seeing conditions (namely 0.5“ and 2.0” FWHM). An important feature of the plot is that mag limit for high-speed satellites does not depend on exposure time but it only improves with a better seeing or a bigger telescope. In the latter case, eq. (\[eq:sn\]) (and the curves in Fig. \[f03\]) offsets by $\Delta R_{\rm lim} = 2.5 \log ({\cal D}/800)$, by expressing ${\cal D}$ in cm.
The [Gaia]{} observations as a benchmark
========================================
Previous arguments make clear that, when observing distant spacecraft, [*on-target*]{} telescope tracking, such as to compensate target motion, is the mandatory requirement for letting any faint target literally “emerge” from the background noise. Compared to the output of eq. (\[eq:sn\]), in fact, the expected improvement in the S/N ratio for the latter case is of the order of $$\left({S\over N}\right)_{\rm track} = \left({S\over N}\right)_{\rm trail} 10^{+0.2\, \Delta \mu_{\rm sat}}.$$
![The spacecraft [Gaia]{}, as detected in the night of Oct 17-18, 2014 with the 1.52m telescope of the Loiano Observatory. The two panels are consecutive 300 sec exposures in “white” light (i.e. CCD with no photometric filter) of the same field with sideral (left) and differential (right) telescope tracking on. The displayed field is $3\times 3$ arcmin across. North is up, East to the left. Pixel size is 0.58 arcsec. See text for a discussion. []{data-label="f04"}](fig04a.ps "fig:"){width="0.46\hsize"} ![The spacecraft [Gaia]{}, as detected in the night of Oct 17-18, 2014 with the 1.52m telescope of the Loiano Observatory. The two panels are consecutive 300 sec exposures in “white” light (i.e. CCD with no photometric filter) of the same field with sideral (left) and differential (right) telescope tracking on. The displayed field is $3\times 3$ arcmin across. North is up, East to the left. Pixel size is 0.58 arcsec. See text for a discussion. []{data-label="f04"}](fig04b.ps "fig:"){width="0.46\hsize"}
As a striking example in this sense, we report here on two recent observing sessions on the [Gaia]{} probe, along its Halo L2 orbit. Observations have been carried out with the Cassini 1.52m telescope of the Loiano Observatory (Italy), in a six-months interval along the night of Oct 17-18, 2014 and on March 26-27, 2015. The telescope was equipped with the BFOSC camera, carrying a EEV $1300\times1340$ px coated and back-illuminated CCD, with a FOV of $12.6\times13.0$ arcmin, and a pixel scale of 0.58 arcsec px$^{-1}$.
In addition to a set of eight “white”-light frames (that is by exposing CCD with no photometric filter), the 2014 frames also consisted of six $V$, $R_c$, and Gunn $z$ exposures. Overall, the spacecraft was tracked along five hours. The 2015 night was just devoted to a one-shot check of [Gaia]{}’s position and apparent luminosity in the $R_c$ and $V$ bands. Both nights had photometric conditions along the observation windows, with an R-band FWHM seeing figure about $1.5^{\prime\prime}$-$1.7^{\prime\prime}$. The relevant data of our observations are summarized in Table \[t01\]. All frames were taken in “on-target” tracking mode, with the only exception of one image (as marked in the table), tracked in sideral mode for the illustrative purpose of Fig. \[f04\].
In the figure we actually report two consecutive 300 sec exposures of the same field, along the Oct 2014 night, with sideral (left panel) and differential (right panel) telescope tracking on. Just at first glance, it is evident that [Gaia]{} can barely be appreciated when trailing across the field, while it clearly stands out when concentrating its light as in the right panel.
[cccll]{} UTC$^{(a)}$ & & Mag$^{(c,d)}$ & Exp.$^{(e)}$\
hh:mm:ss.s & hh:mm:ss.sss & dd:mm:ss.ss & & sec\
\
20:08:55.5 & 01:49:13.459$_{\pm009}$ & +15:30:11.54$_{\pm43}$ & 20.6$_{\pm3}$ w$_R$ & 120 d\
20:15:45.6 & 01:49:14.134$_{\pm049}$ & +15:30:20.50$_{\pm62}$ & 20.7$_{\pm3}$ w$_R$ & 300 d\
20:23:56.1 & 01:49:14.951$_{\pm015}$ & +15:30:31.71$_{\pm21}$ & 21.1$_{\pm3}$ w$_R$ & 300 d\
20:28:55.5 & 01:49:15.434$_{\pm028}$ & +15:30:38.12$_{\pm29}$ & 20.9$_{\pm3}$ w$_R$ & 300 d\
20:34:08.3 & 01:49:15.907$_{\pm016}$ & +15:30:45.32$_{\pm21}$ & 20.6$_{\pm3}$ w$_R$ & 300 d\
20:44:07.5 & 01:49:16.758$_{\pm014}$ & +15:30:57.88$_{\pm13}$ & 20.9$_{\pm3}$ w$_R$ & 300 d\
20:55:32.7 & 01:49:17.695$_{\pm015}$ & +15:31:11.71$_{\pm14}$ & 20.5$_{\pm3}$ w$_R$ & 300 s\
21:13:16.1 & 01:49:18.975$_{\pm017}$ & +15:31:34.35$_{\pm21}$ & 21.0$_{\pm2}$ R$_c$ & 300 d\
21:29:09.9 & 01:49:20.007$_{\pm016}$ & +15:31:52.66$_{\pm16}$ & 22.2$_{\pm2}$ V & 420 d\
21:35:46.8 & 01:49:20.363$_{\pm016}$ & +15:32:00.45$_{\pm54}$ & 20.6$_{\pm4}$ z$_R$ & 300 d\
21:44:25.3 & 01:49:20.896$_{\pm015}$ & +15:32:09.63$_{\pm09}$ & 20.7$_{\pm2}$ R$_c$ & 300 d\
00:50:46.7 & 01:49:29.027$_{\pm005}$ & +15:34:37.50$_{\pm34}$ & 21.0$_{\pm2}$ R$_c$ & 300 d\
00:58:26.2 & 01:49:29.475$_{\pm011}$ & +15:34:40.94$_{\pm25}$ & 20.8$_{\pm3}$ w$_R$ & 300 d\
01:10:54.5 & 01:49:30.163$_{\pm008}$ & +15:34:45.55$_{\pm27}$ & 21.9$_{\pm2}$ V & 600 d\
\
00:00:14.5 & 12:28:27.063$_{\pm013}$ & +04:09:23.99$_{\pm20}$ & 20.7$_{\pm3}$ R$_c$ & 480 d\
00:13:46.8 & 12:28:27.499$_{\pm013}$ & +04:09:02.05$_{\pm20}$ & 21.7$_{\pm3}$ V & 900 d\
\
\
\
\
\
\
Astrometry
----------
Although limited, our observed database allowed us to benchamark the astrometric and photometric reduction procedure, in order to assess the realistic performance in spatially locating the spacecraft at such large distance from Earth and characterize its apparent photometric properties. A first crucial piece of information deals with astrometry.
For the astrometric solution of the CCD images, after standard reduction procedures with [IRAF]{}[^2], we relied on the [WCSTools]{} package[^3] to pick up a reference grid of template stars across the field of view and finally set the World Coordinate System (WCS) of the corresponding image. The 50 brightest objects in each field, as detected by the [IRAF]{} task [STARFIND]{}, have been matched by the [WCSTools IMWCS]{} task with the corresponding HST Guide star Catalogue II (GSC-II) [@russell90][^4] available by default to constrain the WCS across the frame. The WCS is the relationship between the pixel coordinates and the celestial coordinate, and it is written in a standard way in the FITS header. The precise astrometric solution is then refined by means of the [WCSTools SCAT]{} and the [IRAF CCFIND]{} and [CCMAP]{} tasks. In particular, [SCAT]{} retrieves the GSC-II catalogue of the field, while [CCFIND]{} uses the image WCS to convert the celestial coordinates into image pixel coordinates and refine the latter ones by using a centroid algorithm. The matched coordinates are finally used by [CCMAP]{} to compute a new plate solution in the RA and DEC domain by a low-order polynomial fit. The astrometric solution is directly inspected by over-plotting the reference catalogue sources on the astrometrically calibrated image.
Overall, our procedure secured an internal astrometric accuracy of $0.38_{\pm 0.18}$ arcsec (rms) in the photometric centroid determination of [Gaia]{}’s [*individual*]{} observations (see Table \[t01\]).[^5]
Though slightly elongated, the star figures in the telescope images did not prove to severely affect the astrometric solution across the field, provided the trailing intensity of the astrometric calibrators ($I$) does not vary (or just vary in a predictable way) along the exposure time.[^6] This is, actually, a crucial requirement for the photometric barycenter of stellar tracks to consistently tie to the [*luminosity-weighted*]{} time barycenter ($t_o$) of the exposure. By definition, for the latter, we have $$\frac{t_o}{t_{\rm exp}} = \frac{\int \tau I(\tau) d\tau}{\int I(\tau) d\tau}\,
\frac{1}{\int d\tau} \le 1.
\label{eq:9}$$ In case $I(\tau) = {\rm const}$, this relationship delivers the obvious solution $t_o = t_{\rm exp}/2$, so that the spacecraft position should be attributed to the mid-exposure time.
![The angular residuals of [Gaia’s]{} sky path along the night of Oct 17-18, 2014, as seen from the Loiano Observatory (UAI observatory code “598”). The data of Table \[t01\] are compared with the corresponding JPL topocentric ephemeris. Residuals are in arcsec units, both for RA (upper panel) and Dec (lower panel), in the sense “Observed–Computed”, (O-C). The only sideral tracking observation in our sample is singled out with a romb marker in both panels. []{data-label="f05"}](fig05.ps){width="0.9\hsize"}
Clearly, the attainable astrometric resolution ($\theta_{\rm mas}$) also constrains the required accuracy, $\sigma(t_o)$ in setting $t_o$, depending on the spacecraft proper motion: $$\sigma(t_o) \simeq \frac{3.6\,\theta_{\rm mas}}{\zeta} \quad {\rm [sec]}.$$ In the equation, $\theta_{\rm mas}$ is expressed in mas (milliarcsec) unit and $\zeta$ in arcsec hr$^{-1}$, as in Fig. \[f02\]. With the [Gaia]{} typical figures, $t_o$ has to be known within a few seconds, at most. According to the proper-motion constraints of Fig. \[f02\], this is also the accuracy level for tracking interplanetary spacecraft, while a better tuned clock ($\sigma(t_o) \ll 0.2$ sec) is devised, on the contrary, when observing fast-moving ($\zeta \gg 10^3$) Earth- and Moon-orbiting satellites.
### Locating L2 spacecraft
Figure \[f05\] shows that [Gaia]{} position consistently compares with its nominal trajectory along the Oct 17, 2014 night, as predicted by the topocentric ephemeris from the JPL Horizons Solar System Dynamics (SSD) Interface.[^7] The mean coordinate offsets of our data (in the sense “observed–computed”) and their corresponding $\sigma$ uncertainty amount to ($\Delta$RA, $\Delta$Dec) = ($0.10_{\pm 0.08},
-0.09_{\pm 0.10}$) arcsec, which lead to a mean transverse offset $\Delta_{GAIA} = 0.13_{\pm 0.09}$ arcsec. Our observations demonstrate, therefore, that the spacecraft can correctly be located along its expected orbital figure with an angular accuracy $\theta_{mas}$ of the order of 90 mas (see Fig. \[f06\]).[^8] At the reported distance of [Gaia]{} along the night (namely $1.39\,10^6$ km, according to the JPL ephemeris), our measures point to a mean orbital offset of $0.9_{\pm0.6}$ km.
![Arcsec coordinate residuals (in the sense “observed–computed”) of the [Gaia]{} positions along the night of Oct 17, 2014, with respect to the JPL topocentric ephemeris. Mean RA and Dec offsets, together with their 1-$\sigma$ uncertainty, are reported in the plot. When combined, these lead to a mean path offset with respect to the nominal figure of only $\Delta_{GAIA} = 0.13_{\pm 0.09}$ arcsec (big circle in the plot). []{data-label="f06"}](fig06.ps){width="0.87\hsize"}
![Same as Fig. \[f06\] but for a set of astrometric measurements for the night of Oct 14, 2014, taken with the 1.5m Mt. Lemmon (UAI code “G96”) [@kowalski14 dots] and the 1.8m LPL/Spacewatch II (UAI code “291”) telescopes [@tubbiolo14 rombs]. Arcsec coordinate residuals are computed with respect to the correpsonding JPL topocentric ephemeris leading to a mean orbital offset of $\Delta_{GAIA} = 0.16_{\pm 0.11}$ arcsec (big circle in the plot). []{data-label="f07"}](fig07.ps){width="0.87\hsize"}
These figures nicely compare also with other almost parallel sets of observations, taken a few nights earlier (namely on Oct 14, 2014) by the 1.5m Mt. Lemmon [@kowalski14] and the 1.8m LPL Spacewatch II [@tubbiolo14] telescopes. A summary of these data is shown in Fig. \[f07\]. After rejecting one clear outlier in the LPL/ Spacewatch II sample, they indicate, overall, a ($\Delta$RA, $\Delta$Dec) = ($-0.11_{\pm 0.13}$, $-0.11_{\pm 0.10}$) arcsec, leading to a mean orbital offset of $\Delta_{GAIA} = 0.16_{\pm 0.11}$ arcsec (or $1.1_{\pm 0.7}$ km) with respect to the corresponding JPL ephemeris.
Closer to our 2015 observations, a supplementary set of 10 astrometric measurements were provided by @velichko15 in the night of March 10, 2015, at the 2m telescope of the Terskol Observatory (Russia), together with six estimates of @gibson15 with the 1.8m Pan-STARRS telescope in Haleakala (Hawaii, USA) on the nights of March 23 and 24, 2015 (see Fig. \[f09\]).
This coarser set of observations is merged in Fig. \[f11\] with our data and leads, overall, to ($\Delta$RA, $\Delta$Dec) = ($0.21_{\pm 0.06}$, $0.07_{\pm 0.04}$) arcsec, that is a mean orbital offset of $\Delta_{GAIA} = 0.22_{\pm 0.05}$ arcsec (i.e. $1.6_{\pm 0.4}$ km).
![The [Gaia]{} sky path along the March 10-26, 2015 period. The Terskol (UAI code “B18”) [@velichko15 cyan marker], PanSTARRS (UAI code “F51”) [@gibson15 green dots] and Loiano (red dot) observations are superposed to the JPL topocentric ephemeris (nominally for the B18 location, just as a guideline). Note the daily “ripples” of the [Gaia]{} apparent orbit, due to the parallax effect of Earth rotation, that superposes to the overall Lissajous figure on larger scales. []{data-label="f09"}](fig08.ps){width="0.87\hsize"}
### Locating deep-space spacecraft
The brief overview of the [Gaia]{} case convincingly supports the preliminary results of the on-going tracking campaign carried on in the framework of the [Gaia]{} mission plan [see @altmann12 for details], and proves that optical measurements, even carried out with small-class telescopes and under quite standard observing conditions, can easily achieve a superior accuracy when compared to radar tracking, to locate distant spacecraft across the sky. This better score is the obvious consequence of the fact that, for fixed angular resolving power, it must be $$\left(\nu\, D\right)_{\rm radio} \simeq \left(\nu D\right)_{\rm opt}.$$ Therefore, if a radar dish is set to operate at some $X$-band frequency [say 8 GHz, as for the Deep Space Network antennas; @imbriale03], compared to the $\sim 5\,10^{14}$ Hz of a telescope observing in the $V$ band, then a factor of $60,000$ larger antenna is required to achieve the same angular performance of the optical instrument. This means that a 60 km-wide(!) antenna (or radio-interferometric baseline) is needed to offset the (diffraction-limited) performance of a 1m telescope.
![Same as Fig. \[f06\] but for a coarser set of observations taken during March 2015 at the Terskol 2m telescope (triangles), the 1.8m Pan-STARRS telescope (pentagons) and the Loiano 1.52m telescope (rombs, see Table \[t01\]). Arcsec coordinate residuals are computed with respect to the appropriate JPL topocentric ephemeris for each observatory. Error bars are only available for our observations, as from Table \[t01\]. The merged set of data leads to a mean orbital offset of $\Delta_{GAIA} = 0.22_{\pm 0.05}$ arcsec (yellow circle in the plot) along the spanned period. []{data-label="f11"}](fig09.ps){width="0.87\hsize"}
Clearly, one could argue that any ground-based telescope is eventually seeing (and not diffraction) limited, thus restraining its resolving power to a fraction of arcsec, at best. Another important issue also deals with the inherent astrometric accuracy of reference stars to set the absolute astrometric grid. As for the HST Guide Star Catalog GSC-II used here, currently one of the most populated source of data across the sky for astrometric studies, this limit turns to be about 300 mas [@lasker08; @bucciarelli08].[^9]
On the other hand, multiple measurements are possible of the target differential position with respect to a number of reference (astrometric) stars across the field of view, and this eventually leads to improve target astrometric accuracy by a factor $\sqrt{N_{\rm stars}}$. In addition, further improvements could be envisaged in case a series of independent observations could be collected. Definitely, our [Gaia]{} benchmarking showed that an internal accuracy better than 100 mas is a fully attainable goal for standard observations carried out with 2m-class telescopes.
![The expected spatial resolution in detecting distant spacecraft within the Solar System. The transverse component, projected on the sky, is assessed in terms of absolute resolution in kilometers at the different distances from Earth. Two representative values for angular resolution of optical tracking are assumed, namely $\theta_{\mu as} = 300$ mas and 30 mas, as labelled on the plot. []{data-label="f12"}](fig10.ps){width="\hsize"}
As far as even farther distances from Earth are considered, our arguments lead us to conclude that absolute position of man-made probes could be constrained within a nominal accuracy $\Delta_{\rm sat}$ of the order of $$\Delta_{\rm sat} = \frac{d_6\, \theta_{\rm mas}}{206} \quad {\rm [km]},$$ or $$\Delta_{\rm sat} = 0.72\,u\, \theta_{\rm mas} \quad {\rm [km]},$$ assuming to express the geocentric distance in million km ($d_6$) or in AU ($u$, as in eq. \[eq:2\]), respectively. This relationship is displayed in Fig. \[f12\], for values of $\theta_{\rm mas} = 30$ and 300 mas.
According to the figure, one can hope to optically pinpoint deep-space probes around Mars well within a few tens of km. When complemented with radar information (more effective in ranging measurements and Doppler velocity shift), the optical output could therefore be extremely valuable for an effective 3D location of man-made probes at interplanetary distances.
Quite interestingly, Fig. \[f12\] also suggests that, under appropriate observing conditions (i.e. by masking Moon’s overwhelming luminosity), Moon-orbiting spacecraft could be accurately located within a few hundred meters, a precision that could even raise to about 10m in the local framework, for instance when considering station-keeping maneuvering of Earth artificial satellites in geostationary orbit [@montojo11].
Photometry
----------
Both for the 2014 and 2015 observing sessions, photometry in $V$ and $R_c$ bands has been directly calibrated to standard magnitudes by observing two closeby @landolt92 fields. Typical rms uncertainty for these observations is 0.2 mag, as summarized in Table \[t01\]. Although with a much broader passband, the effective wavelength of the CCD response curve happened to closely match that of the $R_c$ filter ($\lambda_o \simeq 6470$ Å), and this eased a rough but still convenient transformation of the “white” instrumental magnitudes of the Oct 2014 session into pseudo-equivalent $R_c$ figures (called $w_R$ in Table \[t01\]). The whole conversion procedure relied on a grid of several field stars in common with the $R_c$ and “white”-light images, and eventually led to a total error of 0.3 mag (rms) for the $w_R$ magnitudes. Just for the sake of comparison, a similar procedure was also applied to the unique Gunn $z$ frame of the 2014 session, leading to a coarser $R_c$ proxy of (i.e. $z_R$ in Table \[t01\]), with a 0.4 mag overall error, given a large difference of the $z$ band effective wavelength ($\lambda_o = 9040$ Å).
![The apparent magnitude of [Gaia]{} in different photometric bands along the Oct 2014 and March 2015 observing runs. In addition to $R_c$ (dots) and $V$ (square) Johnson-Cousins bands, “white”-light (triangles) and Gunn $z$ (romb marker) observations have been converted to the $R_c$ magnitude scale as discussed in the text. Dashed lines mark the mean $R_c$ and $V$ magnitude levels. []{data-label="f13"}](fig11.ps){width="\hsize"}
A plot of all the entries of Table \[t01\] is shown in Fig. \[f13\], versus observing time. Within the photometric accuracy of our observations, along the 2014 session [Gaia]{} displayed a constant apparent luminosity with an average magnitude $\langle R_c \rangle = 20.8_{\pm0.2}$ and $\langle V \rangle = 22.0_{\pm0.2}$. These results are basically confirmed also by the 2015 observations, taken at the same topocentric distance and phase angle,[^10] which indicate $\langle R_c \rangle = 20.7_{\pm0.2}$ and $\langle V \rangle = 21.7_{\pm0.2}$. Overall, the observations point to a color of $(V-R_c) = 1.1_{\pm0.2}$, much redder than the solar value of $(V-R_c)_\odot = 0.4$, as estimated from @pecaut13. By itself, this difference suggests that [Gaia]{} seems to drastically re-process the incident solar flux.
Furthermore, our results also point to a much fainter apparent magnitude than originally expected for the spacecraft [namely $R_c \sim 17$-18, according to @altmann11], based on the direct experience on previous L2 missions (especially [Wmap]{} and [Planck]{}). Yet to a more updated analysis [@altmann14], the reason of this discrepancy remains unclear. Evidently, [Gaia]{}’s faintness and its exceedingly red color have to be related to more inherent reflectance characteristic of the spacecraft structure, especially dealing with the extended Kapton multi-layer insulation (MLI) blankets that cover most of the sun-shield.
To help better investigate the problem, during the preliminary station-keeping maneuvers of Feb 27 and March 7, 2014, the probe has repeatedly been re-oriented, from its nominal attitude configuration at solar aspect angle[^11] $\omega = 45^o$, such as to have its sun-shield directly facing the Sun, that is with $\omega = 0^o$. This greatly brightened [Gaia]{}’s apparent luminosity up to $R_c^{\rm peak} = 14.5_{\pm0.2}$ [@james14; @jacques14; @dupouy14; @dupouy14b].
The “face-on” experiment clearly indicated that [Gaia]{}’s sun-shield reflectance displays an important directional pattern [an effect often referred to as the “opposition surge”, e.g. @warell04] with spacecraft luminosity increasing by $\Delta {\rm mag} = 20.8-14.5 = 6.3_{\pm0.3}$ mag when changing $\omega$ from $45^o$ to $0^o$. This figure has to be compared with the straight Lambertian prediction, that leads to a much shallower magnitude brightening of $$\Delta {\rm mag} = -2.5 \log \left( \frac{\cos 45^o}{\cos 0^o} \right) \sim 0.4~{\rm mag}.
\label{eq:45deg}$$
### Bolometric correction and spacecraft albedo
According to eq. (\[eq:2\]), the expected bolometric magnitude of [Gaia]{} at the relevant distance of our observations is $Bol = 16.9$ mag.[^12] As from eq. (\[eq:3\]), this figure can be linked to the apparent $R_c$ magnitude by correcting for the appropriate bolometric correction ($BC^\prime_R$) and the spacecraft albedo ($\alpha$). Both these quantities directly relates to the reflectance spectrum of the probe along the entire wavelength range, a feature that can only be poorly known [*a priori*]{} and needs to be conveniently assessed “on the fly”. For this task, as a part of the characterization checks during the early phases of the [Gaia]{} mission [@altmann14], narrow-band multicolor photometry was acquired in early 2014 with the [Grond]{} imager at the ESO 2.2m telescope in La Silla. This allowed us to reconstruct the spacecraft reflectance spectrum along the 4500-13000 Å wavelength range and therefrom derive the spacecraft colors during the “face-on” experiments and later along the standard operation phase.
![The [Gaia]{} color properties, as derived from the [Grond]{} reflectance curves of @altmann14, are assessed in the $(B-V)$ vs. $(V-R_c)$ plane. The spacecraft location is compared with the stellar locus for dwarf and giant stars of different spectral type (as labelled on the plot), according to @pecaut13 and and @houdashelt00, respectively. The shaded triangular region on the plot edges the allowed color range of [Gaia]{} (including the experimented “face-on” orientation, as discussed in the text), accounting for the reported variability of the spacecraft reflectance. As a reference, the Sun is also marked on the plot, together with the derived colors of the [Planck]{} probe, according to @altmann14. []{data-label="f14"}](fig12.ps){width="0.97\hsize"}
As far as the $(B-V)$ and the $(V-R_c)$ colors are concerned, the spacecraft location is compared in Fig. \[f14\] with the Sun and the locus for dwarf and giant stars of different spectral type. The empirical compilation of @pecaut13 for MK-class V stars is adopted, for this comparison, together with the theoretical models of giant stars from @houdashelt00. As a further interesting match, in the figure we also added the colors of [Planck]{}, as derived from the original reflectance spectrum of @altmann14. Contrary to [Planck]{}, which resulted to be a quite effective proxy of the Sun, [Gaia]{} roughly behaves like a red dwarf of very late M spectral type, with an even more depleted $B$ luminosity (i.e. “redder” $B-V$ color). In addition, one has to report from the [Grond]{} data, an important (and yet partly unexplained) spectral variability of the probe, with its optical colors actually cornered within the wide region marked in Fig. \[f14\].
![The observed [Gaia]{} $(V-R_c)$ color is contrasted in the plot to constrain the spacecraft bolometric correction. As for Fig. \[f14\], the spacecraft location is compared with the stellar locus for dwarf and giant stars of different spectral type (as labelled on the plot), according to @pecaut13 and and @houdashelt00, respectively, and with the relevant points for the Sun and [Planck]{} probe. []{data-label="f15"}](fig13.ps){width="0.97\hsize"}
On the basis of the reference $(V-R_c)$ color, in Fig. \[f15\] we try to envisage a suitable bolometric correction by comparing, again, with the @pecaut13 and @houdashelt00 stellar calibrations. Differently from [Planck]{}, [Gaia]{}’s color properties point to a much larger correction, that we can tentatively place around a value of $BC^\prime_R \simeq -1.1\pm 0.2$ mag, although with a large uncertainty, further magnified by the spacecraft spectral variability.
With the relevant figures, under Lambertian assumptions, eq. (\[eq:3\]) takes the final form as $$20.8_{\pm 0.2} -0.4_{\pm 0.3} = 16.9 +1.1_{\pm 0.2} -2.5\,\log \alpha,$$ where the apparent $R_c$ magnitude in the l.h. term of the equation has been corrected[^13] to $\omega = 0^o$, according to eq. (\[eq:45deg\]). This relationship eventually leads to a nominal value of the [Gaia]{} albedo $\alpha = 0.11_{\pm 0.05}$.
Strictly speaking, however, this estimate is most likely to be taken as a [*lower limit*]{} for the true Bond albedo, due to sun-shield reflectance anisotropy, as we were discussing before. In fact, by applying our arguments to the real “face-on” spacecraft configuration, one is led to conclude that [Gaia]{} [*could not be brighter*]{} than $R_c^{\rm peak} \ge 16.9_{\pm 0.3} +1.1_{\pm 0.2} = 18_{\pm0.4}$, as originally estimated by @altmann11, but evidently at odds with the observed evidence.
Probing interplanetary distances
================================
When coupled with previous theoretical arguments, the [Gaia]{} observations provide us with a useful tool to consistently size up the required telescope performance in order to detect deep-space probes at even larger distances from Earth, for instance when heading toward other planets of the solar system. If we set a minimum $(S/N)$ threshold for target detection, then eq. (\[eq:sn\]) (with “on-target” telescope tracking, that is for $\Delta \mu_{\rm sat} = 0$) can be solved to obtain the faintest possible magnitude we could reach and, by means of eq. (\[eq:3\]) and (\[eq:2\]), its corresponding maximum geocentric distance.
As probe needs to be under Sun’s illumination to be detected from Earth, this also sets a constraint to the allowed range of the phase angle, depending on spacecraft (geocentric) distance and physical structure. If planar components (i.e. large solar panels etc., either facing Earth or Sun) are the prevailing features of the probe, then the simple geometrical arguments outlined in Sec. 3.2 for the case of [Gaia]{} indicate that its apparent luminosity under different view angles scales as $\cos (\phi)$. This evidently implies that $|\phi| \le 90^o$ for the probe to be visible. Much larger values of $\phi$ may, however, be allowed to roughly spherical shapes, as a cardioid-shaped luminosity law, as $[1+cos (\phi)]/2$, [e.g. @meeus98] holds in this case.
The visibility map for the illustrative case of either a spheric or straight a planar satellite geometry is sketched in the two panels of Fig. \[f16\], respectively. Calculations have been carried out for the [Gaia]{}’s effective area ${\cal A} = \alpha s^2 \simeq 10^5$ cm$^2$ (which assumes a conservative albedo value $\alpha = 0.11$) supposing to detect the spacecraft at a $(S/N)$ ratio better than 5. With the 8m-class (or bigger) telescopes currently in use, one sees that an important region can be explored, well beyond the Earth’s influence sphere (that embodies the Lagrangian L1 and L2 points) reaching, for instance, distant spacecraft along their initial Hohmann transfer to Mars, up to geocentric distances of some 30 million km (roughly 0.2 AU).
Even 2m-class telescopes could however usefully track deep-space probes, for instance during their flyby approach to Earth for gravity sling toward external planets. Successful observations of the Rosetta spacecraft [@glassmeier07], along its 2005 and 2007 gravity-assist maneuvers [e.g. @bedient05; @dillon05; @hill05; @james05; @juels05; @kusnirak05a; @kusnirak05b; @manca05; @stevens05; @ticha05; @birtwhistle07a; @birtwhistle07b; @birtwhistle07c; @bittesini07a; @bittesini07b; @bittesini07c; @donato07; @hill07a; @hill07b; @kowalski07][^14] are a relevant example of such a kind of applications. The current Hayabusa 2 mission [@kuninaka13] could soon provide another intersting case to benchmark the method.
![Same as Fig. \[f16\] but for 2m (green), 8m (dark green) and 40m (pale green) telescopes looking (at a $S/N \sim 5$ or better threshold) at a deep-space spherical probe of [Gaia]{}’s 10$\times$ enhanced effective area (namely ${\cal A} = 10^6$ cm$^2$). Under these more favourable circumstances, note that a VLT-class telescope could yet confidently track a spacecraft along most of its Hohmann trajectory to Mars and Venus, while a 2m telescope could, in general, effectively probe distant spacecraft some 40 million km away from Earth. []{data-label="f17"}](fig15.ps){width="0.9\hsize"}
According to Fig. \[f16\], optical tracking is of lesser point for any space mission toward inner planets (i.e. Venus and Mercury). This is because the orbit geometry leads probes usually to appear from Earth at large phase angles, thus poorly reflecting Sun’s light in our direction. An enhanced spacecraft effective area– either in terms of better reflectance to improve the albedo or directly by increasing the physical size of the target, as likely the case of the forthcoming (manned) missions to Mars– could be the obvious issue in this case, as we show in the illustrative case of Fig. \[f17\]. By the way, a closer look to Fig. \[f16\] also shows that spherical spacecraft would better perform than other probes with more planar surfaces.
Summary & Conclusions
=====================
We assessed in some detail the observing performance of optical telescope-tracking techniques, for accurate characterization of deep-space probes around the Earth-Sun libration point L2 and beyond, toward interplanetary travels. We provided a general estimate of the expected S/N ratio, quantifying the superior improvement of on-target tracking, a mandatory requirement, especially when observing with small-class telescopes.
The current [Gaia]{} mission has been taken as a pilot case for our discussion. In this regard, we reported on fresh optical photometry and astrometric results, that led to angularly locate the spacecraft across the sky within $0.13_{\pm 0.09}$ arcsec, or $0.9_{\pm0.6}$ km. Photometric results also indicate for [Gaia]{} a quite red color, with $(V-R_c) = 1.1_{\pm0.2}$ and an apparent magnitude $R_c = 20.8_{\pm0.2}$, much fainter than expected and dimmed by a large bolometric correction $(Bol-R_c) = -1.1_{\pm0.2}$. These features lead to a lower limit for the Bond albedo $\alpha = 0.11_{\pm 0.05}$ and confirm that the unabsorbed fraction of the incident Sun light is strongly reddened by the MLI blankets, that cover the spacecraft sun-shield.
These observations provided us with the reference figures to consistently assess the detection threshold for any deep-space probe toward inner and outer planets of the Solar System. Waiting for next-generation (E-ELT) telescopes, yet VLT-class instruments could be able to track distant spacecraft, like [Gaia]{}, along their initial Hohmann transfer to Mars, some 30 million km away. As successfully done with the Rosetta spacecraft, we confirm that even 2m-class telescopes could usefully help track deep-space probes, along their Earth’s gravity-assist maneuvers heading interplanetary targets.
We finally demonstrated that optical tracking from 8m ground telescopes could fully flank the standard radar-ranging techniques to probe distant spacecraft along the forthcoming missions to Venus and Mars, providing to deal with (round-shaped) space vehicles of minimum effective area ${\cal A} = \alpha s^2 \ge 10^6$ cm$^2$.
Acknowledgements
================
We would like to acknowledge the timely and very competent review of the two anonymous referees, that greatly helped refine the original version of our study.
This work made extensive use of the [IRAF]{} package, written and supported by the National Optical Astronomy Observatories (NOAO) in Tucson, Arizona. NOAO is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under cooperative agreement with the National Science Foundation.
Our astrometric procedure made reference to the GSC-II Catalog, a joint project of the Space Telescope Science Institute (STScI) and the Osservatorio Astronomico di Torino (OATo), Italy. The STScI is operated by AURA under NASA contract NAS5-26555. The participation of OATo is supported by the Italian Council for Research in Astronomy. Additional support is provided by ESO and the Space Telescope European Coordinating Facility, the International GEMINI project and the European Space Agency (ESA).
Part of the data retrieval has been eased by the DEXTER graphic interface, hosted at the German Astrophysical Virtual Observatory (GAVO) in Heidelberg.
We thank Roberto Gualandi and Italo Foppiani at OABo, Italy, for invaluable help along the telescope observations, and Jessica Mink at the Harvard-Smithsonian Center for Astrophysics of Cambridge MA, USA for her kind help to introduce us to the [WCSTools]{} package.
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E., 2013, ApJS, 208, 9 Qiao, D., Cui, P., Wang, Y., et al. 2014, Advances in Space Research, 54, 2087 Rawal J. J., 1991, EM&P, 54, 283 Roeser S., Demleitner M., Schilbach E., 2010, AJ, 139, 2440 Ruprecht, J. D., Stuart, J. S., Woods, D. F., & Shah, R. Y. 2014, Icarus, 239, 253 Russell, J. L., Lasker, B. M., McLean, B. J., Sturch, C. R., & Jenkner, H. 1990, AJ, 99, 2059 Stevens B. L. 2005, DASO Circ. No. 23, IAU Minor Planet Center, ed. G. V. Williams Ticha J. & Tichy M. 2005, DASO Circ. No. 25, IAU Minor Planet Center, ed. G. V. Williams Tubbiolo, A. F. 2014, DASO Circ. No. 512, IAU Minor Planet Center, ed. G. V. Williams Veis, G. 1963, SSR, 2, 250 Velichko S., Andreev M., Godunova V. 2015, DASO Circ. No. 516, IAU Minor Planet Center, ed. G. V. Williams Warell J., 2004, Icar, 167, 271 Woods, D. F., Lambour, R. L., Faccenda, W. J., et al. 2014, Proc. SPIE, 9149, 91490S Zagouras C., Markellos V. V., 1985, CeMec, 35, 257
[^1]: Although, strictly speaking, $d$ is a topocentric distance, to all extent, for a distant spacecraft in L2 and beyond, it basically coincides with the geocentric distance, as well.
[^2]: See the URLs: [*http://iraf.noao.edu.*]{}
[^3]: See the URLs: [*http://tdc-www.harvard.edu/wcstools.*]{}
[^4]: See the URL: [*http://tdc-www.harward.edu/software/catalogues/ gsc2.html*]{}.
[^5]: For the illustrative scope of our analysis, we relied here on the GSC-II catalog, a default reference for [WCSTools]{} to set the astrometric solution. Other catalogs could, however, be implemented providing to set them in appropriate format. Of these, especially the PPMXL [@roeser10] and CMC-15 [@cmc15] compilations may prove to be viable alternatives to improve accuracy at least in selected sky regions.
[^6]: Among others, relevant change along trailing intensity of stars could either be due to any small drift in the sideral and “on-target” telescope tracking, or to any unperceived tiny cloud crossing the field during exposure etc.
[^7]: See the URL: [*http://ssd.jpl.nasa.gov/horizons.cgi.*]{}
[^8]: Quite consistently, notice that the derived 90 mas rms of the average coordinate residuals of Fig. \[f06\] fully compares with the $(380/\sqrt{N_{\rm obs}})\sim 100$ mas theoretical figure, as for the independent observations of Table \[t01\].
[^9]: The GSC-II provides about 6 reference stars per square arcmin, down to $V \sim 20$, although the reported accuracy refers only to the brightest stars ($V\le 18.5$) eventually used in our study. This figure is expected to drastically improve (down to $\sim30\mu$as for $V \sim 15$ stars) in the forthcoming years, just relying on the [Gaia]{} star survey [@debruijne12].
[^10]: The phase angle is the angular distance between Sun (S) and Earth (E), as seen from the probe (P), that is $\phi = \widehat{\rm SPE}$. For our 2014 and 2015 observations $\phi \simeq 7^o$
[^11]: This is the angle of the incident solar flux with respect to the normal of the satellite planar surface.
[^12]: We adopt $s = 886$ cm as [Gaia]{}’s relevant size to be adopted in eq. (\[eq:2\]). See: [*http://www.esa.int/Our\_Activities/Space\_Science/Gaia/Gaia\_factsheet*]{} for more details.
[^13]: The error bar on the eq. (\[eq:45deg\]) magnitude correction accounts for the $\pm 15^o$ maximum phase angle excursion of the spacecraft along its libration orbit around L2, as seen from Earth.
[^14]: DASO Circulars are made available in electronic form at the URL: [*http://www.minorplanetcenter.net/iau/DASO/DASO.html*]{}.
|
---
abstract: 'We have discussed here the polarization of quantum Hall states in the framework of the hierarchical analysis of IQHE and FQHE in terms of Berry phase. It is observed that we have fully polarized states for the filling factor $\nu=1$ as well as $\nu=\frac{1}{2m+1}$, m being an integer. However, for $\nu=p$ as well as $\nu =\frac{p}{q}$, with $p>1$ and odd, $q$ odd we have partially polarized states and for $\nu=\frac{p}{q}$, $p$ even,'
author:
- 'B. Basu'
- 'S. Dhar'
- 'P. Bandyopadhyay'
title: 'Polarization of Quantum Hall States, Skyrmions and Berry Phase'
---
Introduction
============
The quantum Hall effect (integral and fractional) appears in two-dimensional electron systems in a strong magnetic field. In such a high magnetic field (B $\sim 10$ T)the spin of the electron has no dynamical role and we can assume the electrons to be spinless in these fully polarized quantum Hall states. But when B has relatively smaller value [*i.e.*]{} when the Zeeman splitting is not so large, the corresponding system is not fully polarized. They are sometimes partially polarized and sometimes unpolarized. It is known from experiments that the QH states at filling factors $\nu=4/3, 8/5,
....$ [@2; @3] and at $2/3$ [@4; @5] are unpolarized while the states at $\nu=3/5$ [@5] and $7/5$ [@2] are partially polarized. From numerical computations it is known [@6] that the states with $\nu=\frac{2}{2n+1}$ are unpolarized and with $\nu=\frac{1}{2n+1}$ are fully polarized in the vanishing Zeeman splitting limit.
Wu, Dev and Jain [@7] have studied this problem and reported that all even numerator QH states are unpolarized and all these states with the numerator and denominator odd are partially polarized or fully polarized (in the vanishing Zeeman splitting limit). Later Mandal and Ravishankar [@8] proposed a global model which accounts for all the observed quantum Hall states in terms of an abelian doublet of Chern-Simons gauge fields, with the strength of the Chern-Simons term given by a coupling matrix. Hansonn, Karlhede and Leinaas [@9] proposed a new effective field theory for partially polarized quantum Hall states. They determined the density and polarization for the mean field ground states by couplings to two Chern-Simons gauge fields. They derived a sigma model covariantly coupled to the Chern-Simons field and found mean field solutions which describe partially polarized states.
It is also observed that the low energy excitations in various polarization states are quite different. It is now confirmed that the low energy excitation states for a fully polarized quantum Hall state is a skyrmion whereas for unpolarized states skyrmion excitations are not possible. For partially polarized states also, skyrmion excitations do not seem to occur as the skyrmionic solitons and are found not to be of usual ones. Wu and Sondhi [@10] have shown that in higher Landau levels, skyrmions are not the low energy excitations even at small Zeeman energies. Thus it follows that skyrmions are the quasiparticles only at $\nu=1$, $1/3$ and $1/5$.
In some earlier papers [@rev11; @12], we have analyzed the hierarchy of quantum Hall states from the view point of chiral anomaly and Berry phase. It has been pointed out that this approach embraces in a unified way the whole spectrum of quantum Hall systems with their various characteristic features. Here we propose to study the various polarization states of quantum Hall skyrmions on the basis of this hierarchical model. Also, we shall study the low energy excitation states for various polarization states of quantum Hall fluid.
Polarization of Integer and Fractional Quantum Hall States
==========================================================
In some earlier papers [@rev11], Basu and Bandyopadhyay have studied the whole spectrum of quantum Hall states (IQH) and (FQH) in a unified way from the chiral anomaly and Berry phase approach. It has been shown that the hierarchy of FQH states with filling factor $\nu=p/q$ ($p$ even or odd and $q$ odd) is interpreted in terms of the fact that the Berry phase associated with even number of flux quanta can be removed to the dynamical phase. For $p$ odd and $p>1$, the corresponding state attains a higher Landau level whereas for $p$ even, the system corresponds to particle-hole conjugate states.
In a spherical geometry we consider quantum Hall states in the two dimensional surface of a 3D sphere with a magnetic monopole of strength $\mu$ at the centre. The angular momentum relation is given by $${\bf J} ~ = ~ {\bf r} \times {\bf p} - \mu {\bf \hat{r}} , ~ \mu
~=~ 0 , ~\pm ~1 / 2 , ~\pm ~1 , ~\pm ~ 3 / 2 \cdots.$$ The spherical harmonics incorporating the term $\mu$ have been extensively studied by Fierz [@13] and Hurst [@14]. Following them we write $$\begin{array}{lcl}
Y^{m, \mu}_{\ell} &=& \displaystyle{ {(1+x)}^{-(m - \mu)/2} \dot
{(1-x)}^{-(m +
\mu)/2}}\\ \\
& & \displaystyle {\frac{d^{\ell - m}}{d^{\ell - m}_x} \left[
{(1+x)}^{\ell - \mu} {(1-x)}^{\ell + \mu} \right] e^{im \phi}
e^{-i \mu \chi}}
\end{array}$$ where $x = \cos \theta$ and the quantities $m$ and $\mu$ just represent the eigenvalues of the operators $i
\frac{\partial}{\partial \phi}$ and $i \frac{\partial}{\partial
\chi}$ respectively.
From the description of spherical harmonics we can construct a two-component spinor $\theta = \left( \begin{array}{c}
\displaystyle{u}\\
\displaystyle{v} \end{array} \right)$ where $$\begin{array}{lcl}
u ~=~ Y^{1/2 , 1/2}_{1/2} &=& \displaystyle{sin ~\frac{\theta}{2}
\exp
\left[ i (\phi - \chi) / 2 \right]}\\ \\
v ~=~ Y^{- 1/2 , 1/2}_{1/2} &=& \displaystyle{cos
~\frac{\theta}{2} \exp \left[- i (\phi + \chi) / 2 \right]}
\end{array}$$ Then the $N$-particle wave function for the quantum Hall fluid can be written as $${\psi^{(m)}}_{N} ~=~ \prod_{i < j} {(u_i v_j - u_j v_i)}^{m}$$ where $\nu = \frac{1}{m}$, $m$ being an odd integer.
It is noted that ${\psi^{(m)}}_{N}$ is totally antisymmetric for odd $m$ and symmetric for even $m$. Following Haldane [@14h], we can identify $m$ as $m=J_i + J_j$ for an $N$-particle system where $J_i$ is the angular momentum of the i-th particle. It is evident from eqn. (1) that with ${\bf r} \times {\bf p}=0$ and $\mu=\frac{1}{2}$ we have $J_i (J_j) = \frac{1}{2}$ which gives us $m=1$, the complete filling of the lowest Landau level. From the Dirac quantization condition $e
\mu =\frac{1}{2}$, we note that this state corresponds to $e=1$ describing the IQH state with $\nu=1$.
The next higher angular momentum state can be achieved either by taking ${\bf r} \times {\bf p}=1$ and $\mid\mu \mid=\frac{1}{2}$ (which implies the higher Landau level) or by taking ${\bf r}
\times {\bf p}=0$ and $\mid{\mu_{eff}} \mid=\frac{3}{2}$ implying the ground state for the Landau level. However, with $\mid{\mu_{eff}} \mid=\frac{3}{2}$, we find the filling fraction $\nu=\frac{1}{3}$ which follows from the condition $e \mu
=\frac{1}{2}$ for $\mu=\frac{3}{2}$. Generalizing this we can have $\nu=\frac{1}{5}$ with $\mid{\mu_{eff}} \mid=\frac{5}{2}$.
It is noted that when $\mu$ is an integer, we can have a relation of the form $${\bf J} ~=~ {\bf r} \times {\bf p} - \mu {\bf \hat{r}} ~=~ {\bf
r}^{~\prime} \times {\bf p}^{~\prime}$$ This indicates that the Berry phase which is associated with $\mu$ may be unitarily removed to the dynamical phase. Evidently, the average magnetic field may be considered to be vanishing in these states. The attachment of $2m$ vortices to an electron effectively leads to the removal of Berry phase to the dynamical phase. So, FQH states with $2
\mu_{eff} = 2m + 1$ ($m$ an integer) can be viewed as if one vortex line is attached to the [*electron*]{}. Now we note that for a higher Landau level we can consider the Dirac quantization condition $e \mu_{eff} = \frac{1}{2} n$, with $n$ being a vortex of strength $2 \ell + 1$. This can generate FQH states having the filling factor of the form $\frac{n}{2 \mu_{eff}}$ where both $n$ and $2 \mu_{eff}$ are odd integers. In this case, we can write $2
\mu_{eff}=2m^\prime \pm 1$ with $2m^\prime=2mn$. Indeed, we can write the filling factor as
$$\nu ~=~\frac{n}{2 \mu_{eff}} ~=~ \frac{1}{\frac{2 \mu_{eff}~ \mp
1}{n} \pm \frac{1}{n}} ~=~ \frac{n}{2mn \pm 1}
\label{e1}$$
where $2 \mu_{eff} \mp 1$ is an even integer given by $2m^{~\prime} = 2mn$. This is essentially the Jain classification scheme with the constraint of $n$ being an odd integer.
In this scheme, the FQH states with $\nu$ having the form $$\nu=\frac{n^{~\prime}}{2mn^{~\prime} \pm 1}
\label{e2}$$
with $n^\prime$ an even integer can be generated through particle-hole conjugate states $$\nu ~=~ 1 - \frac{n}{2mn \pm 1} ~=~ \frac{n (2m - 1) \pm 1}{2mn
\pm 1}=\frac{n^{~\prime}}{2mn^{~\prime} \pm 1} \label{e3}$$ where $n(n^\prime)$ is an odd(even) integer. This leads some observed FQH states with
$$\nu ~=~ \frac{p}{q}(p~{\rm even}, q~{\rm odd})~~:~~\frac{2}{3} ,
~\frac{2}{5} , ~\frac{4}{5} , ~\frac{4}{7} , ~\frac{4}{9} ,
~\frac{6}{11} , ~\frac{6}{13} , \cdots \label{e4}$$
Besides. for $\nu = \frac{p}{q}$ with $p > q$ we can think of these states as condensates of one IQH state with $\nu$ an integer and one FQH state with $\nu ~=~ \frac{p}{q}$ ($p < q$). In this category, we have even numerator states such as $\nu=~ \frac{4}{3}
, ~\frac{8}{5}$ and $~\frac{10}{7}$ [@2; @3].
Now to study the polarization states of various FQH systems, we observe that in the lowest Landau level, we have the filling factor $\nu$ given by $\nu= \frac{1}{2 \mu_{eff}} = \frac{1}{2m +
1}$. As we have pointed out that the attachment of $2m$ vortices to an electron leads to the removal of Berry phase to the dynamical phase and this effectively corresponds to the attachment of one vortex(magnetic flux) to an electron, this electron will be a polarized one.
This is also true for $\nu=1$ state, as in this case we have $\mu=\frac{1}{2}$ implying one vortex line (magnetic flux)is attached to an electron.
However, in the higher Landau level, this scenario will change. Indeed, in this case we have $\nu=\frac{n}{2m n + 1}$ with $n>1$ and an odd integer.
Now from the relation (\[e1\]), for $n>1$ and an odd integer $$\nu ~=~\frac{n}{2 \mu_{eff}} ~=~ \frac{1}{\frac{2 \mu_{eff}~ \mp
1}{n} \pm \frac{1}{n}}$$
where $2 \mu_{eff} \mp 1$ is an even integer. We note that as even number of flux units can be accommodated in the dynamical phase we may consider this as $\frac{1}{n}$ flux unit is attached to an electron. This suggests that electrons will not be fully polarized as one full flux unit is not available to it. This will correspond to partially polarized states with $\nu=p/q$ ($p>1$ and odd, $q$ odd) which represents the states like $\frac{3}{5},~\frac{3}{7},~\frac{5}{9}$ and so on. It is observed here that IQH states like $\nu=3,5$ will also exhibit partially polarized states as from the Dirac quantization condition $e \mu
=\frac{n}{2}$, for $n>1$ and odd, the filling factor $\nu=n$ is achieved with $\mu=1/2$. This suggest that one flux unit is shared by an electron in $n$ number of Landau levels so that each electron is attached with $\frac{1}{n}$ flux unit implying a partially polarized state.
Now we consider the states with the filling factor $\nu=p/q$ with $p$ even and $q$ odd. As we have pointed out earlier, this is achieved when we have particle-hole conjugate states given by $\nu=1-\frac{n}{2m n \pm 1}$ with $n$ an odd integer. A hole configuration is described by the complex conjugate of the particle state, the spin orientation of the particle and hole state will be opposite to each other. Thus this will represent an unpolarized state. This gives us the general relation of unpolarized FQH states with filling factor $\nu=p/q$, $p$ even and $q$ odd.
Skyrmion Excitations in Arbitrary Polarized Quantum Hall States
===============================================================
It has been found that in quantum Hall systems deviations from the incompressible filling factor $\nu$ is accomplished by the degradation of the system’s spin polarization. This effect has been observed near $\nu = 1$ in several experiments that directly probe the spin density of the electron gas [@15e; @16e; @17e; @18e] \]. By noticing that the dynamics of quantum Hall system with a spin polarized ground state will follow that of a quantum ferromagnet and that the skyrmion is a charged object of the system, Sondhi et.al.[@19; @20] proposed a phenomenological action which is valid for the long wave length and small frequency limit. In this scheme, the competition between the Zeeman and Coulomb terms sets the size and energy of the skyrmions. Recently, Basu, Dhar and Bandyopadhyay [@21; @22] have proposed a pure sigma model formalism for skyrmions taking resort to spherical geometry where $2D$ electron gas resides on the surface of a $3D$ sphere with a magnetic monopole placed at the centre. To study the system in [*pure sigma model*]{} formalism an $O(4)$ nonlinear sigma model in $3+1$ dimensional manifold was introduced. In this framework, the quartic stability term introduced by Skyrme, known as the Skyrme term, determines the size of the quantum Hall skyrmions and the effect of the Zeeman energy and Coulomb energy is encoded in a certain parameter such that the size and energy are determined from the sigma model Lagrangian with the Skyrme term. Besides, the introduction of the topological $\theta$-term in the Lagrangian helps us to determine the spin and statistics of the skyrmion. The four order parameter fields help us to construct two independent $SU(2)$ algebras which are associated with the two mutually opposite orientations of the magnetization vector which resides on the $2D$ surface of the sphere.
The generalized Lagrangian is taken to be of the form [@21]
$$\begin{array}{lcl}
L~
&=&\displaystyle{~2\pi{{\cal{J}}}^S_\mu~{\cal{A}}_\mu-~\frac{M^2}{16}~Tr(\partial_{\mu}
U^{\dag}\partial_\nu U)-\frac{1}{32 \eta^2}~Tr[\partial_\mu
UU^{\dag},\partial_\nu UU^{\dag}]^2}\\
&&\displaystyle{-\frac{\theta}{16\pi^2}~
{^{*}{\mathcal{F}}_{\mu\nu}}{\mathcal{F}}_{\mu\nu}-\frac{1}{4}{\mathcal{F}}_{\mu\nu}{\mathcal{F}}^{\mu\nu}}\\
\end{array}$$
Here the topological current is defined as
$${{\mathcal{J}}}^S_{\mu}~=~\frac{1}{24 \pi^2}~
\epsilon_{\mu\nu\alpha\beta}
Tr(U^{-1}\partial_{\nu}U)(U^{-1}\partial_{\alpha}U)
(U^{-1}\partial_{\beta}U)$$
where ${\cal{A}}_{\mu}$ is a four-vector gauge field, $\theta=g/c^2$ with $g=\nu e^2/h$, the Hall conductivity and $~^{*}{\mathcal{F}}_{\mu \nu}$ is a Hodge dual given by $$^{*}{\mathcal{F}}_{\mu\nu}
=\frac{1}{2}~\epsilon_{\mu\nu\lambda\sigma}{\mathcal{F}}_{\lambda\sigma}$$ $M$ is a constant of dimension of mass and $\eta$ is a dimensionless coupling parameter.
The $SU(2)$ matrix $U$ is here defined as $$U=n_0 I + \bf{n}.\overrightarrow{\tau}$$ where the chiral fields $n_0$,$n_1$,$n_2$ and $n_3$ satisfy the relation $\sum n_i^2=1$. It may be noted that in $2+1$ dimension, we have the normalized $3$-vector field ${\bf n}$ with $\sum
n^2_i=1$ where $n_i$ corresponds to the local spin direction. In the $O(4)$ model $n_i (i=1,2,3)$ corresponds to this spin direction which live on the $2$-dimensional surface of the sphere where the extra field $n_0$ helps us to consider three [*boost*]{} generators in $(n_0, n_i)$ planes. In view of this, we can consider two types of generators such that the generator $M_k$ rotates the $3$-vector $\bf{n}(x)$ to any chosen axis and the boost generators $N_k$ would mix $n_0$ with the components of $\bf{n}$. We can now construct the following algebra
$$[M_i,M_j]= {i \epsilon_{ijk} M_k}$$ $$[M_i,N_j]= {i \epsilon_{ijk} N_k}$$ $$[N_i,N_j]= {i \epsilon_{ijk} M_k}$$ which is locally isomorphic to the Lie algebra of the $O(4)$ group. This helps us to introduce the left and right generators $$L_i=\frac{1}{2}(M_i-N_i)$$ $$R_i=\frac{1}{2}(M_i+N_i)$$ which satisfy $$[L_i,L_j]= {i \epsilon_{ijk} L_k}$$ $$[R_i,R_j]= {i \epsilon_{ijk} R_k}$$ $$[L_i,R_j]=0$$
Thus the algebra has split into two independent subalgebras each isomorphic to a $SU(2)$ algebra and corresponds to the chiral group $SU(2)_L \otimes SU(2)_R$. The left and right chiral group can now be taken to be associated with two mutually opposite orientations of the magnetization vector which resides on the $2$D surface of the sphere.
The topological charge of quantum Hall skyrmions is given by the winding number $$Z~=~ {\frac{1 }{24
\pi^2}}~\int_{S^3} dS_{\mu}~ \epsilon^{\mu\nu\lambda\sigma}
[(U^{-1}
\partial_{\nu} U)(U^{-1} \partial_{\lambda} U)(U^{-1} \partial_{\sigma} U)]$$ which is related to the homotopy $\pi_3(S^3)=Z$ and the electric charge is given by $\nu e Z$. We observe here that the Pontryagin index given by $$q=2\mu=~-{\frac{1}{16 \pi^2}}~\int Tr ^{*} {\mathcal{F}}_{\mu\nu}
{\mathcal{F}}_{\mu\nu} d^4 x$$
introduces the Berry phase for quantum Hall states as $\mu$ represents magnetic monopole strength. When there is a monopole of strength $\mu$ at the centre, the flux through the sphere is $2
\mu$ and the phase is given by $e^{i\phi_B}$ where $\phi_B=2 \pi
\nu$ (number of flux quanta enclosed by the loop). For $Z=1$, the skyrmion when moving around a closed loop acquires the Berry phase $2\pi\nu N$ where $N$ is the number of skyrmions enclosed by the loop. To find the spin and statistics, we consider a process which exchanges two skyrmions in the rest frame of the other skyrmion. This exchange effectively corresponds to the other skyrmion moving around the first in a half circle and hence it picks up a phase $\pi~\nu$ which is the statistical phase of a skyrmion. For $\nu=1$ it is a fermion and $\nu=1/m$, $m$ being an odd integer it corresponds to an anyon in planar geometry. In general, the spin of the skyrmion having charge $\nu
e Z$ is given by $\nu Z/2$.
Now to study the relevant quasiparticles for arbitrarily polarized states we observe the following possibilities.
1)Fully Polarized States :
In this case the spin orientation is fixed : up($\uparrow$) or down($\downarrow$). So for the chiral group $SU(2)_L \times
SU(2)_R$ only one $SU(2)_L (SU(2)_R)$ group will be operative. From the homotopic relation $\pi_3(SU(2))=Z$, we will have solitons (skyrmions) with charge $\nu e Z$.
2)Partially Polarized States :
In this case it will not be possible to split the $O(4)$ group into two independent left and right chiral groups $SU(2)_L$ or $SU(2)_R$ as each of these represent sharp polarization states. Thus we cannot represent the relevant quasiparticles as skyrmions.
3)Unpolarized States :
In this case, we will have equal number of up and down spin electrons. Now from the homotopic relation $\pi_3(SU(2) \times
SU(2))=\pi_3(SO(4))=0$, we note that there will be no skyrmions.
Thus we find that skyrmions are the relevant quasiparticles only at the filling factor given by $\nu=1$ and $\nu=\frac{1}{2m+1}$ with $m=$ an integer.
Discussion
==========
We have studied here the polarization states of quantum hall fluid at different filling factors from the hierarchical scheme in the framework of Berry phase. It is found that for IQH and FQH states with $\nu=1$ and $\nu=\frac{1}{2m+1}$, $m$ being an integer represent fully polarized states. The states with $\nu=p$ as well as $\nu=p/q,~ p$ odd, $q$ odd with $p>1$ represent partially polarized states and with $\nu=p/q, ~p$ even, $q$ odd with $p>1$ represent unpolarized states. It may be noted that the same result for unpolarized states has also been found from numerical computations [@6] and an exact diagonalization study has revealed that at $\nu=3/5$, the system is partially polarized. These are also in agreement with experiments. Mandal and Ravishankar [@8] have studied the polarization states in terms of an Abelian doublet of Chern-Simons gauge fields where the strength of the Chern-Simons term is given by a coupling matrix in the framework of composite fermion model. Their findings are also consistent with these results.
Wu and Sondhi [@10] have calculated the energies of quasiparticles for odd integer filling factors $\nu=2k+1,~k\geq 1
$ and have observed that skyrmions are not the low energy charged excitations even at small Zeeman energies. Mandal and Ravishankar [@23] have studied the quasiparticles in partially polarized states and have observed that skyrmions in this case are not the usual one whereas for unpolarized states skyrmions do not exist at all. Here from the homotopic analysis we have found that skyrmions are the relevant quasiparticles only for fully polarized states whereas for partially polarized and unpolarized states skyrmionic excitations do not exist at all.
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|
---
abstract: 'Recent discoveries on topological characterization of gapless systems have attracted interest in both theoretical studies and experimental realizations. Examples of such gapless topological phases are Weyl semimetals, which exhibit 3D Dirac cones (Weyl points), and nodal line semimetals, which are characterized by line nodes (two bands touching along a line). Inspired by our previous discoveries that the kicked Harper model exhibits many fascinating features of Floquet topological phases, in this manuscript we consider a generalization of the model, where two additional periodic system parameters are introduced into the Hamiltonian to serve as artificial dimensions, so as to simulate a $3$D system. We observe that by increasing the hopping strength and the kicking strength of the system, many new band touching points at Floquet quasienergies $0$ and $\pi$ will start to appear. Some of them are Weyl points, while the others form line nodes in the parameter space. By taking open boundary conditions along the physical dimension, edge states analogues to Fermi arcs in static Weyl semimetal systems are observed. Finally, by designing an adiabatic pumping scheme, the chirality of the Weyl points and the $\pi$ Berry phase around line nodes can be manifested.'
author:
- Raditya Weda Bomantara
- Gudapati Naresh Raghava
- Longwen Zhou
- Jiangbin Gong
title: Floquet topological semimetal phases of an extended kicked Harper model
---
Introduction
============
One paradigm in studies of quantum chaos is the kicked Harper model (KHM). It was originally introduced in Ref. [@KHM] to model a kicked charge in a magnetic field. Later a whole class of generalized versions of KHM were introduced, which may also be realized via a periodically kicked charge in a magnetic field [@Dana]. Remarkably, more than two decades ago, the KHM was already used to study topological phase transitions in Floquet states and to understand possible connections between large Floquet band Chern numbers, delocalization of Floquet eigenstates, and the underlying classical chaos [@Chaos; @Chaos2]. Refs. [@Chaos; @Chaos2] were perhaps the first two studies illuminating on topological phases of matter in periodically driven systems, a topic of wide interest at present due to the finding of Floquet topological insulators [@FTI]. Studies of topological phase transitions in the KHM opened up stimulating questions of interest to different communities, connecting concepts from different areas such as condensed matter physics and nonlinear dynamics.
The relevance of the KHM to studies of topological phases becomes even more evident in recent years. In particular, it was shown that the KHM and a variant of another paradigm of quantum chaos, namely, the double-kicked rotor model (DKR), have a certain kind of topological equivalence though their spectrum has many distinctively different features [@Hailong]. Furthermore, it was shown that the KHM has drastically different edge states as compared with that associated with the DKR, even though all their bulk Floquet bands have exactly the same Chern numbers [@Derek]. This result is useful for ongoing investigations of the issue of bulk-edge correspondence in Floquet topological states as well as the issue of weak topological insulators.
In previous studies [@Derek], we already showed that adding one periodic system parameter to the conventional KHM Hamiltonian can serve the purpose of introducing an artificial dimension, thus enabling us to study the properties of two-dimensional ($2$D) systems from the perspective of one-dimensional ($1$D) systems. It was found that the quasienergy spectrum of this effective $2$D KHM is able to host as many $2$D Dirac cones as possible at both quasienergy $0$ and quasienergy $\pi$ by simply increasing the kicking strength. This suggests that the KHM might be a good platform to study gapless topological phases, such as Weyl semimetal [@Burkov; @Ferarc; @Hosur2; @Ferarc2] and nodal line semimetal [@Burkov; @Aji; @Mullen; @LN2; @Youngkuk] in Floquet systems. Motivated by these results regarding the KHM, in this paper we propose to study an extended KHM by introducing two periodic system parameters into the Hamiltonian. With two periodic system parameters available, together with the conserved quasimomentum variable due to the translational invariance in the sole physical (lattice) dimension, it is possible to obtain Floquet bands that can be effectively associated with three-dimensional ($3$D) systems. This thus creates an interesting opportunity to explore the KHM from a perspective related to $3$D gapless topological phases of matter, which has recently become an interesting topic [@Parsa; @Shunji].
In order to make both periodic system parameters nontrivial, we propose to introduce modulations to both the lattice hopping term and the kicking potential in the KHM. These modulations are characterized by two phase shift parameters playing the role of two periodic system parameters. The model Hamiltonian can thus also be deemed as a kicked version of a recently proposed off-diagonal Harper model [@Sarma1]. Having two periodic system parameters leads to one direct advantage in understanding the implications of various topological phases for the dynamics. Specifically, it is now possible to execute adiabatic loops in the $2$D parameter space spanned by the two phase shift parameters, by directly and slowly tuning the two system parameters. Therefore, even for a single quasimomentum component of the system, it is possible to realize closed adiabatic loops to manifest the topological phases.
The purpose of this paper is to present a variety of interesting and motivating results for the simplest two-band case of the above-mentioned extended KHM. First of all, we observe the emergence of isolated $3$D Dirac cones (Weyl points) as well as line nodes (two bands touching along a line) in the Floquet bands, at quasienergies $0$ and $\pi$. This strongly suggests that even $1$D driven systems can be useful to study $3$D Floquet topological phases. In particular, the emergence of Weyl points and line nodes at quasienergy $\pi$ is a finding that is absent in static systems. Secondly, by taking advantage of the simplicity of the two-band case, we also analytically show how it is possible to obtain as many Weyl points and line nodes as possible in our extended KHM. Thirdly, when we take open boundary conditions (OBC) along the only physical dimension, we observe a line connecting two Weyl points with opposite chiralities, which is analogues to the so-called Fermi arc [@Ferarc] in static Weyl semimetal systems, at both quasienergies $0$ and $\pi$. Fourthly, we also perform dynamics studies and demonstrate how the chirality of the (Floquet) Weyl points and the $\pi$ Berry phase around the (Floquet) line nodes can be manifested in the adiabatic dynamics of the system. By considering an adiabatic loop with an appropriate initial state in the $2$D parameter space enclosing a Weyl point, its chirality can be captured. Furthermore, by adiabatically driving a Floquet eigenstate at a fixed quasimomentum over one adiabatic cycle, the $\pi$ Berry phase can be observed. These results should be of experimental interest, especially for waveguide and optical lattice realizations of $1$D driven systems.
This paper is organized as follows. In Sec. \[II\], we start by describing the Hamiltonian studied in this paper, which is referred to as the off-diagonal modulated Kicked Harper model (ODKHM), and a possible experimental realization. By focusing on the situation where the Floquet operator associated with this Hamiltonian consists of only two bands, we analytically show the emergence of band touching points in a form of isolated Weyl points or line nodes as the hopping strength and kicking strength are varied. In Sec. \[III\], we present numerically the emergence of localized edge states associated with the Weyl points and line nodes by taking OBC along the physical or artificial dimension of the system. In Sec. \[IV\] A, we design an adiabatic pumping scheme surrounding a Weyl point and show how its chirality can be captured by choosing an appropriate initial state. We further show that by using the same adiabatic pumping scheme and initial state, the chirality of Weyl points at quasienergy $0$ and quasienergy $\pi$ are captured in different manners, which may suggest the distinction between the two types of Weyl point. In Sec. \[IV\] B, we also show a possible way to capture the $\pi$ Berry phase of the line node by adiabatically pumping a Floquet eigenstate at a fixed value of its quasimomentum. Finally, we summarize our results and present possible future directions in Sec. \[V\].
Emergence of Weyl points and line nodes in the ODKHM {#II}
====================================================
Description of the model
------------------------
We start by considering the following Hamiltonian,
$$\begin{split}
&H=\sum_{n=1}^{N-1}\left\lbrace\left[J+\lambda \cos(2\pi \beta_1 n+\phi_y)\right]\, |n+1\rangle \langle n |+h.c.\right\rbrace \\
&+\sum_{n=1}^{N-1}\sum_{j} V \cos(2\pi \beta_2 n+\phi_z) \, |n \rangle \langle n | \, \delta(t-jT)\;,
\end{split}
\label{fund}$$
where $J$ and $\lambda$ are the parameters controlling the hopping strength, $V$ is the kicking strength, $T$ is the kicking period, $n$ is the lattice site index, and $N$ counts the total number of lattice sites. If $\lambda=0$, the above Hamiltonian reduces to the conventional KHM. When $\lambda\neq 0$, however, Eq. (\[fund\]) depends on two periodic parameters $\phi_y$ and $\phi_z$, which can take any value in $[-\pi,\pi)$. Physically, $\phi_y$ and $\phi_z$ represent the phase shift in the hopping and kicking strength modulations, which can be independently tuned. In particular, because of the extra parameter $\phi_y$ which modulates the hopping strength, Eq. (\[fund\]) shall be referred to as the off-diagonal Kicked Harper model (ODKHM). Due to the range of values they can take, the two parameters $\phi_y$ and $\phi_z$ can play the role of quasimomenta along perpendicular directions to simulate a $3$D system. By taking $\beta_1=\beta_2=\frac{1}{2}$, Eq. (\[fund\]) becomes,
$$\begin{split}
&H=\sum_{n=1}^{N-1}\left\lbrace\left[J+(-1)^n\lambda \cos(\phi_y)\right]\, |n+1\rangle \langle n |+h.c.\right\rbrace \\
&+\sum_{n=1}^{N-1}\sum_{j} (-1)^nV \cos(\phi_z) \, |n \rangle \langle n | \, \delta(t-jT)\;.
\end{split}
\label{fund2}$$
Since Eq. (\[fund2\]) is time periodic, we can define a Floquet operator as a one period time-evolution operator from $t=0^{-}$ to $t=T^{-}$, which is given by,
$$U=e^{-\mathrm{i} \sum_{n=1}^{N-1}\left\lbrace\left[J+(-1)^n \lambda \cos(\phi_y)\right] \, |n+1\rangle \langle n |+h.c.\right\rbrace} e^{-\mathrm{i} \sum_{n=1}^{N-1} (-1)^n V \cos(\phi_z)\, |n \rangle \langle n |}\;,
\label{tri}$$
where we have assumed $T=\hbar=1$ for simplicity, and hence $J$, $\lambda$, and $V$ are dimensionless quantities. Under periodic boundary conditions (PBC), Eq. (\[tri\]) will be invariant under translation by two lattice sites. As a consequence, we can perform a Fourier transform to express Eq. (\[tri\]) in terms of quasimomentum $k$. The reduced Floquet operator can then be written as (see Appendix \[derive\] for details),
$$\mathcal{U}(k,\phi_y,\phi_z)= e^{-\mathrm{i} \left[2J\cos(k)\sigma_x +2\lambda \sin(k)\cos(\phi_y)\sigma_y\right]} e^{-\mathrm{i}V \cos(\phi_z) \sigma_z}\;,
\label{mod}$$
where it satisfies $U=\sum_k \mathcal{U}(k,\phi_y,\phi_z) \otimes |k\rangle \langle k |$.
In order to realize Eq. (\[tri\]) experimentally, we could consider a two-stage process defined as follows in the framework of $1$D lattice. When $2m\leq t < 2m+1$, with $m\in \mathbb{Z}$, we turn off any hopping amplitude between two lattice sites while turning on the on-site potential of the lattice sites given by $V_n=(-1)^n V \cos(\phi_z)$. When $2m+1\leq t<2m+2$, we turn off all the on-site potentials, while turning on the hopping amplitude between two lattice sites given by $t_n=J+(-1)^n \lambda \cos(\phi_y)$. Experimentally, this non-uniform hopping amplitude can be realized by using waveguide arrays in the framework of photonic quasicrystals setup [@exp1].
Analysis of the band touching points
------------------------------------
By diagonalizing Eq. (\[mod\]) at given values of $k$, $\phi_y$, and $\phi_z$ and scan these parameters over the whole “Brillouin zone", i.e., $(k,\phi_y,\phi_z)\in [-\pi,\pi)\times [-\pi,\pi)\times [-\pi,\pi)$, we will be able to get an effective quasienergy spectrum of Eq. (\[mod\]), where quasienergy $\varepsilon$ is defined as the phase of the eigenvalue of Eq. (\[mod\]), satisfying $\mathcal{U}(k,\phi_y,\phi_z)|\psi\rangle=e^{-\mathrm{i}\frac{\varepsilon(k,\phi_y,\phi_z) T}{\hbar}}|\psi\rangle$ with $|\psi\rangle$ be an eigenvector of $\mathcal{U}$. Fig. (\[spect1\]) shows the quasienergy spectrum at $J=1$, $\lambda=0.5$, and $V=2$. It suggests that the two bands touch at $(k,\phi_y,\phi_z)=\left(\pm \frac{\pi}{2},\pm \frac{\pi}{2}, \pm \frac{\pi}{2}\right)$. Furthermore, near each of these touching points, the quasienergy dispersion is linear with respect to any two parameters when the other parameter is fixed. In fact, a detailed calculation in the Appendix \[exp\] shows that by expanding Eq. (\[mod\]) around each of these touching points, the effective Hamiltonian is found to be in the form of a Weyl Hamiltonian with chirality $\chi=\pm\mathrm{sgn}(J\lambda V)$ [@Hosur].
![(color online). A typical quasienergy spectrum of the ODKHM under PBC at a fixed (a) $\phi_y=\frac{\pi}{2}$, (b) $\phi_z=\frac{\pi}{2}$, and (c) $k=\frac{\pi}{2}$, under $J=1$, $\lambda=0.5$, $V=2$. Here and in all other figures, plotted quantities are in dimensionless units.[]{data-label="spect1"}](spect1.pdf)
By increasing the value of parameters $J$, $\lambda$, and $V$, we found that more band touching points start to appear at both quasienergy $0$ and quasienergy $\pm \pi$, as illustrated in Fig. (\[spect2\]), an observation which is absent in static systems such as the non-kicked version of the model [@Sarma1; @Sarma2]. In order to characterize all the band touching points, we shall first introduce the following Lemma, whose proof is provided in Appendix \[proof\].
![(color online). A typical quasienergy spectrum of the ODKHM under PBC at a fixed $\phi_y=\frac{\pi}{2}$ (a), $\phi_z=\frac{\pi}{2}$ (b), and $k=\frac{\pi}{2}$ (c), under $J=3$, $\lambda=3$, $V=5$.[]{data-label="spect2"}](spect2.pdf)
Let $\vec{n}$ and $\vec{m}$ be two perpendicular unit vectors, and $a$ and $b$ be functions of three periodic variables $(k,\phi_y,\phi_z)$. The band touching points of the quasienergy spectrum of $e^{\mathrm{i} a \vec{n}\cdot \sigma} e^{\mathrm{i} b \vec{m}\cdot \sigma}$ can only occur if $a$ and $b$ are integer multiples of $\pi$. \[lem\]
In order to use Lemma \[lem\], we need to first turn Eq. (\[mod\]) into the following form,
$$\mathcal{U}(k,\phi_y,\phi_z)= e^{-\mathrm{i} f \hat{r}\cdot \sigma} e^{-\mathrm{i}V \cos(\phi_z) \sigma_z}\;,
\label{mod2}$$
where $f=\sqrt{4J^2\cos^2(k)+4\lambda^2 \cos^2(\phi_y)\sin^2(k)}$, $\hat{r}=\frac{2J}{f} \cos(k)\hat{x}+\frac{2\lambda}{f} \sin(k)\cos(\phi_y)\hat{y}$, and $\hat{x}$, $\hat{y}$, $\hat{z}$ are unit vectors along $x$, $y$, and $z$ directions, respectively. Since $\hat{r} \perp \hat{z}$, Lemma \[lem\] is applicable, and it suggests that the band touching points appear when $f=l_1 \pi$ and $V\cos(\phi_z)=l_2\pi$, where $l_1, l_2 \in \mathbb{Z}$. All possible band touching points that appear in this model can be classified into three types.
### **Type 1**: $l_1=0$ and $l_2\in \mathbb{N}$. {#type-1-l_10-and-l_2in-mathbbn. .unnumbered}
Since the quasienergy is only defined modulo $2\pi$, Eq. (\[mod2\]) implies that if $l_2$ is even, the two bands touch at quasienergy $0$, while if $l_2$ is odd, they touch at quasienergy $\pi$. Furthermore, $l_1=0$ implies that $f=0$, and therefore $k=\pm\frac{\pi}{2}$ and $\phi_y=\pm\frac{\pi}{2}$. In order for the solutions to exist, $|V|\geq l_2\pi$, so that $\phi_z=\arccos\left(\frac{l_2\pi}{V}\right)$. It is therefore clear from this analysis that new band touching points will start to appear as $V$ is increased by $\pi$. Therefore, by controlling the value of $V$, we will be able to generate as many band touching points of this type as possible.
By expanding Eq. (\[mod\]) around one of these band touching points up to first order in $k$, $\phi_y$, and $\phi_z$, we obtain the following,
$$\mathcal{U}(k_x,k_y,k_z)=e^{-\mathrm{i}\left\lbrace l_2\pi- [2J k_x \sigma_x +2\lambda k_y\sigma_y+V\sin(\phi_1)k_z\sigma_z]\right\rbrace}\;,
\label{ham1}$$
where we have defined $k_x=k-\frac{\pi}{2}$, $k_y=\phi_y-\frac{\pi}{2}$, $k_z=\phi_z-\phi_1$, with $\phi_1=\arccos\left(\frac{l_2\pi}{V}\right)$. The details of this calculation is presented in Appendix \[exp\]. According to Eq. (\[ham1\]), the effective Hamiltonian $h_\mathrm{eff}$ associated with the Floquet operator Eq. (\[mod\]), which is defined according to $\mathcal{U}=e^{-\mathrm{i}h_\mathrm{eff}}$, is in the form of a Weyl Hamiltonian with chirality $\chi=-\mathrm{sgn}\left[J\lambda V\sin(\phi_1)\right]$. The analysis therefore shows that this type of band touching points corresponds to a Weyl point. In Sec. \[III\] A, we will show that under OBC along the physical dimension, a line connecting two Weyl points associated with this type of band touching points (Fermi arc) will appear.
### **Type 2**: $l_1\neq 0$, $l_2\in \mathbb{N}$, with $J\geq \frac{l_1 \pi}{2}\geq\lambda$. {#type-2-l_1neq-0-l_2in-mathbbn-with-jgeq-fracl_1-pi2geqlambda. .unnumbered}
By referring to the proof of Lemma \[lem\] presented in Appendix \[proof\], it follows that the two bands touch at quasienergy $\pi$ when either $l_1$ or $l_2$ is odd and the other is even, and they touch at quasienergy $0$ when both $l_1$ and $l_2$ are even or odd. Furthermore, similar to type $1$, in order for the solutions to exist, $|V|\geq l_2\pi$, so that $\phi_z=\arccos\left(\frac{l_2\pi}{V}\right)$. However, since $l_1\neq 0$, $k$ and $\phi_y$ can in general take a continuous range of values satisfying $f=l_1\pi$. By treating $\phi_y$ as an independent variable, the two bands touch if $k$ satisfies the following condition,
$$k=\pm \arccos\left[\sqrt{\frac{l_1^2 \pi^2- 4\lambda^2 \cos(\phi_y)^2}{4J^2-4\lambda^2\cos(\phi_y)^2}}\right]\;.
\label{line}$$
Since $J\geq \frac{l_1\pi}{2}\geq \lambda $, a solution to $k$ exists for any $\phi_y\in [-\pi,\pi)$. As a result, the two bands will touch along a curve as a function of $\phi_y$ and $k$. Fig. \[type23\](b) shows a typical example of the line nodes of this type at $J=3$, $\lambda=1$, and $V=2$, i.e., $l_1=1$ and $l_2=0$. By expanding Eq. (\[mod\]) around $(k,\phi_y,\phi_z)=\left(k_0',\frac{\pi}{2},\frac{\pi}{2}\right)$, where $k_0'=\arccos\left(\frac{\pi}{2J}\right)$, which corresponds to a point where the two bands touch, up to first order in $k$, $\phi_y$, and $\phi_z$, the effective Hamiltonian is found to be (See Appendix \[exp\] for the calculation details),
$$\begin{split}
&h_\mathrm{eff}(k_x,k_y,k_z)=\left[\pi -\Phi(k_x,k_z)\right]\hat{n}\cdot \sigma +\frac{2J k_x \sin(k_0')}{\Phi(k_x,k_z)}\left[-2 \lambda k_y\sin(k_0')+\pi V k_z \right] \sigma_y \;,
\end{split}
\label{lineff}$$
where $k_x=k-k_0'$, $k_y=\phi_y-\frac{\pi}{2}$, $k_z=\phi_z-\frac{\pi}{2}$, $\Phi(k_x,k_z)=\sqrt{V^2 k_z^2+4J^2 k_x^2 \sin^2(k_0')}$, and $\hat{n}=-\frac{Vk_z}{\Phi(k_x,k_z)}\hat{z}+\frac{2J k_x \sin(k_0')}{\Phi(k_x,k_z)}\hat{x}$. The quasienergy associated with this effective Hamiltonian is $\varepsilon \approx \pm \left[\pi - \Phi(k_x,k_z)\right]$.
According to Ref. [@Burkov], line nodes can be topologically characterized by a topological winding number, which is constructed as follows. On $k_z = \frac{2\lambda \sin(k_0')}{\pi V}k_y$ plane, the effective Hamiltonian, as defined in type $1$, depends only on two Pauli matrices and can be written in the following form,
$$h_\mathrm{eff}(k_x,k_z)=h_1(k_x, k_z) \sigma_x+h_2(k_x,k_z) \sigma_z \;.$$
Define a complexified Hamiltonian $h=h_2+\mathrm{i}h_1=|h|e^{\mathrm{i} \phi}$. The topological winding number associated with this line node is defined as,
$$n=\frac{1}{2\pi}\oint d\mathbf{k}\cdot \nabla_\mathbf{k} \phi \;,$$
which can in general take values $0$ or $\pm 1$ (where the sign depends on the direction of the line integral), and is closely related to the Berry phase around a loop surrounding the line node [@Burkov]. By considering a loop parametrized by $\theta$ according to $k_x = r \sin(\theta)$ and $k_z=\frac{r\pi V}{\sqrt{\pi^2 V^2+4\lambda^2 \sin^2(k_0')}} \cos(\theta)$, it follows that $\tan(\phi) = -\frac{2J\sin(k_0')\sqrt{\pi^2 V^2+4\lambda^2 \sin^2(k_0')}}{\pi V^2} \tan(\theta)$. This suggests that as $\theta$ goes from $0$ to $2\pi$, $\phi$ goes from $0$ to $-2\pi$, so that $\phi(2\pi)=-2\pi$ and $\phi(0)=0$. Therefore,
$$n = \frac{1}{2\pi} \int_0^{2\pi} d\theta \frac{d \phi}{d\theta} = \frac{1}{2\pi} \left[\phi(2\pi)-\phi(0)\right]= - 1\;,$$
which shows that the line nodes of this type are topologically nontrivial.
### **Type 3:** $l_1\neq 0$, $l_2\in \mathbb{N}$, with $\lambda \geq \frac{l_1 \pi}{2}$. {#type-3-l_1neq-0-l_2in-mathbbn-with-lambda-geq-fracl_1-pi2. .unnumbered}
Similar to type $2$, the two bands touch at quasienergy $0$ when both $l_1$ and $l_2$ are even or odd, and at quasienergy $\pi$ when either $l_1$ or $l_2$ is even, while the other is odd. Also, similar to both type $1$ and $2$, in order for the solutions to exist, $|V|\geq l_2\pi$, so that $\phi_z=\arccos\left(\frac{l_2\pi}{V}\right)$. By treating $\phi_y$ as an independent variable, $k$ will need to satisfy Eq. (\[line\]) in order for the two bands to touch. However, since a cosine function can only take values between $-1$ and $1$, a solution to $k$ will only exist at some values of $\phi_y$ in $[-\pi,\pi)$. If $J\geq \frac{l_1 \pi}{2}$, a solution to $k$ exists as long as the numerator of Eq. (\[line\]) is $l_1^2 \pi^2- 4\lambda^2 \cos^2(\phi_y)\geq 0$, or $\phi_y \in [-\pi+\phi_0,-\phi_0]\cup [\phi_0, \pi-\phi_0]$, where $\phi_0=\arccos\left(\frac{l_1\pi}{2\lambda}\right)$. If $J\leq \frac{l_1 \pi}{2}$, a solution to $k$ exists as long as the numerator of Eq. (\[line\]) is $l_1^2\pi^2-4\lambda^2\cos^2(\phi_y)\leq 0$. In both cases, the two bands will touch along a closed curve within $k$ and $\phi_y$ Brillouin zone. As an illustration, Fig. \[type23\](a) shows a typical line node of this type at $J=3$, $\lambda=3$, and $V=2$, i.e., $l_1=1$ and $l_2=0$.
![ A typical shape of (a) type 3 and (b) type 2 line nodes. The plot is taken at $J=3$, $V=2$, (a) $\lambda=3$ and (b) $\lambda=1$.[]{data-label="type23"}](line1.pdf "fig:"){width="50.00000%"}![ A typical shape of (a) type 3 and (b) type 2 line nodes. The plot is taken at $J=3$, $V=2$, (a) $\lambda=3$ and (b) $\lambda=1$.[]{data-label="type23"}](line2.pdf "fig:"){width="50.00000%"}
By expanding Eq. (\[mod\]) around $(k,\phi_y,\phi_z)=\left(\frac{\pi}{2},\phi_0,\frac{\pi}{2}\right)$ up to first order in $k$, $\phi_y$, and $\phi_z$, i.e., a point where the two bands touch, the effective Hamiltonian is found to be,
$$h_\mathrm{eff} = \left[\pi -\Phi'(k_y,k_z)\right]\hat{m}\cdot \sigma -\frac{2\lambda k_y \sin(\phi_0)}{\Phi'(k_y,k_z)}\left( \pi V k_z+ 2 J k_x\right) \sigma_x \;,
\label{lineff2}$$
where $k_x=k-\frac{\pi}{2}$, $k_y=\phi_y-\phi_0$, $k_z=\phi_z-\frac{\pi}{2}$, $\Phi'(k_y,k_z)=\sqrt{V^2 k_z^2+4\lambda^2 k_y^2 \sin^2(\phi_0)}$, and $\hat{m}=\frac{2\lambda k_y \sin(\phi_0)}{\psi(k_y,k_z)}\hat{y}-\frac{Vk_z}{\psi(k_y,k_z)}\hat{z}$. The quasienergy associated with this effective Hamiltonian is $\varepsilon \approx \pm \left[\pi - \Phi'(k_y,k_z)\right]$.
We can also calculate the topological winding number in a similar fashion as in type $2$. On $k_z=-\frac{2J}{\pi V}k_x$ plane, the effective Hamiltonian depends only on two Pauli matrices which can be written as,
$$h_\mathrm{eff}(k_y,k_z)=h_1(k_y, k_z) \sigma_y+h_2(k_y,k_z) \sigma_z \;.$$
As before, we define a complexified Hamiltonian $h=h_1+\mathrm{i}h_2=|h|e^{\mathrm{i}\phi}$. By parametrizing $k_y=r\cos(\theta)$ and $k_z=-\frac{r\pi V}{\sqrt{\pi^2V^2+4J^2}}\sin(\theta)$, it follows that $\tan(\phi)=\frac{\pi V^2}{2\lambda \sin(\phi_0)\sqrt{\pi^2V^2+4J^2}}\tan(\theta)$. The topological winding number can then be shown to be $n= 1$, which shows that the line nodes of this type are also topologically nontrivial.
Edge states {#III}
===========
One characterized signature of a topological phase is the existence of localized edge states when the system takes OBC. For example, the band structure of a 2D quantum Hall insulator supports localized chiral edge states which traverse the gap between two bands [@QHI; @QHI2; @QHI3], while that of a spin Hall insulator supports localized helical edge states [@SHI; @SHI2]. Localized edge states can also appear in gapless systems, e.g. Fermi arc in Weyl semimetal [@Burkov; @Ferarc; @Hosur2; @Ferarc2] and surface states in nodal line semimetal [@Burkov; @Mullen; @LN2; @Youngkuk], which indicates that such systems can also be topologically characterized. We have shown in section \[II\] that the ODKHM hosts a number of Weyl points and topologically non-trivial line nodes at quasienergy $0$ and quasienergy $\pi$. In this section, we will investigate the emergence of localized edge states associated with these Weyl points and line nodes. The results are hoped to motivate more studies of bulk-edge correspondence in Floquet systems.
Edge states under OBC along the physical dimension
--------------------------------------------------
Fig. \[2\] shows the quasienergy spectrum of the Floquet operator Eq. (\[tri\]) under OBC with $N=100$, $\lambda=0.5$, $J=1$, and $V=5$. Under the given parameters, it confirms our analysis in section \[II\] that there exists some Weyl points at both quasienergy $0$ and quasienergy $\pi$, corresponding to $l_2=0$ and $l_2=1$. A slice of the full quasienergy spectrum at a fixed $\phi_z=\arccos\left(\frac{\pi}{V}\right)$ as shown in Fig. \[2\](b) shows the existence of a line connecting two Weyl points with opposite chiralities, which is analogous to the Fermi arc in static Weyl semimetal systems. The emergence of this Fermi arc shares a similar topological origin as the edge states of the Su-Schrieffer-Heeger (SSH) model [@SSH]. At a fixed $\phi_z=\arccos\left(\frac{\pi}{V}\right)$, the effective Hamiltonian takes the form,
$$h_\mathrm{eff}=\pi+2J\cos(k)\sigma_x+2\lambda \cos(\phi_y)\sin(k)\sigma_y \;,
\label{inv}$$
which can be mapped to a static SSH model in the form of $h_\mathrm{SSH}=\pi +\sum_n [(J+\delta t_n)|n\rangle \langle n+1| +h.c.]$, where $\delta t_n=(-1)^n\lambda \cos(\phi_y)$, by a suitable Fourier transform. SSH model can be characterized by a $\mathbb{Z}_2$ topological invariant. It is topologically nontrivial when $\delta t_1$ and $\delta t_{N-1}$ are less than $0$, while it is topologically trivial when $\delta t_1$ and $\delta t_{N-1}$ are greater than $0$. In this case, the signs of $\delta t_1$ and $\delta t_{N-1}$ are controlled by the parameter $\phi_y$. This explains why edge states exist only when $-\frac{\pi}{2}<\phi_y<\frac{\pi}{2}$, and thus appear to form a line connecting two Weyl points at $\phi_y=\pm \frac{\pi}{2}$.
![(color online). Quasienergy spectrum of Eq. (\[tri\]) under OBC at a fixed (a) $\phi_y=\frac{\pi}{4}$ and (b) $\phi_z=\arccos\left(\frac{\pi}{V}\right)$. Red and green marks denote edge states localized at the right and left edges, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N=100$, $\lambda=0.5$, $J=1$, and $V=5$.[]{data-label="2"}](weyledge.pdf "fig:"){width="50.00000%"}![(color online). Quasienergy spectrum of Eq. (\[tri\]) under OBC at a fixed (a) $\phi_y=\frac{\pi}{4}$ and (b) $\phi_z=\arccos\left(\frac{\pi}{V}\right)$. Red and green marks denote edge states localized at the right and left edges, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N=100$, $\lambda=0.5$, $J=1$, and $V=5$.[]{data-label="2"}](weylsst.pdf "fig:"){width="60.00000%"}
Fig. \[2\](a) shows a slice of the full quasienergy spectrum at a fixed $\phi_y=\frac{\pi}{4}$, where we observe counter-propagating chiral edge states joining the two bands. These counter-propagating edge states exist for any fixed value of $\phi_y$ in $(-\frac{\pi}{2},\frac{\pi}{2})$. When $|\phi_y|=\frac{\pi}{2}$, the two gaps at quasienergy $0$ and $\pi$ close. By increasing $|\phi_y|$ further, the gaps reopen and the edge states disappear. Furthermore, for any value of $\phi_y$ where the edge states appear, they cross at $\phi_z=\left\lbrace \pm\frac{\pi}{2}, \pm \arccos\left(\frac{\pi}{V}\right), \pm\left[\pi-\arccos\left(\frac{\pi}{V}\right)\right]\right\rbrace$, where the effective Hamiltonian has inversion symmetry and can be written in the form of Eq. (\[inv\]). These edge states crossings are what appears as a Fermi arc joining two Weyl points.
Fig. \[3\] shows another quasienergy spectrum of Eq. (\[tri\]) under OBC with $J=1$, $\lambda=2$, and $V=2$, where the line nodes are predicted to appear. Apart from the counter-propagating edge states and the Fermi arcs we mentioned earlier, there does not seem to exist any additional edge states arising as a consequence of the line nodes. Therefore, in order to be able to capture the topology of the line nodes, we take a look at the system under OBC along an artificial dimension, which will be presented in section \[III\] B.
![(color online). Quasienergy spectrum of Eq. (\[tri\]) under OBC at a fixed (a) $\phi_y=0$ and (b) $\phi_z=\frac{\pi}{2}$. Red and green marks denote edge states localized at the right and left edges, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N=100$, $\lambda=2$, $J=1$, and $V=2$.[]{data-label="3"}](lineedge.pdf "fig:"){width="50.00000%"}![(color online). Quasienergy spectrum of Eq. (\[tri\]) under OBC at a fixed (a) $\phi_y=0$ and (b) $\phi_z=\frac{\pi}{2}$. Red and green marks denote edge states localized at the right and left edges, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N=100$, $\lambda=2$, $J=1$, and $V=2$.[]{data-label="3"}](linesst.pdf "fig:"){width="50.00000%"}
Edge states under OBC along an artificial dimension
---------------------------------------------------
OBC along an artificial dimension may be experimentally realized by decoupling the atomic internal degrees of freedom of the lattice that generate the phase shift $\phi_y$ or $\phi_z$ [@Art1; @Art2]. Mathematically, this can be done by performing a Fourier transform to Eq. (\[mod\]) with respect to $\phi_y$ or $\phi_z$ to get an effective lattice model, then taking OBC along this dimension by considering a finite number of lattice sites. Here, the Fourier transformed lattice site corresponding to $\phi_y$ and $\phi_z$ can be imagined as $y$ and $z$ directions, respectively. We found that even under OBC along $y$ or $z$ direction, there will still be no additional edge states associated with the line nodes. However, by opening the system along a diagonal direction in $y$ and $z$, edge states associated with the line nodes become visible, as shown in Fig. \[4\](c).
To be more explicit, we define $\phi_y'=\frac{\phi_y+\phi_z}{2}$ and $\phi_z'=\frac{\phi_z-\phi_y}{2}$. By performing a Fourier transform with respect to $\phi_y'$, Eq. (\[mod\]) can be transformed into the following form,
$$U_{y'} = e^{-\mathrm{i} \sum_{n_y'=1}^{N_y'-1} \left[J\cos(k) \sigma_x \otimes |n_y' \rangle \langle n_y' | +\frac{\lambda}{2}\sin(k) e^{\mathrm{i}\phi_z'}\sigma_y\otimes |n_y'+1 \rangle \langle n_y' |+h.c. \right] } e^{-\mathrm{i} \sum_{n_y'=1}^{N_y'-1}\left[\frac{V}{2}e^{-\mathrm{i}\phi_z'}\sigma_z\otimes |n_y'+1 \rangle \langle n_y' | +h.c.\right]}\;,
\label{uy}$$
where $n_y'$ represents a lattice site along the diagonal direction. Fig. \[4\](c) is then obtained by plotting the quasienergy spectrum of Eq. (\[uy\]) with $N_y'=100$, $J=1$, $\lambda=3$, and $V=2$, i.e., when line nodes of type $3$ are expected to appear. At a fixed $k=\frac{\pi}{2}$, Fig. \[4\](a) shows the emergence of degenerate $\pi$-modes inside each line node. These degenerate $\pi$-modes are very similar to the Fermi arc, since they correspond to the crossing between two edge states localized at opposite edges, which is illustrated in Fig. \[4\](b).
![(color online). Quasienergy spectrum of Eq. (\[uy\]) under OBC at a fixed (a) $k=\frac{\pi}{2}$ and (b) $\phi_z'=\frac{\pi}{5}$. Red and green marks denote edge states localized at the right and left edge, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N_y'=100$, $\lambda=3$, $J=1$, and $V=2$.[]{data-label="4"}](ciredge3.pdf "fig:"){width="50.00000%"}![(color online). Quasienergy spectrum of Eq. (\[uy\]) under OBC at a fixed (a) $k=\frac{\pi}{2}$ and (b) $\phi_z'=\frac{\pi}{5}$. Red and green marks denote edge states localized at the right and left edge, respectively. The full quasienergy spectrum is presented in (c). The parameters used are $N_y'=100$, $\lambda=3$, $J=1$, and $V=2$.[]{data-label="4"}](ciredgef.pdf "fig:"){width="50.00000%"}
As another illustration, Fig. \[5\](a) and \[5\](b) show the quasienergy spectrum of Eq. (\[uy\]) under OBC at another set of values of the parameters $J$, $\lambda$, and $V$, where line nodes of type $2$ are expected to appear, at a fixed $k=\arccos\left(\frac{\pi}{6}\right)$ and $\phi_z'=1$, respectively. The edge states appearing from a point on a line node and disappearing at another point on the same line node are visible, albeit not as clear as the emergence of edge states when line nodes of type $3$ appear. Nevertheless, the existence of these edge states confirms the fact that both line nodes of type $2$ and type $3$ have topological origins, although this might be associated with a weak topological effect [@WTI; @WTI2; @WTI3], as the emergence of edge states depends on the direction in which we “cut" the system.
![(color online). Quasienergy spectrum of Eq. (\[uy\]) under OBC at a fixed (a) $k=\arccos\left(\frac{\pi}{6}\right)$ and (b) $\phi_z'=1$. Red and green marks denote edge states localized at the right and left edge, respectively. The parameters used are $N_y'=100$, $J=3$, $\lambda=\frac{1}{\arccos\left(\frac{\pi}{6}\right)}$, and $V=\frac{2}{\pi}$.[]{data-label="5"}](ociredge.pdf)
Dynamics studies {#IV}
================
Another way to capture the Floquet band topology is through the quantization of particle transport by means of an adiabatic pumping protocol [@Thouless; @Derek2]. In this section, we will show that the topology of the Weyl points and line nodes appearing in the ODKHM can be manifested by designing certain adiabatic pumping schemes, which are relatively easy to realize experimentally since they only involve tuning the experimental parameters $\phi_y$ and $\phi_z$ without the need to accelerate the particle by changing $k$.
Quantized adiabatic pumping around Weyl points
----------------------------------------------
According to Ref. [@Hosur], the chirality characterizing a Weyl point represents a source of Berry curvature in quasimomentum space. It means that if we form a closed $2$D surface enclosing a Weyl point in quasimomentum space parametrized by two parameters $k_1$ and $k_2$, the Chern number associated with the Brillouin zone formed by $k_1$ and $k_2$ equals the chirality of the Weyl point enclosed.
Therefore, in order to capture the chirality of a Weyl point, we first prepare an initial state as a uniform superposition of the lower-band Floquet eigenstates over all $k$ in the Brillouin zone (Wannier state centered around $n=0$) [@foot1]. That is,
$$| \Psi_0 \rangle = \frac{a}{2\pi}\int_{-\frac{\pi}{a}}^\frac{\pi}{a} |\psi_{-,k}(\phi_y,\phi_z) \rangle dk \;,
\label{is}$$
where $a$ is the lattice constant. Next, an adiabatic pumping protocol is realized by tuning $\phi_y$ and $\phi_z$ adiabatically according to $\phi_y=\phi_{y,0}+r\sin[\theta(t)]$ and $\phi_z=\phi_{z,0}+r\cos[\theta(t)]$, where $\theta(t)$ is a periodic function of time with a period much longer than the kicking period. In this work, we consider a discrete linear sweeping protocol by changing $\theta$ by $\frac{2\pi}{N}$ every time step $\Delta t=1$, i.e., $\theta(t)=\frac{2\pi i}{N}$ for $i-1<t\leq i$ and $0<i\leq N$. Our objective is to evaluate the shift in position expectation value after one adiabatic cycle $t=N$.
Analytically, Eq. (\[chiral\]) derived in Appendix \[quantized\] implies the following two observations:
1. The change in position expectation value of the wave packet ($\Delta \langle X\rangle$) after one adiabatic cycle is quantized by the total chirality of the Weyl points enclosed.
2. Under the same initial state and adiabatic pumping scheme, the chirality of the Weyl points at quasienergy $\pi$ and quasienergy $0$ contribute to $\Delta \langle X \rangle$ in an opposite manner.
The above observations can also be verified numerically, as shown in Fig. \[pump\]. Fig. \[pump\](a) is obtained by setting $\phi_{y,0}=\phi_{z,0}=0$ and $r=\frac{\pi}{4}$ so that the adiabatic loop does not enclose any Weyl points, and $\Delta \langle X \rangle$ after one adiabatic cycle is zero. In Fig. \[pump\](b), $\phi_{y,0}=\phi_{z,0}=\frac{\pi}{2}$ and $r=\frac{\pi}{4}$ so that the adiabatic loop encloses two Weyl points at quasienergy $0$ at $(k,\phi_y,\phi_z)=\left(\pm\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2}\right)$, and $\frac{\Delta\langle X \rangle}{2a}=-2$ [@foot2], which is equal to the total chirality of the Weyl points enclosed. In Fig. \[pump\](c), $\phi_{y,0}=\frac{\pi}{2}$, $\phi_{z,0}=\arccos\left(\frac{\pi}{V}\right)$, and $r=\frac{\pi}{10}$ so that the adiabatic loop encloses two Weyl points at quasienergy $\pi$ at $(k,\phi_y,\phi_z)=\left(\pm\frac{\pi}{2},\frac{\pi}{2},\arccos\left(\frac{\pi}{V}\right)\right)$, and $\frac{\Delta\langle X \rangle}{2a}=2$, which is equal to the total chirality of the Weyl points enclosed multiplied by minus one. An interesting result is observed in Fig. \[pump\](d). Under $\phi_{y,0}=\frac{\pi}{2}$, $\phi_{z,0}=\frac{\pi}{4}+\frac{\arccos\left(\frac{\pi}{V}\right)}{2}$, and $r=\frac{3\pi}{10}-\frac{\arccos\left(\frac{\pi}{V}\right)}{2}$, the adiabatic loop encloses two Weyl points at quasienergy $0$ at $(k,\phi_y,\phi_z)=\left(\pm\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2}\right)$ and two other Weyl points at quasienergy $\pi$ at $(k,\phi_y,\phi_z)=\left(\pm\frac{\pi}{2},\frac{\pi}{2},\arccos\left(\frac{\pi}{V}\right)\right)$. Although all the Weyl points enclosed have the same chirality $-1$, $\frac{\Delta\langle X \rangle}{2a}=0$ rather than $4$. This suggests that the adiabatic pumping scheme we considered responds to the chirality of the Weyl points at quasienergy $0$ and quasienergy $\pi$ in the opposite manner. These numerical results agree with the two observations earlier based on analytical derivation.
![The shift in position expectation value after one adiabatic cycle along a loop in the artificial dimensions that (a) does not enclose any Weyl point, (b) encloses two Weyl points at quasienergy $0$, (c) encloses two Weyl points at quasienergy $\pi$ and (d) encloses two Weyl points at quasienergy $0$ and two other Weyl points at quasienergy $\pi$. The parameters used are $J=1$, $\lambda=0.5$, $V=2$ for (a) and (b) and $V=5$ for (c) and (d).[]{data-label="pump"}](plotpump.pdf)
$\pi$ Berry phase around line nodes
-----------------------------------
As suggested in Sec. \[II\] B, the line nodes appearing in the ODKHM can be characterized by a topological winding number, which is closely related to the Berry phase around a loop enclosing a line node. This $\pi$ Berry phase can also be manifested via the following adiabatic pumping scheme. First, we prepare a Floquet eigenstate at a fixed $k=\frac{\pi}{2}$ as an initial state. Then, we drive it adiabatically along a closed loop in the artificial dimensions parametrized by $\theta$ according to $\phi_y=\frac{\pi}{2}+\frac{\pi}{5}\cos\left[\theta(t)\right]$ and $\phi_z=\frac{\pi}{2}+\frac{\pi}{5}\sin\left[\theta(t)\right]$. Fig. \[piberry\] shows the Berry phase at $J=1$ and $V=2$ as $\lambda$ is varied from $0.5$ to $2.9$. It shows that when $0.5<\lambda<\frac{\pi}{2}$, the Berry phase is $0$ as the loop does not enclose any line node at these values of $\lambda$. When $\lambda>\frac{\pi}{2}$ and therefore line nodes start to appear, the Berry phase jumps to $\pi$.
We note also that the $\pi$ Berry phase obtained above when $\lambda>\frac{\pi}{2}$ is contributed only due to the singularity of the line node, i.e., the point where the two quasienergies become degenerate, and hence due to its topology. This is because the contribution from the integral of the Berry curvature over the area of the loop, excluding the singularity due to the line node, is $0$. To show this, we start by rewriting Eq. (\[mod\]) as follows,
$$\mathcal{U}\left(k=\frac{\pi}{2},k_y,k_z\right)=e^{2\mathrm{i} \lambda \sin(k_y)\sigma_y}e^{\mathrm{i}V\sin(k_z)\sigma_z}\;,
\label{med}$$
where $k_y=\frac{\pi}{2}+\phi_y$ and $k_z=\frac{\pi}{2}+\phi_z$. We then apply a unitary transformation to Eq. (\[med\]) to obtain,
$$\mathcal{U'}\left(k=\frac{\pi}{2},k_y,k_z\right)=e^{\mathrm{i} \lambda \sin(k_y)\sigma_y}e^{\mathrm{i}V\sin(k_z)\sigma_z}e^{\mathrm{i} \lambda \sin(k_y)\sigma_y}\;.
\label{med2}$$
Since $\mathcal{U}\left(k=\frac{\pi}{2},k_y,k_z\right)$ and $\mathcal{U'}\left(k=\frac{\pi}{2},k_y,k_z\right)$ are related by a unitary transformation, they share the same eigenvalues, while the eigenvectors are related by
$$\left|u\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle =e^{\mathrm{i}\lambda \sin(k_y)\sigma_y}\left|u'\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle \;.$$
Furthermore, Eq. (\[med2\]) implies that $\mathcal{U'}\left(k=\frac{\pi}{2},k_y,k_z\right)=\mathcal{U'}^\dagger\left(k=\frac{\pi}{2},-k_y,-k_z\right)$. It therefore follows that if $\left|u'^+\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle$ and $\left|u'^-\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle$ are eigenvectors to $\mathcal{U'}\left(k=\frac{\pi}{2},k_y,k_z\right)$ that correspond to quasienergy $+\varepsilon\left(k=\frac{\pi}{2},k_y,k_z\right)$ and $-\varepsilon\left(k=\frac{\pi}{2},k_y,k_z\right)$ respectively,
$$\left|u'^\pm\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle = \left|u'^\mp\left(k=\frac{\pi}{2},-k_y,-k_z\right)\right\rangle \;.$$
In terms of the eigenvectors of $\mathcal{U}\left(k=\frac{\pi}{2},k_y,k_z\right)$,
$$\left|u^\pm\left(k=\frac{\pi}{2},k_y,k_z\right)\right\rangle=e^{2\mathrm{i}\lambda \sin(k_y)\sigma_y}\left|u^\mp\left(k=\frac{\pi}{2},-k_y,-k_z\right)\right\rangle \;.$$
Therefore, the Berry curvature at $\left(k,\phi_y,\phi_z\right)=\left(\frac{\pi}{2}, \frac{\pi}{2}+k_y,\frac{\pi}{2}+k_z\right)$ can be shown to be
$$\mathcal{B}_\pm(k_y,k_z)=-\mathcal{B}_\pm(-k_y,-k_z) \;,$$
which is odd in $k_y$ and $k_z$, and thus its integral over the area, excluding the singularity due to the line node, enclosed by the loop we defined earlier is $0$.
![Berry phase acquired by a Floquet eigenstate (at a fixed $k=\frac{\pi}{2}$) adiabatically driven along a closed loop in the artificial dimensions as a function of $\lambda$ at $J=1$ and $V=2$.[]{data-label="piberry"}](Berryline.pdf)
Concluding remarks {#V}
==================
In this study, we have shown that even by focusing only on the two-band case, the Floquet states of the ODKHM are able to host as many Weyl points and line nodes as possible. By taking OBC along the physical dimension of the system, dispersionless edge states in a form of a line connecting two Weyl points with opposite chiralities are observed, which is analogue to the Fermi arc in static systems. The emergence of these edge states can be connected with a type of $1$D topological invariant, such as that used to characterize the topology of the SSH model. Although the edge states associated with the line nodes cannot be observed when the system is open along its physical dimension, they become visible if instead we open the system along its artificial dimension.
By preparing a Wannier state centered around $n=0$ as the initial state, and driving it adiabatically along a closed loop in its artificial dimensions, which can be achieved by simply changing the experimental parameters $\phi_y$ and $\phi_z$, the change in position expectation value $\Delta \langle X \rangle$ of the wave packet is quantized by the total chirality of the Weyl points enclosed. We further observed that under the same initial state and adiabatic pumping scheme, the chirality of the Weyl points at quasienergy $0$ and quasienergy $\pi$ contributes to $\Delta \langle X \rangle$ in an opposite manner. Lastly, the $\pi$ Berry phase of the line nodes is verified by adiabatically driving a Floquet eigenstate along a closed loop in the artificial dimensions.
This paper should therefore add new knowledge on the studies of the topological semimetal phases in Floquet systems. Following this study, there are many possible future directions which deserve further explorations. For example, given that the two-band ODKHM has a very rich Floquet band structure, it is expected that considering the ODKHM with more than two bands may give rise to more interesting novel phenomena to observe. Furthermore, we might also consider a continuous driven version of this model by replacing the kicking strength in Eq. (\[fund\]) with a harmonic driving amplitude [@Longwen], which is more feasible to realize experimentally. In addition, it will be interesting to further explore the difference between Weyl points at quasienergy $0$ and quasienergy $\pi$, for example in terms of their edge states behaviour or chiral anomaly [@Hosur; @Burkov2]. Lastly, since we can practically control the number of edge states in the two-band ODKHM, it will be a good platform to study interaction effects and edge state transfer [@Doublon] in Floquet systems.
[**Acknowledgements:**]{} We thank Hailong Wang and Dario Poletti for helpful discussions. This work is supported by Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112).
[**Appendix**]{}
This Appendix has four sections. In Appendix A, we present detailed derivations of Eq. (4) in the main text. This is followed by Appendix B, where Lemma $1$ is proved. In Appendix C, we derive the approximate Floquet operator around different types of band touching points. In Appendix D, we establish the relation between the quantized adiabatic pumping of Wannier states and the chirality of the Weyl nodes. Whenever possible, we use the same notation as in the main text.
Derivation of Eq. (\[mod\]) {#derive}
===========================
Define $|a_n \rangle = |2 n \rangle$ and $|b_n \rangle = |2n-1 \rangle$. In terms of $|a_n \rangle$ and $|b_n \rangle$, Eq. (\[tri\]) can be written as,
$$U=e^{-\mathrm{i} H_\mathrm{OFF}}e^{-\mathrm{i} H_\mathrm{ON}}\;,$$
where,
$$\begin{aligned}
\label{off} H_\mathrm{OFF} &=& \sum_{n} \left\lbrace \left[J+\lambda \cos(\phi_y)\right]|b_{n-1} \rangle \langle a_n |+ \left[J-\lambda \cos(\phi_y)\right]|a_n \rangle \langle b_n | + h.c\right\rbrace , \\
H_\mathrm{ON} &=& \sum_{n} \left[ V \cos(\phi_z) |a_n\rangle \langle a_n | - V \cos(\phi_z) |b_n\rangle \langle b_n |\right] . \label{on}\end{aligned}$$
Next, introduce Fourier transformed quantities,
$$\begin{aligned}
|a_n \rangle &=& \frac{1}{\sqrt{\frac{N}{2}}} \sum_k e^{-\mathrm{i} k (2n) c} |a_k\rangle , \\
|b_n \rangle &=& \frac{1}{\sqrt{\frac{N}{2}}} \sum_k e^{-\mathrm{i} k (2n-1) c} |b_k\rangle ,
\end{aligned}$$
where $c$ is the lattice constant, which can be taken to be $1$ for simplicity. Eq. (\[off\]) and (\[on\]) become,
$$\begin{aligned}
H_\mathrm{OFF} &=& \sum_k \left[ (J+\lambda \cos(\phi_y))e^{-\mathrm{i} k}|b_k \rangle \langle a_k |+ (J-\lambda \cos(\phi_y))e^{-\mathrm{i} k}|a_k \rangle \langle b_k | + h.c\right] \nonumber \\
& \hat{=} & \sum_k \left[\begin{array}{cc} 0 & 2 J \cos(k)-2\mathrm{i}\lambda \sin(k) \cos(\phi_y) \\ 2 J \cos(k)+2\mathrm{i}\lambda \sin(k) \cos(\phi_y) & 0 \\ \end{array}\right] \otimes |k \rangle \langle k |\nonumber\\
&=& \sum_k \left(2 J \cos(k) \sigma_x+2\lambda \sin(k)\cos(\phi_y) \sigma_y \right)\otimes |k\rangle \langle k | \nonumber \\
&=& \sum_k h_\mathrm{OFF}(k) \otimes |k \rangle \langle k | , \\
H_\mathrm{ON}&=& \sum_{k} V \cos(\phi_z) |a_k\rangle \langle a_k | - V \cos(\phi_z) |b_k\rangle \langle b_k | \nonumber \\
&\hat{=}& \sum_k \left[\begin{array}{cc} V\cos(\phi_z) & 0 \\ 0 & -V \cos(\phi_z) \\ \end{array}\right] \otimes |k\rangle \langle k | \nonumber \\
&=& \sum_k V \cos(\phi_z) \sigma_z\otimes |k \rangle \langle k | \nonumber \\
&=& \sum_k h_\mathrm{ON}(k) \otimes |k\rangle \langle k |.\end{aligned}$$
$k$ is therefore a conserved quantity, and Eq. (\[tri\]) becomes,
$$\begin{aligned}
U &=& \sum_k \left[e^{-\mathrm{i}h_\mathrm{OFF}(k)}e^{-\mathrm{i}h_\mathrm{ON}(k)}\right]\otimes |k\rangle \langle k | \nonumber \\
&=& \sum_k \mathcal{U}(k,\phi_y,\phi_z) \otimes |k \rangle \langle k |,\end{aligned}$$
where $\mathcal{U}(k,\phi_y,\phi_z)$ is given by Eq. (\[mod\]).
Proof of Lemma 1 {#proof}
================
We start by using Euler’s Formula, $$\begin{aligned}
e^{\mathrm{i} a \vec{n}\cdot \sigma} e^{\mathrm{i} b \vec{m}\cdot \sigma} &=& \left[\cos(a)+\mathrm{i} \sin(a) \vec{n}\cdot \sigma \right] \left[\cos(b)+\mathrm{i}\sin(b) \vec{m}\cdot \sigma \right] \\
&=& \cos(a)\cos(b)-\sin(a)\sin(b)\vec{n}\cdot\vec{m} +\mathrm{i} \left[ \sin(a)\cos(b) \vec{n}\cdot \sigma\right. \\
&& \left.+\sin(b)\cos(a) \vec{m}\cdot \sigma +\sin(a)\sin(b)\vec{m}\times\vec{n}\cdot \sigma \right] \\
&=& e^{\mathrm{i}\theta \vec{r}\cdot \sigma},
\end{aligned}$$
where $r$ is some unit vector and $\theta = \arccos\left[\cos(a)\cos(b)-\sin(a)\sin(b)\vec{n}\cdot\vec{m}\right]$. Since $\vec{n}$ and $\vec{m}$ are perpendicular, $\theta = \arccos\left[\cos(a)\cos(b)\right]$. Band touching points correspond to $\theta$ being an integer multiple of $\pi$. Since cosine function is bounded in $\left[ -1, 1\right]$, this is only possible if both $\cos(a)$ and $\cos(b)$ are $\pm 1$. This proves Lemma \[lem\].
Expansion of the Floquet operator around band touching points
=============================================================
Around band touching points of type $1$ (Weyl points)
-----------------------------------------------------
As explained in the main text, one of the band touching points occurs at $(k,\phi_y,\phi_z)=\left(\frac{\pi}{2},\frac{\pi}{2},\phi_1\right)$, where $\phi_1=\arccos\left(\frac{l_2\pi}{V}\right)$. Let $k_x=k-\frac{\pi}{2}$, $k_y=\phi_y-\frac{\pi}{2}$, and $k_z=\phi_z-\phi_1$. Eq. (\[mod\]) then becomes,
$$\mathcal{U}(k_x,k_y,k_z) = e^{\mathrm{i} (2J k_x \sigma_x +2\lambda k_y\sigma_y)} e^{\left[-\mathrm{i} l_2 \pi +\mathrm{i}V\sin(\phi_1)k_z\right]\sigma_z} \;.
\label{c1}$$
The second exponential can be written as $e^{-\mathrm{i} l_2 \pi \sigma_z}e^{\mathrm{i}V\sin(\phi_1)k_z\sigma_z}$ since $\sigma_z$ commutes with itself. Furthermore, according to Euler’s formula, $e^{-\mathrm{i} l_2 \pi \sigma_z}=\cos(l_2\pi)+\mathrm{i}\sin(l_2\pi)\sigma_z=\pm 1=e^{-\mathrm{i}l_2\pi}$, where the sign depends on whether $l_2$ is an even or odd integer. In both cases, the resulting exponential commutes with all Pauli matrices, and Eq. (\[c1\]) becomes,
$$\begin{aligned}
\mathcal{U}(k_x,k_y,k_z) &=& e^{-\mathrm{i}l_2 \pi}e^{\mathrm{i} (2J k_x \sigma_x +2\lambda k_y\sigma_y)} e^{\mathrm{i}V\sin(\phi_1)k_z\sigma_z} \\
&\approx & e^{-\mathrm{i}l_2 \pi}e^{\mathrm{i} \left[2J k_x \sigma_x +2\lambda k_y\sigma_y+V\sin(\phi_1)k_z\sigma_z\right]} \\
&=& e^{-\mathrm{i} \left\lbrace l_2\pi- \left[2J k_x \sigma_x +2\lambda k_y\sigma_y+V\sin(\phi_1)k_z\sigma_z\right]\right\rbrace} \\\end{aligned}$$
Around line nodes of type $2$ or $3$
------------------------------------
Since the derivation of the effective Hamiltonian near a line node of type $2$ and $3$ is similar, we will present the derivation of Eq. (\[lineff\]) only in this section. Let $k_x=k-k_0'$, $k_y=\phi_y-\frac{\pi}{2}$, and $k_z=\phi_z-\frac{\pi}{2}$, where $k_0'=\arccos\left(\frac{\pi}{2J}\right)$. Eq. (\[mod\]) then becomes,
$$\mathcal{U}(k_x,k_y,k_z) = e^{\mathrm{i} \left\lbrace\left[-\pi + 2J k_x\sin(k_0')\right] \sigma_x +2\lambda k_y\sin(k_0')\sigma_y\right\rbrace} e^{\mathrm{i}V\sin(\phi_1)k_z\sigma_z} \;.
\label{c2}$$
By using Euler’s formula, Eq. (\[c2\]) can be written as,
$$\begin{aligned}
\mathcal{U}(k_x,k_y,k_z) &=& \cos(\theta)\cos(V k_z)+\mathrm{i} \left[ \cos(\theta)\sin(Vk_z)\sigma_z\right. \nonumber\\
&& +\frac{\sin(\theta)}{\theta}\left\lbrace -2\lambda k_y\sin(k_0') \sin(Vk_z)+ \left[2Jk_x\sin(k_0')-\pi\right]\cos(Vk_z)\right\rbrace\sigma_x\nonumber \\
&& +\frac{\sin(\theta)}{\theta}\left\lbrace 2\lambda k_y\sin(k_0') \cos(Vk_z)+\left[2Jk_x\sin(k_0')-\pi\right]\sin(Vk_z)\right\rbrace \sigma_y\left.\right] \nonumber \\
&=& e^{\mathrm{i}\frac{\theta'}{\sin(\theta')}\vec{r}\cdot \sigma} \nonumber \\
&=& e^{-\mathrm{i}h_\mathrm{eff}} ,\end{aligned}$$
where $\theta=\sqrt{\left[\pi-2Jk_x\sin(k_0')\right]^2+4\lambda^2k_y^2\sin^2(k_0')}$ and $\theta' = \arccos\left[\cos(\theta)\cos(Vk_z)\right]$. Up to first order in $k_x$, $k_y$, and $k_z$, $\theta' \approx \pi-\sqrt{V^2k_z^2+4J^2k_x^2\sin^2(k_0')}=\pi-\Phi(k_x,k_z)$, $\sin(\theta')\approx -\Phi(k_x,k_z)$, $\sin(\theta)\approx-2Jk_x\sin(k_0')$, and $\frac{\theta'}{\theta}\approx 1-\frac{\Phi(k_x,k_z)-2Jk_x\sin(k_0')}{\pi}$. It therefore follows that,
$$\begin{aligned}
h_\mathrm{eff} &=& -\frac{\theta'}{\sin(\theta')}\vec{r}\cdot \sigma \nonumber \\
&=& \left[\pi-\Phi(k_x,k_z)\right]\left[-\frac{Vk_z}{\Phi(k_x,k_z)}\sigma_z+\frac{2Jk_x\sin(k_0')}{\Phi(k_x,k_z)}\sigma_x\right] \nonumber \\
&& +\frac{2Jk_x\sin(k_0')}{\Phi(k_x,k_z)}\left[-2\lambda k_y\sin(k_0')+\pi Vk_z\right]\sigma_y \nonumber \\
&=& \left[\pi-\Phi(k_x,k_z)\right]\hat{n}\cdot\sigma +\frac{2Jk_x\sin(k_0')}{\Phi(k_x,k_z)}[-2\lambda k_y\sin(k_0')+\pi Vk_z]\sigma_y ,\end{aligned}$$
as given in the main text. The derivation of Eq. (\[lineff2\]) follows exactly the same step as above with $\sigma_y\rightarrow \sigma_x$, $\sigma_x\rightarrow \sigma_y$, $\lambda k_y\sin(k_0')\rightarrow Jk_x$, and $Jk_x\sin(k_0')\rightarrow \lambda k_y \sin(\phi_0)$.
Analytical derivation of the quantized particle transport around Weyl points {#quantized}
============================================================================
The purpose of this section is to show how chirality emerges when an adiabatic pumping scheme elucidated in Sec. \[IV\] A around a Weyl point is performed. A typical effective Hamiltonian around a Weyl point in the ODKHM takes the following form,
$$H_\mathrm{eff}=l\pi - v_x k_x\sigma_x -v_y k_y\sigma_y -v_z k_z \sigma_z\;.
\label{Weyl}$$
Since quasienergy is only defined mod $2\pi$, odd $l$ corresponds to a Weyl point at quasienergy $\pi$ while even $l$ corresponds to a Weyl point at quasienergy $0$. By parametrizing $k_y=r\cos(\theta)$ and $k_z=r\sin(\theta)$, the lower-band eigenstates as a function of $k,\theta\in [-\pi,\pi)$ associated with the above effective Hamiltonian at quasienergy $\pi$ and $0$ respectively can be explicitly calculated in the $\sigma_y$ representation as follow,
$$\begin{aligned}
|\pi (k_x,\theta) \rangle &\hat{=}& \frac{1}{\sqrt{2r_0^2+2v_x^2 k_x^2+2 v_x k_x \sqrt{r_0^2+v_x^2 k_x^2}}}\left( \begin{array}{c} r_0 e^{-\mathrm{i} \phi} \\ - v_x k_x - \sqrt{r_0^2+v_x^2 k_x^2}\end{array}\right) , \\
|0 (k_x,\theta) \rangle &\hat{=}& \frac{1}{\sqrt{2r_0^2+2v_x^2 k_x^2- 2v_x k_x \sqrt{r_0^2+v_x^2 k_x^2}}}\left( \begin{array}{c} r_0 e^{-\mathrm{i} \phi} \\ - v_x k_x + \sqrt{r_0^2+v_x^2 k_x^2}\end{array}\right) ,\end{aligned}$$
where $r_0 = r\sqrt{v_y^2 \cos^2(\theta)+ v_z^2 \sin^2(\theta)}$ and $\tan(\phi) = \frac{v_z}{v_y}\tan(\theta)$. By preparing an initial state in a form of Eq. (\[is\]) with $k_x$, $k_y$, and $k_z$ replace $k$, $\phi_y$, and $\phi_z$ respectively, both the population weight of the initial state and the quasienergy spectrum of Eq. (\[Weyl\]) are invariant under reflection about $k_x=0$. Ref. [@Hailong] then suggests that the shift in position expectation value after one adiabatic cycle is given by,
$$\begin{aligned}
\Delta \langle X \rangle_{\pi(0)} &=& \frac{a \mathrm{i}}{2\pi} \int_{-\frac{\pi}{a}}^\frac{\pi}{a} dk_x \int_0^{2\pi} d\theta \left[ \langle \partial_{k_x}\pi(0) | \partial_\theta \pi(0) \rangle - c.c. \right] \nonumber \\
&=& \frac{a}{2\pi} \int_{-\frac{\pi}{a}}^\frac{\pi}{a} dk_x \int_0^{2\pi} d\theta \frac{d\phi}{d\theta} \frac{d}{dk_x}\left[\frac{r_0^2}{2r_0^2+2v_x^2 k_x^2+(-) 2v_x k_x \sqrt{r_0^2+v_x^2 k_x^2}}\right]\nonumber \\
&=& \frac{a}{2\pi} \int_0^{2\pi} d\theta \frac{d\phi}{d\theta} \left\lbrace \int_{-\frac{\pi}{a}}^\frac{\pi}{a} dk_x \frac{d}{dk_x}\left[\frac{r_0^2}{2r_0^2+2v_x^2 k_x^2+(-) 2v_x k_x \sqrt{r_0^2+v_x^2 k_x^2}}\right] \right\rbrace. \label{save}\end{aligned}$$
Let’s first focus on the term inside the square bracket. Since we can take $r$ as small as possible, and hence $r_0$ as small as possible, we can assume $\frac{\pi}{a}\gg r_0$, so that upon performing the integration,
$$\begin{aligned}
\int_{-\frac{\pi}{a}}^\frac{\pi}{a} dk_x \frac{d}{dk_x}\left[\frac{r_0^2}{2r_0^2+2v_x^2 k_x^2+(-) 2v_x k_x \sqrt{r_0^2+v_x^2 k_x^2}}\right] &=& \frac{r_0^2}{2r_0^2+2v_x^2 \frac{\pi^2}{a^2}+(-) 2v_x \frac{\pi}{a} \sqrt{r_0^2+v_x^2 \frac{\pi^2}{a}}}\\
&&-\frac{r_0^2}{2r_0^2+2v_x^2 \frac{\pi^2}{a^2}-(+) 2v_x \frac{\pi}{a} \sqrt{r_0^2+v_x^2 \frac{\pi^2}{a}}} \\
&\approx & -(+) \frac{4 r_0^2 \frac{v_x\pi}{a} \sqrt{r_0^2+v_x^2 \frac{\pi^2}{a}}}{\left(2r_0^2+2\frac{v_x^2\pi^2}{a}\right)^2-4\frac{v_x^2\pi^2}{a}\left(r_0^2+v_x^2 \frac{\pi^2}{a}\right)} \\
&\approx & -(+) \mathrm{sgn}(v_x)\end{aligned}$$
Therefore, Eq. (\[save\]) becomes,
$$\begin{aligned}
\Delta \langle X \rangle_{\pi(0)} &=& -(+) \frac{a}{2\pi} \int_0^{2\pi} d\theta \frac{d\phi}{d\theta} \mathrm{sgn}(v_x) \nonumber \\
&=& -(+) \frac{a}{2\pi} \mathrm{sgn}(v_x) \left[\phi(\theta =2\pi)-\phi (\theta =0)\right] \nonumber \\
&=& -(+) a \mathrm{sgn}(v_x) \mathrm{sgn}(v_z)\mathrm{sgn}(v_y) \nonumber \\
&=& -(+) a\mathrm{sgn}(v_xv_yv_z) \nonumber \\
&=& +(-) a \chi .
\label{chiral}\end{aligned}$$
We have thus showed that the change in position expectation value after one adiabatic cycle is quantized by the lattice constant times the chirality of the Weyl point enclosed. Although in the derivation above we assumed that the radius $r$ of the loop in the artificial dimensions is very small, the above result holds for arbitrary $r$ or even arbitrary shape of the loop enclosing the Weyl point, provided there are no other singularities inside the loop we constructed. This is because chirality can also be interpreted as the total Berry flux passing through a given closed surface. In analogy to Gauss Law in electrostatics, the total flux should not depend on the shape of the closed surface. Therefore, as long as the initial state fully populates all quasimomenta $k$ (so that $k$ and $\theta$ form a closed surface) and has a reflection symmetry about $k_x=0$, the above result is expected to hold.
\[exp\]
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|
---
abstract: 'We implement a quantum error correction algorithm for bit-flip errors on the topological toric code using deep reinforcement learning. An action-value Q-function encodes the discounted value of moving a defect to a neighboring site on the square grid (the action) depending on the full set of defects on the torus (the syndrome or state). The Q-function is represented by a deep convolutional neural network. Using the translational invariance on the torus allows for viewing each defect from a central perspective which significantly simplifies the state space representation independently of the number of defect pairs. The training is done using experience replay, where data from the algorithm being played out is stored and used for mini-batch upgrade of the Q-network. We find performance which is close to, and for small error rates asymptotically equivalent to, that achieved by the Minimum Weight Perfect Matching algorithm for code distances up to $d=7$. Our results show that it is possible for a self-trained agent without supervision or support algorithms to find a decoding scheme that performs on par with hand-made algorithms, opening up for future machine engineered decoders for more general error models and error correcting codes.'
author:
- Philip Andreasson
- Joel Johansson
- Simon Liljestrand
- Mats Granath
bibliography:
- 'Bib.bib'
title: Quantum error correction for the toric code using deep reinforcement learning
---
Introduction
============
Much of the spectacular advances in machine learning using artificial neural networks has been in the domain of supervised learning were deep convolutional networks excel at categorizing objects when trained with big annotated data sets[@krizhevsky2012imagenet; @lecun2015deep; @goodfellow2016deep]. A different but also more challenging type of problem is when there is no a priori solution key, but rather a dynamic environment through which we want to learn to navigate for an optimal outcome. For these types of problems reinforcement learning (RL) [@sutton2018reinforcement] combined with deep learning has had great success recently when applied to problems such as computer and board games[@tesauro1995temporal; @mnih2013playing; @mnih2015human; @silver2017mastering]. The super-human performance achieved by deep reinforcement learning has revolutionized the field of artificial intelligence and opens up for applications in many areas of science and technology.
In physics the use of machine learning has seen a great deal of interest lately[@arsenault2014machine; @van2017learning; @carrasquilla2017machine; @carleo2017solving; @gao2017efficient]. The most natural type of application of neural networks is in the form of supervised learning where the deep network can capture correlations or subtle information in real or artificial data. The use of deep reinforcement learning may be less obvious in general as the type of topics addressed by RL typically involve some sort of “intelligent” best strategy search, contrary to the deterministic or statistical models used in physics. In this paper we study a type of problem where artificial intelligence is applicable, namely the task of finding a best strategy for error correction of a topological quantum code; the potential basic building blocks of a quantum computer. In the field of quantum computing, smart algorithms are needed for error correction of fragile quantum bits[@PhysRevA.52.R2493; @PhysRevLett.77.793; @nielsen2002quantum; @terhal2015quantum]. Reinforcement learning has been suggested recently as a tool for quantum error correction and quantum control[@melnikov2018active; @PhysRevX.8.031084; @PhysRevX.8.031086], where an agent learns to manipulate a quantum device that functions in an imperfect environment and with incomplete information. Under the umbrella term “Quantum Machine Learning” there are also interesting prospects of utilizing the natural parallelization of quantum computers for machine learning itself[@biamonte2017quantum], but we will be dealing here with the task of putting (classical) deep learning and AI at the service of quantum computing.
Due to the inherently fragile nature of quantum information a future universal quantum computer will require quantum error correction.[@PhysRevA.52.R2493; @PhysRevLett.77.793; @nielsen2002quantum; @terhal2015quantum] Perhaps the most promising framework is to use topological error correcting codes.[@kitaev2003fault; @dennis2002topological; @raussendorf2007topological; @fowler2012surface; @kelly2015state] Here, logical qubits consisting of a large set of entangled physical qubits are protected against local disturbances from phase or bit flip errors as logical operations require global changes. Local stabilizer operators, in the form of parity checks on a group of physical qubits, provide a quantum non-demolition diagnosis of the logical qubit in terms of violated stabilizers; the so-called syndrome. In order for errors not to proliferate and cause logical failure, a decoder, that provides a set of recovery operations for correction of errors given a particular syndrome, is required. As the syndrome does not uniquely determine the physical errors, the decoder has to incorporate the statistics of errors corresponding to any given syndrome. In addition the syndrome itself may be imperfect, due to stabilizer measurement errors, in which case the decoder must also take that into account.
In the present work we consider Kitaev’s toric code[@kitaev2003fault; @dennis2002topological; @fowler2012surface] which is a stabilizer code formulated on a square lattice with periodic boundary conditions (see Figure \[fig:model\] and Section \[sec:toric\]). We will only consider bit-flip errors which correspond to syndromes with one type of violated stabilizer that can be represented as plaquette defects on the lattice (see Figure \[fig:errorchain\]). The standard decoder for the toric code is the Minimum Weight Perfect Matching (MWPM) or Blossom algorithm[@edmonds1965paths; @fowler2015minimum; @PhysRevA.90.032326] that works by finding the pairwise matching of syndrome defects with shortest total distance, corresponding to the minimal number of errors consistent with the syndrome. The decoder problem is also conceptually well suited for reinforcement learning, similar in spirit to a board game; the state of the system is given by the syndrome, actions correspond to moving defects of the syndrome, and with reward given depending on the outcome of the game. By playing the game, the agent improves its error correcting strategies and the decoder is the trained agent that provides step by step error correction. As in any RL problem the reward scheme is crucial for good performance. The size of the state-action space is also a challenge, to provide the best action for each of a myriad syndromes, but this is exactly the problem addressed by recent deep learning approaches to RL.[@tesauro1995temporal; @mnih2013playing; @mnih2015human; @silver2017mastering]
We find that by setting up a reward scheme that encourages the elimination of the syndrome in as few operations as possible within the deep Q-learning (or deep Q-network, DQN)[@mnih2013playing; @mnih2015human] formalism we are able to train a decoder that is comparable in performance to MWPM. Although the present algorithm does not outperform the latter we expect that it has the potential to be more versatile when addressing depolarizing noise (with correlated bit and phase flip errors), measurement noise giving imperfect syndromes, or varying code geometries. Compared to the MWPM algorithm the RL algorithm also has the possible advantage that it provides step by step correction whereas the MWPM algorithm only provides information on which defects should be paired, making the former more adaptable to the introduction of additional errors.
In concurrent work by @sweke2018reinforcement an application of reinforcement learning to error correction of the toric code was implemented. That work focuses on the important issue of imperfect syndromes as well as depolarizing noise and used an auxiliary “referee decoder” to assist the RL decoder. In the present work we consider the simpler but conceptually more direct problem of error correction on a perfect syndrome and with only bit flip error. Also in contrast to [@sweke2018reinforcement] we study the actual “toric” code, rather than the code with boundaries. Clearly the toric code will be harder to implement experimentally but nevertheless provides a well understood standard model. It also provides a simplification from the fact that on a torus only the relative positions of syndrome defects are relevant which reduces the state space complexity that decoder agent has to master. By focusing on this minimal problem we find that we can make a rigorous benchmark on the RL decoder showing near optimal performance.
Finding better performing decoders has been the topic of many studies, using methods such as renormalization group[@duclos2010fast; @Duclos-Cianci:2014:FRG:2638670.2638671], cellular automata[@herold2015cellular; @kubica2018cellular], and a number of neural network based decoders[@PhysRevLett.119.030501; @PhysRevX.8.031084; @krastanov2017deep; @varsamopoulos2017decoding; @baireuther2018machine; @breuckmann2018scalable; @chamberland2018deep; @maskara2018advantages; @ni2018neural; @PhysRevLett.122.200501]. The decoder presented in this paper does not outperform state of the art decoders, it’s value lies in showing that it is possible to use reinforcement learning to achieve excellent performance on a minimal model. Given that deep reinforcement learning is arguably the most promising AI framework it holds prospect for future versatile self-trained decoders that can adapt to different error scenarios and code architectures.
The outline of the paper is the following. In the [*Background*]{} section we give a brief but self-contained summary of the main features of the toric code including the basic structure of the error correction and a similar summary of one-step Q-learning and deep Q-learning. The following section, [*RL Algorithm*]{}, describes the formulation and training of the error correcting agent. In the [*Results*]{} section we shows that we have trained the RL agent up to code distance $d=7$ with performance which is very close to the MWPM algorithm. We finally conclude and append details of the asymptotic fail rate for small error rates as well as the neural network architecture and the RL and network hyperparameters.
Background
==========
Toric code\[sec:toric\]
-----------------------
The basic construction of the toric code is a square lattice with a spin-$\frac{1}{2}$ degree of freedom on every bond, the physical qubits, and with periodic boundary conditions, as seen in Figure \[fig:model\].[@kitaev2003fault; @dennis2002topological][^1] (An alternative rotated lattice representation with the qubits on sites is also common in the literature.) The model is given in terms of a Hamiltonian $$H=-\sum_{\alpha}\hat{P}_\alpha-\sum_{\nu}\hat{V}_{\nu}\,,$$ where $\alpha$ runs over all plaquettes and $\nu$ over all vertices (sites). The stabilizers are the plaquette operators $\hat{P}_\alpha=\prod_{i\in\alpha}\sigma^z_i$ and the vertex operators $\hat{V}_\nu=\prod_{i\in\nu}\sigma^x_i$, where $\sigma^z$ and $\sigma^x$ are the Pauli matrices. (Where, in the $\sigma^z$ basis, $\sigma^z|\uparrow/\downarrow\rangle=\pm 1 |\uparrow/\downarrow\rangle$ and $\sigma^x|\uparrow/\downarrow\rangle=|\downarrow/\uparrow\rangle$.) The stabilizers commute with each other and the Hamiltonian thus block diagonalizing the latter. On a $d\times d$ lattice of plaquettes $d^2-1$ plaquette operators are linearly independent (e.g. it is not possible to have a single $-1$ eigenvalue with all other $+1$) and correspondingly for the vertex operators. With $2d^2$ physical qubits and $2d^2-2$ stabilizers the size of each block is $2^{2d^2}/2^{2d^2-2}=4$, corresponding in particular to a ground state which is 4-fold degenerate. These are the states that will serve as the logical qubits. (More precisely, given the 4-fold degeneracy it is a qudit or base-4 qubit.)
![\[fig:model\] A $d=5$ toric code with rings indicating the physical qubits and grey showing the periodic boundary conditions. a) Plaquette (green) and vertex (red) stabilizer operators, as products of $\sigma^z$ and $\sigma^x$ Pauli matrices. b) A single vertex operator can be represented as a loop flipping the qubits that the it crosses. c) Two neighboring vertex operators make up a larger loop. d) The logical operators $\bar{X}_{1/2}$ (red) and $\bar{Z}_{1/2}$ (green) consist of loops winding the torus and are not representable in terms of products of vertex or plaquette operators.](new_lattice_figures.pdf)
To derive the ground state consider first the plaquette operator in the $\sigma^z$-basis; clearly a ground state must have an even number of each spin-up and spin-down on every plaquette to be a $+1$ eigenstate of each plaquette operator. Let’s consider the state with all spin-up $|\uparrow\uparrow\uparrow\cdots\rangle$; acting with a vertex operator on this flips all the spins around the vertex (see Fig. \[fig:model\]b) giving a state still in ground state sector of the plaquette operators as an even number of spins are flipped on the plaquettes surrounding the vertex. (As is also clear from the fact that all the stabilizer operators commute.) The $+1$ eigenstate of that particular vertex operator is thus the symmetric superposition of the two states. A convenient way to express the operation of one or several adjacent vertex operators is in turns of loop traversing the flipped spins. Such loops (fig. \[fig:model\]b-c) generated from products of vertex operators will always be topologically trivial loops on the surface of the torus since they are just constructed by merging the local loop corresponding to a single vertex operator. Successively acting with vertex operators on the states generated from the original $|\uparrow\uparrow\uparrow\cdots\rangle$ we realize that the ground state is simply the symmetric superposition of all states that are generated from this by acting with (trivial) loops $|\text{GS}_0\rangle=\sum_{i\in \text{all trivial loops}}\text{loop}_i|\uparrow\uparrow\uparrow\cdots\rangle$.
To generate the other ground states we consider the operators $\bar{X}_1$ and $\bar{X}_2$ (Fig. \[fig:model\]d) which are products of $\sigma^x$ corresponding to the two non-trivial loops that wind the torus. (Deformations of these loops just correspond to multiplication by trivial loops and is thus inconsequential.) Correspondingly there are non-trivial loops of $\sigma^z$ operators $\bar{Z}_1$ and $\bar{Z}_2$. The four ground states are thus the topologically distinct states $\{|\text{GS}_0\rangle,\bar{X}_1|\text{GS}_0\rangle,\bar{X}_2|\text{GS}_0\rangle,\bar{X}_2\bar{X}_1|\text{GS}_0\rangle\}$ distinguished by their eigenvalues of $\bar{Z}_1$ and $\bar{Z}_2$ being $\pm 1$. For a torus with $d\times d$ plaquettes there are $2d^2$ physical qubits and the code distance, i.e. minimum length of any logical operator ($\bar{X}_i$ or $\bar{Z}_i$), is $d$.
### Error correction
Errors in the physical qubits will take the state out of the ground state sector and thereby mask the encoded state. The task of the error correction procedure is to move the system back to the ground state sector without inadvertently performing a logical operation to change the logical qubit state. A $\sigma^x$ error on a physical qubit corresponds to a bit-flip error. On the toric code this gives rise to a pair of defects (a.k.a. quasiparticles or anyons) in the form of neighboring plaquettes with $-1$ eigenvalues of the plaquette stabilizers. Similarly a $\sigma^z$ error corresponds to a phase-flip error which gives rise to a pair of neighboring $-1$ defects on two vertices. A $\sigma^y=i\sigma^x\sigma^z$ simultaneously creates both types of defects. A natural error process is to assume that $X,Y,Z$ errors occur with equal probability, so called depolarizing noise. This however requires to treat correlations between $X$ and $Z$ errors and the simpler uncorrelated noise model is often used, which is what we will consider in this work, focusing on bit-flip errors and corresponding plaquette defects. Here $X$ and $Z$ errors occur independently with probability $p$ whereas $Y=XZ$ errors occur with probability $p^2$. Correcting independent $X$ and $Z$ errors is completely equivalent (with defects either on plaquettes or on vertices) and it is therefore sufficient to formulate an error correcting algorithm for one type of error. (For actual realizations of the physical qubits the error process may in fact be intermediate between these two cases[@PhysRevLett.120.050505].) Regardless of noise model and type of error an important aspect of the error correction of a stabilizer formalism is that the entanglement of the logical qubit states or its excitations does not have to be considered explicitly as errors act equivalently on all states that belong to the same stabilizer sector.
![\[fig:errorchain\] Bit-flip errors (red ’X’) and possible error correction bit-flips (blue ’X’). (a) Two neighboring errors and the corresponding error chain (red line) and syndrome (red dots). (b) Visualized in terms of the syndrome with error chain and two possible correction chains (blue) as expressed explicitly in (c) and (d). The error chain plus the correction chain in (d) constitutes a non-trivial loop and a logical bit-flip operation (as in Figure \[fig:model\]d), thus a failed error correction, in contrast to the trivial loop in (c).](error_chains.pdf)
A crucial aspect of quantum error correction is that the actual bit-flip errors cannot be measured without collapsing the state into a partial basis and destroying the qubit. What can be measured without destroying the logical qubit are the stabilizers, i.e. for bit-flip error the parity of the plaquette operators. The complete set of incorrect ($-1$) plaquettes makes up the syndrome of the state. The complete set of bit-flip errors will produce a unique syndrome as the end-points of strings of bit-flip errors. The converse however is not true, which is what makes the task challenging. In order to do the error correction we need to suggest a number of physical bits that should be flipped in order to achieve the pair-wise annihilation of the defects of the syndrome. Consider a single pair of defects which have been created by a particular chain of errors. (See Figure \[fig:errorchain\].) The error correction needs to suggest a correction string connecting the two defects. If this is done properly the correction string and the error string form a trivial loop, thus returning the qubit to the original state. If instead the correction string and the error string together make up a non-trivial loop that winds the torus we have eliminated the error syndrome but changed the state of qubit (corresponding to a logical bit-flip), thus failed the task of correcting the error.
For the uncorrelated noise model it can be shown, by mapping to the random bond Ising model, that for $d\rightarrow\infty$ there is a critical threshold $p_c\approx 0.11$ below which the most probable correction chains to complement the error chain will with certainty form trivial loops, while for $p>p_c$ non-trivial loops occur with finite probability.[@dennis2002topological] For a finite system, the sharp transition is replaced by a cross-over, as seen in Figure \[fig:results\], where for increasing $d$ the fraction of successful error correction evolves progressively towards 1 for $p<p_c$, and to $1/4$ (thus completely unpredictable) for $p>p_c$.
For the uncorrelated noise model on the torus the most likely set of error chains between pairs of defects which is consistent with a given syndrome would be one that corresponds to the smallest number of total bit flips, i.e. the shortest total error chain length. Thus, a close to optimal algorithm for error correction for this system is the Minimum Weight Perfect Matching (MWPM) algorithm[@edmonds1965paths]. (This algorithm is also near optimal for the problem with syndrome errors as long as it is still uncorrelated noise[@dennis2002topological; @fowler2015minimum].) The MWPM algorithm for the perfect syndrome corresponds to reducing a fully connected graph, with an even number of nodes and with edges specified by the inter-node distances, to the set of pairs of nodes that minimize the total edge length. This algorithm can be implemented efficiently[@kolmogorov2009blossom] and we will use this as the benchmark of our RL results. In fact, as we will see, the RL algorithm that we formulate amounts to solving the MWPM problem. In this sense the work presented in this paper is to show the viability of the RL approach to this problem with the aim for future generalizations to other problems where MWPM is sub-optimal, such as for depolarizing noise or more general error models.
Q-learning
----------
Reinforcement learning is a method to solve the problem of finding an optimal policy of an agent acting in a system where the actions of the agent causes transitions between states of the system.[@sutton2018reinforcement] The policy $\pi(s,a)$ of an agent describes (probabilistically perhaps) the action $a$ to be taken by the agent when the system is in state $s$. In our case the state will correspond to a syndrome, and an action to moving a defect one step. The optimal policy is the one that gives the agent maximal return (cumulative discounted reward) over the course of its interaction with the system. Reward $r_{t+1}$ is given when the system transitions from state $s_t\rightarrow s_{t+1}$ such that the return starting at time $t$ is given by $R_t=r_{t+1}+\gamma r_{t+2}+\gamma^2 r_{t+3}+\cdots$. Here $\gamma\leq 1$ is the discounting factor that quantifies how we want to value immediate versus subsequent reward. As will be discussed in more detail, in the work presented in this paper a constant reward $r=-1$ will be given for each step taken, so that in practice the optimal policy will be the one that minimizes the number of actions, irrespectively of the value of $\gamma$. (Although in practice, even here the value of $\gamma$ can be important for the convergence of the training.)
One way to represent the cumulative reward depending on a set of actions and corresponding transitions is by means of an action-value function, or Q-function. This function $Q(s,a)$ quantifies the expected return when in state $s$ taking the action $a$, and subsequently following some policy $\pi$. In one-step Q-learning we quantify $Q$ according to $Q(s,a)=r+\gamma\max_{a'}Q(s',a')$, with $s\xrightarrow{a}s'$, which corresponds to following the optimal policy according to our current estimate of $Q$. In order to learn the value of the Q-function for all states and actions we should explore the full state-action space, with the policy given by taken action $a$ according to $\max_a Q(s,a)$ eventually guaranteed to converge to the optimal policy. However, an unbiased exploration gets prohibitively expensive and it is therefore in general efficient to follow an $\epsilon$-greedy policy which with probability $(1-\epsilon)$ takes the optimal action based on our current estimate of $Q(s,a)$ but with probability $\epsilon$ takes a random action. From what we have learned by this action we would update our estimate for $Q$ according to $$\label{Q_update}
Q(s,a)\leftarrow Q(s,a)+\alpha[(r+\gamma\max_{a'}Q(s',a'))-Q(s,a)]\,,$$ where $\alpha<1$ is a learning rate. This procedure is then a trade-off between using our current knowledge of the $Q$ function as a guide for the best move to avoid spending extensive time on expensive moves but also exploring to avoid missing out on rewarding parts of the state-action space.
### Deep Q-learning
![\[fig:network\]Structure of the deep Q-network. The input layer is a $d\times d$ matrix corresponding to the “perspective” $P$, of one defect of the syndrome. (Using translational symmetry on the torus, any defect can be placed at the center.) The output layer gives the action value $Q(P,a,\theta)$ of moving the central defect to any of the four neighboring plaquettes $a=U,D,R,L$, given the current training state of network parameters $\theta$. The hidden layers consist of a convolutional layer (of which a $3\times 3$ filter is indicated on the input layer) and several fully connected layers. (For details, see Appendix.) Successively scanning all defects using the same network gives the full action value function of the syndrome. ](neural_network.pdf)
For a large state-action space it is not possible to store the complete action-value function. (Disregarding symmetries, for a $d\times d$ system with $N_S$ defects, the state space has size $d^2\choose N_S$, $\sim 10^{13}$ for $p\approx 10$% and $d=7$.) In deep Q-learning[@mnih2015human], the action-value function is instead represented by a deep neural network with the input layer corresponding to some representation of a state and the output layer corresponding to the value of the possible actions. The idea is that similarities in the value of different regions of the state-action space may be stored in an efficient way by the deep network. Parametrizing the $Q$-function by means of neural network we write $Q(s,a,\theta)$, where $\theta$ represents the complete set of weights and biases of the network. (We use a convolutional network with $\sim 10^6$ parameters for the $d=7$ problem.) As outlined in more detail in the following sections the latter can be trained using supervised learning based on a scheme similar to one step Q-learning.
RL Algorithm \[RLsection\]
==========================
The decoder presented in this paper is a neural network-based agent optimized using reinforcement learning to observe toric code syndromes and suggesting recovery chains for them step by step. The agent makes use of a deep convolutional neural network, or Q-network, (see Fig. \[fig:network\]) to approximate Q values of actions given a syndrome.
In a decoding session, a syndrome $S$ corresponding to the coordinates of $N_S$ defects $e_i$ ($i=1,...,N_S$) is fed to the algorithm as input. The syndrome is the state of the system as visible to the agent. The syndrome at any time step is that generated by accumulated actions of the agent on the syndrome given by the initial random distribution of bit-flips. There is also a hidden state corresponding to the joint set of initial and agent flipped qubits. After the complete episode resulting in a terminal state with an empty syndrome, an odd number of non-trivial loops (in either $X_1$ or $X_2$) indicates a failed error correction. In the algorithm used in this work however, the success/fail information does not play any explicit role in the training, except as external verification of the performance of the agent. Instead reward $r=-1$ is given at every step until the terminal state regardless of whether the error correcting string(s) gave rise to an unwanted logical operation. Taking the fewest number of steps to clear the syndrome is thus the explicit target of the agent, corresponding to actuating the MWPM algorithm. (An alternative formulation with different dependence on $\gamma$ would be to reward $+1$ at the terminal step.)
It would seem very natural to base the RL reward scheme on the success/failure information from the hidden state. However, we found it difficult to converge to a good agent based on this, for the following reason: given a particular starting syndrome, consistent with a distribution of different error strings, most of these are properly corrected by the MWPM algorithm whereas a minority are not. As the syndrome is all that the agent sees, it has no chance to learn to distinguish between these two classes, thus trying to use it for training will only obscure the signal. Nevertheless, for future more advanced tasks, such as dealing with noise biased towards bit or phase flips or with spatial variations it will probably be necessary to explore the use of the fail/success information for the reward scheme.
State-space formulation
-----------------------
Due to the periodic boundary conditions of the code, the syndrome can be represented with an arbitrary plaquette as its center. Centering a defect $e_i$, we define the [*perspective*]{}, $P_i$, of that defect, consisting of the relative positions of all other defects in the syndrome. The set of all perspectives given a syndrome we define as an [*observation*]{}, $O$, as exemplified in Figure \[fig:perspective\]. (The syndrome, observation and perspective all contain equivalent information but represented differently.)
![\[fig:perspective\]State formulation. The toric code syndrome, defines an “observation” that contains the centralized “perspectives” for each defect.](centralize.pdf)
The agent will be given the option of moving any defect one plaquette in any direction (left, right, up, or down), corresponding to performing a bit flip on one of the physical qubits enclosing the plaquette containing the defect. Clearly the total number of available actions varies with the number of defects, which is inconvenient if we want to represent the Q-function in terms of a neural network. In order for the Q network to have a constant-sized output regardless of how many defects are present in the system, each perspective in the observation is instead sent individually to the Q network. Thus, $Q(P,a,\theta)$ represents the value of moving the central defect $a=L,R,U,D$, given the positions of all other defects specified by the perspective $P$, for network parameters $\theta$. The network with input and output is represented graphically in Figure \[fig:network\]. The full Q-function corresponding to a syndrome is given by $\{Q(P,a,\theta)\}_{P\in O}$. When the Q value of each action for each defect has been obtained, the choice of action and defect is determined by a greedy policy. The new syndrome is sent to the algorithm and the procedure is repeated until no defects remain.
Training the neural network\[training\]
---------------------------------------
![\[fig:flowchart2\] Flowchart of the training procedure. A learning sequence consists of an acting stage followed by a learning stage. $(P,a,r)$ is (perspective,action,reward) corresponding to what defect is acted on with what action and giving what reward. $O'$ is the observation (see Fig. \[fig:perspective\]) corresponding to the resulting syndrome. A pseudocode representation is given in Algorithm \[alg:our\]. (SGD is Stochastic Gradient Descent.)](training_new.pdf)
Training of the decoder agent was done using the Deep Q Network (DQN) algorithm [@mnih2015human]. This algorithm utilizes the technique of experience replay in which the experience acquired by the agent is stored as transition tuples in a memory buffer. When updating the Q network (given by parameters $\theta$), a mini-batch of random samples is drawn from this memory buffer. By taking random samples of experience, the temporal correlation of the data is minimized, resulting in a more stable training procedure of the neural network. To further increase the stability of the training, the DQN algorithm makes use of a target Q network (with parameters $\theta_T$) to compute update targets. The target Q network is periodically synchronized with the updated Q network.
A training sequence begins with an acting stage, where a syndrome is sent to the agent, which uses the Q network $\theta$ to suggest a defect perspective, $P$, and an action, $a$. An $\epsilon$-greedy policy is used by the agent, meaning that it will suggest the action with the highest Q-value with probability $(1-\epsilon)$. Otherwise a random action is suggested. The action is performed on the defect, $e$, corresponding to $P$, resulting in a reward, $r$, and a new observation, $O'$, derived from the resulting syndrome. The whole transition is stored as a tuple, $T = (P, a, r, O')$, in a memory buffer. After this, the training sequence enters the learning stage using (mini-batch) stochastic gradient descent. First, a random sample of transitions, $\{T_i = (P_i, a_i, r_i, O'_i)\}_{i=1}^N$, of a given batch size, $N$, is drawn with replacement from the memory buffer. (Here the discrete $C_4$ rotational symmetry of the problem is enforced by including all four rotated versions of the same tuple.) The training target value for the Q-network is given by $$\label{training_target}
y_i=r_i + \gamma \max_{P'\in O'_i;a'}Q(P',a',\theta_T)\,,$$ where $\gamma$ is the discount factor and where the more slowly evolving target network parametrized by $\theta_T$ is used to predict future cumulative award. After this, gradient descent is used to minimize the discrepancy between the targets of the sample and the Q network predictions for it, upgrading the network parameters schematically according to $-\nabla_\theta\sum_i(y_i-Q(P_i,a_i,\theta))^2\,$. A new training sequence is then started, and with some specified rate, the weights of the target network, $\theta_T$, are synchronized with the Q network $\theta$. A pseudocode description of the procedure is presented in algorithm \[alg:our\] and an illustration of the different components and procedures of the training algorithm and how they relate to each other is found in Figure \[fig:flowchart2\].
Get observation $O$ from syndrome Calculate $Q(P,a,\theta)$ using $Q$-network for all perspectives $P\in O$. Choose which defect $e$ to move with action $a$ using $\epsilon$-greedy policy $P \gets $ perspective of defect $e$ Perform action $a$ on defect $e$ $r \gets $ reward from taking action $a$ on defect $e$ $O' \gets $ observation corresponding to new syndrome Store transition tuple $T = (P, a, r, O')$ in memory buffer Draw a random sample of transition tuples Construct targets $y_i$ using target network $\theta_T$ and reward $r_i$ according to Eqn. \[training\_target\]. Update $Q$-network parameters $\theta$ Every $n$ iterations, synchronize the target network with network, setting $\theta_T=\theta$
\[alg:our\]
Result
======
Data sets with a fixed error rate of 10% were generated to train the agent to operate on a code of a specified size. The syndromes in a data set is fed one at a time to the agent, which operates on it until no errors remain. The data sets also contain information about the physical qubit configuration (the hidden state) of the lattice, which (as discussed in section \[RLsection\]) is used to check the success rate of the decoder. This is compared to the performance of the MWPM decoder on the same syndromes [@kolmogorov2009blossom]. The operation of the trained decoder is similar to the cellular automaton decoders[@herold2015cellular; @kubica2018cellular] in the sense of providing step by step actions based on the current state of the syndrome. This also means that it could be implemented in parallel with the error generation process by continuously adapting to the introduction of new errors.
![\[fig:results\]Error correction success rate $p_s$ of the converged agents versus bit-flip error rate $p$, for system size $d=3,5,7$, and compared to the corresponding results using MWPM (lines). (The MWPM decoder for $d=30$ is included as a reference for the approach to large $d$.)](largeErrorResult.pdf)
The proficiency of the well converged agents are shown in figures \[fig:results\] and \[fig:results\_smallp\] as compared to the MWPM performance. Given our specified reward scheme, which corresponds to using as few operations as possible, we achieve near optimal results with a performance which is close to that of the MWPM decoder. For small error rates $p_L\rightarrow 0$ it is possible to derive an exact expression for the MWPM fail rate $p_L$ (see Appendix \[app\_error\] and [@fowler2012surface; @fowler2013optimal]) by explicitly identifying the dominant type of error string. We have checked explicitly that our Q-network agent is equivalent to MWPM for these error strings and thus gives the same asymptotic performance.
For larger system size $d=9$ we have only been partially successful, with good performance for small error rates, but sub-MWPM performance for larger error rates. Given the exponential growth of the state space this is perhaps not surprising, but by scaling up the hardware and the corresponding size of the manageable Q-network we anticipate that larger code distances would be achievable within the present formalism.
As a demonstration of the operation of the trained agent and the corresponding Q-network we present in Figure \[fig:Q\_values\] the action values $Q(S,a)$ for two different syndromes. (As discussed previously, $Q(S,a)=\{Q(P,a,\theta)\}_{P\in O}$, where $O$ is the observation, or set of perspectives, corresponding to the syndrome $S$.) The size of the arrows are proportional to the discounted return $R$ of moving a defect one initial step in the direction of the arrow and then following the optimal policy. In Fig. \[fig:Q\_values\]a, the values are written out explicitly. The best (equivalent) moves have a return $R=-3.57$ which corresponds well to the correct value $R=-1-\gamma-\gamma^2-\gamma^3=-3.62$ for following the optimal policy to annihilate the defects in four steps, with reward $r=-1$ and discount rate $\gamma=.95$. Figure \[fig:Q\_values\]b shows a seemingly challenging syndrome where the fact that the best move does not correspond to annihilating the two neighboring defects is correctly captured by the Q-network.
One interesting aspect of the close to MWPM performance of the fully trained agent is the ability of the Q-network to suggest good actions independently of how many defects are in the syndrome. A $d=7$ system with $p=10$% would start out with a syndrome with maybe 20 defects, which is successively pair-wise reduced down to two and finally zero defects, all based on action-values given by the same Q-network ($\theta$). The network is thus surprisingly versatile and capable, given the enormous reduction of the number of adjustable parameters compared to representing and training the full Q-value function as a table.
![\[fig:Q\_values\]Action value function produced by the Q-network for two different syndromes and code distance $d=7$. The magnitude of the arrows indicate the expected return from taking a next step along the arrow and after that following the optimal policy. The optimal policy for the next move corresponds to the action with the biggest arrow(s). In (a) the expected return is written out explicitly, where the best moves are consistent with the constant reward of $-1$ per step and discounting rate $\gamma=0.95$ used in the training.](Qvalues.pdf)
Conclusions
===========
In conclusion, we have shown how to implement deep reinforcement learning for quantum error correction on the toric code for moderate size systems using uncorrelated bit-flip (or phase-flip) noise. By training an agent to find the shortest paths for the error correction chains we are able to achieve accuracy close to that using a Minimum Weight Perfect Matching decoder. In order to accomplish this we used the deep Q-network formalism that encodes the action-value function by means of an artificial neural network.[@mnih2013playing; @mnih2015human] The construction also made good use of the translational invariance on the torus to be able to efficiently reduce the state space representation.
For future work it will be interesting to see how the formalism generalizes to more advanced noise models, imperfect measurements, as well as more general topological codes. Work in progress[@Eliasson] indicates that the formalism is in fact readily extended to handle depolarizing noise on the toric code by allowing for the full set of $X$, $Y$, and $Z$ qubit actions. By learning to account for correlations between plaquette and vertex errors super-MWPM performance can be achieved. Also using larger and better adapted convolutional networks allow for somewhat larger system sizes to be addressed. Nevertheless, given the exponential growth of the state-action space it is clear that going much beyond the code distances presented in this paper will require parallelization of the training[@horgan2018distributed] as well as massive networks using state of the art hardware, similarly to what is used to achieve super-human performance for board games and computer games.[@silver2017mastering; @mnih2015human] In the longer perspective the main potential of a deep reinforcement learning approach to quantum error correction lies in the fact that is arguably the most promising implementation of AI. Future developments in that area thus opens up also for powerful and flexible machine engineered quantum decoders.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Niklas Forsström, Gustav Karlsson, and Elias Hannouch for contributing to the early stages of this work. We also thank Austin Fowler for valuable discussions. Source code can be found at this url: <https://github.com/phiandre/ToricCodeRL>
\[app\_error\]Small error rate
==============================
As discussed by Fowler [*et al.*]{}[@fowler2012surface; @fowler2013optimal] the likely operating regime of surface code is in the limit of small error rate $p\ll 1$. In addition, in the limit $p\rightarrow 0$ we can derive an exact expression for the rate of logical failure under the assumption of MWPM error correction, thus providing a solid benchmark for our RL algorithm. Such expressions were derived for the surface code in [@fowler2013optimal] and here we derive the corresponding expression for bit-flip errors in the toric code.
Consider first the case of code distance $d$ with $d\in$ odd, which is what we have assumed in the present work. (Using odd $d$ gives an additional simplification of the Q-learning set-up from the fact that any plaquette can be considered the center of the lattice.) As a reminder, the error formulation we use is that every physical qubit has a probability $p$ of bit-flip error, and probability $1-p$ of no error. (In contrast to [@fowler2013optimal] we don’t consider $\sigma^y$ errors, which would give rise to both bit-flip and phase-flip errors.) For very low $p$, we only need consider states with the minimal number of bit-flip errors that may cause a logical failure. One can readily be convinced (from a few examples) that such states are ones where a number $\lceil d/2\rceil$ (e.g. $\lceil 7/2\rceil=4$) of errors are placed along the path of the shortest possible non-trivial (logical) loops. The latter are $d$ sites long, and on the torus there are $2d$ such loops. For such a state MWPM will always fail, because it will provide a correction string which has $\lfloor d/2\rfloor$ bit-flips rather than the $\lceil d/2\rceil$ flips needed to make a successful error correction. The former correction string, together with the initial error string, will sum to one of the non-trivial (shortest length) loops and give rise to a logical bit-flip. The fail-rate $p_L$, i.e. the fraction of logical fails of all generated syndromes, is thus to lowest order in $p$ and for odd $d$ given by $$p_L=2d\binom{d}{\lceil d/2\rceil}p^{\lceil d/2\rceil}\,.$$ Here $2d$ is the number of shortest non-trivial loops, $\binom{d}{\lceil d/2\rceil}$ is the number of ways of placing the errors on such a loop, and $p^{\lceil d/2\rceil}$ is the lowest order term in the probability ($p^{\lceil d/2\rceil}(1-p)^{2d^2-\lceil d/2\rceil}$) of any particular state with $\lceil d/2\rceil$ errors.
Considering d even (for reference), the corresponding minimal fail scenario has $d/2$ errors on a length $d$ loop. Here the MWPM has a 50% chance of constructing either a non-trivial or trivial loop, thus giving the asymptotic fail rate $p_L=d\binom{d}{d/2}p^{d/2}$.
\[app\]Network architecture and training parameters
===================================================
The reinforcement learning agent makes use of a deep convolutional neural network to approximate the Q values for the possible actions of each defect. The network (see Fig. \[fig:network\]) consists of an input layer which is $d\times d$ matrix corresponding to a perspective (binary input, 0 or 1, with 1 corresponding to a defect), and a convolutional layer followed by several fully-connected layers and an output layer consisting of four neurons, representing each of the four possible actions. All layers have ReLU activation functions except the output layer which has simple linear activation.
\# Type Size \# parameters
---- ---------- ------------------------ ---------------
0 Input 5x5
1 Conv. 512 filters; 3x3 size;
2-2 stride 5 120
2 FC 256 neurons 524 544
3 FC 128 neurons 32 896
4 FC 64 neurons 8 256
5 FC 32 neurons 2 080
6 FC (out) 4 neurons 132
573 028
: \[tab:d5\] Network architecture d=5. FC=Fully connected
\# Type Size \# parameters
---- ---------- ------------------------ ---------------
0 Input 7x7
1 Conv. 512 filters; 3x3 size;
2-2 stride 5 120
2 FC 256 neurons 1 179 904
3 FC 128 neurons 32 896
4 FC 64 neurons 8 256
5 FC 32 neurons 2 080
6 FC (out) 4 neurons 132
1 228 388
: \[tab:d7\] Network architecture d=7.
The network architecture is summarized in Table \[tab:d5\] and \[tab:d7\]. We also included explicitly a count of the number of parameters (weights and biases) to emphasize the huge reduction compared to tabulating the Q-function. The latter requires of the order $d^2\choose N_S$ entries, for $N_s$ defects, where $N_s$ will also vary as the syndrome is reduced, with initially $N_S\sim 4pd^2$ as each isolated error creates a defect pair and there are $2d^2$ physical qubits.
![\[fig:validate\]Early training convergence of the Q network agent. Success rate $P_s$ versus number of iterations. One iteration corresponds to annihilating all the defects of a single syndrome. (The very early below $1/4$ success rate is an artifact of using a max count for the number of error correcting steps for the validation.)](training_validation_500.pdf)
In Figure \[fig:validate\] we also provide an example of the initial convergence of the algorithm for lattice size $d\times d$, with $d=3,5,7$. Here, each iteration corresponds to solving one syndrome and making the corresponding number of mini-batch training sessions from the experience buffer, as explained in section \[training\]. A constant set of syndromes is used for the testing so that fluctuations correspond to actual performance variations of the agent.
In Table \[tab:hyper\] we list the hyperparameters related to the Q-learning and experience replay set-up, as well as the neural network training algorithm used. The full RL algorithm is coded in Python using Tensorflow and Keras for the Q-network. A single desktop computer was used, with training converging over a matter of hours (for $d=3$) to days (for $d=7$).
Parameter Value
---------------------------- ----------------------
discount rate $\gamma$ 0.95
reward $r$ -1/step; 0 at finish
exploration $\epsilon$ 0.1
max steps per syndrome 50
mini batch size, $N$ 32
target network update rate 100
memory buffer size 1 000 000
optimizer ’Adam’
learning rate 0.001
beta$_1$ 0.9
beta$_2$ 0.999
decay 0.0
: \[tab:hyper\] Hyperparameters
[^1]: Figures in this section were inspired by lecture notes [@Browne.lecture].
|
---
abstract: 'It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.'
address:
- ' Andreas Savas-Halilaj Department of Mathematics Section of Algebra & Geometry University of Ioannina 45110 Ioannina, Greece '
- ' Knut Smoczyk Institut für Differentialgeometrie and Riemann Center for Geometry and Physics Leibniz Universität Hannover Welfengarten 1 30167 Hannover, Germany '
author:
- '<span style="font-variant:small-caps;">Andreas Savas-Halilaj</span>'
- '<span style="font-variant:small-caps;">Knut Smoczyk</span>'
title: Lagrangian mean curvature flow of Whitney spheres
---
[^1]
Introduction
============
Let $\complex{m}={\mathbb{R}}^m\times{\mathbb{R}}^m$ be the $m$-dimensional complex euclidean space and denote by $\omega$ the standard Kähler 2-form on $\C^m$. An immersion $F:L^m\to\C^m$ of an $m$-dimensional manifold $L^m$ is called *Lagrangian*, if $F^\ast\omega=0$. It is well known that there do not exist embedded Lagrangian spheres in $\C^m$ for $m>1$. However, there are plenty of immersed Lagrangian spheres in $\C^m$. A remarkable family of such Lagrangian immersions of $\Sf^m$ in $\C^m$ is given by the *Whitney spheres* (Figure \[fig whitney\]) $\W_{R,C}:\Sf^m\subset{\mathbb{R}}\times\real{m}\to\C^m$ defined by the maps
![A projection of the Whitney 2-sphere.[]{data-label="fig whitney"}](whitney.png)
$$\W_{R,C}(x_0,x_1,\dots,x_m)=\frac{R}{1+x_0^2}(x_1,\dots,x_m;x_0x_1,\dots,x_0x_m)+C,$$ where here $R$ is a positive constant and $C$ is a point in $\C^m$. We will refer to the quantities $R$ and $C$ as the *radius* and the *center* of the Whitney sphere. Therefore, up to dilations and translations, all Whitney spheres can be identified with $\W=\W_{1,0}$. These spheres admit a single double point at the origin of $\C^m$.
From the definition of $\W$ one immediately observes that the Whitney sphere is invariant under the full group of isometries in $\mathbb{O}(m)$, where the action of ${\rm A}\in\mathbb{O}(m)$ on $\C^m$ is defined by ${\rm A}(x,y):=({\rm A}x,{\rm A}y)$ for all $x,y\in{\mathbb{R}}^m$. Whitney spheres are therefore in a sense the most symmetric Lagrangian spheres that can be immersed into $\C^m$. There do not exist any compact Lagrangian submanifolds in euclidean space with strictly positive Ricci curvature. This follows from the theorem of Bonnet-Myers, the Gauß equation for the curvature tensor and from the fact that the first Maslov class $m_1$ of the Lagrangian submanifold satisfies $m_1=[\Theta_H/\pi],$ where $\Theta_H$ denotes its mean curvature one-form. The Whitney spheres have strictly positive Ricci curvature except at the intersection point, where it vanishes. More precisely, the Ricci curvature of a Whitney sphere can be diagonalized in the form $$\label{eq ricwhitney}
\operatorname{Ric}=\operatorname{diag}\big({2(m-1)}r^2, mr^2,\dots,mr^2\big),$$ where $r$ is the distance of the point to the origin. Lagrangian submanifolds that are invariant under the full action of $\mathbb{O}(m)$ are often called *equivariant* and they have been studied by many authors; see for example [@anciaux; @chen; @groh; @gssz; @viana].
An equivariant Lagrangian submanifold can be described in terms of its corresponding *profile curve* that is defined as the intersection of the Lagrangian submanifold with the plane $$\big\{(\alpha_1,0,\dots, 0;\alpha_2,0,\dots,0)\in\C^m:\alpha_1,\alpha_2\in{\mathbb{R}}\big\}.$$ This profile curve is point symmetric with respect to the origin. For the Whitney sphere one obtains the *figure eight curve* $z:\Sf^1\to\C$ (see Figure \[fig figure eight\]) given by
![The figure eight curve is the profile curve for the Whitney sphere.[]{data-label="fig figure eight"}](figureeight1.png)
$$z(s):=\frac{1}{1+\cos^2(s)}\bigl(-\sin(s),\sin(s)\cos(s)\bigr).$$ The behavior of the Whitney sphere under the mean curvature flow differs significantly from that of a round sphere in euclidean space. In the matter of fact, the submanifolds that evolve from the Whitney sphere cannot be Whitney spheres anymore; they do not simply scale by a time dependent dilation, like round spheres in the euclidean space.
This can be easily seen, because such a behavior would imply that the Whitney spheres are self-shrinking Lagrangian submanifolds with a trivial Maslov class. But as has been shown by Smoczyk [@smoczyk1 Theorem 2.3.5] such self-shrinkers do not exist. More generally, it was later shown by Neves [@neves1] that Lagrangian submanifolds with trivial first Maslov class, so especially Lagrangian spheres of dimension $m>1$, will never develop singularities of [type-I]{}. It seems that type-II singularities are generic singularities in the Lagrangian mean curvature flow, even in the non-zero Maslov class case. For some recent results we refer to [@evans; @smoczyk3]. We would like to mention that not very much is known about the classification of type-II singularities of the mean curvature flow, particularly in higher codimensions. Possible candidates are translating solitons, special Lagrangians, products of these types and various other types of eternal solutions. There exist a number of classification results for translating solitons; see for example [@joyce; @martin; @tian; @xin].
It is therefore in particular clear that a Whitney sphere must develop a type-II singularity. The main objective of this paper is to classify its singularity in more detail. We will prove the following result.
\[thma\] Let $F_0:\Sf^m\to\C^m$, $m>1$, be a smooth equivariant Lagrangian immersion such that the Ricci curvature satisfies $$\label{eq ricci}
\operatorname{Ric}\ge c r^2\hspace{-3pt}\cdot\hspace{-2pt}\operatorname{g},$$ where $c>0$ is a constant, $r$ denotes the distance to the origin and $\operatorname{g}$ is the induced metric. Then the Lagrangian mean curvature flow $$F:\Sf^m\times[0,T)\to\C^m,\qquad\frac{dF}{dt}=H,\qquad F(\cdot ,0)=F_0(\cdot)$$ exists for a maximal finite time $T>0$. The Lagrangian submanifolds stay equivariant and satisfy an inequality $\operatorname{Ric}\ge \varepsilon r^2\cdot\operatorname{g}$, where $\varepsilon>0$ is a time independent constant. The flow develops a type-II singularity and its blow-up converges to a product of a grim reaper (see Figure $\ref{fig grim}$) with a flat $(m-1)$-dimensional Lagrangian subspace in $\C^{m-1}$.
![The grim reaper $\{(x,y)\in\C:\cos y=e^{-x}\}$ is a translating soliton of the mean curvature flow. It moves with speed $1$ in direction of the positive $x$-axis.[]{data-label="fig grim"}](grim.png)
From equation it follows that the Whitney spheres satisfy with $c=m$ and therefore our theorem includes this important case. Since the mean curvature flow is isotropic, it follows that the condition of being equivariant is preserved under the flow. That there exists a finite maximal time interval on which a smooth solution exists is also well known and by the aforementioned result by Neves it is also known that the singularity must be of type-II. The difficulty here is to understand the singularity in more detail. To achieve this, we will need the condition on the Ricci curvature which in view of equation is very natural. To our knowledge this is the first result on the Lagrangian mean curvature flow where a convexity assumption can be used to classify its singularities.
In dimension $m=1$ the equivariance is meaningless, since the flow reduces to the usual curve shortening flow. In this case the behavior of the figure eight curve was studied among others by Altschuler [@altschuler] and Angenent [@angenent]. In particular it was shown that the figure eight curve develops a type-II singularity that after blow-up converges to a grim reaper.
Very recently Viana [@viana] studied the equivariant curve shortening flow for a class $\mathcal{C}$ of antipodal invariant figure eight curves with only one self-intersection which is transversal and located at the origin. In particular he proved that the tangent flow at the origin is a line of multiplicity two, if the curve $\gamma\subset\mathcal{C}$ satisfies at least one the following two conditions: i) $\gamma\cap \mathbb{S}^1(R)$ has at most 4 points for every $R>0$; ii) $\gamma$ is contained in a cone of opening angle $\pi/2$. We would like to mention that neither the class of curves in Viana’s paper nor our class of curves contain each other. For example we allow profile curves with more self-intersections or profile curves that are not contained in a cone of opening angle $\pi/2$. The essential condition in our paper is the positivity of the Ricci curvature away from the origin. Moreover the methods in both papers are substantially different from each other.
Equivariant Lagrangian Submanifolds
===================================
Basic facts {#BF}
-----------
Consider a regular curve $z=(u,v):(-\delta,\delta)\to\C$, $\delta>0$, denote by $S$ the standard embedding of $\mathbb{S}^{m-1}$ into $\real{m}$ and define the map $F:M^m:=(-\delta,\delta)\times\mathbb{S}^{m-1}\to\C^m$ by $$F(s,x):=\big(u(s)S(x),v(s)S(x)\big).$$ The curve $z$ is called the [*profile curve*]{} of $F$. Observe that $F$ gives rise to an immersion if its profile curve does not pass through the origin. Let us denote by $r$ the distance function with respect to the origin of $\C^m$. One can easily check that, if $\{e_1,\dots,e_{m-1}\}$ is a local orthonormal frame field of the sphere, then the vector fields $$E_0=|z'|^{-1}\partial_s,\,\, E_1=r^{-1} e_1,\dots,E_{m-1}=r^{-1}e_{m-1},$$ form a local orthonormal frame on $M^m$ with respect to the Riemannian metric $\gind$ induced by $F$. Moreover, the vector fields $$\nu=-J\cdot DF(E_0),\,\,\nu_1=J\cdot DF(E_1),\dots,\,\,\nu_{m-1}=J\cdot DF(E_{m-1}),$$ where $J$ is the complex structure of $\C^m$, form a local orthonormal frame field of the normal bundle of $F$. Hence $F$ is a Lagrangian submanifold. The shape operators with respect to the above mentioned orthonormal frames have the form $$A^{\nu}=-\left(
\begin{matrix}
k & 0 & \cdots & 0 \\
0 & p & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & p\\
\end{matrix}
\right)
\quad \text{and}\quad
A^{\nu_i}=-\left(
\begin{matrix}
0 & p & \cdots & p \\
p & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
p & 0 & \cdots & 0\\
\end{matrix}
\right),$$ for any $i\in\{1,\dots,m\}$, where $k$ denotes the curvature of the curve $z$ and $p=\langle z,\nu\rangle r^{-2}$.
One can readily check that the mean curvature vector $H$ and the squared norm $|A|^2$ of the second fundamental form of $F$ are given by $$\label{eq sec}
H=-\big(k+(m-1)p\big)\nu\quad\text{and}\quad |A|^2=k^2+3(m-1)p^2,$$
respectively. In particular, one obtains that $$\label{eq pinch}
|A|^2-\frac{3}{m+2}|H|^2=\frac{m-1}{m+2}(k-3p)^2.$$ Moreover, from the Gau[ß]{} equation follows:
\[lemm ricc\] The eigenvalues of the Ricci curvature of an equivariant Lagrangian submanifold are given by $$(m-1)p(k-p)\quad\text{and}\quad p(k-p+(m-2)p),$$ where the latter occurs with multiplicity $m-1$.
As can be easily computed, for the figure eight curve (see Figure \[fig figure eight\]) we have $$k=3r=3p.$$ This fact and Lemma \[lemm ricc\] imply that on the Whitney sphere the eigenvalues of the Ricci curvature are $$2(m-1)r^2\quad\text {and}\quad mr^2,$$ where the latter has multiplicity $m-1$. Observe also (compare with ) that for the Whitney spheres $$|A|^2-\frac{3}{m+2}|H|^2=0.$$ It is well known that for any Lagrangian submanifold one always has the pinching inequality $$|A|^2-\frac{3}{m+2}|H|^2\ge 0.$$ Ros and Urbano [@ru Corollary 3] showed that equality holds on a compact Lagrangian submanifold $L\subset\C^{m}$, if and only if $L$ coincides with a Whitney sphere; see also [@bcm] for another proof.
Equivariant Lagrangian spheres
------------------------------
Now let $z:(-\delta,\delta)\to\C$ be a real analytic regular curve which is passing through the origin only once and such that $z(0)=0$. Then for $m>1$ one can easily verify that the tangent space of $F$, defined as above, smoothly extends over the set $\{0\}\times{\mathbb{S}}^{m-1}$ and gives rise to a smooth equivariant Lagrangian submanifold if and only if the curve $z$ is point symmetric. Note that in this case, the curvature $k$ of $z$ at the origin must be necessarily zero.
On the other hand, if $F:{\mathbb{S}}^m\to\C^m$, $m>1$, is a real analytic equivariant Lagrangian immersion, then it is not difficult to see that its profile curve intersects itself at the origin only once. Since immersions are locally embeddings, each arc of the profile curve passing through the origin must be point-symmetric; because the profile curve is point-symmetric itself, it can only admit two such arcs. This does not hold for $m=1$ since the orbits in this case are not connected.
We summarize this in the following lemma.
\[lemma profile\] If $m>1$ and $F:{\mathbb{S}}^m\to\C^m$ is a real analytic equivariant Lagrangian immersion, then the profile curve $\gamma$ can be parametrized by a point-symmetric real analytic regular curve $z:[-\pi,\pi]\to\C$ such that $z^{-1}(0)=\{-\pi,0,\pi\}$ and $z(-s)=-z(s)$, for all $s\in [-\pi,\pi]$.
\[rem butterfly\] Let us make some comments.
(a) The [*butterfly curve*]{} (Figure \[fig butterfly\]) is an analytic point symmetric curve but not each arc passing through the origin is point-symmetric. Therefore it
![The butterfly curve.[]{data-label="fig butterfly"}](butterfly.png)
cannot be the profile curve of an equivariant Lagrangian sphere, if $m>1$.\
(b) In view of the last lemma it is sufficient to analyze the behavior of the arc $\gamma_\ell:=z([0,\pi])$. Additionally, after rotating the curve we can assume without loss of generality that $z(s)=(x(s),y(s))$ satisfies $x(s)<0$ and $ y(s)>0$ for sufficiently small $s>0$. Throughout the paper we will always make this assumption. The arc $\gamma_\ell$ then starts and ends at the origin and is nonzero elsewhere. It does not have to be embedded and it might wind around the origin, e.g. the arc in Figure \[fig arc\] generates a profile curve of an immersed Lagrangian sphere.
![An immersed arc $\gamma_\ell$, generating a profile curve of an immersed Lagrangian sphere.[]{data-label="fig arc"}](arc.png)
The signed distance function
----------------------------
Since we are interested in equivariant Lagrangian spheres, we may for the rest of the paper assume that $z:\Sf^1\to\C$ is a real analytic profile curve satisfying the conditions in Lemma \[lemma profile\], i.e. we identify $\Sf^1$ with the interval $[-\pi,\pi]$ and assume that $z:[-\pi,\pi]\to\C$ is a closed real analytic curve that satisfies $z(-s)=-z(s),$ for all $s\in[-\pi,\pi]$ and $z^{-1}(0)=\{-\pi,0,\pi\}$. For such curves we may introduce the signed distance function $$r(s):=\operatorname{sign} (s)\cdot|z(s)|.$$
\[lemm crucial1\] Let $r$ be the signed distance function of a real analytic curve $z:[-\pi,\pi]\to\C$ as above. Then $r$ is real analytic and satisfies $$|\nabla r|^2=1-r^2p^2\quad\text{and}\quad\Delta r=rp(p-k).$$
At points where the curve $z$ is not passing through the origin, the analyticity of $r$ is clear. To see that the distance function $r$ is real analytic at points $s_0\in\{-\pi,0,\pi\}$ we represent the curve $z$ locally around $s_0$ as the graph over its tangent line at $s_0$, i.e. locally $z$ can be represented as the graph of an analytic function $y:(-\delta,\delta)\to\real{}$ with $y(0)=y'(0)=0$. This implies $y(s)=s^2\vartheta (s)$ with another real analytic function $\vartheta$. Then the distance function $r$ is locally given by $$r(s)=\operatorname{sign}(s)\sqrt{s^2+s^4\vartheta^2(s)}=s\sqrt{1+s^2\vartheta^2(s)}$$ which clearly is real analytic. Using the frames introduced in Section \[BF\] we see $$2r\nabla r=\nabla r^2= \nabla|z|^2=2\langle \nabla_{E_0}z,z\rangle E_0=2|z'|^{-1}\langle z,z'\rangle E_0.$$ From the decomposition $$\frac{z}{r}=\frac{\langle z,z'\rangle}{r|z'|}\frac{z'}{|z'|}+\frac{\langle z,\nu\rangle}{r}\nu$$ we deduce $$|\nabla r|^2=1-r^2p^2.$$ Since $\Delta z=-k\nu$ we obtain $$2|\nabla r|^2+2r\Delta r=\Delta r^2=2\langle z,\Delta z\rangle+2|\nabla z|^2=-2kr^2p+2$$ which implies the second equation.
Note that the analyticity of $r$ and $|\nabla r|^2=1-r^2p^2$ already imply that $r^2p^2$ must be analytic. From [@gssz Lemma 3.3] we also know that $\lim_{r\to 0}p=0$ since the curvature $k$ at the intersection point must vanish. We will now improve these results further, in fact we will show that $p$ itself is a real analytic function defined on the whole curve.
\[lemm crucial\] Let $\gamma$ be a point-symmetric real analytic arc that passes through the origin only once.
1. There exist a constant $r_0>0$ and a local point symmetric real analytic parametrization $z$ of $\gamma$ in the interval $(-r_0,r_0)$ such that for all $r\in(-r_0,r_0)$ $$|z(r)|=|r|\quad\text{and}\quad z(r)=\sum_{n=0}^\infty z^{(2n+1)}(0)\frac{r^{2n+1}}{(2n+1)!}.$$ In particular $r$ is the signed distance function of $z$ and $|z'(0)]=1$.\
2. The functions $pr^{-1}$ and $(k-3p)r^{-3}$ are real analytic on $\gamma$. In particular $$\label{eq lim s}
\lim_{r\to 0}p=0,\quad \lim_{r\to 0}\frac{k-3p}{r^2}=0.$$
3. If $\gamma$ is not flat, then the function $k/p$ is well defined and real analytic in a neighborhood of $r=0$ and $$\lim_{r\to 0}\frac{k}{p}=2\ell+3,$$ where $2\ell+1$ is the order of the zero of $p$ at $r=0$.
<!-- -->
(a) Observe that $|\nabla r|^2=1-r^2p^2$ and $\lim_{r\to 0}p=0$ imply that at the origin the gradient of the distance function $r$ is non-zero. Hence from the inverse function theorem, the real analyticity of $\gamma$ and the point-symmetry of the arc we conclude that $\gamma$ can locally be parameterized by a Taylor expansion $$z(r)=\sum_{n=0}^\infty z^{(2n+1)}(0)\frac{r^{2n+1}}{(2n+1)!}$$ such that $|z(r)|=|r|$. From $$\left|\frac{z(r)}{r}\right|=1\quad\text { and }\quad\lim_{r\to 0}\frac{z(r)}{r}=z'(0)$$ we get $|z'(0)|=1$.\
(b) Applying part (a), the curve $z$ can be represented locally around the origin as a Taylor series in terms of the signed distance. Since $z$ is an odd function in $r$, its derivative $z'$ must be even. The same holds for the unit normal $\nu$. Hence there exists a Taylor series for $\nu$ which has the form $$\nu(r)=\sum_{n=0}^\infty a_{2n}r^{2n},$$ where $a_{2n}$ are complex numbers. Since $$z'(0)=\lim_{r\to 0}\frac{z}{r}$$ is the unit tangent vector to $z$ at the origin, we must have $$\langle z'(0),a_0\rangle=\langle z'(0),\nu(0)\rangle=0.$$ Therefore $$\langle z(r),\nu(r)\rangle=\sum_{n=0}^\infty b_{2n+1}r^{2n+1},$$ where $$b_1=\langle z'(0), a_0\rangle=0.$$ So $\langle z,\nu\rangle$ has a zero at $r=0$ of order at least $3$. This proves that $pr^{-1}=\langle z,\nu\rangle r^{-3}$ is real analytic on $\gamma$.\
For the second claim we first observe that $$r\nabla p=(k-2p)\nabla r.$$ Representing everything in terms of functions in $r$, we deduce $$k-3p=rp'-p=r^2\left(\frac{p}{r}\right)'.$$ But since $pr^{-1}$ is even, its derivative must have a zero at $r=0$ and hence the function $k-3p$ must have a zero at $r=0$ of order at least $3$. This proves the second claim.\
(c) Since $p$ is real analytic we can represent $p$ locally around $r=0$ as a Taylor series in $r$. Let $2\ell+1$, $\ell\in\mathbb{N}_0$, be the order of the zero of $p$ at $r=0$. Then locally $$p(r)=\sum_{n=\ell}^\infty c_{2n+1}r^{2n+1}$$ with coefficients $c_{2n+1}$. Because $k=rp'+2p$, we see that the zero of $k$ at $r=0$ has the same order $2\ell+1$. If $\gamma$ is not flat, then $p(r)\neq 0$ for sufficiently small $r\neq 0$. Therefore the quotient $k/p$ is well defined and analytic for such $r$ and the isolated singularity at $r=0$ is removable. From $k=rp'+2p$ we then immediately obtain $$\lim_{r\to 0}\frac{k}{p}=\lim_{r\to 0}\frac{rp'}{p}+2=2\ell+3.$$
This completes the proof.
The next lemma will be important in our proofs.
\[est1\] For an equivariant Lagrangian sphere in $\C^m$, $m>1$, the following conditions are equivalent.
1. The Ricci curvature satisfies $\operatorname{Ric}\ge \varepsilon r^2\cdot\operatorname g$ for some positive constant $\varepsilon>0$.
2. There exist constants $\varepsilon_1>0$ and $\varepsilon_2>0$ such that the estimates $p\ge\varepsilon_1\,r$ and $k-p\ge \varepsilon_2\,r$ hold on the profile curve.
Recall from Lemma \[lemm ricc\] that the eigenvalues of the Ricci tensor are given by $$(m-1)p(k-p)\quad\text{and}\quad p(k-p)+(m-2)p^2,$$ where the second eigenvalue has multiplicity $m-1$. Therefore condition (b) clearly implies (a). To show the converse, observe at first that $\operatorname{Ric}\ge \varepsilon\,r^2\cdot \operatorname{g}$ implies $$\frac{p}{r}\cdot\frac{k-p}{r}\ge \frac{\varepsilon}{m-1}.$$ Hence the functions $r^{-1}p$ and $(k-p)r^{-1}$ are nonzero and have the same sign. Taking into account Lemma \[lemm crucial\](b), we deduce $$\begin{aligned}
\frac{\varepsilon}{m-1}&\le&\lim_{r\to 0}\left(\frac{p}{r}\cdot\frac{k-p}{r}\right)\\
&=&\lim_{r\to 0}\left(\frac{p}{r}\cdot\frac{k-3p}{r}\right)+2\lim_{r\to 0}\frac{p^2}{r^2}\\
&=&2\lim_{r\to 0}\frac{p^2}{r^2}.
\end{aligned}$$ Therefore the function $r^{-1}p$ is bounded and non-zero everywhere. Since the profile curve is closed, there exists always a point where $r^{-1}p$ is positive. Hence there exist positive constants $\varepsilon_1,\varepsilon_2$ with $r^{-1}p\ge \varepsilon_1$ and $(k-p)r^{-1}\ge \varepsilon_2$. This completes the proof.
Evolution equations
===================
As was shown in [@gssz], the mean curvature flow of equivariant Lagrangian submanifolds is fully determined by the evolution of their profile curves $z_t:\Sf^1\to\C$, $t\in[0,T)$, under the *equivariant curve shortening flow* (ECSF) given by $$\tag{ECSF}\label{equiv}
\frac{dz}{dt}=-(k+(m-1)p)\nu,$$ where $k(s,t)$ is the curvature, $\nu(s,t)$ the *outward*[^2] directed unit normal of the curve $z_t$ at the point $z_t(s):=z(s,t)$ and $T>0$ is the maximal time of existence of the solution.
Let us define the function $$h:=k+(m-1)p.$$ The statements of the next Lemma were already shown in [@smoczyk2 Lemma 2.2, Lemma 2.3] and [@gssz Lemma 3.12] for curves not passing through the origin. In the sequel the computations of various evolution equations are considered at points where $r\neq 0$, even though some of the quantities might extend smoothly to the origin.
\[lemma 4\] Under the equivariant curve shortening flow the following evolution equations for $z$, the normal $\nu$, the curvature $k$, the induced length element $d\mu$, the distance function $r$ and the driving term $h$ hold. $$\dt d\mu=-hk \, d\mu,\quad\frac{d\nu}{dt}=\nabla h,\quad\frac{d k}{dt}=\lap h+hk^2,$$ $$\frac{d z}{dt}=\Delta z-(m-1)p\nu,\quad \frac{dr}{dt}=\lap r-mrp^2$$ and $$\begin{aligned}
\frac{dh}{dt}&=&\lap h+(m-1)\langle\nabla h,\nabla \log r\rangle\nonumber\\
&&+h\left\{-(m-1)|\nabla \log r|^2+\frac{1}{m}h^2+\frac{m-1}{m}(k-p)^2\right\}.
\end{aligned}$$
Additionally, for any real number $\alpha$ we set $$p_\alpha:=r^{-1-\alpha}p,\quad h_\alpha:=r^{-1-\alpha}h,\quad q_\alpha:=h_\alpha-mp_\alpha=(k-p)r^{-1-\alpha}.$$
\[lemma 4b\] The functions $p_\alpha$, $h_\alpha$ and $q_\alpha$ satisfy the following evolution equations. $$\begin{aligned}
\frac{dp_\alpha}{dt}&=&\Delta p_\alpha+(2\alpha+m+1)\langle\nabla p_\alpha,\nabla \log r\rangle
+2(q_\alpha-2p_\alpha){|\nabla \log r|^2}\nonumber\\
&&+p_\alpha\Big\{\alpha(\alpha+m)|\nabla \log r|^2+m(\alpha+2)p^2+(k-p)^2\Big\},\\
\frac{dh_\alpha}{dt}&=&\Delta h_\alpha+(2\alpha+m+1)\langle\nabla h_\alpha,\nabla \log r\rangle+2phq_\alpha\\
&&+h_\alpha\Big\{\alpha(\alpha+m)|\nabla \log r|^2+m(\alpha+2)p^2+ (k-p)^2\Big\},\\
\frac{dq_\alpha}{dt}&=&\Delta q_\alpha+(2\alpha+m+1)\langle\nabla q_\alpha,\nabla \log r\rangle\nonumber\\
&&+\Big\{\big(\alpha(\alpha+m)-2m\big)q_\alpha+4mp_\alpha\Big\}|\nabla \log r|^2\\
&&+q_\alpha\Big\{m(\alpha+2)p^2+ (k-p)^2+2ph\Big\}.
\end{aligned}$$
For $p_\alpha$ we can use the evolution equation in the proof of [@gssz Lemma 3.26]. This gives in a first step $$\begin{aligned}
\label{qa}
&&\frac{dp_\alpha}{dt}=\Delta p_\alpha+(m-1)\langle\nabla p_\alpha,\nabla \log r\rangle
+2(\alpha+2)q_\alpha|\nabla \log r|^2\\
&&+p_\alpha\left\{-\Bigl((\alpha+2)(\alpha+4)+2m\Bigr)|\nabla \log r|^2+\frac{m(\alpha+2)}{r^2}+(k-p)^2 \right\}.\nonumber
\end{aligned}$$ From the identity $$\label{eq gradpa}
\nabla p_\alpha=(q_\alpha-(\alpha+2)p_\alpha)\nabla \log r$$ we deduce that $$2(\alpha+2)q_\alpha|\nabla \log r|^2=2(\alpha+2)\langle\nabla p_\alpha,\nabla \log r\rangle+2(\alpha+2)^2p_\alpha|\nabla \log r|^2.$$ Moreover we have $$\frac{m(\alpha+2)}{r^2}=m(\alpha+2)(|\nabla \log r|^2+p^2).$$ Substituting the last two equations into and taking into account $$2\langle\nabla p_\alpha,\nabla\log r\rangle=2(q_\alpha-2p_\alpha)|\nabla\log r|^2-2\alpha p_\alpha|\nabla \log r|^2,$$ we obtain the evolution equation for $p_\alpha$.
Let us now compute the evolution equation of $h_\alpha$. From the evolution equations of $r$ and $h$ we get $$\begin{aligned}
\frac{dh_\alpha}{dt}
&=&-(\alpha+1)\frac{h}{r^{2+\alpha}}\Bigl(\Delta r-mrp^2\Bigr)\nonumber\\
&&+\frac{1}{r^{1+\alpha}}\left\{\lap h+(m-1)\langle\nabla h,\nabla \log r\rangle\phantom{\frac{1}{r}}\right.\nonumber\\
&&+\left.h\Bigl(-(m-1) |\nabla \log r|^2+\frac{1}{m}h^2+\frac{m-1}{m}(k-p)^2\Bigr)\right\}\nonumber\\
&=&-(\alpha+1)\frac{h}{r^{2+\alpha}}\,\Delta r+\frac{1}{r^{1+\alpha}}\,\lap h+\frac{m-1}{r^{1+\alpha}}\langle\nabla h,\nabla \log r\rangle\label{eq h1}\\
&&\hspace{-20pt}+h_\alpha\Bigl(-(m-1) |\nabla \log r|^2+\frac{h^2}{m}+\frac{m-1}{m}(k-p)^2+m(\alpha+1)p^2\Bigr).\nonumber
\end{aligned}$$ Moreover we have $$\begin{aligned}
\Delta h_\alpha&=&-(\alpha+1)\frac{h}{r^{2+\alpha}}\,\Delta r+(\alpha+1)(\alpha+2)h_\alpha|\nabla \log r|^2\nonumber\\
&&+\frac{1}{r^{1+\alpha}}\Bigl(\Delta h-2(\alpha+1)\langle\nabla h,\nabla \log r\rangle\Bigr).
\end{aligned}$$ Substituting the last equation into we get $$\begin{aligned}
\frac{dh_\alpha}{dt}
&=&\Delta h_\alpha+\frac{2\alpha+m+1}{r^{1+\alpha}}\langle\nabla h,\nabla\log r\rangle\nonumber\\
&&-(\alpha^2+3\alpha+m+1) h_\alpha|\nabla \log r|^2\nonumber\\
&&+h_\alpha\Bigl(\frac{1}{m}h^2+\frac{m-1}{m}(k-p)^2+m(\alpha+1)p^2\Bigr).\label{eq h2}
\end{aligned}$$ Note that $$\begin{aligned}
&&\frac{1}{m}h^2+\frac{m-1}{m}(k-p)^2+m(\alpha+1)p^2\\
&=&\frac{1}{m}(h^2-(k-p)^2)+(k-p)^2+m(\alpha+2)p^2-mp^2\\
&=&2p(k-p)+(k-p)^2+m(\alpha+2)p^2.
\end{aligned}$$ Then the evolution equation for $h_\alpha$ follows from , $$h_\alpha\cdot 2p(k-p)=2phq_\alpha$$ and from the identity $$\label{eq gradh}
\frac{1}{r^{1+\alpha}}\langle\nabla h,\nabla \log r\rangle= \langle\nabla h_\alpha,\nabla \log r\rangle+(\alpha+1)h_\alpha |\nabla \log r|^2.$$ Finally, the evolution equation for $q_\alpha=h_\alpha-mp_\alpha$ follows directly from those of $p_\alpha$ and $h_\alpha$.
\[est2\] Let the dimension $m$ of the equivariant Lagrangian spheres be at least two.
1. Suppose $\inf_{t=0}r^{-1}p>0$. Then $\inf_{t=t_0}p_\alpha$ is attained for any $\alpha>0$ and any $t_0\in[0,T)$ and $$\min_{t=t_0}p_\alpha\ge \min_{t=0}p_\alpha.$$
2. Suppose $\inf_{t=0}r^{-1}h>0$. Then $\inf_{t=t_0}h_\alpha$ is attained for any $\alpha>0$ and any $t_0\in[0,T)$ and $$\min_{t=t_0}h_\alpha\ge \min_{t=0}h_\alpha.$$
3. Suppose $\inf_{t=0}r^{-1}p>0$ and $\inf_{t=0}(k-p)r^{-1}>0$. Then $\inf_{t=t_0}p_\alpha$ and $\inf_{t=t_0}q_\alpha$ are attained for any $\alpha>0$ and any $t_0\in[0,T)$ and $$\min_{t=t_0} q_\alpha\ge \min\left\{\min_{t=0} q_\alpha,\min_{t=0} p_\alpha/2\right\}.$$
Since $m>1$, Lemma \[lemma profile\] implies that for all $t\in[0,T)$ the profile curves are given by point-symmetric functions $z:[-\pi,\pi]\to\C$ such that $z^{-1}(0)=\{-\pi,0,\pi\}$. In particular, locally around each of the zeros of $z$ we can introduce a signed distance function $r$ and consequently from the real analyticity of $z$ we get a Taylor expansion of $p$, $k$, $h$ in terms of $r$ at these points.
(a) By assumption $\inf_{t=0}r^{-1}p>0$. Thus the first coefficient $c_1$ in the Taylor expansion $$p(r)=\sum _{n=0}^\infty c_{2n+1}r^{2n+1}$$ is positive. Since the coefficients smoothly depend on $t$ and $\alpha>0$, the function $p_\alpha$ will tend to $+\infty$ at points on the curve with $r=0$ on some maximal time interval $[0,\tau)$ where $0<\tau\le T$. Therefore, for any $t_0\in[0,\tau)$ the infimum $\inf_{t=t_0}p_\alpha$ will be attained at some point $s_0$ with $r(s_0)>0$. At such a point we get $$\nabla p_\alpha(s_0)=0\quad\text{and}\quad \Delta p_\alpha(s_0)\ge 0.$$ Since gives at $s_0$ $$0=\langle\nabla p_\alpha,\nabla \log r\rangle=\big(q_\alpha-(2+\alpha)p_\alpha\big){|\nabla \log r|^2},$$ we conclude from Lemma \[lemma 4\] that at $s_0$ $$\frac {d p_\alpha}{dt}\ge p_\alpha\big\{\alpha(\alpha+m+2)|\nabla \log r|^2+m(\alpha+2)p^2+(k-p)^2\big\}.$$ So $p_\alpha$ cannot admit a positive minimum that is decreasing in time. Consequently, $$\inf_{[0,\tau)}p_\alpha\ge\min_{t=0}p_\alpha>0.$$ Since this holds for any $\alpha>0$, we may let $\alpha$ tend to zero and obtain that the first coefficient $c_1$ in the Taylor expansion of $p$ is non-decreasing in time and hence strictly positive for all $t$. In particular $\tau$ must coincide with $T$ and our estimate holds for all $t_0< T$.\
(b) Similarly as above, from the assumption $\alpha>0$ and $\inf_{t=0}r^{-1}h>0$, we conclude that the function $h_\alpha$ will tend to $+\infty$ at points on the curve with $r=0$ on some maximal time interval $[0,\tau)$ where $0<\tau\le T$. In addition for any $t_0\in[0,\tau)$ the infimum $\inf_{t=t_0}h_\alpha$ will be attained at some point $s_0$ with $r(s_0)>0$. Observe that $$2phq_\alpha=2p(k-p)h_\alpha$$ and $$m(\alpha+2)p^2+(k-p)^2+2p(k-p)=k^2+\bigl(m(\alpha+2)-1\bigr)p^2.$$ Then from Lemma \[lemma 4\] we deduce that at a minimum $s_0$ of $h_\alpha$ we have $$\frac {dh_\alpha}{dt}\ge
h_\alpha\big\{\alpha(\alpha+m)|\nabla \log r|^2+k^2+\bigl(m(\alpha+2)-1\bigr)p^2\big\}.$$ Therefore $h_\alpha$ cannot attain a decreasing positive minimum at points where $r> 0$. The rest of the proof is the same as in part (a).\
(c) First note that $$m(\alpha+2)p^2+(k-p)^2+2ph=\bigl(m(\alpha+4)-1\bigr)p^2+k^2\ge 0.$$ Therefore, at a point of a positive minimum of $q_\alpha$ we conclude from part (a) that $$\big(\alpha(\alpha+m)-2m\big)\min_{t=t_0}q_\alpha+4mp_\alpha\ge 2m\big(2\min_{t=0} p_\alpha-\min_{t=t_0} q_\alpha\big).$$ Hence, if $\min_{t=t_0} q_\alpha$ is positive but smaller than $\min_{t=0} p_\alpha$, then the minimum cannot be decreasing. Consequently, we obtain the desired lower bound for the minimum of $q_\alpha$.
This completes the proof.
For $\alpha= 0$, Lemma \[est1\] and Lemma \[est2\] imply the following estimate on the Ricci curvature.
\[lemm ricci\] Let $F_0:\Sf^m\to\C^m$, $m>1$, be an equivariant Lagrangian immersion such that the Ricci curvature satisfies $\operatorname{Ric}\ge cr^2\cdot\gind,$ where $c$ is a positive constant. Then for all $t_0\in[0,T)$ we have $$\inf_{t=t_0}\frac{p}{r}\ge\inf_{t=0}\frac{p}{r}>0,\quad\inf_{t=t_0}\frac{k-p}{r}\ge\inf_{t=0}\frac{k-p}{r}>0.$$ In particular the Ricci curvature of the Lagrangian submanifolds evolving under the mean curvature flow satisfies $\operatorname{Ric}\ge \varepsilon r^2\cdot\gind$, with a positive constant $\varepsilon$ not depending on $t$.
Angles and areas
----------------
Recall that the profile curve $z:[-\pi,\pi]\to\C^m$ of a Lagrangian sphere is point symmetric with respect to the origin. So it can be divided into the union of the two arcs $\gamma_\ell=z\big([0,\pi]\big)$ and $-\gamma_\ell=z\big([-\pi,0]\big)$. It is clear that it is sufficient to study one of these arcs under the equivariant curve shortening flow.
Let $\alpha_0$ denote the angle between the unit normal $\nu$ of $\gamma_\ell$ at $s=0$ and the unit vector $e_1=\partial/\partial x$. Without loss of generality we may assume that $\alpha_0\in(0,\pi/2)$, so that as in Remark \[rem butterfly\](b) for sufficiently small $s$ we have $x(s)<0$ and $y(s)>0$. Then for any $s\in[0,\pi]$ we define the [*Umlaufwinkel*]{} $$\alpha(s):=\alpha_0+\int\limits_0^sk(s)d\mu(s),$$ Then for all $s\in[0,\pi]$ the function $\alpha$ measures the angle between the unit normal $\nu$ and $e_1$ modulo some integer multiple of $2\pi$.
![The relation between the angles $\alpha$, $\beta$ and $\phi$ is $\phi=\alpha+\beta$.[]{data-label="fig angles"}](angles.png)
Similarly we define the [*polar angle*]{} by $$\phi(s):=\phi_0+\int\limits_0^sp(s)d\mu(s).\label{eq phi}$$ where $$\phi_0:=\alpha_0+\pi/2.$$ Then for any $s\in[0,\pi]$ the quantity $\phi(s)$ is the angle between $e_1$ and the normalized position vector $z(s)/r(s)$, again up to some integer multiple of $2\pi$. For $s\to 0$ the normalized position vector converges to the unit tangent vector $e$ at $s=0$.
In addition we define $$\beta(s):=\phi(s)-\alpha(s).$$ Then up to multiples of $2\pi$ the angle $\beta$ is the angle between the unit normal $\nu(s)$ and the normalized position vector $z(s)/r(s)$, i.e. $\cos\beta=rp$ (compare with Figure \[fig angles\]). In particular $\beta_0:=\beta(0)=\pi/2$.
The condition $k>0$ implies that the Umlaufwinkel $\alpha$ is a strictly increasing function in $s$. The polar angle $\phi$ is strictly increasing, if $p>0$. In view of $$\beta=\phi-\alpha,\quad d\alpha=kd\mu\quad\text {and} \quad d\phi=pd\mu$$ we conclude that the condition $k-p>0$ implies that the angle $\beta$ is strictly decreasing in $s$. From this, Lemma \[est1\] and Lemma \[est2\] we conclude the following result.
\[lemm opening\] Let $F_0:\Sf^m\to\C^m$, $m>1$, be an equivariant Lagrangian immersion such that the Ricci curvature satisfies $\operatorname{Ric}\ge cr^2\cdot\gind,$ where $c$ is a positive constant. Then, under the equivariant curve shortening flow, the angles $\alpha$ and $\phi$ remain strictly increasing functions on $[0,\pi]$ for all $t\in[0,T)$, whereas the angle $\beta$ stays strictly decreasing on $[0,\pi]$.
Let $z:[-\pi,\pi]\to\C$ be a profile curve of an equivariant Lagrangian sphere $F_0:\Sf^m\to\C^m$, $m>1$. We define the [*opening angle*]{} $\phi_\ell$ of the loop $\gamma_\ell:=z([0,\pi])$ to be $$\phi_\ell:=\phi(\pi)-\phi(0)=\int\limits_0^\pi p(s)d\mu(s)=\int\limits_{\gamma_\ell}pd\mu.$$
Since the opening angle evolves in time we want to understand its behavior in more detail. To achieve this we will apply the divergence theorem several times. Note that the divergence theorem implies for any smooth function $f$ the identity $$\label{eq div}
\int\limits_{\gamma_\ell}\operatorname{div}\left(\nabla f-\frac{f}{r}\nabla r\right)d\mu=0.$$
According to Lemma \[lemma 4\] $d\mu$ is evolving by $$\dt d\mu=-hk d\mu.$$ Then a straightforward computation shows that for any constant $\varepsilon$ the $1$-form $r^\varepsilon p d\mu$ satisfies $$\nonumber \dt (r^\varepsilon p d\mu)=\Bigl(\operatorname{div}(r^{\varepsilon-1}h\nabla r)-\varepsilon r^{\varepsilon-2}h\Bigr) d\mu.$$ For $\varepsilon=2$ and $\varepsilon=0$ we derive the evolution equations $$\begin{aligned}
\dt(\langle z,\nu\rangle d\mu)
&=&\Bigl(\operatorname{div}(rh\nabla r)-2h\Bigr) d\mu,\nonumber\\ \dt(pd\mu)
&=&\Bigl(\operatorname{div}(h\nabla \log r)\Bigr) d\mu.\nonumber\\$$ From the last equation we can now proceed to compute the evolution of the opening angle $\phi_\ell$. We get $$\begin{aligned}
\dt\phi_\ell&=&\dt\int\limits_{\gamma_\ell}pd\mu=\int\limits_{\gamma_\ell}\operatorname{div}(h\nabla \log r)d\mu\label{eq evolphi}\\
&=&\int\limits_{\gamma_\ell}\operatorname{div}\left(\frac{k-3p}{r}\nabla r\right)d\mu+(m+2)\int\limits_{\gamma_\ell}\operatorname{div}\left(\frac{p}{r}\nabla r\right)d\mu,\nonumber\\
&=&-(m+2)\left(\lim_{s\searrow 0}\frac{p(s)}{r(s)}+\lim_{s\nearrow \pi}\frac{p(s)}{r(s)}\right),\nonumber\end{aligned}$$ where we have used Lemma \[lemm crucial\](b), $$\lim_{r\to 0}|\nabla r|^2=1$$ and the divergence theorem in the last step. Now let $c_0(t)$ denote the first order coefficient in the Taylor expansion of the function $p$ in terms of $r$ at the point $s=0$, i.e. $$p(r,t)=c_0(t)r+\sum_{n=1}^\infty a_{2n+1}(t)r^{2n+1}$$ for some smooth time dependent functions $a_{2n+1}(t)$. Likewise let $c_\pi(t)$ denote the first order coefficient in the Taylor expansion of $p$ in terms of $r$ at the point $s=\pi$, i.e. $$p(r,t)=c_\pi(t)r+\sum_{n=1}^\infty b_{2n+1}(t)r^{2n+1}$$ for some smooth time dependent functions $b_{2n+1}(t)$.
A direct consequence of this and Lemma \[lemm ricc\] is the following conclusion
The opening angle $\phi_\ell$ of the loop $\gamma_\ell$ evolves by $$\dt\phi_\ell=-(m+2)\bigl(c_0(t)+c_\pi(t)\bigr),$$ where $c_0(t)$, respectively $c_\pi(t)$ denote the first order coefficients in the Taylor expansion of $p$ around the points $s=0$ respectively $s=\pi$. In particular, if the initial Ricci curvature satisfies $\operatorname{Ric}\ge cr^2\cdot\gind$ for some positive constant $c$, then in dimension $m>1$ we obtain $$\dt\phi_\ell\le-2(m+2)\inf_{t=0}\frac{p}{r}<0.$$
From the last lemma we see that the initial condition $\operatorname{Ric}\ge cr^2\cdot\gind$ on the Ricci curvature implies that the opening angle is decreasing in time. This gives another geometric interpretation of our assumption in the main theorem.
As above, suppose we have parametrized $z$ in such a way that $\nu$ is the outward unit normal along $\gamma_\ell$. Let $A_\ell$ denote the area (with sign) enclosed by the loop $\gamma_\ell$. By the divergence theorem we must have $$A_\ell=\frac{1}{2}\int\limits_{\gamma_\ell}\langle z,\nu\rangle d\mu.$$ Therefore $$\frac{d A_\ell}{dt}=\frac{1}{2}\int\limits_{\gamma_\ell}\operatorname{div}(rh\nabla r)d\mu
-\int\limits_{\gamma_\ell}h d\mu.$$ Applying the divergence theorem once more and taking into account that the quantity $rh\nabla r$ tends to zero as $r$ tends to zero, we get $$\label{eq evolarea}
\frac{d A_\ell}{dt}=-\int\limits_{\gamma_\ell}h d\mu.$$ On the other hand we have $$\label{eq h}
\int\limits_{\gamma_\ell}hd\mu=\int\limits_{\gamma_\ell}(k-p)d\mu+m\int\limits_{\gamma_\ell}pd\mu=\int\limits_{\gamma_\ell}(k-p)d\mu+m\phi_\ell.$$ The evolution equation for $k$ and $d\mu$ in Lemma \[lemma 4\] imply $$\dt\int\limits_{\gamma_\ell}kd\mu=\int\limits_{\gamma_\ell}\Delta h\, d\mu\overset{\eqref{eq div}}{=}\int\limits_{\gamma_\ell}\operatorname{div}(h\nabla\log r)\, d\mu\overset{\eqref{eq evolphi}}{=}\dt\int\limits_{\gamma_\ell}pd\mu.$$ Hence $$\dt\int\limits_{\gamma_\ell}(k-p)d\mu=0$$ and from , we obtain $$\frac{d^2}{dt^2}A_\ell=-m\dt\phi_\ell=-m(m+2)\Bigl(c_0(t)+c_\pi(t)\Bigr).$$ In particular the initial condition $\operatorname{Ric}\ge cr^2\cdot\gind$ implies that in dimension $m>1$ the enclosed area $A_\ell$ of the loop $\gamma_\ell$ is a strictly decreasing and strictly convex function in $t$.
Rescaling the singularity
=========================
In this section we will rescale the singularities and prove Theorem A. To this end let us first recall some general facts about singularities.
The maximal time $T$ of existence of a smooth solution to the mean curvature flow of a compact submanifold $M^m\subset\real{n}$ must be finite. The following general theorem is well known and shows how one can analyze forming singularities of the mean curvature flow by parabolic rescalings around points where the norm of the second fundamental form attains its maximum; for details see [@chen-he], [@tao Section 2.16] and the references therein.
\[blow\] Let $F:M\times [0,T)\to {\mathbb{R}}^n$ be a solution of the mean curvature flow, where $M$ is compact, connected and $0<T<\infty$ is the maximal time of existence of a smooth solution. There exists a point $x_\infty\in M$ and a sequence $\{(x_j,t_j)\}_{j\in\natural{}}$ of points in $M\times[0,T)$ with $\lim x_j=x_\infty$, $\lim t_j=T$ such that $$|A(x_j,t_j)|=\max_{M\times[0, t_j]}|A(x,t)|=:a_j\to\infty.$$
Consider the family of maps $F_j:M\times[L_j,R_j)
\to{\mathbb{R}}^n$, $j\in\natural{}$, given by $$F_j(x,\tau):=F_{j,\tau}(x):=a_j\big(F(x,\tau/{a^{2}_j}+t_j)-\Gamma_j\big),$$ where $L_j:=-a^2_jt_j$, $R_j:=a^2_j(T-t_j)$ and $\Gamma_j:=F(x_j,t_j)$. Then the following holds:
1. The family of maps $\{F_j\}_{j\in\natural{}}$ evolve in time by the mean curvature flow. The norm $|A_j|$ of the second fundamental form of $F_j$ satisfies the equation $$|A_j(x,\tau)|=a^{-1}_j|A(x,\tau/a_j^2+t_j)|.$$ Moreover, for any $\tau\le 0$ we have $|A_j(x,\tau)|\le 1$ and $|A_j(x_j,0)|=1$ for any $j\in\natural{}$.
2. For any fixed $\tau\le 0$, the sequence of pointed Riemannian manifolds $$\big\{M,F^*_{j,\tau}\langle\,\cdot,\cdot\rangle,\Gamma_j\big\}_{j\in\natural{}}$$ smoothly subconverges in the Cheeger-Gromov sense to a connected complete pointed Riemannian manifold $(M_\infty,g_\infty,\Gamma_\infty)$ that does not depend on the choice of $\tau$.
3. There is an ancient solution $F_\infty:M_\infty\times(-\infty,0]\to{\mathbb{R}}^n$ of the mean curvature flow, such that for each fixed time $\tau\le 0$, the sequence $\{F_{j,\tau}\}_{j\in\natural{}}$ smoothly subconverges in the Cheeger-Gromov sense to $F_{\infty,\tau}$. Additionally, $|A_{F_\infty}|\le 1$ and $|A_{F_\infty}(x_\infty,0)|=1.$
4. If the singularity is of type-II, then $R_j\to\infty$ and the limiting flow $F_\infty$ can be constructed on the whole time axis $(-\infty,\infty)$ and thus gives an eternal solution of the mean curvature flow.
The following lemma will be crucial for the classification of the blow-up curves.
\[lemm ray\] Let $z:[0,\infty)\to\C$ be a real analytic curve, parametrized by arc length $\sigma$ such that $z(0)=0$, $k\ge p\ge 0$ and $$\int\limits_0^\infty p(\sigma)d\sigma<\infty.$$ Then $p\equiv 0$ and $z$ is a ray.
As in let $$\phi(\sigma)=\phi_0+\int\limits_0^\sigma p(\sigma)d\sigma$$ be the polar angle and define $$\phi_\infty:=\phi_0+\int\limits_0^\infty p(\sigma)d\sigma.$$ Since $p\ge 0$, $\phi$ is an increasing function.
We will show $\phi\equiv\phi_0$, hence $p\equiv 0$, and distinguish two cases.
[**Case 1.**]{} Suppose there exists $\sigma_0\in[0,\infty)$ such that $\phi(\sigma_0)=\phi_\infty$. Then from the monotonicity of $\phi$ we get $\phi(\sigma)=\phi_\infty$ for all $\sigma\ge \sigma_0$ and then the real analyticity implies $\phi(\sigma)=\phi_\infty$ for all $\sigma\in[0,\infty)$. But then in particular $\phi_0=\phi_\infty$ and $p\equiv 0$.
[**Case 2.**]{} In this case we have $\phi(\sigma)<\phi_\infty$ for all $\sigma\in[0,\infty)$. Since $\phi$ is monotone and bounded, there exists $\sigma_0\in[0,\infty)$ such that $z_{|[\sigma_0,\infty)}$ can be represented as a graph over the flat line passing through the origin in direction of $v=(\cos\phi_\infty,\sin\phi_ \infty)$. After a rotation around the origin we may assume without loss of generality that $\phi_\infty=\pi$ such that $z_{|[\sigma_0,\infty)}$ can be represented as the graph of a real analytic function $y:(-\infty,s_0]\to\real{}$, $s_0<0$, where $y$ is a strictly decreasing concave function, because $k\ge p\ge 0$ (see Figure \[fig limit\]).
On the interval $(-\infty,s_0]$ the function $rp=\cos\beta$ is given by $$\cos\beta=\frac{sy'-y}{\sqrt{(1+(y')^2)(s^2+y^2)}}.$$ Since $y$ is concave, $s<0$ and $(sy'-y)'=sy''\ge 0$, we conclude that for $s<s_0$ we must have $$\label{eq estimate 1}
y'(s)\ge \frac{y(s)}{s} +\frac{s_0y'(s_0)-y(s_0)}{s}.$$ Since $y'\le 0$ and $y''\le 0$ we observe that $$a:=\lim_{s\to-\infty}y'(s)$$ exists and is non-positive. From inequality we conclude that $$a\ge\lim_{s\to-\infty}\frac{y(s)}{s}.$$ Now for $\tau<s_0$ let $$\epsilon(s):=y'(\tau)s+b(\tau)$$ be the tangent line to the graph of the function $y$ passing through the point $z(\tau)=(\tau,y(\tau))$ (see Figure \[fig limit\]).
![The angle $\beta$ between the normal vector and the position vector tends to $\pi/2$ for $s\to-\infty$.[]{data-label="fig limit"}](limit.png)
Since $y$ is concave we derive $y(s)\le \epsilon(s),$ for all $s\le s_0.$ Therefore we obtain the estimate $$\label{eq estimate 2}
\frac{y(s)}{s}\ge y'(\tau)+\frac{b(\tau)}{s},\quad\text{ for all }s\le \tau.$$ Passing to the limit we obtain that $$\lim_{s\to-\infty}\frac{y(s)}{s}\ge y'(\tau).$$ Since the above inequality holds for any $\tau<s_0$ we may let $\tau$ tend to $-\infty$ which implies $$\lim_{s\to-\infty}\frac{y(s)}{s}\ge a.$$ Thus we have shown $$\lim_{s\to-\infty}(y'(s)-y(s)/s)=0$$ and this yields $$\lim_{s\to-\infty}\cos\beta(s)=0.$$ We will now see that this implies that $\cos\beta\equiv 0$ on the whole curve $z$. First note that the graphical property implies that for sufficiently large $\sigma$ the distance function $r$ is a strictly increasing function in terms of the arc length parameter $\sigma$, so that $r'=\partial r/\partial\sigma>0$. Since $(\cos\beta)'=(rp)'=(k-p)r'$ we conclude that the function $\cos\beta$ is increasing for sufficiently large $\sigma$. Because $\cos\beta=rp\ge 0$ and $\lim_{\sigma\to\infty}\cos\beta(\sigma)=0$, this implies $\cos\beta(\sigma)=0$ for all sufficiently large $\sigma$ and then the real analyticity also implies the same for all $\sigma\in[0,\infty)$. Hence $p\equiv 0$ and $\phi\equiv\phi_0$ as claimed.
[**Proof of Theorem A.**]{} Suppose that $F:{\mathbb{S}}^m\times[0,T)\to\C^m$ is the evolution by Lagrangian mean curvature flow of a Lagrangian sphere satisfying the assumptions of Theorem A. Let $$a_j=\max_{{\mathbb{S}}^m\times[0, t_j]}|A|,$$ where $\{t_j\}_{j\in\natural{}}$ is an increasing sequence with $\lim_{j\to\infty}t_j=T$. Following Proposition \[blow\], take a sequence $\{x_j\}_{j\in\natural{}}$ in $\Sf^m$ such that $$|A|(x_j,t_j)=a_j.$$ Let $$F_j(x,\tau)=a_j\bigl(F(x,\tau/a_j^2+t_j)-F(x_j,t_j)\bigr)$$ be the rescaled Lagrangian spheres and denote the distance between the rescaled blow-up point (the origin) and the rescaled intersection point $-a_jF(x_j,t_j)$ by $$r_j:=a_jr(x_j,t_j).$$ [**Claim.**]{} $\limsup_{j\to\infty}r_j=\infty$.
Indeed, suppose to the contrary that $\limsup_{j\to\infty} r_j=r_\infty<\infty.$ Then, following Proposition \[blow\], we obtain an eternal [*equivariant*]{} Lagrangian mean curvature flow $F_\infty$ generated by limit profile curves $z_\infty:M_\infty\to\C$, where $M_\infty$ is a complete pointed Riemannian manifold of dimension one. Since $F_\infty$ is an eternal solution and eternal solutions in euclidean space are known to be non-compact, $M_\infty$ is diffeomorphic to $\real{}$. The limit profile curves $z_\infty$ evolve again by an equivariant curve shortening flow with a different center, which without loss of generality can be assumed to be the origin. Because $k$ and $p$ have the same scaling behavior, the limit flow still satisfies $$rk\ge rp\ge 0.$$ For any fixed time $\tau\in\real{}$ there exists a local parametrization of $z_\infty(M_\infty\times\{\tau\})$ by a real analytic arc $z:[0,\infty)\to\C$, parametrized by arc length $\sigma$ such that $z(0)=0$ and $k\ge p\ge 0$ and $k\not\equiv 0$, because $|A|^2=k^2+3(m-1)p^2$. Moreover, since by Lemma \[lemm opening\] the opening angle is decreasing, we conclude that $$\int\limits_0^\infty p(\sigma)d\sigma<\infty.$$ Applying Lemma \[lemm ray\] and using the point-symmetry we obtain that $z_\infty(M_\infty\times\{\tau\})$ must be straight line, which is a contradiction. This proves the claim.
Instead of taking a blow-up sequence for $F$ we may, equivalently, consider a blow-up sequence for the corresponding profile curves $z$. Therefore consider a sequence of points $\{s_j\}_{j\in\natural{}}$ in $\Sf^1$ such that $$\bigl(k^2+(m-1)p^2\bigr)(s_j,t_j)=a_j^2.$$ Following the statements of Proposition \[blow\], let us define the family of curves $z_j:\Sf^1\times[L_j,R_j)\to\C$, $j\in\natural{}$, given by $$z_j(s,\tau):=z_{j,\tau}(s):=a_j\big(z(s,\tau/{a^{2}_j}+t_j)-\gamma_j\big),$$ where $L_j=-a^2_jt_j$, $R_j=a^2_j(T-t_j)$ and $\gamma_j=z(s_j,t_j)$. One can easily verify that the curves $\{z_j\}_{j\in\natural{}}$ evolve by the equation $$\frac{dz_j}{d\tau}=-\left(k_j+(m-1)\frac{\langle z_j+a_j\gamma_j,\nu_j\rangle}{|z_j+a_j\gamma_j|^2}\right)\nu_j,$$ where $$k_j(s,\tau)=a^{-1}_jk(s,\tau/a_j^2+t_j)$$ is the signed curvature and $$\nu_j(s,\tau)=\nu(s,\tau/a_j^2+t_j)$$ is the outer unit normal of the rescaled curve $z_{j,\tau}$ at $s$. Note that with the notation from above we have $$r_j=a_j|\gamma_j|=a_jr(s_j,t_j).$$ From Cauchy-Schwarz’ inequality we obtain $$\left|\frac{\langle z_j+a_j\gamma_j,\nu_j\rangle}{|z_j+a_j\gamma_j|^2}\right|\le\frac{1}{|z_j+a_j\gamma_j|}.$$ For large $r_j$ we can thus estimate $$\left|\frac{\langle z_j+a_j\gamma_j,\nu_j\rangle}{|z_j+a_j\gamma_j|^2}\right|\le\frac{1}{r_j-|z_j|}.$$ So for each $s\in{\mathbb{S}}^1$ where $z_j(s,\tau)$ converges as $j\to\infty$ we conclude that $$\lim_{j\to\infty}\frac{\langle z_j+a_j\gamma_j,\nu_j\rangle}{|z_j+a_j\gamma_j|^2}=0.$$ Since the intersection point after rescaling tends to infinity, this and the point-symmetry show that the limit flow $z_\infty$ in this case is an eternal and weakly convex non-flat solution of the curve shortening flow. It is well known by results of Altschuler [@altschuler] and Hamilton [@hamilton] that such a solution is a grim reaper. It is clear that the parabolic rescaling of the Lagrangian spheres then converge in the Cheeger-Gromov sense to the product of a grim reaper with a flat Lagrangian space. $\square$
[^1]: Supported by DFG SM 78/6-1
[^2]: Here *outward* means that $\{\nu,z'\}$ forms a positively oriented basis of $\C$.
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abstract: 'In the framework of a holographic QCD approach we study an influence of matters on the deconfinement temperature, $T_c$. We first consider quark flavor number ($N_f$) dependence of $T_c$. We observe that $T_c$ decreases with $N_f$, which is consistent with a lattice QCD result. We also delve into how the quark number density $\rho_q$ affects the value of $T_c$. We find that $T_c$ drops with increasing $\rho_q$. In both cases, we confirm that the contributions from quarks are suppressed by $1/N_c$, as it should be, compared to the ones from a gravitational action (pure Yang-Mills).'
author:
- Youngman Kim
- 'Bum-Hoon Lee'
- Siyoung Nam
- Chanyong Park
- 'Sang-Jin Sin'
title: Deconfinement phase transition in holographic QCD with matter
---
Introduction
============
To understand QCD phase structure at finite temperature and/or density has been an fascinating theme in hadron physics. Some of studies so far are basically based on phenomenological models or effective field theories of QCD such as chiral perturbation theory (ChPT) and Nambu-Jona-Lasinio (NJL) model. Lattice QCD has been a powerful tool for QCD phase diagram at finite temperature and, recently, at finite density.
Recently interesting developments in AdS/CFT [@adscft] to study strongly interacting system such as QCD, which goes under the name of AdS/QCD, have been made. Confinement is realized with an infrared (IR) cut off $z_{IR}$ in AdS space [@polchinski], and flavors are introduced by adding extra probe branes [@karch]. More phenomenological approaches were also suggested to construct a holographic model dual to QCD, for example, [@EKSS; @PR; @Brodsky]. The deconfinement temperature is estimated in a Hawking-Page type transition analysis in cutoff AdS space in [@Herzog:2006ra]. In this analysis, the contribution from mesons (quarks) are not considered, since they are suppressed by $1/N_c$ compared to the gravitational part, and consequently, $T_c$ may correspond to that of pure Yang-Mills. The quark (baryon) chemical potential is introduced through AdS/CFT to study physics of dense matter [@dAdSQCD].
In the present work, we consider contributions from mesons, motivated largely by lattice QCD results, to see flavor $N_f$ and quark number density dependence of $T_c$. As it is well know from many lattice QCD results so far, the details of the transition strongly depends on the number of quark flavors. The transition temperature with no dynamical quarks is roughly $270~{\rm MeV}$, for instance see [@Karsch], while with dynamical quarks it is around $170~{\rm MeV}$ [@KLP; @Aoki]. In addition, $T_c$ will decreases with increasing $N_f$ [@KLP]. Apart from the temperature axis in the QCD phase diagram, lattice QCD is now investigating the QCD equation of state at nonzero chemical potential [@FK; @AEH]. In [@FK], it is shown that the transition temperature decreases with the baryon chemical potential up to $1~{\rm GeV}$. Finally, we remark that $T_c$ in the present work denotes the critical temperature of the deconfinement (first order) phase transition, while at low density there is a cross over as observed in lattice QCD [@FK; @Aoki; @Bernard]. We attribute this difference to the large $N_c$ nature of the Hawking-Page type transition.
A Hawking-Page type transition with quark flavors
=================================================
A study [@Herzog:2006ra] based on a Hawking-Page type transition in the AdS/QCD models calculated the deconfinement temperature. In the analysis, the contribution from mesons are not considered, since they are suppressed by $1/N_c$ compared to the gravitational part: the gravitational coupling scales as $\kappa \approx g_s \approx 1/N_c$, and the contribution from the mesons scales only as $N_c$. Here we briefly summarize the analysis of [@Herzog:2006ra] done in the hard wall model [@EKSS; @PR]. In the hard wall model, the AdS space is compactified such that $z_0<z<z_{IR}$, where $z_0\rightarrow 0$. The value of $z_{IR}$ is fixed by the rho-meson mass ($m_\rho$) at zero temperature: $m_\rho (\simeq 770~{\rm MeV})\simeq3\pi/(4z_{IR})$$\rightarrow$ $1/z_{IR}\simeq 320~{\rm
MeV}$[@EKSS; @PR].
The Euclidean gravitational action given by $$\label{action1}
S_{grav} ~=~ -\frac{1}{2\kappa^2} \int d^5x \sqrt{g}\left(\textrm{R}+\frac{12}{L^2}\right)$$ where $\kappa^2 = 8\pi G_5$ and $L$ is the length scale of the $AdS_5$, there are two relevant solutions for the equations of motion derived from the above action.The one is cut-off thermal AdS(tAdS) with the line element $$ds^2=\frac{L^2}{z^2}\left(d\tau^2+dz^2+d\vec{x}^2_3\right),$$ where the radial coordinate runs from the boundary of tAdS space $z=0$ to the cut-off $z_{IR}$, which corresponds to an infrared cut-off in energies proportional to $1/z_{IR}$ from the point of view of the boundary dual theory. The other solution is cut-off AdS black hole(AdSBH) with the line element $$ds^2=\frac{L^2}{z^2}\left(f(z)d\tau^2+\frac{dz^2}{f(z)}+d\vec{x}^2_3\right)$$ where $f(z)=1-(z/z_h)^4$ and $z_h$ is the horizon of the black hole. Note that there will be no cut-off in this space for $z_{IR}<z_h$. The Hawking temperature of the black hole solution is $T = 1/(\pi z_h)$ which is given by regularizing the metric near the horizon. In the tAdS case, the periodicity in the Euclidean time-direction is fixed by comparing two geometries at an UV cut-off $\epsilon$ where the periodicity of the time-direction in both cases is locally the same. Then, the time periodicity of tAdS is given by $$\beta = \pi z_h \sqrt{f(\epsilon)}.$$ Now we calculate the action density $V$, which is defined by the action divided by the common volume factor of $R^3$. The regularized action density of the tAdS is given by $$V_1(\epsilon) = \frac{4L^3}{\kappa^2} \int^{\beta '}_{0} d\tau \int^{z_{IR}}_{\epsilon}\frac{dz}{z^5}\, ,$$ and that of the AdSBH is given by $$V_2(\epsilon) = \frac{4L^3}{\kappa^2}\int^{\pi z_h}_{0} d\tau \int^{\bar{z}}_{\epsilon}\frac{dz}{z^5}$$ where $\bar{z} = min(z_{IR},z_h)$. Then, the difference of the regularized actions is given by $$\Delta V_g = \lim_{\epsilon\rightarrow 0}\left[V_2(\epsilon) -V_1(\epsilon)\right]= \left\{\begin{array}{ll} \frac{L^3 \pi z_h}{\kappa^2}
\frac{1}{2z_h^4} & z_{IR} < z_h\\ \\
\frac{L^3 \pi z_h}{\kappa^2} \left( \frac{1}{z_{IR}^4} - \frac{1}{2z_h^4}\right)
& z_{IR} > z_h.\end{array}
\right.$$ This is the result of [@Herzog:2006ra] in the hard wall model. When $\Delta V_g$ is positive(negative), tAdS (the black hole) is stable. Thus, at $\Delta V_g=0$ there exists a Hawking-Page transition. In the first case $z_{IR} < z_h$, there is no Hawking-Page transition and the thermal AdS is always stable. In the second case $z_{IR} > z_h$, the Hawking-Page transition occurs at $$\label{tempads}
T_0 = 2^{1/4}/(\pi z_{IR})$$ and at low temperature $T < T_0$ (at high temperature $T > T_0$) the thermal AdS (the AdS black hole) geometry becomes a dominant background.
Now, we include the bulk matter into the theory. The action for mesons is given by $$\label{action2}
S_{matter} ~=~ M_5 \int d^5x \sqrt{g}~ \textrm{Tr} \left[\frac{1}{2}|D_\mu \Phi|^2
+ \frac{1}{2} M_{\Phi}^2|\Phi^2| +\frac{1}{4}\left(F_{L}^2 +F_{R}^2\right)\right]$$ where $L^2 M_{\Phi}^2=-3$, $D_\mu \Phi=\partial_\mu \Phi +i A_{L\mu} \Phi -i\Phi A_{R\mu}$, $A_{L,R} = A_{L,R}^a t^a$ and $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu +i[A_\mu,A_\nu]$. The vector fields and the axial vector fields are defined by $V=(A_L +A_R)/\sqrt{2}$ and $A=(A_L -A_R)/\sqrt{2}$ respectively. For later convenience, we present some relations here: $$\frac{1}{\kappa^2}=\frac{1}{8\pi G_5}, ~~~~ \frac{1}{G_5}=\frac{32N^2_c}{\pi L^3},
~~ ~~ \textrm{and}~~~~M_5=\frac{N_c}{12\pi^2 L},$$ where the second relation comes from Ref. [@Csaki:2006ji]. In this section, we study the quark flavor number dependence of the critical temperature of the Hawking-Page type transition at zero density. Here we assume that there is no condensate in the vector and axial-vector channel, and so we turn off the vector gauge fields temporarily until next section. On thermal AdS background, the solution of the equation of motion for a scalar field is given by $$v(z) = a z + b z^3 \, ,$$ where $v(z)=<\Phi>$. According to the AdS/CFT correspondence, the coefficient $a$ is a mass of the boundary quark and $b$ corresponds to chiral condensate $<\bar qq>$. When considering the AdS black hole background, the boundary theory becomes a finite temperature field theory. The equation of motion of the scalar field is $$\left[\partial_z^2 -\frac{4-f}{zf}\partial_z +\frac{3}{z^2f}\right]v(z) =0$$ and the solution of the equation is given by [@GY; @Kim:2006ut] $$\label{sinbh}
v(z) = a \cdot _2F_1\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2},
\frac{z^4}{z_h^4}\right) z + \tilde b \cdot _2F_1\left(\frac{3}{4}, \frac{3}{4},
\frac{3}{2}, \frac{z^4}{z_h^4} \right) z^3 .$$ For simplicity, we take the chiral limit where the quark mass is zero, $a=0$. We assume that chiral symmetry restoration and the deconfinement take place at the same temperature, and so the chiral condensate is zero in the deconfined phase. Since AdS black hole background corresponds to a deconfined phase of the boundary theory, we have $\tilde b=0$. At low temperature described by the thermal AdS background, the boundary theory corresponds to a confined phase, so $b\neq 0$. Therefore, the scalar field solutions in two backgrounds are given by $v_{tAdS}\simeq b z^3$ and $v_{BH}\simeq 0$.
Now we calculate the matter contribution to the Hawking-Page transition. For the thermal AdS, the scalar field contribution to the action is $$V_{1m} = \frac{L^3\pi z_h}{2}\cdot 3M_5 N_f b^2z^2_{IR}.$$ In the black hole background, since $v_{BH}\simeq 0$ there is no contribution of the scalar field, i.e. $V_{2m}=0$. Therefore, the values of $\Delta V_m$ of the matter part for $z_{IR} < z_h$ and $z_{IR} > z_h$ are the same : $$\Delta V_m = -\frac{L^3 \pi z_h}{\kappa^2}\cdot \frac{b_t^2 L^2 N_f z_{IR}^2}{32N_c}.$$ Note that the parameter $b$ is given by $$a \simeq 0 ~~\textrm{and}~~ b \simeq \frac{\xi}{L z_{IR}^3}$$ where $\xi \simeq4$ [@PR]. Finally, the total difference of the action in each background becomes $$\Delta V = \left\{\begin{array}{ll}\frac{L^3 \pi z_h}{2\kappa^2}\left[ \frac{1}{z_h^4}
- \left(\frac{N_f}{N_c}\right)\frac{\xi^2}{16 z_{IR}^4}\right] & z_{IR} < z_h\\ \\
\frac{L^3 \pi z_h}{2\kappa^2} \left[ \frac{2}{z_{IR}^4} - \frac{1}{z_h^4}
- \left(\frac{N_f}{N_c}\right)\frac{\xi^2}{16 z_{IR}^4}\right] & z_{IR} > z_h.
\end{array} \right.$$ If we define the critical temperature of the pure $AdS_5$ gravity (pure Yang-Mills) as $T_0 =\frac{1}{\pi z_h}
=\frac{2^{1/4}}{\pi z_{IR}}$ following (\[tempads\]) the critical temperatures modified by mesons is given by $$T = \left\{\begin{array}{ll} {T}_0 \left(\frac{\xi^2}{32}
\frac{N_f}{N_c}\right)^{\frac{1}{4}} & z_{IR} <z_h\\ \\
{T}_0 \left(1
-\frac{\xi^2}{32}\frac{N_f}{N_c}\right)^{\frac{1}{4}} & z_{IR}>z_h.\end{array}\right.$$ In the first case $z_{IR} <z_h$, a Hawking-Page type transition occurs, which seems unphysical, since it implies that there is a phase transition at low temperature. Unfortunately, we have no clear understanding on this phase transition. If we include, however, the back-reaction of the matter field on the background and work on a deformed AdS metric, then the unphysical phase transition may disappear.
In the second case $z_{IR}>z_h$, at high (low) temperature the AdS black hole (the thermal AdS) is again stable, but due to the matter field, the critical temperature is smaller than that of the pure AdS gravity theory, which is consistent with an observation made in [@KLP].
A Hawking-Page type transitions at finite density
=================================================
In this section, we consider the quark number density dependence on the deconfinement temperature. In QCD, quark chemical potential introduced as $\mu_q\bar\psi\gamma_0\psi$, and so according to an AdS/CFT dictionary we need to introduce a bulk U(1) field in AdS$_5$ whose boundary value is $\mu_q$. To this end, we follow [@DH] and generalize the symmetry to U$(N_f)\times $U$(N_f)$. The equation of motion for the time component of the U(1) vector field is given by $$\partial_z\left[\frac{1}{z}\partial_z V_{\tau}(z)\right] = 0\, ,$$ and a solution of the equation of motion is given by $$V_{\tau} = c_1 + c_2 z^2\, .$$ Since the factors $g^{\tau\tau}g^{zz}$ in the equation of motion for tAdS and AdSBH backgrounds are same, we conclude that the equations of motion of both backgrounds are same and the corresponding forms of the solutions are also same. That is, we can use the same form of the solution $V_{\tau}$ in both background tAdS and AdSBH. According to the AdS/CFT correspondence, the coefficient of the non-normalizable term, $c_1$, is proportional to coupling with the dual operator of the boundary theory. Since the time component of the U(1) vector field is dual to the quark number current, $c_1$ must correspond to the quark chemical potential. Meanwhile, the coefficient of the normalizable term, $c_2$, corresponds to the expectation value of the dual operator so that $c_2$ is interpreted as the quark number density, $c_2=12\pi^2\rho_q/N_c$ [@DH].
Following the same procedure used in the previous section, we arrive at $$V_{v1} = \pi z_h M_5 N_f L^5 c_2^2 z_{IR}^2$$ for the tAdS and $$V_{v2}=\left \{\begin{array}{ll} \pi z_h M_5 N_f L^5 c_2^2 z_{h}^2 & z_h<z_{IR}\\
\\\pi z_h M_5 N_f L^5 c_2^2 z_{IR}^2 & z_h>z_{IR} \end{array} \right.$$ for AdSBH. From these results, the differences of the action reads $$\Delta V_{v}=\left\{\begin{array}{ll} -\pi z_h M_5 N_f L^5 c_2^2(z^2_{IR}- z_{h}^2)
& z_h<z_{IR}\\ \\0 & z_h>z_{IR} \end{array} \right.$$ The final result for $z_h<z_{IR}$ is $$\label{hp2}
\Delta V = \frac{L^3 \pi z_h}{\kappa^2}\left[ \frac{1}{z_{IR}^4} -\frac{1}{2z_h^4}
-\frac{L^4 N_f c^2_2}{48N_c}\left(z^2_{IR}- z_{h}^2\right)\right]$$ When $\Delta V < 0$, the dominant geometry is a AdS black hole and the boundary theory corresponding to this geometry is in a confined phase. As in Ref. [@Herzog:2006ra], in the pure gravity theory, the critical temperature is given by $T_0 = \frac{2^{1/4}}{\pi z_{IR}}$. So when considering the quark density, using the relation $T = 1/(\pi z_h)$, the critical temperature $T_c$ is determined by solving the following equation $$\label{eTc}
0~=~ T^4 - \frac{2}{\pi^4} \left( \frac{1}{z_{IR}^4}
-\frac{L^4 N_f }{48N_c} c^2_2 \left( z^2_{IR} - \frac{1}{\pi^2 T^2} \right) \right)\Bigg{|}_{T =T_c} .$$ In (\[eTc\]), the last term $z^2_{IR}- z_{h}^2$, which is from quark number density, is positive, and so the critical temperature at finite density is always lower than that of the pure gravity theory. Now, we consider a case where $z_h\ll z_{IR}$ to see the quark number dependence of $T_c$ clearly. We note here that typically $z_h^2/z_{IR}^2\approx 0.6$.
In this case, the critical temperature is given by $$\label{Tc}
T_c (\rho_q) = \frac{2^{1/4}}{\pi} \left( \frac{1}{z_{IR}^4}
-\frac{L^4 N_f z^2_{IR}}{48N_c} c^2_2 \right)^{1/4},$$ which is a consistent result with the lattice QCD data. Note here again that $c_2\sim \rho_q$ [@DH]. In Fig. \[TcF\], we plot Eq. (\[eTc\]), together with the lattice result [@FK; @FKtalk], where $R\equiv T_c(\rho_q)/T_0$ and $\bar\rho_q =\rho_q z_{IR}^3$. Our result in Fig. \[TcF\] is consistent with lattice QCD results at low density, [*i.e*]{}, see [@FK].
Finally, we combine the results obtained in the previous and in this sections. The final form of $\Delta V$ is given by $$\Delta V = \left\{
\begin{array}{ll}
\frac{L^3 \pi z_h}{2\kappa^2}\left[ \frac{1}{z_h^4} - \left(\frac{N_f}{N_c}\right)
\frac{\xi^2}{16 z_{IR}^4}\right] & z_{IR} < z_h\\ \\ \frac{L^3 \pi z_h}{2\kappa^2}
\left[ \frac{2}{z_{IR}^4} - \frac{1}{z_h^4}- \left(\frac{N_f}{N_c}\right)
\frac{\xi^2}{16 z_{IR}^4} -\frac{L^4 N_f c^2_2}{48N_c}\left(z^2_{IR}- z_{h}^2\right)
\right] & z_{IR} > z_h. \end{array} \right.$$
Discussion
==========
We have studied the effects of matters on the deconfinement transition in the context of a Hawking-Page type analysis. We observed, as it should be, that the corrections from mesons to the deconfinement temperature are suppressed by $1/N_c$. We found that $T_c$ decreases with the number of quark flavor $N_f$. At finite density, we obtained the density dependence of $T_c$, and it shows a similar behavior calculated in lattice QCD.
We remark here that, as it is well known, in the presence of the dynamical quarks the Polyakov loop is no longer a good order parameter to describe the deconfinement transition [@BU]. In lattice QCD, however, it has been established that the Polyakov loop could be served as an approximate order parameter for the deconfinement transition. While, one of the key quantities that connects the Hawking-Page transition to the deconfinement is the Polyakov loop. In the present work, we assume that the the gravity description of confinement/deconfinement through the Hawking-Page transition is still viable with dynamical quarks. It may be interesting to see how the expectation value of the Polyakov loop behaves in the gravity side, when the dynamical quarks in the fundamental representation are present.
The work of Bum-Hoon Lee, Siyoung Nam and Chanyong Park was supported by the Science Research Center Program of the Korea Science and Engineering Foundation through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021.
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abstract: 'In the space $L_2({{\mathbf R}}^d)$ we consider the Schrödinger operator $H_\gamma=-\Delta+ V({{\mathbf x}})\cdot+\gamma W({{\mathbf x}})\cdot$, where $V({{\mathbf x}})=V(x_1,x_2,\dots,x_d)$ is a periodic function with respect to all the variables, $\gamma$ is a small real coupling constant and the perturbation $W({{\mathbf x}})$ tends to zero sufficiently fast as $|{{\mathbf x}}|\rightarrow\infty$. We study so called virtual bound levels of the operator $H_\gamma$, that is those eigenvalues of $H_\gamma$ which are born at the moment $\gamma=0$ in a gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0=-\Delta+ V({{\mathbf x}})\cdot$ from an edge of this gap while $\gamma$ increases or decreases. For a definite perturbation $(W({{\mathbf x}})\ge 0)$ we investigate the number of such levels and an asymptotic behavior of them and of the corresponding eigenfunctions as $\gamma\rightarrow 0$ in two cases: for the case where the dispersion function of $H_0$, branching from an edge of $(\lambda_-,\lambda_+)$, is non-degenerate in the Morse sense at its extremal set and for the case where it has there a non-localized degeneration of the Morse-Bott type. In the first case in the gap there is a finite number of virtual eigenvalues if $d<3$ and we count the number of them, and in the second case in the gap there is an infinite number of ones, if the codimension of the extremal manifold is less than $3$. For an indefinite perturbation we estimate the multiplicity of virtual bound levels. Furthermore, we show that if the codimension of the extremal manifold is at least $3$ at both edges of the gap $(\lambda_-,\,\lambda_+)$, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, that is $\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset$ for a small enough $|\gamma|$.'
author:
- Leonid Zelenko
title: '**Virtual bound levels in a gap of the essential spectrum of the Schrödinger operator with a weakly perturbed periodic potential**'
---
[**Mathematics Subject Classification 2000,**]{} Primary: 47F05, Secondary: 47E05, 35Pxx
[**Keywords.**]{} Schrödinger operator, perturbed periodic potential, coupling constant, virtual bound levels, asymptotic behavior of virtual bound levels
Introduction {#sec:introduction}
============
Background {#subsec:background}
----------
In this paper we consider the Schrödinger operator $$\label{dfHgm}
H_\gamma=-\Delta+ V({{\mathbf x}})\cdot+\gamma W({{\mathbf x}})\cdot$$ acting in the space $L_2({{\mathbf R}}^d)$, where $V({{\mathbf x}})=V(x_1,x_2,\dots,x_d)$ is a periodic function with respect to all the variables and satisfying some mild condition which will be pointed below, the function $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$ and $\gamma$ is a small real coupling constant. In what follows we shall impose on the perturbation $ W({{\mathbf x}})$ some conditions, which mean that it tends to zero sufficiently fast as $|{{\mathbf x}}|\rightarrow\infty$ in an integral sense. We study so called virtual bound levels of the operator $H_\gamma$, that is those eigenvalues of $H_\gamma$ which are born at the moment $\gamma=0$ in a gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0=-\Delta+ V({{\mathbf x}})\cdot$ from an edge of this gap while $\gamma$ increases or decreases. In physics they are called “resonance levels” or “trapped levels”.
A wide literature is devoted to the study of discrete spectrum in gaps of the essential spectrum. In [@Rof] and [@Rof1] for the one-dimensional case (d=1) the tests for finiteness and infiniteness of the number of eigenvalues of the operator $H_1=H_\gamma\vert_{\gamma=1}$ in the gaps of the spectrum of the unperturbed operator $H_0$ (the impurity spectrum) were obtained. In [@Zl1] in the one-dimensional case a local dilative perturbation of the periodic potential $V(x)$ was studied, which under some condition adds an infinite number of eigenvalues in a gap of the spectrum of the unperturbed operator. In [@Ros] the asymptotic formula was obtained for the counting function of negative eigenvalues of the operator $H_1$ (for $d=1$) accumulating to the bottom of its essential spectrum in the case where $V({{\mathbf x}})\equiv 0$ (a perturbation of the motion of a “free” electron). In the general case (a perturbation of the motion of an electron in a periodic lattice) the analogous asymptotic formula was obtained in [@Zel], [@Zel1], [@Khr], [@Khr1], [@Rai] and [@Schm] for the counting function of eigenvalues accumulating to an edge of a gap of the essential spectrum of the operator $H_1$. The asymptotic behavior of the discrete spectrum of the operator $H_\gamma$ in a gap of its essential spectrum is well studied for a “strong coupling”, that is for $|\gamma|\rightarrow\infty$ ([@Bi2], [@Bi-So], [@Sob]).
But there is a comparatively small number of results concerning the behavior of the discrete part of the spectrum of $H_\gamma$ for a “weak coupling”, that is for $|\gamma|\rightarrow 0$. In the paper of M. Sh. Birman [@Bi1] (1961) a variational approach has been worked out for the study of birth of negative eigenvalues under a small perturbation in the case $V({{\mathbf x}})\equiv 0$. In the $1970$’s in a series of papers the asymptotic behavior as $\gamma\rightarrow 0$ of negative eigenvalues and the corresponding eigenfunctions of the Schrödinger operator $H_\gamma$ was studied for $V({{\mathbf x}})\equiv 0$ with the help of analytical methods ([@Re-Si], [@S], [@S1], [@Kl], [@B-G-S]). These investigations were based on the explicit form of the Green function for the unperturbed operator $H_0=-\Delta$ and on the Birman-Schwinger principle, which describes the discrete spectrum of the perturbed operator in the gaps of the spectrum of the unperturbed one with the help of so called Birman-Schwinger operator, defined by (\[BrmSchwop\]) ([@Bi3], [@Sc], [@Re-Si], [@S]).
The interest in this subject was renewed in the last two decades. In [@Wei] T. Weidl has developed the Birman approach for the study of the existence of virtual eigenvalues for a wide class of elliptic differential operators of high order and even for indefinite perturbations. In the papers [@Ar-Zl1] and [@Ar-Zl2] the negative virtual eigenvalues were studied for the perturbation $(-\Delta)^l+\gamma W({{\mathbf x}})\cdot$ of the polyharmonic operator $(-\Delta)^l$ with the help of an analytical method. The Green function of the unperturbed operator $(-\Delta)^l$ was not constructed there explicitly, but by using the Fourier transform (the “momentum representation”) a representation for the resolvent $((-\Delta)^l-\lambda I)^{-1}$ of the unperturbed operator was obtained near the bottom $\lambda=0$ of its spectrum, which permit to get asymptotic formulas for the negative virtual eigenvalues of the perturbed operator with the help of the Birman-Schwinger principle. Observe that for the unperturbed operator $(-\Delta)^l$ the dispersion function (dependence of the energy $\lambda$ on the momentum ${{\mathbf p}}$) has the form $\lambda=|{{\mathbf p}}|^{2l}$, hence it has one minimum point ${{\mathbf p}}=0$. Thanks this fact, for a short range perturbation $W({{\mathbf x}})$ (that is, or it has a compact support, or tends to zero sufficiently fast as ${{\mathbf x}}\rightarrow\infty$) the perturbed operator $(-\Delta)^l+\gamma W({{\mathbf x}})\cdot$ has a finite number of negative virtual eigenvalues. In particular (for $l=1$), this property holds for the perturbation of the Laplacian. But in [@Ch-M] for $d=2$ an axially symmetric Hamiltonian describing a spin-orbit interaction was considered such that the minima set of its dispersion function is a circle (non-localized degeneration of the dispersion function). By the use of the method of separating variables it was shown in [@Ch-M] that in the presence of an arbitrarily shallow axially symmetric potential well $W({{\mathbf x}})$ an infinite number of negative eigenvalues of the perturbed Hamiltonian appear. With the physical point of view in this case an infinite number of electrons have the minimal energy level of the unperturbed Hamiltonian, hence an infinite number of bound energy levels can appear beneath this level in the presence of some small impurities in the system. For a more general situation the analogous result was established in [@Br-G-P] and [@Pan] by using the variational method. In [@H-S] a general situation of unperturbed Hamiltonian (“kinetic energy”) is considered such that the minima set of its dispersion function is a submanifold of codimension one. The existence of an infinite number of virtual negative eigenvalues is proved there and asymptotic formulas are obtained for them and corresponding eigenfunctions for a small coupling constant. The method used in [@H-S] is close to one used in [@Ar-Zl1], [@Ar-Zl2].
In [@P-L-A-J] with the help of the variational method the existence of virtual eigenvalues in gaps of the essential spectrum of the operator $H_\gamma$, defined by (\[dfHgm\]), is proved. But this method does not permit to investigate the asymptotic behavior of virtual eigenvalues and of the corresponding eigenfunctions for $\gamma\rightarrow 0$.
Description of methods and results {#subsec:description}
----------------------------------
In the present paper we investigate the number of virtual eigenvalues in a gap of the essential spectrum of the operator $H_\gamma$ and obtain asymptotic formulas for them and for the corresponding eigenfunctions for $\gamma\rightarrow 0$. The method we use is close to one used in [@Ar-Zl1], [@Ar-Zl2] and [@H-S], but instead of the Fourier transform we apply the so called Gelfand-Fourier-Floquet transform (\[unit\]) (a “quasi-momentum representation”of the Hamiltonian), which realizes a unitary equivalence between the unperturbed operator $H_0=-\Delta+V({{\mathbf x}})\cdot$ and a direct integral over the Brillouin zone $B$ of a family of operators $\{H({{\mathbf p}})\}_{{{\mathbf p}}\in B}$, generated on a fundamental domain $\Omega$ of the lattice of periodicity of $V({{\mathbf x}})$ by the operation $-\Delta+V({{\mathbf x}})\cdot$ and some cyclic boundary conditions ([@Gel], [@Wil], [@Kuch], [@Zl]). In this situation the role of the dispersion function for the operator $H_0$ plays the dependence of the energy $\lambda$ (the eigenvalue of $H({{\mathbf p}})$) on the quasi-momentum ${{\mathbf p}}$: $\lambda=\lambda({{\mathbf p}})$. Like in [@Ar-Zl1], [@Ar-Zl2], our considerations are based on a representation of the resolvent $(H_0-\lambda I)^{-1}$ of the unperturbed operator near the edge $\lambda=\lambda_+$ $(\lambda=\lambda_-)$ of a gap $(\lambda_-,\,\lambda_+)$ of its spectrum as a sum of a singular (w.r.t. $\lambda$) part and a regular remainder. We consider this representation in two cases of behavior of the dispersion function $\lambda({{\mathbf p}})$ branching from the edge $\lambda_+$ $(\lambda_-)$ of the gap $(\lambda_-,\,\lambda_+)$ and having the extremal set $F^+=\lambda^{-1}(\lambda_+)$ $(F^-=\lambda^{-1}(\lambda_-))$: the case of a non-degenerate edge $\lambda_+$ $(\lambda_-)$ (Proposition \[prrprres\]) of the Morse’s type in the sense that at any point of the set $F^+\;(F^-)$ the Hessian operator of $\lambda({{\mathbf p}})$ is non-degenerate, and the case of a non-localized degeneration of the Bott-Morse type (Proposition \[prrprres1\]) in the sense that the set $F^+\;(F^-)$ is a smooth submanifold of a non-zero dimension and at each point of $F^+\;(F^-)$ the Hessian operator of $\lambda({{\mathbf p}})$ is non-degenerate along the normal subspace to $F^+\;(F^-)$. In both cases we assume that $\lambda({{\mathbf p}})$ is a simple eigenvalue of the operator $H({{\mathbf p}})$ for any ${{\mathbf p}}\in F^+\;({{\mathbf p}}\in F^-)$[^1]. As it is clear, if the edge $\lambda_+$ $(\lambda_-)$ of a gap $(\lambda_-,\,\lambda_+)$ is non-degenerate, then the set $F^+\;(F^-)$ is finite. Notice that the representation of the resolvent of $H_0$ in the non-degenerate case given by Proposition \[prrprres\] is close to one given by Corollary 4.2 of [@Ger], but in our case we estimate also the integral kernel of the remainder of this representation. These representations of the resolvent enable us to extract a singular portion from the Birman-Schwinger operator (Propositions \[rprBirSchw\], \[rprBirSchwdeg\]), which yields the leading terms of the asymptotic formula for the virtual eigenvalues of $H_\gamma$ as $|\gamma|\rightarrow 0$. It is known that in the one-dimensional case ($d=1$) the edges of all the spectral gaps of the operator $H_0$ are non-degenerate ([@Titch]). As it was shown in [@Kir-S] (Theorem 2.1), the bottom of the spectrum of the operator $H_0$ is non-degenerate even in the multi-dimensional case. Taking the periodic potential in the form $V({{\mathbf x}})=\sum_{k=1}^dV_k(x_k)\;({{\mathbf x}}=(x_1,x_2,\dots,x_d))$ and using the method of separating variables, it is not difficult to construct an example of the multi-dimensional operator $H_0$ having a finite spectral gap with non-degenerate edges.
In Theorem \[thnondegedg\] we count the number of virtual eigenvalues of $H_\gamma$ in a spectral gap $(\lambda_-,\,\lambda_+)$ of $H_0$ being born from its non-degenerate edge $\lambda_+\;(\lambda_-)$ and obtain asymptotic formulas for them as $|\gamma|\rightarrow 0$ for a definite perturbation (that is, $W({{\mathbf x}})\ge 0$ on ${{\mathbf R}}^d$) under the assumption that $W({{\mathbf x}})$ tends to zero sufficiently fast in an integral sense. For $d=1$ there is only one virtual eigenvalue being born from the edge $\lambda_+\;(\lambda_-)$ of the spectral gap of $H_0$ for $\gamma<0\;(\gamma>0)$ and the leading term of the asymptotic formula for the distance between this virtual eigenvalue and the edge $\lambda_+\;(\lambda_-)$ has the order $O(\gamma^2)$ as $\gamma\uparrow 0\;(\gamma\downarrow 0)$. For $d=2$ the number of virtual eigenvalues being born from the edge $\lambda_+\;(\lambda_-)$ of the spectral gap of $H_0$ for $\gamma<0\;(\gamma>0)$ coincides with the number of points of the extremal set $F^+\;(F^-)$ of the dispersion function branching from $\lambda_+\;(\lambda_-)$ and the leading terms of the asymptotic formulas for the distances between these virtual eigenvalues and the edge $\lambda_+\;(\lambda_-)$ has an exponential order as $\gamma\uparrow 0\;(\gamma\downarrow 0)$. Furthermore, Theorem \[thnondegedg\] claims that for $d\le 2$ the eigenfunctions corresponding to the virtual eigenvalues of $H_\gamma$ converge in some sense as $\gamma\uparrow 0\;(\gamma\downarrow 0)$ with the rate $O(\gamma)$ to linear combinations of Bloch functions of $H_0$ corresponding to the energy level $\lambda_+\;(\lambda_-)$ and the quasi-momenta from $F^+\;(F^-)$. For $d\ge 3$ there is a threshold for the birth of the impurity spectrum of $H_\gamma$ in the spectral gap $(\lambda_-,\,\lambda_+)$ of $H_0$ for $\gamma<0\;(\gamma>0)$, that is for a small enough $\gamma<0\;(\gamma>0)$ there is no eigenvalue of $H_\gamma$ in $(\lambda_-,\lambda_+)$. Furthermore, for $d=2$ Theorem \[thnondegedg\] yields an asymptotic formula of Lieb-Thirring type for the sum of inverse logarithms of distances between the virtual eigenvalues and the edge $\lambda_+\;(\lambda_-)$, and its leading term is expressed explicitly via the perturbation $W({{\mathbf x}})$ and spectral characteristics of the unperturbed operator $H_0$ at this edge: Bloch functions and effective masses of electrons at the energy level $\lambda_+\;(\lambda_-)$.
Theorem \[thdegedg\] describes the birth of virtual eigenvalues of $H_\gamma$, under a definite perturbation, from the edge $\lambda_+\;(\lambda_-)$ of a spectral gap $(\lambda_-,\,\lambda_+)$ of $H_0$, which has a non-localized degeneration of the Morse-Bott type. The character of this birth depends on the codimension $\rm{codim}(F^+)\;(\rm{codim}(F^-))$ of the extremal submanifold $F^+\;(F^-)$ of the dispersion function branching from $\lambda_+\;(\lambda_-)$. For $\rm{codim}(F^+)\le
2\;(\rm{codim}(F^-)\le 2)$ there is an infinite number of virtual eigenvalues of $H_\gamma$ being born from this edge, and the asymptotic behavior of them as $\gamma\uparrow 0\;(\gamma\downarrow
0)$ for $\rm{codim}(F^+)=1\;(\rm{codim}(F^-)=1)$ and $\rm{codim}(F^+)=2\;(\rm{codim}(F^-)=2)$ is analogous to one in the non-degenerate case for $d=1$ and $d=2$ respectively, and furthermore, the asymptotic behavior of the corresponding eigenfunctions is analogous to one in the non-degenerate case for $d\le 2$. If $\rm{codim}(F^+)\ge 3\;(\rm{codim}(F^-)\ge
3)$, then for $\gamma<0\;(\gamma>0)$ there is a threshold for the birth of the impurity spectrum of $H_\gamma$ in the spectral gap $(\lambda_-,\,\lambda_+)$ of $H_0$ like in the non-degenerate case for $d\ge 3$. Furthermore, for $\rm{codim}(F^+)\le 2\;(\rm{codim}(F^-)\le 2)$ Theorem \[thdegedg\] yields a weak version of the asymptotic formula of Lieb-Thirring type, mentioned above, for the sum of square roots of distances between virtual eigenvalues and the edge $\lambda_+\;(\lambda_-)$ if $\rm{codim}(F^+)=1\;(\rm{codim}(F^-)=1)$, and for the sum of inverse logarithms of ones if $\rm{codim}(F^+)=2\;(\rm{codim}(F^-)=2)$. The leading terms of these formulas have the form of the integrals over $F^+\;(F^-)$, whose integrands are expressed explicitly via the perturbation $W({{\mathbf x}})$ and spectral characteristics of the unperturbed operator $H_0$ at the edge $\lambda_+\;(\lambda_-)$, mentioned above (merely in this case the effective masses are computed in the directions normal to the extremal submanifold).
Theorem \[thestmult\] treats the case of an indefinite perturbation ($W({{\mathbf x}})$ may change the sign) and the non-degenerate edge $\lambda_+\;(\lambda_-)$. It yields an estimate of the multiplicity of virtual eigenvalues if $d\le 2$, and for $d\ge 3$ it claims the existence of a threshold for the birth of the impurity spectrum from this edge.
Theorem \[ththresholddeg\] claims the existence of a threshold for the birth of the impurity spectrum from the degenerate edge $\lambda_+\;(\lambda_-)$ for an indefinite perturbation, if $\rm{codim}(F^+)\ge 3\;(\rm{codim}(F^-)\ge 3)$.
In the Appendix we prove Theorem \[thmainApp\], which yields some kind of an elliptic regularity result: under a mild condition for the periodic potential $V({{\mathbf x}})$ it claims that if a branch of eigenvalues $\lambda({{\mathbf p}})$ of the family of operators $H({{\mathbf p}})$, mentioned above, is holomorphic and a branch of corresponding eigenfunctions $b(\cdot,{{\mathbf p}})$ is holomorphic in the $L_2(\Omega)$-norm, then the latter branch is holomorphic in the $C(\Omega)$-norm. For $d\le 3$ this claim follows immediately from results of the paper [@Wil], but for $d\ge 4$ the arguments used there fail and we use a modification of them. Corollary \[cormainApp\] of Theorem \[thmainApp\] is used in the proof of the main results.
The paper is organized as follows. After this Introduction, in Section \[sec:notation\] we introduce some basic notation, in Section \[sec:preliminaries\] (Preliminaries) we recall some known facts concerning the operator $H_0$ with the periodic potential and define the notion of a virtual eigenvalue. In Section \[sec:mainres\] we formulate the main results. In Section \[sec:prmainres\] we prove the main results. In Section \[sec:resunprop1\] we obtain the representation of the resolvent of the operator $H_0$, mentioned above. Section \[sec:appendix\] is the Appendix. We add the label “A” to numbers of claims and formulas from the Appendix.
Basic notation {#sec:notation}
==============
${{\mathbf x}}\cdot{{\mathbf y}}\;({{\mathbf x}},{{\mathbf y}}\in{{\mathbf R}}^d)$ is the canonical inner product in the real vector space ${{\mathbf R}}^d$; $|{{\mathbf x}}|=\sqrt{{{\mathbf x}}\cdot{{\mathbf x}}}$ is the Euclidean norm in ${{\mathbf R}}^d$;
$S^{d}$ is the $d$-dimensional unit sphere: $S^d=\{{{\mathbf x}}\in{{\mathbf R}}^{d+1}:\;|{{\mathbf x}}|=1\}$; $s_d$ is the $d$-dimensional volume of $S^{d}$;
${{\mathbf T}}^d=\times_{k=1}^d S^1$ is the $d$-dimensional torus;
${{\mathbf Z}}$ is the ring of integers;
${{\mathbf Z}}^d=\times_{k=1}^d {{\mathbf Z}}$;
${{\rm Dom}}(f)$ is the domain of a mapping $f$;
$(f,g)\;(f,g\in{{\mathcal H}})$, $\Vert f\Vert$ are the inner product and the norm in a complex Hilbert space ${{\mathcal H}}$ (in particular, in $L_2({{\mathbf R}}^d)$); the norm of linear bounded operators, acting in ${{\mathcal H}}$, is denoted in the same manner;
$L_{2,0}({{\mathbf R}}^d)$ is the set of all functions from $L_2({{\mathbf R}}^d)$ having compact supports;
If $A$ is a closed linear operator acting in a Hilbert space ${{\mathcal H}}$, then:
$\ker(A)$ is the kernel of $A$, i.e. $\ker(A)=\{x\in{{\mathcal H}}:\;Ax=0\}$;
$\mathrm{Im}(A)$ is the image of $A$;
$\sigma(A)$ is the spectrum of $A$;
${{\mathcal R}}(A)$ is the resolvent set of $A$, i.e. ${{\mathcal R}}(A)={{\mathbf C}}\setminus\sigma(A)$;
$R_\lambda(A)\;\;(\lambda\in{{\mathcal R}}(A))$ is the resolvent of $A$, i.e. $R_\lambda(A)=(A-\lambda I)^{-1}$.
${{\mathcal B}}(E)$ is the Banach space of linear bounded operators, acting in a Banach space $E$.
Some specific notation will be introduced in what follows.
Preliminaries {#sec:preliminaries}
=============
Spectral characteristics of the unperturbed operator {#subsec:spectchar}
----------------------------------------------------
Consider the unperturbed operator $H_0=-\Delta+ V({{\mathbf x}})\cdot$. We assume that the potential $V({{\mathbf x}})$ is periodic on the lattice $\Gamma=\{{{\mathbf l}}\in{{\mathbf R}}^d\;|\;{{\mathbf l}}=(l_1T_1,l_2T_2,\dots,l_dT_d),\;{{\mathbf k}}=(l_1,l_2,\dots,l_d)\in{{\mathbf Z}}^d\}$ ($T_k>0\,(k=1,2,\dots,d)$), that is $V({{\mathbf x}}+{{\mathbf l}})=V({{\mathbf x}})$ for any ${{\mathbf x}}\in{{\mathbf R}}^d$ and ${{\mathbf l}}\in\Gamma$. For the simplicity we shall assume that $T_1=T_2=\dots=T_d=1$, that is $\Gamma={{\mathbf Z}}^d$. Denote by $\Omega$ the fundamental domain of the lattice $\Gamma$: $\Omega:=\times_{k=1}^d[-1/2,1/2]$. Furthermore, assume that $$\label{condperpotent}
V\in\left\{\begin{array}{ll}
L_2(\Omega), &\rm{if}\quad d\le 3\\
\bigcup_{q>\frac{d}{2}}L_q(\Omega), &\rm{if}\quad d\ge 4.
\end{array}\right.$$ By claim (ii) of Proposition \[propselfadj\], the operator $H_0$ with the domain $W_2^2({{\mathbf R}}^d)$ is self-adjoint and bounded below.
For any ${{\mathbf p}}\in{{\mathbf T}}^d$ consider the operator $H({{\mathbf p}})$, generated by the operation $h=-\Delta+V({{\mathbf x}})\cdot$ in the Hilbert space ${{\mathcal H}}_{{{\mathbf p}}}$ of functions $u\in L_{2,loc}({{\mathbf R}}^d)$ satisfying the condition $$\label{Htau}
u({{\mathbf x}}+{{\mathbf l}})=\exp(i{{\mathbf p}}\cdot{{\mathbf l}})u({{\mathbf x}})\quad \forall\;{{\mathbf x}}\in{{\mathbf R}}^d,\;
{{\mathbf l}}\in\Gamma$$ with the inner product and the norm, defined by $$\label{dfinnprd}
(f,g)_2=\int_\Omega f({{\mathbf x}})\overline{g({{\mathbf x}})}\,d{{\mathbf x}}\;(f,g\in{{\mathcal H}}_{{\mathbf p}}),\quad
\|f\|_2=\sqrt{(f,f)_2}.$$ The domain of $H({{\mathbf p}})$ is the linear set ${{\mathcal D}}_{{\mathbf p}}(\Gamma)=W_{2,loc}^2({{\mathbf R}}^d)\cap{{\mathcal H}}_{{{\mathbf p}}}$. By claim (i) of Proposition \[propselfadj\] and Proposition \[prcompres\], the operator $H({{\mathbf p}})$ is self-adjoint, bonded below uniformly w.r. to ${{\mathbf p}}\in{{\mathbf T}}^d$ and its spectrum is discrete. As it is easy to check, the operator $(E_{{\mathbf p}}u)({{\mathbf x}}):=\exp(-i{{\mathbf p}}\cdot{{\mathbf x}})u({{\mathbf x}})$, acting from ${{\mathcal H}}_{{\mathbf p}}$ onto ${{\mathcal H}}_0=L_2({{\mathbf R}}^d/\Gamma)$, realizes a unitary equivalence between the operator $H({{\mathbf p}})$ and the operator $$\label{b16}
\tilde H({{\mathbf p}})=-\Delta_{{\mathbf p}}+V({{\mathbf x}})\cdot,$$ with the domain $W_2^2({{\mathbf R}}^d/\Gamma)$, where $$\label{b17}
\Delta_{{\mathbf p}}=\sum_{j=1}^d\left(D_j +ip_j\right)^2.$$ Let $\lambda_1({{\mathbf p}})\le\lambda_2({{\mathbf p}})\le\dots\lambda_n({{\mathbf p}})\le\dots$ be the eigenvalues of the operator $\tilde H({{\mathbf p}})$ (counting their multiplicities), and $e_1({{\mathbf x}},{{\mathbf p}}),e_2({{\mathbf x}},{{\mathbf p}}),\dots,e_n({{\mathbf x}},{{\mathbf p}}),\dots$ be the corresponding eigenfunctions of this operator which form an orthonormal basis in the space $L_2({{\mathbf R}}^d/\Gamma)$. Using the physical terminology, we shall call the vector ${{\mathbf p}}$ the [*quasi-momentum*]{} and each branch of the eigenvalues $\lambda_n({{\mathbf p}})$ will be called the [*dispersion function*]{}. It is known that $\lambda_n({{\mathbf p}})$ are continuous functions on ${{\mathbf T}}^d\;$ and $\sigma(H_0)=\bigcup_{{{\mathbf p}}\in{{\mathbf T}}^d}\sigma(\tilde
H({{\mathbf p}}))=\bigcup_{{{\mathbf p}}\in{{\mathbf T}}^d}\{\lambda_n({{\mathbf p}})\}_{n=1}^\infty$ ([@Gel], [@Eas], [@Eas1] [@Kuch], [@Zl]). We shall consider also the [*Bloch function*]{} corresponding to a dispersion function $\lambda_l({{\mathbf p}})$ and a quasi-momentum ${{\mathbf p}}\in{{\mathbf T}}^d$: $$\label{Bloch1}
b_l({{\mathbf x}},{{\mathbf p}})=\exp(i{{\mathbf p}}\cdot{{\mathbf x}})e_l({{\mathbf x}},{{\mathbf p}}).$$ It is clear that for each natural $l$ $b_l({{\mathbf x}},{{\mathbf p}})$ is an eigenfunction of the operator $H({{\mathbf p}})$, corresponding to its eigenvalue $\lambda_l({{\mathbf p}})$ and for any fixed ${{\mathbf p}}\in{{\mathbf T}}^d$ the sequence $\{b_l({{\mathbf x}},{{\mathbf p}})\}_{l=1}^\infty$ forms an orthonormal basis in the space ${{\mathcal H}}_{{\mathbf p}}$.
It is easy to check that for any ${{\mathbf p}}\in{{\mathbf T}}^d$ $JH({{\mathbf p}})J=H(-{{\mathbf p}})$, where $J$ is the conjugation operator $(Jf)(f)({{\mathbf x}}):=\overline {f({{\mathbf x}})}$ (the property of a “time reversibility”). Hence in this case $\sigma(H({{\mathbf p}}))=\sigma(H(-{{\mathbf p}}))$ and for any $\mu\in\sigma(H({{\mathbf p}}))$ the corresponding eigenprojections $Q_\mu({{\mathbf p}})$ of $H({{\mathbf p}})$ and $Q_\mu(-{{\mathbf p}})$ of $H(-{{\mathbf p}})$ are connected in the following manner: $Q_\mu(-{{\mathbf p}})=JQ_\mu J({{\mathbf p}})$. The same property is valid for the operator $\tilde H({{\mathbf p}})$.
Assume that $(\lambda_-,\,\lambda_+)$ is a gap of $\sigma(H_0)$, that is for some $j\ge 1$ $\;\lambda_-=\max_{{{\mathbf p}}\in{{\mathbf T}}^d}\lambda_j({{\mathbf p}})<\lambda_+=\min_{{{\mathbf p}}\in{{\mathbf T}}^d}\lambda_{j+1}({{\mathbf p}})$, and $\lambda_+=\min_{{{\mathbf p}}\in{{\mathbf T}}^d}\lambda_1({{\mathbf p}})$, $\lambda_-=-\infty$ for $j=0$. Since $j$ will be fixed in our considerations, we shall denote $\lambda^-({{\mathbf p}}):=\lambda_j({{\mathbf p}})$ and $\lambda^+({{\mathbf p}}):=\lambda_{j+1}({{\mathbf p}})$. In other words, $\lambda^-({{\mathbf p}})$ and $\lambda^+({{\mathbf p}})$ are the dispersion functions branching from the edges $\lambda_-$ and $\lambda_+$ respectively. We shall denote by $b^-({{\mathbf x}},{{\mathbf p}})$ and $b^+({{\mathbf x}},{{\mathbf p}})$ the eigenfunctions of $H({{\mathbf p}})$ (Bloch functions), corresponding to $\lambda^-({{\mathbf p}})$ and $\lambda^+({{\mathbf p}})$ respectively, i.e. $b^-({{\mathbf x}},{{\mathbf p}})=b_j({{\mathbf x}},{{\mathbf p}})$ and $b^+({{\mathbf x}},{{\mathbf p}})=b_{j+1}({{\mathbf x}},{{\mathbf p}})$. In the analogous manner we denote $e^-({{\mathbf x}},{{\mathbf p}})=e_j({{\mathbf x}},{{\mathbf p}})$ and $e^+({{\mathbf x}},{{\mathbf p}})=e_{j+1}({{\mathbf x}},{{\mathbf p}})$.
Consider the following subsets of ${{\mathbf T}}^d$: $$\begin{aligned}
&&F^-:=\{{{\mathbf p}}\in{{\mathbf T}}^d:\;\lambda^-({{\mathbf p}})=\lambda_-\},\\
&&F^+:=\{{{\mathbf p}}\in{{\mathbf T}}^d:\;\lambda^+({{\mathbf p}})=\lambda_+\},
$$ which are the extremal sets of the functions $\lambda^-({{\mathbf p}})$ and $\lambda^+({{\mathbf p}})$ respectively. Assume that for the edge $\lambda_-\;(\lambda_->-\infty)$ or for the edge $\lambda_+$ of the gap $(\lambda_-,\,\lambda_+)$ the condition is fulfilled:
\(A) The edge $\lambda_+\;(\lambda_-)$ is non-degenerate in the Morse’s sense, that is
\(a) it is simple in the sense that for any ${{\mathbf p}}_0\in F^+\;({{\mathbf p}}_0\in
F^-)$ the number $\lambda^+({{\mathbf p}}_0)=\lambda_+\;(\lambda^-({{\mathbf p}}_0)=\lambda_-)$ is a simple eigenvalue of the operator $H({{\mathbf p}}_0)$, that is $(\lambda^+({{\mathbf p}}_0)<\lambda_{j+2}({{\mathbf p}}_0))$, if $j\ge 0$ ($\lambda^-({{\mathbf p}}_0)>\lambda_{j-1}({{\mathbf p}}_0)$, if $j\ge 2$);
\(b) for any ${{\mathbf p}}_0\in F^+\;({{\mathbf p}}_0\in F^-)$ the second differential $d^2\lambda^+({{\mathbf p}}_0)\;(d^2\lambda^-({{\mathbf p}}_0))$ is a positive-definite (negative-definite) quadratic form.
Observe that if condition (A)-(a) is satisfied for the edge $\lambda_+\;(\lambda_-)$, then by claim (i) of Corollary \[cormainApp\], for any ${{\mathbf p}}_0\in F^+\;({{\mathbf p}}_0\in F^-)$ there exists a neighborhood ${{\mathcal O}}^+({{\mathbf p}}_0)\;({{\mathcal O}}^-({{\mathbf p}}_0))$ of ${{\mathbf p}}_0$ such that the function $\lambda^+({{\mathbf p}})\;(\lambda^-({{\mathbf p}}))$ is real-analytic in ${{\mathcal O}}^+({{\mathbf p}}_0)\;({{\mathcal O}}^-({{\mathbf p}}_0))$ (hence in particular there exists the second differential of this function, taking part in the condition (A)-(b)), and furthermore, the corresponding branch of eigenfunctions $b^+({{\mathbf x}},{{\mathbf p}})\;(b^-({{\mathbf x}},{{\mathbf p}}))$ of $H({{\mathbf p}})$ can be chosen such that for each fixed ${{\mathbf p}}\in{{\mathcal O}}^+({{\mathbf p}}_0)\;({{\mathbf p}}\in{{\mathcal O}}^-({{\mathbf p}}_0))$, $\Vert b^+(\cdot,{{\mathbf p}}) \Vert_2=1$ ($\Vert b^-(\cdot,{{\mathbf p}}) \Vert_2=1$), the function $b^+({{\mathbf x}},{{\mathbf p}})\;(b^-({{\mathbf x}},{{\mathbf p}}))$ is continuous and the mapping ${{\mathbf p}}\rightarrow b^+({{\mathbf x}},{{\mathbf p}})\in C(\Omega)\;({{\mathbf p}}\rightarrow b^-({{\mathbf x}},{{\mathbf p}})\in
C(\Omega))$ is real-analytic in ${{\mathcal O}}^+({{\mathbf p}}_0)\;({{\mathcal O}}^-({{\mathbf p}}_0))$. It is clear that the corresponding eigenfunction $e^+({{\mathbf x}},{{\mathbf p}})\;(e^-({{\mathbf x}},{{\mathbf p}}))$ of the operator $\tilde H({{\mathbf p}})$ has the same properties.
It is clear that if the condition (A) is satisfied for $\lambda_+\;(\lambda_-)$, then the set $F^+\;(F^-)$ is finite, that is $F^+=\{{{\mathbf p}}_1^+,{{\mathbf p}}_2^+,\dots,{{\mathbf p}}_{n_+}^+\}$ $(F^-=\{{{\mathbf p}}_1^-,{{\mathbf p}}_2^-,\dots,{{\mathbf p}}_{n_-}^-\})$. In particular, it is known ([@Titch]) that in the case where $d=1$ condition (A) is always satisfied, all the functions $\lambda_l({{\mathbf p}})\,(l=1,2,\dots)$ are even, $n_+=n_-=1$ and or $p_1^+=p_1^-=0$, or $p_1^+=p_1^-=\pi$.[^2] Hence in this case we shall write $p_1$ instead of $p_1^+$ and $p_1^-$. Denote $$\label{dfmkpl}
m_k^+=\big(J\,\rm{Hes}_{{{\mathbf p}}_k^+}(\lambda^+)\big)^{-1}\quad
\Big(m_k^-=-\big(J\,\rm{Hes}_{{{\mathbf p}}_k^-}(\lambda^-))\big)^{-1}\Big)$$ where $$\begin{aligned}
\label{Hess}
&&J\,\rm{Hes}_{{\mathbf p}}(\lambda^+)=\rm{det}\left(\frac{\partial^2\lambda^+({{\mathbf p}})}{\partial
p_\mu\partial p_\nu}|_{{{\mathbf p}}={{\mathbf p}}_k^+}\right)_{\mu,\nu=1}^d\quad\\
&&\left(J\,\rm{Hes}_{{\mathbf p}}(\lambda^-)=\rm{det}\left(\frac{\partial^2\lambda^-({{\mathbf p}})}{\partial
p_\mu\partial
p_\nu}|_{{{\mathbf p}}={{\mathbf p}}_k^-}\right)_{\mu,\nu=1}^d\right).\nonumber\end{aligned}$$ In particular, in the case where $d=1$ $$\label{dfmupl}
(m^+_1)^{-1}=\frac{d^2\lambda^+(p)}{dp^2}|_{p=p_1}\quad
\left((m^-_1)^{-1}=-\frac{d^2\lambda^-(p)}{dp^2}|_{p=p_1}\right)$$ $(p_1\in\{0,\pi\})$. With the physical point of view the quantity $m_k^+\;(m_k^-)$ is (up to a physical constant multiplier) the modulus of the determinant of the effective-mass tensor, that is the product of effective masses (in the principal directions) of an electron having the quasi-momentum ${{\mathbf p}}_k^+\;({{\mathbf p}}_k^-)$ at the energy level $\lambda_+\;(\lambda_-)$.
Along with the Bloch functions $b^+({{\mathbf x}},{{\mathbf p}})$, $b^-({{\mathbf x}},{{\mathbf p}})$ we shall consider the [*weighted Bloch functions*]{} $$\label{dfvlxp}
v^+({{\mathbf x}},{{\mathbf p}})=\sqrt{W({{\mathbf x}})}b^+({{\mathbf x}},{{\mathbf p}}),\quad
v^-({{\mathbf x}},{{\mathbf p}})=\sqrt{W({{\mathbf x}})}b^-({{\mathbf x}},{{\mathbf p}}),$$ corresponding to them, with $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$, and denote $$\label{dfvlpl}
v_k^+({{\mathbf x}})=v^+({{\mathbf x}},{{\mathbf p}}_k^+),\quad v_l^-({{\mathbf x}})=v^-({{\mathbf x}},{{\mathbf p}}_l^-).$$ In particular, in the case where $d=1$ $$\label{dfvlpld1}
v_1^+(x)=v^+(x,p_1),\quad v_1^-(x)=v^-(x,p_1)\quad
(p_1\in\{0,\pi\}).$$
A part of our results concerns the case of degenerate edges of the gap of the spectrum of the unperturbed operator $H_0$, where condition (A)-(b) does not fulfilled. Let $\mathrm{Hes}_{{\mathbf p}}(f)$ be the Hessian operator of a function $f:\,{{\mathbf T}}^d\rightarrow{{\mathbf R}}$ at a point ${{\mathbf p}}\in{{\mathbf T}}^d$, that is this is a linear operator acting in the tangent space $T_{{\mathbf p}}({{\mathbf T}}^d)$ to the torus ${{\mathbf T}}^d$ at the point ${{\mathbf p}}$ and defined by $$\label{dfHes}
\forall\;s,t\in T_{{\mathbf p}}({{\mathbf T}}^d):\quad d^2f({{\mathbf p}})[s,t]=\rm{Hes}_{{\mathbf p}}(f)s\cdot
t.$$ The more general condition than (A), which we shall consider, is following:
\(B) The function $\lambda^+\;(\lambda^-)$ and the set $F^+\;(F^-)$ satisfy the Morse-Bott type conditions ([@Ban-Hur]):
\(a) condition (A)-(a) is satisfied;
\(b) the set $F^+\;(F^-)$ consists of a finite number of disjoint connected components: $F^+=\bigcup_{k=1}^{n_+}F^+_k\;(F_-=\bigcup_{k=1}^{n_-}F^-_k)$, such that each of $F^+_k\;(F^-_k)$ is a $C^\infty$-smooth submanifold of ${{\mathbf T}}^d$ of the dimension $d_k^+\;(d_k^-)$;
\(c) for any point ${{\mathbf p}}\in F^+\;({{\mathbf p}}\in F^-)$ the normal Hessian $$\begin{aligned}
\label{dfnrmHes}
&&JN\,\rm{Hes}_{{\mathbf p}}(\lambda^+):=\det\big(Hes_{{\mathbf p}}(\lambda^+)\vert_{N_{{\mathbf p}}}\big)\\
&&\big(JN\,
\rm{Hes}_{{\mathbf p}}(\lambda^-):=\det\big(Hes_{{\mathbf p}}(\lambda^-)\vert_{N_{{\mathbf p}}}\big)\big)\nonumber\end{aligned}$$ is not equal to zero. Here $N_{{\mathbf p}}\subseteq T_{{\mathbf p}}({{\mathbf T}}^d)$ is a normal subspace to $F^+\;(F^-)$ at ${{\mathbf p}}$, that is $N_{{\mathbf p}}^+=T_{{\mathbf p}}({{\mathbf T}}^d)\ominus
T_{{\mathbf p}}(F^+)\;(N_{{\mathbf p}}^-=T_{{\mathbf p}}({{\mathbf T}}^d)\ominus T_{{\mathbf p}}(F^-))$.
If $d_k^+\ge 1\;(d_k^-\ge 1)$ for at least one $k$, we shall say that the edge $\lambda_+\;(\lambda_-)$ is [*degenerate in the Morse-Bott sense*]{}.
Denote $$\begin{aligned}
\label{dfmplp}
&&m^+({{\mathbf p}})=\big(JN\,\rm{Hes}_{{\mathbf p}}(\lambda^+)\big)^{-1}\quad({{\mathbf p}}\in
F^+)\\
&&\Big(m^-({{\mathbf p}})=-\big(JN\,\rm{Hes}_{{\mathbf p}}(\lambda^-)\big)^{-1}\quad({{\mathbf p}}\in
F^-)\Big).\nonumber\end{aligned}$$ In the similar manner as above the quantity $m^+({{\mathbf p}})\;(m^+({{\mathbf p}}))$ is (up to a physical constant multiplier) the product of effective masses (in the principal directions normal to $F^+\;(F^-)$ ) of an electron having the quasi-momentum ${{\mathbf p}}\in F^+\;({{\mathbf p}}\in F^-)$ at the energy level $\lambda_+\;(\lambda_-)$.
Assuming that condition (A)-(a) is satisfied and taking ${{\mathbf p}}_0\in
F^+$, consider the integral kernel ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ of the eigenprojection $Q^+({{\mathbf p}})$ of $H({{\mathbf p}})$, corresponding to the dispersion function $\lambda^+({{\mathbf p}})$ branching from the edge $\lambda_+$ and defined in a neighborhood ${{\mathcal O}}^+({{\mathbf p}}_0)$ of ${{\mathbf p}}_0$. We shall call it the [*eigenkernel*]{} of $H({{\mathbf p}})$, corresponding to $\lambda^+({{\mathbf p}})$. By claims (ii) and (iii) of Corollary \[cormainApp\], after a suitable choice of ${{\mathcal O}}^+({{\mathbf p}}_0)$ this kernel acquires the form ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=b^+({{\mathbf x}},{{\mathbf p}})\overline{b^+({{\mathbf s}},{{\mathbf p}})}$, it does not depend on the choice of a branch of Bloch functions $b^+({{\mathbf x}},{{\mathbf p}})$ having the properties mentioned above and the mapping ${{\mathbf p}}\rightarrow{{\mathcal Q}}^+(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ is real-analytic in ${{\mathcal O}}^+({{\mathbf p}}_0)$. The analogous properties has the eigenkernel ${{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ of $H({{\mathbf p}})$, corresponding to $\lambda^-({{\mathbf p}})$. It is clear that the eigenkernel of $\tilde H({{\mathbf p}})$, corresponding to $\lambda^+({{\mathbf p}})\;(\lambda^-({{\mathbf p}}))$ has the form $$\begin{aligned}
\label{connecteigkern}
&&\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\exp(-i{{\mathbf p}}\cdot({{\mathbf x}}-{{\mathbf s}})){{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\\
&&(\tilde{{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\exp(-i{{\mathbf p}}\cdot({{\mathbf x}}-{{\mathbf s}})){{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}}))\nonumber\end{aligned}$$ and it has the same properties as ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\;({{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}}))$, but in addition it is $\Gamma$-periodic w.r.t. ${{\mathbf x}}$ and ${{\mathbf s}}$. We shall consider also the [*weighted eigenkernels*]{} $$\begin{aligned}
\label{dfWxsp}
&&{{\mathcal Q}}_W^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\sqrt{W({{\mathbf x}})}{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{W({{\mathbf s}})},\\
&&{{\mathcal Q}}_W^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\sqrt{W({{\mathbf x}})}{{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{W({{\mathbf s}})}.\nonumber\end{aligned}$$
The notion of a virtual eigenvalue {#subsec:notionsfacts}
----------------------------------
Before formulating the main results, let us recall some notions and facts from [@Ar-Zl1]. Consider an operator $H_\gamma=H_0+\gamma
W$ acting in a Hilbert space ${{\mathcal H}}$, where $H_0$ and $W$ are self-adjoint operators and $\gamma$ is a real coupling constant. We assume that the following conditions are satisfied:
\(1) $(\lambda_-,\,\lambda_+)\;\;(-\infty\le \lambda_-<\lambda_+\le
+\infty)$ is a gap of the spectrum $\sigma(H_0)$ of the unperturbed operator $H_0$.
\(2) The operator $W$ is bounded[^3] and for some $\lambda_0\in{{\mathcal R}}(H_0)$ the operator $R_{\lambda_0}(H_0)|W|^{\frac{1}{2}}$ is compact.
By Proposition \[BrScwspct\] of the present paper, the set $\sigma(H_\gamma)\cap (\lambda_-,\lambda_+)$ consists of at most countable number of eigenvalues having finite multiplicities which can cluster only to the edges $\lambda_+$ and $\lambda_-$: $\lambda_-<\dots\le\rho_{-k}(\gamma)\le\dots\le\rho_{-1}(\gamma)
\le\rho_0(\gamma)\le\rho_1(\gamma)\le\dots
\le\rho_k(\gamma)\le\dots<\lambda_+$ (each eigenvalue is repeated according to its multiplicity).
\[gone1\] [Let $\rho(\gamma)\in (\lambda_-,\lambda_+)$ be a branch of eigenvalues of the operator $H_\gamma$ which enters the gap $(\lambda_-,\,\lambda_+)$ of $\sigma(H_0)$ across the edge $\lambda_+<+\infty$ at the moment $\gamma=0$ as $\gamma$ decreases (increases) from $0$ to a negative (positive) value. We call it a]{} [*branch of virtual eigenvalues*]{} [of the operator $H_\gamma$ created in $(\lambda_-,\lambda_+)$ at the edge $\lambda_+$. This means that $(-\bar\gamma,\,0)\subseteq{{\rm Dom}}(\rho)$ $\big((0,\,\bar\gamma)\subseteq{{\rm Dom}}(\rho)\big)$ for some $\bar\gamma>0$ and $ \lim_{\gamma\rightarrow
0}\rho(\gamma)=\lambda_+$. In the analogous manner we define a]{} [*branch of virtual eigenvalues*]{} [of the operator $H_\gamma$ created in $(\lambda_-,\lambda_+)$ at the edge $\lambda_->-\infty$]{}.
Formulation of main results {#sec:mainres}
===========================
The birth of virtual eigenvalues from a non-degenerate edge {#subsec:mainresbirthnondeg}
-----------------------------------------------------------
Let us return to the Schrödinger operator $H_\gamma=H_0+\gamma W$ ($H_0=-\Delta+ V({{\mathbf x}})\cdot$, $W=W({{\mathbf x}})\cdot$, ${{\mathcal H}}=L_2({{\mathbf R}}^d)$), considered in Section \[sec:preliminaries\]. Recall that $V({{\mathbf x}})$ is measurable, bounded and periodic on some lattice $\Gamma$. In this section we consider the case of a definite perturbation and of a non-degenerate edge of a gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0$, that is at least one of the edges $\lambda_+$ or $\lambda_-$ satisfies condition (A) of Section \[subsec:spectchar\]. Consider the finite rank operator $$\label{dfGW}
G_W^+=\sum_{k=1}^{n_+}\sqrt{m_k^+}\,(\,\cdot,\,v_k^+)v_k^+\quad\Big(G_W^-=
\sum_{k=1}^{n_-}\sqrt{m_k^-}\,(\,\cdot,\,v_k^-)v_k^-\Big),$$ $m_k^+\;(m_k^+)$ is defined by (\[dfmkpl\])-(\[Hess\]) and $v_k^+({{\mathbf x}})\;(v_k^-({{\mathbf x}}))$ is the weighted Bloch function defined by (\[dfvlpl\]), (\[dfvlxp\]) and (\[Bloch1\]). We shall show in what follows (Lemma \[lmGWnondeg\]) that $G_W^+\;(G_W^-)$ has $n_+\;(n_-)$ positive eigenvalues $$\label{eigvalGWnondeg}
\nu_1^+\ge \nu_2^+\ge\dots\ge
\nu_{n_+}^+>0\;\big(\nu_1^-\ge\nu_2^-\ge\dots\ge \nu_{n_-}^->0\big)$$ (counting their multiplicities), which are eigenvalues of the matrix $$\begin{aligned}
\label{Gram}
&&\left((m_k^+m_l^+)^{\frac{1}{4}}(v_l^+,v_k^+)\right)_{k,l=1}^{n^+}\\
&&\left(\left(
(m_k^-m_l^-)^{\frac{1}{4}}(v_l^-,v_k^-)\right)_{k,l=1}^{n^-}\right).\nonumber\end{aligned}$$ Let $g_1^+({{\mathbf x}}),g_2^+({{\mathbf x}}),\dots,g_{n_+}^+({{\mathbf x}})\;\big(g_1^-({{\mathbf x}}),g_2^-({{\mathbf x}}),\dots,g_{n_+}^-({{\mathbf x}})\big)$ be an orthonormal sequence of eigenfunctions of the operator $G_W^+\;(G_W^-)$ corresponding to its eigenvalues (\[eigvalGWnondeg\]).
If $d\le 2 $, we shall impose on the non-negative perturbation $W({{\mathbf x}})$ the following conditions of its fast decay as $|{{\mathbf x}}|\rightarrow\infty$: for $d=1$ $$\label{cndWd1}
\int_{-\infty}^\infty\int_{-\infty}^\infty
W(x)(x-s)^2W(s)\,dx\,ds<\infty$$ and for $d=2$ $$\label{cndWd2}
\int_{{{\mathbf R}}^2}\int_{{{\mathbf R}}^2}
W({{\mathbf x}})(\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)^2W({{\mathbf s}})\,d{{\mathbf x}}\,d{{\mathbf s}}<\infty.$$
Our result about virtual eigenvalues in the non-degenerate case is following:
\[thnondegedg\] Assume that the unperturbed potential $V({{\mathbf x}})$ satisfies the strengthened version of condition (\[condperpotent\]): $$\label{Hold}
V\in\left\{\begin{array}{ll}
L_2(\Omega), &\rm{if}\quad d=1\\
\bigcup_{q>d}L_q(\Omega), &\rm{if}\quad d\ge 2,
\end{array}\right.$$ the perturbation $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$, $\lim_{|{{\mathbf x}}|\rightarrow\infty} W({{\mathbf x}})=0$, $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$ and $W({{\mathbf x}})>0$ on a set of positive measure. Furthermore, assume that if the edge $\lambda_+\;(\lambda_-)$ is non-degenerate, for $d=1$ the condition (\[cndWd1\]) is satisfied and for $d=2$ the condition (\[cndWd2\]) is satisfied. Then
\(i) for $\gamma<0$ the operator $H_\gamma$ can have in $(\lambda_-,\lambda_+)$ a virtual eigenvalue only at the edge $\lambda_+$;
\(ii) if the edge $\lambda_+$ is non-degenerate, $d\le 2$ and $\gamma<0$, the operator $H_\gamma$ has in $(\lambda_-,\lambda_+)$ virtual eigenvalues at $\lambda_+$ having the properties:
\(a) if $d=1$, there is a unique virtual eigenvalue $\rho^+_1(\gamma)$ in $(\lambda_-,\lambda_+)$ , having the following asymptotic representation for $\gamma\uparrow 0$: $$\begin{aligned}
\label{asympteigvd1}
\sqrt{\lambda_+-\rho^+_1(\gamma)}=|\gamma|\big(\sqrt{m^+_1}\frac{\|v^+_1\|^2}{\sqrt{2}}+O(\gamma)\big),\end{aligned}$$ where $m^+_1$ is defined by (\[dfmupl\]-a) and $v_1^+(x)$ is defined by (\[dfvlpld1\]-a);
\(b) if $d=2$, there are $\;n_+$ virtual eigenvalues $$\label{virteignondeg}
\rho_1^+(\gamma)\le\rho_2^+(\gamma)\le\dots\le \rho_{n_+}^+(\gamma)$$ in $(\lambda_-,\lambda_+)$ (counting their multiplicities), and the following asymptotic representation is valid for them for $\gamma\uparrow 0$: $$\begin{aligned}
\label{asympteigvd2}
\left(\ln\left(\frac{1}{\lambda_+-\rho_k^+(\gamma)}\right)\right)^{-1}=
|\gamma|\Big(\frac{\nu_k^+}{2\pi}+O(\gamma)\Big)\quad(k=1,2,\dots,n_+),\end{aligned}$$ and furthermore, the asymptotic formula of Lieb-Thirring type is valid for $\gamma\uparrow 0$: $$\begin{aligned}
\label{asymptLbThrd2}
\sum_{k=1}^{n_+}\left(\ln\left(\frac{1}{\lambda_+-\rho_k^+(\gamma)}\right)\right)^{-1}=
\frac{|\gamma|}{2\pi}\sum_{k=1}^{n_+}\|v_k^+\|^2\sqrt{m_k^+}+O(\gamma^2);\end{aligned}$$
\(iii) if the edge $\lambda_+$ is non-degenerate and $d\le 2$, the eigenfunctions corresponding to the virtual eigenvalues, considered above, have the properties:
\(a) if $d=1$, there exists $\bar\gamma>0$ such that for any $\gamma\in[-\bar\gamma,0)$ it is possible to choose an eigenfunction $\psi^+_{\gamma,1}({{\mathbf x}})$ of the operator $H_\gamma$ corresponding to its eigenvalue $\rho^+_1(\gamma)$ such that $\Vert\sqrt{W}\psi^+_{\gamma,1}- g_1^+\Vert=O(\gamma)$ for $\gamma\uparrow 0$, where $g^+_1=\frac{v^+_1}{\Vert
v^+_1\Vert}$;
\(b) if $d=2$ and $m(j)\;(j\in\{1,2,\dots,n_+\})$ is the multiplicity of an eigenvalue $\nu_j^+$ of the operator $G_W^+$ and $\rho_{l(j)}^+(\gamma)\le\rho_{l(j)+1}^+(\gamma)\le\dots\le\rho_{l(j)+m(j)-1}^+(\gamma)$ $(l(j)\in\{1,2,\dots,n_+\})$ is the group of virtual eigenvalues of $H_\gamma$, for which $\lim_{\gamma\uparrow
0}\ln\left(\frac{1}{\lambda_+-\rho_k^+(\gamma)}\right)|\gamma|=\frac{2\pi}{\nu_j^+}$, then there exists $\bar\gamma>0$ such that for any $\gamma\in[-\bar\gamma,0)$ there are numbers $\gamma_0(\gamma)=\gamma$, $\{\gamma_k(\gamma)\}_{k=1}^{m(j)-1}$ having the properties: $\;\gamma-\gamma_k(\gamma)=O(\gamma^2)$ as $\gamma\uparrow 0$, for each $k\in\{0,1,\dots,m(j)-1\}$ the number $\rho_{l(j)}^+(\gamma)$ is an eigenvalue of the operator $H_{\gamma_k(\gamma)}$, and it is possible to choose a basis $$\psi_{\gamma,\,l(j)}^+({{\mathbf x}}),\psi_{\gamma,\,l(j)+1}^+({{\mathbf x}}),\dots,\psi_{\gamma,\,l(j)+m(j)-1}^+({{\mathbf x}})$$ in the linear span of eigenspaces of all the operators $H_{\gamma_k(\gamma)}\;(k\in\{0,1,\dots,m(j)-1\})$, corresponding to their eigenvalue $\rho_{l(j)}^+(\gamma)$, for which the property $$\label{asympteigvec1}
\Vert\sqrt{W}\psi_{\gamma,k}^+-g_k^+\Vert=O(\gamma) \quad
\rm{as}\quad \gamma\uparrow 0$$ is valid for any $k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\}$;
\(iv) if the edge $\lambda_+$ is non-degenerate, $d\ge 3$, $W\in
L_1({{\mathbf R}}^d)$ and $\gamma<0$ the operator $H_\gamma$ has in $(\lambda_-,\lambda_+)$ no virtual eigenvalue at $\lambda_+$;
\(v) for $\gamma>0$ the claims (i)-(iv) are valid with $\lambda_-$, $n_-$, $\rho_k^-(\gamma)$, $\nu_k^-$, $m_k^-$, $v_k^-({{\mathbf x}})$, $\psi_{\gamma,k}^-({{\mathbf x}})$, $g_k^-({{\mathbf x}})$, $\gamma>0$ and $\gamma\downarrow 0$ instead of, respectively, $\lambda_+$, $n_+$, $\rho_k^+(\gamma)$, $\nu_k^+$, $m_k^+$, $v_k^+({{\mathbf x}})$, $\psi_{\gamma,k}^+({{\mathbf x}})$, $g_k^+({{\mathbf x}})$, $\gamma<0$ and $\gamma\uparrow 0$;
\(vi) if both edges $\lambda_+$ and $\lambda_-$ are non-degenerate, $d\ge 3$ and $W\in L_1({{\mathbf R}}^d)$, there is a threshold for the birth of the impurity spectrum in the gap $(\lambda_-,\,\lambda_+)$, that is $\sigma(H_\gamma)\cap(\lambda_-,\lambda_+)=\emptyset$ for a small enough $|\gamma|$.
The birth of virtual eigenvalues from a degenerate edge {#subsec:mainresbirthdeg}
-------------------------------------------------------
Assume that at least one of the edges $\lambda_+$ or $\lambda_-$ of a gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0$ satisfies the condition (B) of Section \[subsec:spectchar\] such that the dimension of at least one of the connected components of the extremal set $F^+=(\lambda^+)^{-1}(\lambda_+)\;(F_-=(\lambda^-)^{-1}(\lambda_-))$ of the dispersion function $\lambda^+({{\mathbf p}})\;(\lambda^-({{\mathbf p}}))$, branching from the edge $\lambda_+\;(\lambda_-)$, is non-zero. In this case we have a non-localized degeneration of the dispersion function $\lambda_{j+1}({{\mathbf p}})\;(\lambda_j({{\mathbf p}}))$ at the edge $\lambda_+\;(\lambda_-)$ of the forbidden zone $(\lambda_-,\lambda_+)$. For the simplicity we shall assume in this section that the following condition is satisfied for at least one of the edges of $(\lambda_-,\lambda_+)$:
\(C) The edge $\lambda_+\;(\lambda_-)$ of a gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0$ satisfies condition (B) of Section \[subsec:spectchar\] with $n_+=1\,(n_-=1)$ (that is $F^+\,(F^-)$ is a connected smooth submanifold of ${{\mathbf T}}^d$) and $d_+=\dim(F^+)\ge
1\;(d_-=\dim(F^-)\ge 1)$.
Consider the integral operator acting in $L_2({{\mathbf R}}^d)$ $$\label{dfintopGW}
G_W^+f=\int_{{{\mathbf R}}^d}{{\mathcal G}}_W^+({{\mathbf x}},{{\mathbf s}})f({{\mathbf s}})\,d{{\mathbf s}}\quad\Big(G_W^-f=\int_{{{\mathbf R}}^d}{{\mathcal G}}_W^-({{\mathbf x}},{{\mathbf s}})f({{\mathbf s}})\,d{{\mathbf s}}\Big)$$ with $$\begin{aligned}
\label{dfkerGW}
&&{{\mathcal G}}_W^+({{\mathbf x}},{{\mathbf s}})=\int_{F^+}{{\mathcal Q}}_W^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}})\\
&&
\Big({{\mathcal G}}_W^-({{\mathbf x}},{{\mathbf s}})=\int_{F^-}{{\mathcal Q}}_W^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{m^-({{\mathbf p}})}\,dF({{\mathbf p}})\Big),
\nonumber\end{aligned}$$ where ${{\mathcal Q}}_W^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\;({{\mathcal Q}}_W^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}}))$ is the weighted eigenkernel, corresponding to $\lambda^+({{\mathbf p}})\;(\lambda^-({{\mathbf p}}))$ and defined by (\[dfWxsp\]), $m^+({{\mathbf p}})\;(m^-({{\mathbf p}}))$ is defined by (\[dfmplp\]) and $dF({{\mathbf p}})$ is the volume form on the submanifold $F^+\,(F^-)$. In what follows we shall prove (Lemma \[lmspecGWpl\]) that if $W\in L_1({{\mathbf R}}^d)$, $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$ and $W({{\mathbf x}})> 0$ on a set of the positive measure, then the integral operator $G_W^+\,(G_W^-)$ is self-adjoint, nonnegative, belongs to the trace class and has an infinite number of positive eigenvalues $$\label{eigvalGW}
\nu_1^+\ge\nu_2^+\ge\dots\ge\nu_n^+\ge\dots\;(\nu_1^-\ge\nu_2^-\ge\dots\ge\nu_n^-\ge\dots)$$ (each eigenvalue is repeated according to its multiplicity). Let $$g_1^+({{\mathbf x}}),g_2^+({{\mathbf x}}),\dots,g_n^+({{\mathbf x}}),\dots\;\big(g_1^-({{\mathbf x}}),g_2^-({{\mathbf x}}),\dots,g_n^-({{\mathbf x}}),\dots\big)$$ be an orthonormal sequence of eigenfunctions of the operator $G_W^+\;(G_W^-)$ corresponding to its eigenvalues (\[eigvalGW\]).
If $d-d_+\le 2\;(d-d_-\le 2)$, we shall impose on the non-negative perturbation $W({{\mathbf x}})$ the following conditions: for $d-d_+=1\;(d-d_-=1)$ $$\label{cndWd1deg}
\int_{{{\mathbf R}}^d}\int_{{{\mathbf R}}^d} W({{\mathbf x}})({{\mathbf x}}-{{\mathbf s}})^2W({{\mathbf s}})\,d{{\mathbf x}}\,d{{\mathbf s}}<\infty$$ and for $d-d_+=2\;(d-d_-=2)$ $$\label{cndWcod2}
\int_{{{\mathbf R}}^d}\int_{{{\mathbf R}}^d}
W({{\mathbf x}})(\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)^2W({{\mathbf s}})\,d{{\mathbf x}}\,d{{\mathbf s}}<\infty$$
Our result about virtual eigenvalues in the degenerate case is following:
\[thdegedg\] Assume that the unperturbed potential $V({{\mathbf x}})$ satisfies the condition (\[Hold\]), the perturbation $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$, $\lim_{|{{\mathbf x}}|\rightarrow\infty}\\ W({{\mathbf x}})=0$, $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$ and $W({{\mathbf x}})>0$ on a set of positive measure. Furthermore, assume that if the edge $\lambda_+\;(\lambda_-)$ satisfies the condition (C), for $d-d_+=1\;(d-d_-=1)$ the condition (\[cndWd1deg\]) is satisfied and for $d-d_+=2\;(d-d_-=2)$ the condition (\[cndWcod2\]) is satisfied. Then
\(i) for $\gamma<0$ the operator $H_\gamma=H_0+\gamma W\cdot$ can have in $(\lambda_-,\lambda_+)$ a virtual eigenvalue only at the edge $\lambda_+$;
\(ii) if the edge $\lambda_+$ satisfies the condition (C), $d-d_+\le
2$ and $\gamma<0$, the operator $H_\gamma$ has in $(\lambda_-,\lambda_+)$ an infinite number of virtual eigenvalues $\rho_1^+(\gamma)\le\rho_2^+(\gamma)\le\dots\rho_n^+(\gamma)\le\dots$ at $\lambda_+$, and moreover the following asymptotic representation is valid for $\gamma\uparrow 0$: $$\label{asympteigvd2deg}
\Psi\big(\lambda_+-\rho_n^+(\gamma)\big)=|\gamma|\big(\nu_n^++O(\gamma)\big),$$ where $$\Psi(s)=\left\{\begin{array}{ll}
\frac{(2\pi)^d}{\sqrt{2}\pi}\sqrt{s},&\rm{if}\quad d-d_+=1,\\
(2\pi)^{d-1}\Big(\ln\big(\frac{1}{s}\big)\Big)^{-1},&\rm{if}\quad
d-d_+=2
\end{array}\right.$$ $(s\in(0,\;\lambda_+-\lambda_-))$, and furthermore, the asymptotic formula of Lieb-Thirring type is valid: $$\begin{aligned}
\label{asymptLbThrd2deg}
&&\lim_{n\rightarrow\infty}\lim_{\gamma\uparrow
0}\frac{1}{|\gamma|}\sum_{k=1}^n
\Psi\big(\lambda_+-\rho_k^+(\gamma)\big)=\nonumber\\
&&\int_{F^+}\int_{{{\mathbf R}}^d}{{\mathcal Q}}_W^+({{\mathbf s}},{{\mathbf s}},{{\mathbf p}})\;d{{\mathbf s}}\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}});\end{aligned}$$
\(iii) if the edge $\lambda_+$ satisfies the condition (C) and $d-d_+\le 2$, the eigenfunctions corresponding to the virtual eigenvalues, considered above, have the property: if $\;m(j)$ is the multiplicity of an eigenvalue $\nu_j^+$ of the operator $G_W^+$ and $\rho_{l(j)}^+(\gamma)\le\rho_{l(j)+1}^+(\gamma)\le\dots\le\rho_{l(j)+m(j)-1}^+(\gamma)$ is the group of virtual eigenvalues of $H_\gamma$, for which $\lim_{\gamma\uparrow
0}\frac{1}{|\gamma|}\Psi\big(\lambda_+-\rho_k^+(\gamma)\big)=\nu_j^+$ $(k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\})$, then there exists $\bar\gamma>0$ such that for any $\gamma\in[-\bar\gamma,0)$ there are numbers $\gamma_0(\gamma)=\gamma$, $\{\gamma_k(\gamma)\}_{k=1}^{m(j)-1}$ such that they and the eigenvectors of the operators $H_{\gamma_k(\gamma)}\;(k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\})$, corresponding to their eigenvalue $\rho_{l(j)}^+(\gamma)$, have the same asymptotic properties as in claim (iii-b) of Theorem \[thnondegedg\];
\(iv) if the edge $\lambda_+$ satisfies the condition (C), $d-d_+\ge
3$, $W\in L_1({{\mathbf R}}^d)$ and $\gamma<0$, the operator $H_\gamma$ has in $(\lambda_-,\lambda_+)$ no virtual eigenvalue at $\lambda_+$;
\(v) for $\gamma>0$ the claims (i)-(iv) are valid with $d_-$, $\lambda_-$, $\rho_k^-(\gamma)$, $\nu_k^-$, ${{\mathcal Q}}_W^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, $m^-({{\mathbf p}})$, $\psi_{\gamma,k}^-({{\mathbf x}})$, $g_k^-({{\mathbf x}})$, $\gamma>0$ and $\gamma\downarrow 0$ instead of, respectively, $d_+$, $\lambda_+$, $\rho_k^+(\gamma)$, $\nu_k^+$, ${{\mathcal Q}}_W^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, $m^+({{\mathbf p}})$, $\psi_{\gamma,k}^+({{\mathbf x}})$, $g_k^+({{\mathbf x}})$, $\gamma<0$ and $\gamma\uparrow
0$;
\(vi) if both edges $\lambda_+$ and $\lambda_-$ satisfy the condition (C), $d-d_+\ge 3$, $d-d_-\ge 3$ and $W\in L_1({{\mathbf R}}^d)$, there is a threshold for the birth of the impurity spectrum in the gap $(\lambda_-,\,\lambda_+)$, that is $\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset$ for a small enough $|\gamma|$.
The estimate of the multiplicity of virtual eigenvalues for an indefinite perturbation in the case of a non-degenerate edge {#subsec:mainresestmult}
---------------------------------------------------------------------------------------------------------------------------
In the case of an indefinite perturbation $W({{\mathbf x}})$ we only can, under some conditions, estimate from above the multiplicity of virtual eigenvalues of $H_\gamma$ in the gap $(\lambda_-,\,\lambda_+)$ and, in particular, establish the existence of a threshold for the birth of the impurity spectrum for $d\ge 3$. Denote by $W_+({{\mathbf x}})$ and $W_-({{\mathbf x}})$ the positive and negative parts of $W({{\mathbf x}})$, that is $W_+({{\mathbf x}})=\frac{1}{2}(W({{\mathbf x}})+|W({{\mathbf x}})|)$ and $W_-({{\mathbf x}})=\frac{1}{2}(W({{\mathbf x}})-|W({{\mathbf x}})|)$. We shall consider the following conditions: in the case where $d=1$ $$\label{cndWpld1}
\int_{-\infty}^\infty\int_{-\infty}^\infty
W_+(x)(x-s)^2W_+(s)\,dx\,ds<\infty,$$ $$\label{cndWmnd1}
\int_{-\infty}^\infty\int_{-\infty}^\infty
W_-(x)(x-s)^2W_-(s)\,dx\,ds<\infty,$$ and in the case where $d=2$ $$\label{cndWpld2}
\int_{{{\mathbf R}}^2}\int_{{{\mathbf R}}^2}
W_+({{\mathbf x}})(\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)^2W_+({{\mathbf s}})\,d{{\mathbf x}}\,d{{\mathbf s}}<\infty,$$ $$\label{cndWmnd2}
\int_{{{\mathbf R}}^2}\int_{{{\mathbf R}}^2}
W_-({{\mathbf x}})(\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)^2W_+({{\mathbf s}})\,d{{\mathbf x}}\,d{{\mathbf s}}<\infty.$$
Recall that if the edge $\lambda_+\;(\lambda_-)$ is non-degenerate, then the extremal set $F^+\;(F^-)$ of the dispersion function $\lambda^+({{\mathbf p}})\;(\lambda^-({{\mathbf p}}))$, branching from this edge, is finite and we denote by $n_+\;(n_-)$ the number of its points.
The following result is valid:
\[thestmult\] Let $(\lambda_-,\lambda_+)$ be a gap of the spectrum of the unperturbed operator $H_0$. Assume that the unperturbed potential $V({{\mathbf x}})$ satisfies the condition (\[Hold\]), the perturbation $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$ and $\lim_{|{{\mathbf x}}|\rightarrow\infty} W({{\mathbf x}})=0$. Let us take $\delta\in(0,\,\lambda_+-\lambda_-)$. Then
\(i) for $d=1$: if the edge $\lambda_+$ is non-degenerate and condition (\[cndWpld1\]) is satisfied, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ all the virtual eigenvalues of the operator $H_\gamma$, being born from the edge $\lambda_+$ and lying in $(\lambda_+-\delta,\;\lambda_+)$, are simple; if the edge $\lambda_-$ is non-degenerate and condition (\[cndWmnd1\]) is satisfied, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ all the virtual eigenvalues of the operator $H_\gamma$, being born from the edge $\lambda_-$ and lying in $(\lambda_-,\;\lambda_-+\delta)$ are simple;
\(ii) for $d=2$: if the edge $\lambda_+$ is non-degenerate the condition (\[cndWpld2\]) is satisfied, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ the multiplicity of each virtual eigenvalue of the operator $H_\gamma$, being born from the edge $\lambda_+$ and lying in $(\lambda_+-\delta,\;\lambda_+)$ is not bigger than $n_+$; if the edge $\lambda_-$ is non-degenerate and condition (\[cndWmnd2\]) is satisfied, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ the multiplicity of each virtual eigenvalue of the operator $H_\gamma$, being born from the edge $\lambda_-$ and lying in $(\lambda_-,\;\lambda_-+\delta)$, is not bigger than $n_-$;
\(iii) for $d\ge 3$: if the edge $\lambda_+$ is non-degenerate and $W_+\in L_1({{\mathbf R}}^d)$, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ no eigenvalue of the operator $H_\gamma$ lies in $(\lambda_+-\delta,\;\lambda_+)$; if the edge $\lambda_-$ is non-degenerate and $W_-\in L_1({{\mathbf R}}^d)$, then there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ no eigenvalue of the operator $H_\gamma$ lies in $(\lambda_-,\lambda_-+\delta)$;
\(iv) for $\gamma>0$ all the above claims are valid with $\lambda_-$, $\lambda_+$, $W_-({{\mathbf x}})$, $W_+({{\mathbf x}})$, $\gamma\in (0,\bar\gamma)$ $(\lambda_-,\;\lambda_-+\delta)$, $(\lambda_+-\delta,\lambda_+)$, $n_-$ and $n_+$ instead of, respectively, $\lambda_+$, $\lambda_-$, $W_+({{\mathbf x}})$, $W_-({{\mathbf x}})$, $\gamma\in (-\bar\gamma,0)$, $(\lambda_+-\delta,\lambda_+)$, $(\lambda_-,\;\lambda_-+\delta)$, $n_+$ and $n_-$;
\(v) for $d\ge 3$: if both edges $\lambda_-$ and $\lambda_+$ are non-degenerate and $W\in L_1({{\mathbf R}}^d)$, then there is a threshold for the birth of the impurity spectrum in the gap $(\lambda_-,\,\lambda_+)$, that is $\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset$ for a small enough $|\gamma|$.
A threshold for the birth of virtual eigenvalues for an indefinite perturbation in the case of a degenerate edge {#subsec:mainresthresdeg}
----------------------------------------------------------------------------------------------------------------
\[ththresholddeg\] Let $(\lambda_-,\,\lambda_+)$ be a gap of the spectrum of the unperturbed operator $H_0$. Assume that the unperturbed potential $V({{\mathbf x}})$ satisfies the condition (\[Hold\]), the perturbation $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$ and $\lim_{|{{\mathbf x}}|\rightarrow\infty} W({{\mathbf x}})=0$. Let us take $\delta\in(0,\,\lambda_+-\lambda_-)$. Then
\(i) if the edge $\lambda_+$ satisfies condition (C) of Section \[subsec:mainresbirthdeg\], $d-d_+\ge 3$ and $W_+\in L_1({{\mathbf R}}^d)$, there exists $\bar\gamma>0$ such that for any $\gamma\in
(-\bar\gamma,0)$ no eigenvalue of the operator $H_\gamma$ lies in $(\lambda_+-\delta,\;\lambda_+)$; if the edge $\lambda_-$ satisfies condition (C), $d-d_-\ge 3$ and $W_-\in L_1({{\mathbf R}}^d)$, there exists $\bar\gamma>0$ such that for any $\gamma\in (-\bar\gamma,0)$ no eigenvalue of the operator $H_\gamma$ lies in $(\lambda_-,\;\lambda_-+\delta)$;
\(ii) for $\gamma>0$ claim (i) is valid with $d_-$, $d_+$, $\lambda_-$, $\lambda_+$, $W_-({{\mathbf x}})$, $W_+({{\mathbf x}})$, $\gamma\in
(0,\bar\gamma)$ $(\lambda_-,\;\lambda_-+\delta)$ and $(\lambda_+-\delta,\lambda_+)$ instead of, respectively, $d_+$, $d_-$, $\lambda_+$, $\lambda_-$, $W_+({{\mathbf x}})$, $W_-({{\mathbf x}})$, $\gamma\in
(-\bar\gamma,0)$, $(\lambda_+-\delta,\lambda_+)$ and $(\lambda_-,\;\lambda_-+\delta)$;
\(iii) if condition (C) is satisfied for both edges $\lambda_-$ and $\lambda_+$, $d-d_+\ge 3$, $d-d_-\ge 3$ and $W\in L_1({{\mathbf R}}^d)$, there is a threshold for the birth of the impurity spectrum in the gap $(\lambda_-,\lambda_+)$, that is $\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset$ for a small enough $|\gamma|$.
Proof of main results {#sec:prmainres}
=====================
General results on the birth of virtual eigenvalues {#subsec:genres}
---------------------------------------------------
Let $H_0$,$\;H_\gamma=H_0+\gamma W$ and $(\lambda_-,\lambda_+)$ are the same as in Section \[subsec:spectchar\]. For $\lambda\in(\lambda_-,\lambda_+)$ consider the operator $$\label{BrmSchwop}
X_W(\lambda)=W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}},$$ which is called in the literature the [*Birman-Schwinger operator*]{} ([@S1], [@Sc]). Here $$\begin{aligned}
\label{dfsqrtW}
W^{1/2}x=\left\{\begin{array}{ll}
W(|W||_{(\ker(W))^{\perp}})^{-1/2}x,& \rm{if}\; x\in
(\ker(W))^{\perp},\\
0,& \rm{if}\; x\in \ker(W),
\end{array}\right.\end{aligned}$$ hence $W=|W|^{\frac{1}{2}}W^{\frac{1}{2}}$. In [@S1] (Lemma 7.1) a connection between the spectrum of the operator pencil $I+\gamma
X_W(\lambda)\,(\lambda\in(-\infty, 0))$ and the spectrum of $H_\gamma$ in the gap $(-\infty,0)$ of $\sigma(H_0)$ was established for the case where ${{\mathcal H}}=L_2({{\mathbf R}}^d)$ and $H_0=-\Delta$. For the case of a definite perturbation $(W\ge 0)$ of a periodic potential the analogous result was obtained in [@Bi2] (Proposition 1.5). The following claim is a generalization of this claim to the case of an indefinite perturbation. We also have indicated an operator which realizes a linear isomorphism between $\ker(H_\gamma-\lambda I)$ and $\ker(I+\gamma X_W(\lambda))$.
\[BrScwspct\] Assume that for some $\lambda_0\in{{\mathcal R}}(H_0)$ the operator $R_{\lambda_0}(H_0)|W|^{\frac{1}{2}}$ is compact and $(\lambda_-,\,\lambda_+)$ is a gap of the spectrum of the operator $H_0$. Then the set $\sigma_\gamma(\lambda_-,\lambda_+)=\sigma(H_\gamma)\cap(\lambda_-,\lambda_+)$ consists of at most countable number of eigenvalues of finite multiplicity of the operator $H_\gamma$ and they can cluster only to the edges of the gap $(\lambda_-,\,\lambda_+)$. Furthermore, $\sigma_\gamma(\lambda_-,\lambda_+)$ coincides with the spectrum of the pencil of operators $\Phi(\gamma)=I+\gamma
X_W(\lambda)\,(\lambda\in(\lambda_-,\lambda_+))$. Moreover, the operator function $X_W(\lambda)$ is holomorphic in ${{\mathcal R}}(H_0)$ in the operator norm, each of the operators $X_W(\lambda)$ is compact and any point $\lambda\in\sigma_\gamma(\lambda_-,\lambda_+)$ is an eigenvalue of the pencil $\Phi(\gamma)$ such that the operator $W^{1/2}\vert_{E_\lambda}$ realizes a linear isomorphism between the subspaces $E_\lambda=\ker(H_\gamma-\lambda I)$ and $L_\lambda=\ker(I+\gamma X_W(\lambda))$, hence $\dim(E_\lambda)=\dim(L_\lambda)$.
Assume that $\lambda\in{{\mathcal R}}(H_0)$. The subspace $E_\lambda$ is the set of solutions of the equation $H_0x-\lambda x+\gamma Wx=0$, which is equivalent to the equation $x+\gamma R_\lambda(H_0)Wx=0$. Denote $y=P_Wx$, $z=(I-P_W)x$, where $P_W$ is the orthogonal projection on the subspace ${{\mathcal H}}_W={{\mathcal H}}\ominus\ker(W)$. Since $W(I-P_W)=0$, the last equation is equivalent to the system $$\label{systeq}
y+\gamma P_WR_\lambda(H_0)Wy=0, \quad z+\gamma\Xi y=0,$$ where $$\label{dfXi}
\Xi=(I-P_W)R_\lambda(H_0)W.$$ Hence the operator $\Psi=(I-\gamma\Xi)\vert_{G_\lambda}$ realizes a linear isomorphism between the subspace $G_\lambda$ of solutions of equation (\[systeq\]-a) and the subspace $E_\lambda$. As it is clear, $\Psi^{-1}=P_W\vert_{E_\lambda}$. Consider the operator $S=W^{1/2}\vert_{{{\mathcal H}}_W}$. Since $\ker(W^{1/2})=\ker(W)$, ${{\mathcal H}}_W$ is an invariant subspace for the operator $W^{1/2}$. Hence $S$ maps injectively ${{\mathcal H}}_W$ into itself. Let us show that $S(G_\lambda)=L_\lambda$. Assume that $y\in G_\lambda$, that is $y$ is a solution of the equation (\[systeq\]-a). Multiplying the both sides of (\[systeq\]-a) by $W^{1/2}$, we get that $v=Sy=W^{1/2}y$ is a solution of the equation $v+\gamma
W^{1/2}P_WR_\lambda(H_0)|W|^{1/2}v=0$. Since $W^{1/2}(I-P_W)=0$, the last equation is equivalent to $$\label{kerXW}
v+\gamma W^{1/2}R_\lambda(H_0)|W|^{1/2}v=0.$$ This means that $v\in\ker(I+\gamma X_W(\lambda))=L_\lambda$, hence $S(G_\lambda)\subseteq L_\lambda$. Let us prove the inverse inclusion. Assume that $v\in L_\lambda$, that is $v$ is a solution of equation (\[kerXW\]). Then we see that $v\in
\mathrm{Im}(W^{1/2})\subseteq H_W$. Hence there exists $y\in H_W$ such that $v=Sy=W^{1/2}y$ and $W^{1/2}y+\gamma
W^{1/2}R_\lambda(H_0)Wy=0$. The last equality and $W^{1/2}(I-P_W)=0$ imply that $Sy+\gamma SP_WR_\lambda(H_0)Wy=0$. Since the operator $S$ is injective, we get that $y$ satisfies equation (\[systeq\]-a), that is $y\in G_\lambda$. We have shown that $L_\lambda\subseteq S(G_\lambda)$. Since we have proved above the inverse inclusion, we get: $L_\lambda= S(G_\lambda)$. Observe that $W^{1/2}(I-P_W)=0$. Thus, the operator $S\Psi^{-1}=W^{1/2}P_W\vert_{E_\lambda}=W^{1/2}\vert_{E_\lambda}$ realizes a linear isomorphism between $E_\lambda$ and $L_\lambda$. Hence, in particular, $\dim(E_\lambda)=\dim(L_\lambda)$.
The compactness of the operator $R_{\lambda_0}(H_0)|W|^{\frac{1}{2}}$, the boundedness of the operator $W^{\frac{1}{2}}$ and the Hilbert identity $R_{\lambda}(H_0)-R_{\lambda_0}(H_0)=(\lambda-\lambda_0)R_{\lambda}(H_0)R_{\lambda_0}(H_0)$ imply that for any $\lambda\in{{\mathcal R}}(H_0)$ the operator $X_W(\lambda)$ is compact and it is a holomorphic on ${{\mathbf C}}\setminus\sigma(H_0)$ operator function w.r.t. the operator norm. Then, as it is known, $\alpha(\lambda)=\dim(\ker(I+\gamma X_W(\lambda)))<\infty$ for any $\lambda\in{{\mathcal R}}(H_0)$ and $\alpha(\lambda)$ takes a constant value $m$ at any point $\lambda\in{{\mathcal R}}(H_0)$, except a set of isolated points in ${{\mathbf C}}\setminus\sigma(H_0)$ at which $\alpha(\lambda)>m$ ([@Gh-Kr], Chapt. 1, Sect. 5, Theorem 5.4). On the other hand, since $H_0$ is self-adjoint, for $\Im(\lambda)\neq 0$ the estimate $\Vert X_W(\lambda)\Vert\le\Vert W\Vert\Vert
R_{\lambda_0}(H_0)\Vert\le \frac{\Vert W\Vert}{|\Im(\lambda)|)}$ is valid, hence for a large enough $|\Im(\lambda)|$ the operator $I+\gamma X_W(\lambda)$ is continuously invertible. This means that $m=0$. Therefore, in particular, $\alpha(\lambda)=0$ for any $\lambda\in(\lambda_-,\lambda_+)$, except a set $\Lambda\subset(\lambda_-,\lambda_+)$ of isolated in $(\lambda_-,\lambda_+)$ points. Taking into account that $\dim(E_\lambda)=\dim(L_\lambda)$ for $\lambda\in(\lambda_-,\lambda_+)$, we obtain that each point of $\Lambda$ is an isolated eigenvalue of the operator $H_\gamma$ of a finite multiplicity. On the other hand, if $\lambda\in(\lambda_-,\lambda_+)\setminus\Lambda$, the operator $I+\gamma X_W(\lambda)$ is continuously invertible, because the operator $X_W(\lambda)$ is compact. Hence $\mathrm{Im}(I+\gamma
X_W(\lambda))={{\mathcal H}}$. Therefore, in particular, for any $f\in{{\mathcal H}}$ the equation $v+\gamma X_W(\lambda)v=W^{1/2}P_Wf$ has a solution $v$ which, as it is clear, belongs to $\mathrm{Im}(W^{1/2})$. Hence the vector $y=\big((W^{1/2}\vert_{{{\mathcal H}}_W}\big)^{-1}v\in{{\mathcal H}}_W$ satisfies the equation $W^{1/2}y+\gamma X_W(\lambda)W^{1/2}y=W^{1/2}P_Wf$. Since the operator $(W^{1/2}\vert_{{{\mathcal H}}_W}$ is injective, the last equality implies that $y+\gamma P_WR_\lambda(H_0)Wy=P_Wf$. Hence the vector $x=(I-\gamma\Xi)y+(I-P_W)f$ satisfies the equation $x+R_\lambda(H_0)W=f$, which is equivalent to $H_\gamma x-\lambda
x=f$. Recall that the operator $\Xi$ is defined by (\[dfXi\]). Thus, we have proved that for any $\lambda\in(\lambda_-,\lambda_+)\setminus\Lambda$ $\;\mathrm{Im}(H_\gamma-\lambda I)={{\mathcal H}}$. Since $H_\gamma$ is self-adjoint, this means that $(\lambda_-,\lambda_+)\setminus\Lambda\subseteq{{\mathcal R}}(H_\gamma)$. Thus, the set $\Lambda$ which is the spectrum of the pencil $X_W(\lambda)\;(\lambda\in(\lambda_-,\lambda_+))$, coincides with the part of the spectrum of the operator $H_\gamma$ lying in $(\lambda_-,\lambda_+)$ and it consists of isolated eigenvalues of $H_\gamma$ having finite multiplicities.
In what follows we shall assume that the condition of Proposition \[BrScwspct\] is fulfilled.
Consider the case where $W\ge 0$, hence $X_W(\lambda)=W^{\frac{1}{2}}R_\lambda(H_0)W^{\frac{1}{2}}$. Let us recall some notions and results from [@Ar-Zl1] used in this section. By Proposition \[BrScwspct\], for each $\lambda\in(\lambda_-,\lambda_+)$ the operator $X_W(\lambda)$ is compact. Then since it is self-adjoint, its spectrum consists of at most a countable number of real eigenvalues which can cluster only to the point $0$. Furthermore, each its non-zero eigenvalue has a finite multiplicity.
\[gde1\]([@Ar-Zl1]) [For any fixed $\lambda\in(\lambda_-,\lambda_+)$ let us number all the positive eigenvalues ${\mu_k^+(\lambda)}_{k\in{{\mathbf N}}}$ of the operator $X_W(\lambda)$ in the non-increasing ordering $\mu_1^+(\lambda)\ge\mu_2^+(\lambda)\ge\dots\ge\mu_k^+(\lambda)\ge
\dots$ and all the negative ones ${\mu_k^-(\lambda)}_{k\in{{\mathbf N}}}$ in the non-decreasing ordering $\mu_1^-(\lambda)\le\mu_2^-(\lambda)\le\dots\le\mu_k^-(\lambda)\le
\dots$ (each eigenvalue is repeated according to its multiplicity). So, by such ordering we have chosen one-valued branches of eigenvalues of the operator function $X_W(\lambda)$. We call these branches the]{} [*characteristic branches*]{} [(positive and negative) of the operator $H_0$ with respect to the operator $W$ on a gap $(\lambda_-,\,\lambda_+)$ of $\sigma(H_0)$]{}.
\[rembranch\] As it was shown in [@Ar-Zl1] (Proposition 3.7), if $W\ge 0$ the functions $\mu_k^+(\lambda)$ and $\mu_k^-(\lambda)$ are continuous and increasing. Since they can “go to zero” at some points of the gap $(\lambda_-,\,\lambda_+)$, each of them has its domain ${{\rm Dom}}(\mu_k^+)$ and ${{\rm Dom}}(\mu_k^-)$ which have the form ${{\rm Dom}}(\mu_k^+)=(\eta_k^+,\lambda_+),\;\eta_k^+\in[\lambda_-,\lambda_+]$, ${{\rm Dom}}(\mu_k^-)=(\lambda_-,\eta_k^-),\;\eta_k^-\in[\lambda_-,\lambda_+]$ and the following property is valid: $\eta_k^+\in(\lambda_-,\lambda_+)\Rightarrow
\lim_{\lambda\downarrow\eta_k^+}\mu_k^+(\lambda)=0$, $\eta_k^-\in(\lambda_-,\lambda_+)\Rightarrow
\lim_{\lambda\uparrow\eta_k^-}\mu_k^-(\lambda)=0$. Furthermore, the sequence $\{\eta_k^+\}$ is non-decreasing and the sequence $\{\eta_k^-\}$ is non-increasing.
Denote $\tilde\mu_k^+=\lim_{\lambda\uparrow
\lambda_+}\mu_k^+(\lambda)$, $\;\tilde\mu_k^-=\lim_{\lambda\downarrow \lambda_-}\mu_k^-(\lambda)$ (the values $+\infty$ and $-\infty$ are allowed for these limits).
\[gde2\] [Consider $l(\lambda_-), l(\lambda_+)\in{{\mathbf Z}}_+\cup\{+\infty\}$ defined by the conditions: $$\begin{aligned}
&&-\infty=\tilde\mu_1^-=\tilde\mu_2^-=\dots=\tilde\mu_{l(\lambda_-)}^-
<\tilde\mu_{l(\lambda_-)+1}^-\le\tilde\mu_{l(\lambda_-)+2}^-\le\dots,\\
&& {{\rm if}}\quad\lambda_->-\infty,
\end{aligned}$$ $$\begin{aligned}
&& +\infty=\tilde\mu_1^+=\tilde\mu_2^+=\dots=\tilde\mu_{l(\lambda_+)}^+
>\tilde\mu_{l(\lambda_+)+1}^+\ge\tilde\mu_{l(\lambda_+)+2}^+\ge\dots,\\
&& {{\rm if}}\quad\lambda_+<+\infty.\end{aligned}$$ We call $l(\lambda_-)$ and $l(\lambda_+)$ the]{} [*asymptotic multiplicities of the edges $\lambda_-$ and $\lambda_+$*]{} [of the gap $(\lambda_-,\,\lambda_+)$ of $\sigma(H_0)$ with respect to the operator $W$ and denote them $
l(\lambda_-)=M(\lambda_-,H_0,W),\;\;l(\lambda_+)=M(\lambda_+,H_0,W).
$ The branches $\{\mu_k^-(\lambda)\}_{k=1}^{l(\lambda_-)}$, $\{\mu_k^+(\lambda)\}_{k=1}^{l(\lambda_+)}$ are called the]{} [*main characteristic branches*]{} [of the operator $H_0$ with respect to the operator $W$ near the edges $\lambda_-$ and $\lambda_+$ respectively.]{}
We shall use the following results from [@Ar-Zl1]:
\[ordl\] The set $\sigma(H_\gamma)\cap(\lambda_-,\lambda_+)$ has the representation: $$\begin{aligned}
\sigma(H_\gamma)\cap(\lambda_-,\lambda_+)=\left\{\begin{array}{ll}
\bigcup_{k=1}^\infty\{\rho_k^+(\gamma)\} &{{\rm for}}\;\;\gamma<0,\\
\bigcup_{k=1}^\infty\{\rho_k^-(\gamma)\} &{{\rm for}}\;\;\gamma>0,
\end{array}\right.\end{aligned}$$ where $\rho_k^+(\gamma)=(\mu_k^+)^{-1}\left(-\frac{1}{\gamma}\right)$, $\rho_k^-(\gamma)=(\mu_k^-)^{-1}\left(-\frac{1}{\gamma}\right)$ and $(\mu_k^+)^{-1},\;\;(\mu_k^-)^{-1}$ are the inverses of the functions $\mu_k^+(\lambda),\;\mu_k^-(\lambda)$, the positive and negative characteristic branches of $H_0$ with respect to $V$ on the gap $(\lambda_-,\,\lambda_+)$.
\[gpbr\] Assume that the edge $\lambda_+$ of the gap $(\lambda_-,\,\lambda_+)$ of $\sigma(H_0)$ is finite.
\(i) If $l(\lambda_+)=M(\lambda_+, H_0,W)<\infty$, then for $\gamma<0$ the operator $H_\gamma$ has exactly $l(\lambda_+)$ branches $\lambda_-<\rho_1^+(\gamma)\le\rho_2^+(\gamma)\le
\dots\le\rho_{l(\lambda_+)}^+(\gamma)$ of virtual eigenvalues near the edge $\lambda_+$ of $(\lambda_-,\,\lambda_+)$. Furthermore, for any $1\le k\le l(\lambda_+)$ and $\gamma\in{{\rm Dom}}(\lambda_k^+)$, $$\label{frm}
\rho_k^+(\gamma)=(\mu_k^+)^{-1}\left(-\frac{1}{\gamma}\right),$$ where $(\mu_k^+)^{-1}$ is the inverse of the function $\mu_k^+(\lambda)$ (the main characteristic branch of $H_0$ with respect to $W$ near the edge $\lambda_+$). If $\;l(\lambda_+)<\infty$ and $\lambda_-=-\infty$, then ${{\rm Dom}}(\rho_k^+)=(-\infty,0)\;\;\forall k\in\{1,2,\dots,l(b)\}$. Hence the operator $H_\gamma$ has at least $l(\lambda_+)$ eigenvalues in the gap $(-\infty,\,\lambda_+)$ for any $\gamma<0$.
\(ii) If $M(\lambda_+,H_0,W)=\infty$, then for $\gamma<0$ the number of the branches of eigenvalues $\{\lambda_k^+(\gamma)$ of the operator $H_\gamma$, which enter the gap $(\lambda_-,\,\lambda_+)$ across the edge $\lambda_+$, is infinite, each of them is a virtual eigenvalue and the property is valid for them $(-\theta_k,
0)\subseteq{{\rm Dom}}(\rho_k^+),\; {\rm where}\;\;\theta_k\uparrow
+\infty\;\;{{\rm for}}\;\;k\rightarrow\infty$. The latter fact means that the operator $H_\gamma$ has an infinite number of eigenvalues in the gap $(\lambda_-,\,\lambda_+)$ for any $\gamma<0$. These eigenvalues cluster to the edge $\lambda_+$ only and formula (\[frm\]) with $l(\lambda_+)=\infty$ is valid for them.
\(iii) If $\gamma<0$, for the branches of virtual eigenvalues of the operator $H_\gamma$ at the edge $\lambda_-$ of $(\lambda_-,\,\lambda_+)$ the analogous claims are valid like in the case of the edge $\lambda_+$.
In order to get in what follows an asymptotic representation for virtual eigenvalues, we need the following claim:
\[lmasymp\] If $\mu:\,(a,b)\rightarrow{{\mathbf R}}$ be an increasing continuous function having the asymptotic representation for $\lambda\uparrow b$: $\mu(\lambda)=\frac{A}{f(\lambda)}+O(1)$, where $A>0$ and $f:\,(a,b)\rightarrow{{\mathbf R}}$ is a decreasing continuous function such that $\lim_{\lambda\uparrow b}f(\lambda)=0$, then
\(a) there is $\bar\gamma>0$ such that for any $\gamma\in(0,\bar\gamma]$ the equation $\mu(\lambda)=\frac{1}{\gamma}$ has in $(a,b)$ a unique solution $\lambda=\rho(\gamma)$;
\(b) the asymptotic representation $f(\rho(\gamma))=\gamma(A+O(\gamma))$ is valid for $\gamma\downarrow
0$.
Claim (a) follows from the monotony and continuity of $\mu$ and the fact that $\lim_{\lambda\uparrow b}\mu(\lambda)=\infty$. Then in view of the equality $\frac{A}{f(\rho(\gamma))}+O(1)=\frac{1}{\gamma}$, we have: $f(\rho(\gamma)=\gamma A+\gamma f(\lambda(\gamma)O(1)$, hence $f(\rho(\gamma)=O(\gamma)$ for $\gamma\downarrow 0$. The last two equalities imply the desired asymptotic representation.
Perturbation of a compact operator {#subsec:pertcompoper}
----------------------------------
In what follows we shall use the following result about a perturbation of a compact operator in a Hilbert space.
\[prpertcompoper\] Let $\Phi(t)\;(t\in[0,\delta]\,\delta>0)$ be a family of linear compact operators acting in a Hilbert space ${{\mathcal H}}$ such that $$\label{rprPhit}
\Phi(t)=\Phi(0)+t\theta(t),$$ where $$\label{cndtht}
\bar\theta=\sup_{t\in[0,\delta]}\Vert\theta(t)\Vert<\infty$$ and the operator $\Phi(0)$ has positive eigenvalues. Let $\mu_1^0\ge\mu_2^0\ge\dots\ge\mu_N^0\;(N\le\infty)$ be such eigenvalues of $\Phi(0)$ arranged counting their multiplicities and $e_1^0,e_2^0,\\\dots,e_N^0$ be an orthonormal system of eigenvectors corresponding to them. Then
\(i) if $N<\infty$, there exists $\tilde\delta\in[0,\delta]$ such that
\(a) for any $t\in[0,\tilde\delta]$ the operator $\Phi(t)$ has $N$ positive eigenvalues $\mu_1(t)\ge\mu_2(t)\ge\dots\ge\mu_N(t)$ (counting their multiplicities) having the asymptotic representation $$\label{asymptlmu}
\mu_k(t)=\mu_k^0+O(t)\quad\rm{as}\quad t\rightarrow
0\quad(k=1,2,\dots, N))$$ and all the rest of positive eigenvalues of $\Phi(t)$ (if they exist) have the asymptotic representation $$\label{asymptlmurest}
\mu_k(t)=O(t)\quad\rm{as}\quad t\rightarrow 0\quad(k> N);$$
\(b) if $m(j)$ is the multiplicity of the positive eigenvalue $\mu_j^0$ of the operator $\Phi(0)\;(j\in\{1,2,\dots,N\})$ and $$\label{groupeigval}
\mu_{l(j)}(t)\ge\mu_{l(j)+1}(t)\ge\dots\ge\mu_{l(j)+m(j)-1}(t)\quad(l(j)\in\{1,2,\dots,N\})$$ is the group of the eigenvalues of the operator $\Phi(t)$ which tend to $\mu_j^0$ as $t\rightarrow 0$, then for any $t\in[0,\tilde\delta]$ it is possible to choose an orthonormal basis $e_k(t)\;(k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\})$ in the subspace spanned by eigenvectors of $\Phi(t)$ corresponding to these eigenvalues, such that for any $k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\}$ $\Vert
e_k(t)-e_k^0\Vert=O(t)$ for $t\rightarrow 0$.
\(ii) If $N=\infty$, for any natural $n$ there exists $\tilde\delta\in[0,\delta]$ such that for any $t\in[0,\tilde\delta]$ the operator $\Phi(t)$ has $n$ positive eigenvalues $\mu_1(t)\ge\mu_2(t)\ge\dots\ge\mu_n(t)$ (counting their multiplicities). Furthermore, these eigenvalues and eigenvectors, corresponding to them, have the same properties as in claim (i) except (\[asymptlmurest\]), with the number $n$ instead of $N$.
\(i) In view of (\[rprPhit\]) and (\[cndtht\]), $\Phi(0)-\bar\theta tI\le\Phi(t)\le\Phi(0)+\bar\theta tI$ for any $t\in[0,\delta]$. Hence by Lemma 3.4 of [@Ar-Zl1] ( a modification of the comparison theorem on the base of the minimax principle), if $\tilde
\delta<\min\{\delta,\frac{\tilde\mu_N^0}{\bar\theta}\}$, then for any $t\in[0,\tilde\delta]$ the operator $\Phi(t)$ has $N$ positive eigenvalues $\mu_1(t)\le\mu_2(t)\le\dots\le\mu_N(t)$ (counting their multiplicities), for which the estimates $$\label{esttlmukt}
\forall\;t\in[0,\tilde\delta]:\quad\mu_k^0-\bar\theta
t\le\mu_k(t)\le\mu_k^0+\bar\theta t\quad (k=1,2,\dots, N)$$ are valid and all the rest of positive eigenvalues of $\Phi(t)$ (if they exist) satisfy the estimates $\mu_k(t)\le\bar\theta t\quad
(k>N)$ for any $t\in[0,\tilde\delta]$. These estimates imply the desired asymptotic formulas (\[asymptlmu\]) and (\[asymptlmurest\]). Claim (i-a) is proven.
Let $\tilde\mu_1^0>\tilde\mu_2^0>\dots>\tilde\mu_{\tilde
N}^0\;(\tilde N\le N)$ be all the mutually different positive eigenvalues of the operator $\Phi(0)$. The sum of their multiplicities is equal to $N$. Let $E_1^0,E_2^0,\dots,E_{\tilde
N}^0$ be the eigenspaces of $\Phi(0)$ corresponding to these eigenvalues. In order to prove the second part of claim (i), we shall show that there exists an operator function $U(\cdot):\,[0,\tilde\delta]\rightarrow{{\mathcal B}}({{\mathcal H}})$ taking unitary values such that for any $j\in\{1,2,\dots \tilde N\}$ the subspace $E_j(t)=U(E_j^0)$ is spanned by the eigenspaces of those eigenvalues of the operator $\Phi(t)$ which tend to the eigenvalue $\tilde
\mu_j^0$ of the operator $\Phi(0)$ as $t\rightarrow 0$ and moreover, $$\label{estdiffIandU}
\Vert I-U(t)\Vert=O(t)\quad \rm{for}\quad t\rightarrow 0;$$ Using (\[esttlmukt\]), we can choose $r>0$ such that for any $j\in\{1,2,\dots N\}$ and $t\in[0,\tilde\delta]$ the circle $C_r^j$ in the complex plane with the radius $r$ and the center at the eigenvalue $\tilde \mu_j^0$ lies in the resolvent set of the operator $\Phi(t)$ and surrounds the group of eigenvalues (\[groupeigval\]) of the operator $\Phi(t)$. Consider the total orthogonal projection $P_j(t)$ on the subspace $E_j(t)$ spanned by the eigenspaces of the above group of eigenvalues of $\Phi(t)$: $P_j(t)=-\frac{1}{2\pi i}\oint_{C_r^j}R_\lambda(\Phi(t))\,d\lambda$. In particular, $P_j^0=P_j(0)$ is the orthogonal projection on the eigenspace $E_j^0$ of the operator $\Phi(0)$ corresponding to its eigenvalue $\tilde\mu_j^0$. Then $$\label{diffproj}
P_j(t)-P_j^0=-\frac{1}{2\pi
i}\oint_{C_r^j}\big(R_\lambda(\Phi(t))-R_\lambda(\Phi(0))\big)\,d\lambda.$$ From (\[rprPhit\]) we get easily the representation $R_\lambda(\Phi(t))-R_\lambda(\Phi(0))=-tR_\lambda(\Phi(0))\\\theta(t)R_\lambda(\Phi(t))$. Then choosing $\tilde\delta>0$ such that $$\forall\;t\in[0,\tilde\delta],\quad\forall\;\lambda\in\bigcup_{j=1}^NC_r^j:\quad
\Vert
tR_\lambda(\Phi(0))\theta(t)\Vert\le\tilde\delta\bar\theta\Vert
R_\lambda(\Phi(0))\Vert<1$$ and using standard arguments, we obtain from representation (\[diffproj\]) that $$\exists\;C>0,\quad\forall\;t\in[0,\tilde\delta],\quad\forall\;j\in\{1,2,\dots
\tilde N\}:\quad\Vert P_j(t)-P_j^0\Vert\le Ct.$$ Then after a suitable restriction of the interval $[0,\tilde\delta]$ the inequality $\Vert P_j(t)-P_j^0\Vert\le 1$ is valid for any $t\in[0,\tilde\delta]$, which implies easily that the operator $Q(t)=P_j(t)\vert_{E_j^0}$ realizes a linear isomorphism between the subspaces $E_j^0$ and $E_j(t)$. Since for any $x\in E_j^0\;$ $\Vert
Q(t)x-x\Vert\le\Vert P_j(t)-P_j^0\Vert\Vert x\Vert\le Ct\Vert
x\Vert$, then for the sequence of vectors $e_k(t)=Q(t)e_k^0\;(k\in\{k(j),k(j)+1,\dots,k(j)+m(j)-1\})$ claim (i-b) is valid.
\(ii) Claim (ii) follows straightforwardly from all the arguments used in the proof of claim (i).
The case of non-degenerate edges of a gap of the spectrum of the unperturbed operator {#subsec:nondegedg}
-------------------------------------------------------------------------------------
In what follows we need some results about the Birman-Schwinger operator corresponding to the Schrödinger operator $H_0=-\Delta+V({{\mathbf x}})\cdot$ with a periodic potential $V({{\mathbf x}})$ and the perturbation $W({{\mathbf x}})$ of this potential (see Section \[sec:preliminaries\]).
### Compactness of Birman-Schwinger operator {#subsubsec:compBirmSchwing}
\[compBirSchw\] If the periodic potential $V({{\mathbf x}})$ satisfies the condition (\[condperpotent\]) and the perturbation $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$ and $\lim_{|{{\mathbf x}}|\rightarrow\infty} W({{\mathbf x}})=0$, then for some $\lambda_0\in{{\mathbf R}}\setminus\sigma(H_0)$ the operator $R_{\lambda_0}|W|^{\frac{1}{2}}$ is compact, hence for the operator $H_\gamma$ and the Birman-Schwinger operator $X_W(\lambda)=W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}}$ all the claims of Proposition \[BrScwspct\] are valid.
By claim (ii) of Proposition \[propselfadj\], the operator $H_0$ is self-adjoint, bounded below and its domain is $W_2^2({{\mathbf R}}^d)$. It is enough to prove that the operator $|W|^{\frac{1}{2}}R_{\lambda_0}(H_0)^{\frac{1}{2}}=(R_{\lambda_0}(H_0)^{\frac{1}{2}}|W|^{\frac{1}{2}})^\star$ is compact for some $\lambda_0<\inf(\sigma(H_0))$, that is the set $$\label{defK}
K=\{u\in W_2^2({{\mathbf R}}^d)\;\vert\;\big((H_0-\lambda_0I) u,u\big)\le 1\}$$ is compact in the semi-norm $\sqrt{(|W|u,u)}$. By claims (ii) and (iii) of Lemma \[lmestVuOm\] and claims (ii) and (iii) of Lemma \[lmestVuRd\], it is possible to choose $\lambda_0<\inf(\sigma(H_0))$ such that for some $m,\,m_1>0$ and any $u\in W_2^2({{\mathbf R}}^d)$, ${{\mathbf l}}\in\Gamma$ $\int_{\Omega_{{\mathbf l}}}(H_0u-\lambda_0u)\bar u\,d{{\mathbf x}}\ge
m\int_{\Omega_{{\mathbf l}}}(|\nabla u|^2+|u|^2)\,d{{\mathbf x}}$ and $\big((H_0-\lambda_0I)u,u\big)\ge m_1(\|\nabla u\|^2+\|u\|^2)$, where $\Omega_{{\mathbf l}}=\Omega+\{{{\mathbf l}}\}$ (recall that $\Gamma$ is the lattice of periodicity of $V({{\mathbf x}})$). Then $\int_{G_N}|W({{\mathbf x}})||u({{\mathbf x}})|^2\,d{{\mathbf x}}\le m\sup_{{{\mathbf x}}\in
G_N}|W({{\mathbf x}})|\big((H_0-\lambda_0I)u,u\big)$, where $G_N=\bigcup_{|{{\mathbf l}}|_\infty\ge N}\Omega_{{\mathbf l}}$ and $|{{\mathbf l}}|_\infty=\max_{1\le j\le d}|({{\mathbf l}})_j|$. These circumstances and the compactness of the embedding of $W_2^1({{\mathbf R}}^d\setminus G_N)$ into $L_2({{\mathbf R}}^d\setminus G_N)$ imply that the set $K$, defined by (\[defK\]), is compact in the semi-norm $\sqrt{(|W|u,u)}$.
### Representation of Birman-Schwinger operator in the case of a non-degenerate edge {#subsubsec:reprBirmSchwingnondeg}
\[rprBirSchw\] Let $(\lambda_-,\,\lambda_+)$ be a gap of the spectrum of the unperturbed operator $H_0$. Assume that $W({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$, $W\in L_1({{\mathbf R}}^d)$ and $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$. Let us take $\delta\in(0,\,(\lambda_+-\lambda_-))$.
\(i) If $d=1$, the edge $\lambda_+$ is non-degenerate and condition (\[cndWd1\]) is satisfied, then for $\lambda\in((\lambda_+-\delta,\;\lambda_+)$ the representation is valid: $$\label{rprXwd1}
X_W(\lambda)=\Phi_+(\lambda)+\Theta_+(\lambda),$$ where $\Phi_+(\lambda))$ is a rank one operator of the form $$\label{dfPhipld1}
\Phi_+(\lambda)=\frac{\sqrt{m^+_1}}{\sqrt{2(\lambda_+-\lambda)}}(\cdot,v^+_1)v^+_1,$$ $m_1^+$ and $v^+_1$ are defined by (\[dfmupl\]) and (\[dfvlpld1\]) and $\Theta_+(\lambda)$ is a bounded operator in $L_2({{\mathbf R}})$ such that $$\begin{aligned}
\label{bndthtpl}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\Theta_+(\lambda)\|<\infty\end{aligned}$$
\(ii) If $d=2$, the edge $\lambda_+$ is non-degenerate and condition (\[cndWd2\]) is satisfied, then for $\lambda\in(\lambda_+-\delta,\;\lambda_+)$ the representation (\[rprXwd1\]) is valid, where $\Theta_+(\lambda)$ is a bounded operator in $L_2({{\mathbf R}}^2)$ such that condition (\[bndthtpl\]) is satisfied, $\Phi_+(\lambda)$ is a finite rank operator of the form $$\label{dfPhipld2}
\Phi_+(\lambda)=\frac{1}{2\pi}\ln\left(\frac{1}{\lambda_+-\lambda}\right)G_W^+,$$ and $G_W^+$ is defined by (\[dfGW\]), in which $v_k^+$ is defined by (\[dfvlpl\]), (\[dfvlxp\]) and (\[Bloch1\]);
\(iii) If $d\ge 3$ and the edge $\lambda_+$ is non-degenerate, then $$\label{bndXw}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|X_W(\lambda)\|<\infty.$$
\(iv) The claims analogous to (i)-(iii) are valid with $\lambda_-$, $(\lambda_-,\;\lambda_-+\delta)$, $m_1^-$, $G_W^-$ and $v_k^-$ instead of, respectively, $\lambda_+$, $(\lambda_+-\delta,\;\lambda_+)$, $m_1^+$, $G_W^+$ and $v_k^+$.
Using claim (i) of Proposition \[prrprres\], we get: $X_W(\lambda)=\sqrt{W}K(\lambda)\sqrt{W}+\sqrt{W}\Theta(\lambda)\sqrt{W}$, where, in view of (\[bndtht2\]) and the boundedness of $W({{\mathbf x}})$, $$\begin{aligned}
\label{bndtht1}
&&\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\sqrt{W}\Theta(\lambda)\sqrt{W}\|\le\sup_{{{\mathbf x}}\in{{\mathbf R}}^d}|W({{\mathbf x}})|
\nonumber\times\\
&&\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\Theta(\lambda)\|<\infty.\end{aligned}$$ Assume that $d=1$ and $\lambda\in(\lambda_+-\delta,\;\lambda_+)$. Observe that the multiplication operator $\sqrt{W}\cdot$ maps the set $L_{2,0}({{\mathbf R}})$ into itself. In view of claim (ii,a) of Proposition \[prrprres\], $\sqrt{W}K(\lambda)\sqrt{W}\vert_{L_{2,0}({{\mathbf R}}^d)}$ is an operator with the integral kernel $$\sqrt{W(x)}F^+(x,s,\lambda)\sqrt{W(s)}+\sqrt{W(x)}\tilde
K^+(x,s,\lambda)\sqrt{W(s)},$$ and $$(\Phi_+(\lambda)f)(x)=\frac{1}{2\mu^+\sqrt{\lambda_+-\lambda}}
\int_{-\infty}^\infty\sqrt{W(x)}b^+(x,p_1)\overline{b^+(s,p_1)}\sqrt{W(s)}f(s)\,ds.$$ is the integral kernel of the operator $\sqrt{W(x)}
F^+(x,s,\lambda)\sqrt{W(s)}$. Since $W\in L_1({{\mathbf R}})$ and, in view of claim (i) of Theorem \[thmainApp\], the function $b^+(x,p_1)$ is bounded on ${{\mathbf R}}$, then the function $v^+_1(x)$, defined by (\[dfvlpld1\]), belongs to $L_2({{\mathbf R}})$. Hence the operator $\Phi_+(\lambda)$ can be written in the form (\[dfPhipld1\]). Furthermore, in view of (\[cndK1pl\]), there is $M>0$ such that for any $\lambda\in(\lambda_+-\delta,\;\lambda_+)$ and $x,s\in{{\mathbf R}}$ $\sqrt{W(x}|\tilde K^+(x,s,\lambda)|\sqrt{W(s)}\le
M(1+|x-s|)\sqrt{W(x)}\sqrt{W(s)}$. Then, in view of condition (\[cndWd1\]), $$\begin{aligned}
&&\sup_{\lambda\in((\lambda_++\lambda_-)/2,\;\lambda_+)}
\int_{-\infty}^\infty\int_{-\infty}^\infty W(x)|\tilde K^+(x,s,\lambda)|^2W(s)\,dx\,ds\le\\
&&M^2\int_{-\infty}^\infty\int_{-\infty}^\infty W(x)(1+|x-s|)^2W(s)\,dx\,ds<\infty.\end{aligned}$$ This means that for any $\lambda\in(\lambda_+-\delta,\;\lambda_+)$ the operator $\tilde K^+(\lambda)$ with the integral kernel $\sqrt{W(x)}\tilde K^+(x,s,\lambda)\sqrt{W(s)}$ belongs to the Hilbert-Schmidt class (hence it is bounded in $L_2({{\mathbf R}})$), and furthermore $\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\tilde
K^+(\lambda)\|<\infty$. Then, in view of (\[bndtht1\]) and the density of $L_{2,0}({{\mathbf R}})$ in $L_2({{\mathbf R}})$, we obtain representation (\[rprXwd1\]), where the operator $\Theta_+(\lambda)=\tilde
K^+(\lambda)+\sqrt{W}\Theta(\lambda)\sqrt{W}$ has the property (\[bndthtpl\]). So, we have proved claim (i). Claims (ii) and (iii) are proved analogously using claims (ii,b) and (i) of Proposition \[prrprres\]. Claim (iv) is proved in the same manner as the previous ones.
### Linear independence of weighted Bloch functions {#subsubsec:linindepBloch}
For the proof of Theorem \[thnondegedg\] we need three lemmas.
\[lmBloch\]([@Kuch]) A finite sequence of Bloch functions $$\label{seqBloch}
\{b_k({{\mathbf x}})=\exp(i{{\mathbf p}}_k\cdot{{\mathbf x}})e_k({{\mathbf x}})\}_{k=1}^N,$$ corresponding to mutually different quasi-momenta ${{\mathbf p}}_k\in{{\mathbf T}}^d\,(k=1,2,\dots,N)$, is linearly independent. Recall that the functions $e_k({{\mathbf x}})$ are $\Gamma$-periodic.
Consider the functions $b_k({{\mathbf x}})$ as distributions from the space $\mathcal{S}^\prime$, which is dual to the Schwartz space $\mathcal{S}$. As it is known, the Gelfand-Fourier-Floquet transform $$\label{unit}
\hat
f({{\mathbf x}},{{\mathbf p}})=(Uf)({{\mathbf x}},{{\mathbf p}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\sum_{{{\mathbf l}}\in\Gamma}\exp(i{{\mathbf p}}\cdot{{\mathbf l}})f({{\mathbf x}}-{{\mathbf l}})$$ maps the space $\mathcal{S}^\prime$ onto the space of distributions $\mathcal{L}^\prime$, which is dual to the set $\mathcal{L}$ of all $C^\infty$-sections of the direct integral $\int_{{{\mathbf T}}^d}^\oplus{{\mathcal H}}_{{\mathbf p}}\,d{{\mathbf p}}$ ([@Kuch]). Recall that ${{\mathcal H}}_{{{\mathbf p}}}$ is the Hilbert space of functions $u\in L_{2,loc}({{\mathbf R}}^d)$ satisfying the condition (\[Htau\]) with the inner product defined by (\[dfinnprd\]-a). Using the $\Gamma$-periodicity of $e_k({{\mathbf x}})$, we have: $$\begin{aligned}
\label{transpsi}
&&\hskip-12mm\hat
b_k({{\mathbf x}},{{\mathbf p}})=(Ub_k)({{\mathbf x}},{{\mathbf p}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\sum_{{{\mathbf l}}\in\Gamma}
\exp(i{{\mathbf p}}\cdot{{\mathbf l}}+{{\mathbf p}}_k\cdot({{\mathbf x}}-{{\mathbf l}}))e_k({{\mathbf x}})=\nonumber\\
&&\hskip-12mm\frac{\exp(i{{\mathbf p}}\cdot{{\mathbf x}})}{(2\pi)^{\frac{d}{2}}}\sum_{{{\mathbf l}}\in{{\mathbf Z}}^d}
\exp(i({{\mathbf p}}-{{\mathbf p}}_k)\cdot{{\mathbf l}})e_k({{\mathbf x}})=\frac{\exp(i{{\mathbf p}}\cdot{{\mathbf x}})}{(2\pi)^{\frac{d}{2}}}\,\delta({{\mathbf p}}-{{\mathbf p}}_k)
e_k({{\mathbf x}})\end{aligned}$$ $(k=1,2,\dots,N)$. Here $\delta({{\mathbf p}})$ is the Dirac delta function. Assume, on the contrary, that $\{b_k({{\mathbf x}})\}_{k=1}^N$ are linearly dependent. Then $$b_{k_0}({{\mathbf x}})=\sum_{k\in\{1,2,\dots,N\}\setminus\{k_0\}}c_kb_k({{\mathbf x}}),$$ hence $$\hat
b_{k_0}({{\mathbf x}},{{\mathbf p}})=\sum_{k\in\{1,2,\dots,N\}\setminus\{k_0\}}c_k\hat
b_k({{\mathbf x}},{{\mathbf p}}).$$ In view of (\[transpsi\]), the support of the distribution in l.h.s. of the last equality is $\{{{\mathbf p}}_{k_0}\}$, but the distribution in its r.h.s. vanishes in a neighborhood of ${{\mathbf p}}_{k_0}$. We have a contradiction. Thus, the sequence $\{b_k({{\mathbf x}})\}_{k=1}^N$ is linearly independent.
\[lmlinidep\] Assume that the unperturbed potential $V({{\mathbf x}})$ satisfies condition (\[Hold\]). Let $\{b_k({{\mathbf x}})\}_{k=1}^N$ be a finite sequence of Bloch functions of the form (\[seqBloch\]) corresponding to the same energy level $\lambda$ of the unperturbed Hamiltonian $H_0=-\Delta+V({{\mathbf x}})\cdot$ and having mutually different quasi-momenta ${{\mathbf p}}_k\in{{\mathbf T}}^d\,(k=1,2,\dots,N)$. If $W\in L_1({{\mathbf R}}^d)$, $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$ and $W({{\mathbf x}})>0$ on a set of positive measure, then the sequence of functions $v_k({{\mathbf x}})=\sqrt{W({{\mathbf x}})}b_k({{\mathbf x}})\,(k=1,2,\dots,N)$ belongs to $L_2({{\mathbf R}}^d)$ and it is linearly independent.
Since all the Bloch functions are bounded on ${{\mathbf R}}^d$, the inclusion $v_k\in L_2({{\mathbf R}}^d)$ is obvious. Since $b_k({{\mathbf x}})$ correspond to the same energy level $\lambda$, they are solutions of the Schrödinger equation $H_0b=\lambda b$ belonging to $W_{2,\,loc}^2({{\mathbf R}}^d)$. Assume, on the contrary, that $\{v_k({{\mathbf x}})\}_{k=1}^N$ are linearly dependent, that is there exist constants $c_1,c_2,\dots,c_N$ such that $\sum_{k=1}^N|c_k|>0$ and $\sum_{k=1}^N
c_kv_k({{\mathbf x}})=\sqrt{W({{\mathbf x}})}\sum_{k=1}^Nc_kb_k({{\mathbf x}})=0$ for almost all ${{\mathbf x}}\in{{\mathbf R}}^d$. The latter equality implies that the function $b({{\mathbf x}})=\sum_{k=1}^Nc_kb_k({{\mathbf x}})$ vanishes on a set of positive measure. On the other hand, since by Lemma \[lmBloch\] the functions $b_k({{\mathbf x}})$ are linearly independent, then the function $b({{\mathbf x}})$ is a non-trivial solution of the equation $H_0b=\lambda b$ belonging to $W_{2,\,loc}^2({{\mathbf R}}^d)$. Observe that its first generalized derivatives $w_j=D_jb\,(j=1,2,\dots d)$ belong to $W_{2,\,loc}^1({{\mathbf R}}^d)$ and satisfy the equations $\Delta w_j+\lambda
w_j=D_j(V({{\mathbf x}})b)$. Observe that by claim (i) of Theorem \[thmainApp\], the function $b({{\mathbf x}})$ is continuous on ${{\mathbf R}}^d$. Then, in view of condition (\[Hold\]) and by the elliptic regularity theorem ([@Gil-Tr], Theorem 8.22), the function $b({{\mathbf x}})$ is locally $C^{1,\alpha}\;(\alpha\in(0,1))$. Then it follows from Theorem 1.7. of [@Har-Sim] that the set of zeros of $b({{\mathbf x}})$ (the nodal set) has a locally finite $d-1$-dimensional Hausdorff measure, hence it has a zero Lebesgue measure. We have a contradiction. Thus, the sequence $\{v_k({{\mathbf x}})\}_{k=1}^N$ is linearly independent.
It is not difficult to prove the following consequence of Lemma \[lmlinidep\]:
\[lmGWnondeg\] Under all the conditions of Lemma \[lmlinidep\] the finite rank operator $G=\sum_{k=1}^N\theta_k\,(\,\cdot,\,v_k)v_k$ ($\theta_k>0$ for any $k\in\{1,2,\dots,N\}$) is non-negative, it has exactly $N$ positive eigenvalues (counting their multiplicities) and they coincide with the eigenvalues of the matrix $\left((\theta_k\theta_l)^{\frac{1}{2}}(v_l,v_k)\right)_{k,l=1}^N$.
### Proof of Theorem \[thnondegedg\] {#subsubsec:proofthnondegedg}
Since each positive characteristic branch $\mu_k^+(\lambda)$ of the operator $H_0$ w.r. to $W$ on the gap $(\lambda_-,\,\lambda_+)$ is an increasing function (see Remark \[rembranch\]), then by Proposition \[ordl\] each eigenvalue $\rho_k^+(\gamma)$ of the operator $H_\gamma$ appearing in the gap $(\lambda_-,\,\lambda_+)$ for $\gamma<0$ increases, hence it cannot approach the edge $\lambda_-$ as $\gamma\uparrow 0$. Hence it cannot be a virtual eigenvalue of $H_\gamma$ at the edge $\lambda_-$, that is claim (i) is valid (see Definitions \[gone1\], \[gde1\] and \[gde2\]).
Observe that, as it is easy to show, for $d\le 2$ each of the conditions (\[cndWd1\]) and (\[cndWd2\]) imply that $W\in
L_1({{\mathbf R}}^d)$. On the other hand, by claim (i) of Theorem \[thmainApp\], the Bloch functions $b_k^+({{\mathbf x}})$ are bounded on ${{\mathbf R}}^d$. Hence the corresponding weighted Bloch functions $v_k^+({{\mathbf x}})=\sqrt{W({{\mathbf x}})}b_k^+({{\mathbf x}})$ belong to $L_2({{\mathbf R}}^d)$.
We now turn to the proof of the claim (ii-a) (the case where $d=1$). Recall that for any fixed $\lambda\in(\lambda_-,\lambda_+)$ $\{\mu_k^+(\lambda)\}_{k=1}^{m_+}(\lambda)$ is the set of all positive eigenvalues of the Birman-Schwinger operator $X_W(\lambda)$ arranged in the non-decreasing ordering $\mu_1^+(\lambda)\ge\mu_2^+(\lambda)\ge\dots\ge\mu_{N_+}^+(\lambda)>0$, were each $\mu_k^+(\lambda)$ is repeated according to its multiplicity (the value $N_+=\infty$ is allowed). Denote $t=\sqrt{\lambda_+-\lambda}$ and $\Phi(t)=tX_W(\lambda_+-t^2)$. In view of the representation (\[rprXwd1\]), definition (\[dfPhipld1\]) and the relation (\[bndthtpl\]) (Proposition \[rprBirSchw\]), for some $\delta>0$ and any $t\in[0,\delta]$ $\Phi(t)=\Phi(0)+t\theta(t)$, where $\Phi(0)=\frac{\sqrt{m^+}(\,\cdot\,,v_1^+)v_1^+}{\sqrt{2}}$ and $\sup_{t\in[0,\delta]}\Vert\theta(t)\Vert<\infty$. As it is clear, the one-rank operator $\Phi(0)$ has the unique non-zero eigenvalue $\mu_1^0=\frac{\sqrt{m^+}\|v_1^+\|^2}{\sqrt{2}}$ and it is positive. Then by claim (i-a) of Proposition \[prpertcompoper\], there exists $\tilde\delta\in[0,\delta]$ such that for any $t\in[0,\tilde\delta]$ the operator $\Phi(t)$ has a unique simple positive eigenvalue $\mu_1(t)$ having the asymptotic representation $\mu_1(t)=\mu_1^0+O(t)$ as $t\rightarrow 0$ and all the rest of positive eigenvalues of $\Phi(t)$ (if they exist) have the asymptotic representation $\mu_k(t)=O(t)$ as $t\rightarrow
0\;(k>1)$. Returning from the variable $t$ to the variable $\lambda$ and taking into account that all the eigenvalues of $X_W(\lambda)$ are obtained from the eigenvalues of $\Phi(t)$ via multiplication by $\frac{1}{t}=\frac{1}{\sqrt{\lambda_+-\lambda}}$, we obtain that for some $\hat\delta\in(0,\lambda_+-\lambda_-)$ and for any $\lambda\in[\lambda_+-\hat\delta,\lambda_+)$ the operator $X_W(\lambda)$ has a unique positive eigenvalue $\mu_1^+(\lambda)$ having the asymptotic representation $$\label{asymptmu}
\mu_1^+(\lambda)=\frac{\sqrt{m^+}\|v_1^+\|^2}{\sqrt{2(\lambda_+-\lambda)}}+O(1)\quad
\rm{for}\quad\lambda\uparrow\lambda_+$$ (hence $\lim_{\lambda\uparrow\lambda_+}\mu_1^+(\lambda)=\infty$) and all the rest of its positive eigenvalues $\mu_k^+(\lambda)\;(k>1)$ (if they exist) are bounded in $(\lambda_+-\hat\delta,\lambda_+)$. This means that $\mu_1^+(\lambda)$ is a unique main characteristic branch of $H_0$ w.r. to $W$ near the edge $\lambda_+$ of the gap $(\lambda_-,\,\lambda_+)$, hence the corresponding asymptotic multiplicity $M(\lambda_+,H_0,W)$ of $\lambda_+$ is one (see Definition \[gde2\]). By Proposition \[gpbr\] the latter circumstances mean that for $\gamma<0$ there exists a unique branch $\rho_1^+(\gamma)$ of virtual eigenvalues of the operator $H_\gamma$ at the edge $\lambda_+$ (see Definition \[gone1\]) and it is the solution of the equation $\mu_1^+(\lambda)=\frac{1}{|\gamma|}$ for a sufficiently small $|\gamma|$. From (\[asymptmu\]) and Lemma \[lmasymp\], we obtain the desired asymptotic formula (\[asympteigvd1\]). Claim (ii-a) is proven.
Let us prove claim (ii-b) (the case where $d=2$). By Proposition \[rprBirSchw\], in this case the finite rank operator $\Phi_+(\lambda)$, taking part in the representation (\[rprXwd1\]) for $X_W(\lambda)$ has the form (\[dfPhipld2\]), where $G_W^+$ is defined by (\[dfGW\]). Like above, let $\mu_k^+(\lambda)\;(k=1,2,\dots,N_+)$ be positive characteristic branches of $H_0$ w.r.t. $W$ in the gap $(\lambda_-,\,\lambda_+)$, that is they are all the positive eigenvalues of the operator $X_W(\lambda)$. Denote $$\label{dfPhit}
t=t(\lambda)=\Big(\ln\left(\frac{1}{\lambda_+-\lambda}\right)\Big)^{-1}\quad
\rm{and}\quad \Phi(t)=tX_W\big(\lambda_+-\exp(-t^{-1})\big).$$ In view of representation (\[rprXwd1\]), definition (\[dfPhipld2\]) and condition (\[bndthtpl\]), for some $\delta>0$ and any $t\in[0,\delta]$ representation (\[rprPhit\]) is valid with $\Phi(0)=\frac{1}{2\pi}G_W^+$ and $\theta(t)$ satisfying the condition (\[cndtht\]). Then by claim (i-a) of Proposition \[prpertcompoper\] and Lemma \[lmGWnondeg\], there exists $\tilde\delta\in[0,\delta]$ such that for any $t\in[0,\tilde\delta]$ the operator $\Phi(t)$ has $n_+$ positive eigenvalues $\mu_1(t)\ge\mu_2(t)\ge\dots\ge\mu_{n_+}(t)$ (counting their multiplicities), having the asymptotic representation $\mu_k(t)=\frac{1}{2\pi}\nu_k^++O(t)\;(k=1,2,\dots,n_+)$ for $t\rightarrow 0$ and all the rest of positive eigenvalues of $\Phi(t)$ (if they exist) have the asymptotic representation $\mu_k(t)=O(t)$ for $t\rightarrow 0\;(k>n_+)$. Recall that $\nu_k^+$ are the positive eigenvalues of the operator $G_W^+$. Returning from the variable $t$ to the variable $\lambda$, we obtain that there exists $\hat\delta\in (0,\lambda_+-\lambda_-)$ such that $\mu_1^+(\lambda),\mu_2^+(\lambda),\dots,\mu_{n_+}^+(\lambda)$ exist for any $\lambda\in[\lambda_+-\hat\delta,\lambda_+)$, they have the asymptotic representation $$\label{asympmud2}
\mu_k^+(\lambda)=\frac{1}{2\pi}\ln\left(\frac{1}{\lambda_+-\lambda}\right)\nu_k^++O(1)\quad
\rm{for}\quad\lambda\uparrow\lambda_+\quad (k=1,2,\dots,n_+),$$ (hence $\lim_{\lambda\uparrow\lambda_+}\mu_k^+(\lambda)=\infty$ for $k=1,2,\dots,n_+$), and for $k>n_+$ $\mu_k^+(\lambda)$ are bounded in $(\lambda_+-\hat\delta,\lambda_+)$ (if they exist). This means that $\mu_k^+(\lambda)\;(k=1,2,\dots,n_+)$ is the set of all main characteristic branches of $H_0$ w.r. to $W$ near the edge $\lambda_+$ of $(\lambda_-,\lambda_+)$, hence $M(\lambda_+,H_0,W)=n_+$. By Proposition \[gpbr\] the latter circumstances mean that for $\gamma<0$ there exist exactly $n_+$ branches $\rho_1^+(\gamma)\le\rho_2^+(\gamma)\le\dots\le\rho_{n_+}^+(\gamma)$ of virtual eigenvalues of the operator $H_\gamma$ near the edge $\lambda_+$ and each of them is the solution of the equation $\mu_k^+(\lambda)=\frac{1}{|\gamma|}$ for any $\gamma\in[-\bar\gamma,0)$ and for some $\bar\gamma>0$. Using Lemma \[lmasymp\], we obtain from (\[asympmud2\]) the desired asymptotic formula (\[asympteigvd2\]). Formula (\[asymptLbThrd2\]) follows from (\[asympteigvd2\]) and the fact that $\mathrm{tr}(G_W^+)=\sum_{k=1}^{n_+}\nu_k^+$. Claim (ii-b) is proven.
We shall prove only claim (iii-b) (the case where $d=2$), because claim (iii-a) (the case where $d=1$) is proved analogously. For any fixed $j\in\{1,2,\dots,n_+\}$ consider the group of virtual eigenvalues $$\begin{aligned}
&&\rho_{l(j)}^{\,+}(\gamma)\le\rho_{l(j)+1}^{\,+}(\gamma)\le\dots\le\rho_{l(j)+m(j)-1}^{\,+}(\gamma)\\
&&(l(j)\in\{1,2,\dots,n_+\},\;\gamma\in[-\bar\gamma,0))\nonumber\end{aligned}$$ of the operator $H_\gamma$ in $(\lambda_-,\,\lambda_+)$, for which the quantities $\frac{t(\rho_k^+(\gamma))}{|\gamma|}$ $(k\in\{l(j),l(j)+1,\dots,l(j))+m(j)-1\})$ tend as $\gamma\uparrow
0$ to the eigenvalue $\frac{\nu_j^+}{2\pi}$ of the operator $\Phi(0)$, whose multiplicity is $m(j)$. Recall that the function $t(\lambda)$ and the operator function $\Phi(t)$ are defined by (\[dfPhit\]) and $g_1^+({{\mathbf x}}),g_2^+({{\mathbf x}}),\dots,g_{n_+}^+({{\mathbf x}})$ is an orthonormal sequence of eigenfunctions of the operator $G_W^+=2\pi\Phi(0)$ corresponding to its positive eigenvalues $\nu_1^+,\nu_2^+,\dots, \nu_{n_+}^+$. Denote $\tilde\rho(\gamma)=\rho_{l(j)}^{\,+}(\gamma)$. Let $\tilde\mu_0^+(\gamma)=\mu_{l(j)}^+\big(\tilde\rho(\gamma)\big)>
\tilde\mu_1^+(\gamma)>\dots>\tilde\mu_{n(\gamma)}^+(\gamma)$ $(n(\gamma)\le m(j)-1)$ be all the mutually different numbers from the sequence $\{\mu_{l(j)+k}^+\big(\tilde\rho(\gamma)\big)\}_{k=0}^{m(j)-1}$. These numbers are positive eigenvalues of the self-adjoint operator $X_W\big(\tilde\rho(\gamma)\big)$. Let $L_0(\gamma),
L_1(\gamma),\dots, L_{n(\gamma}(\gamma)$ be the eigenspaces of $X_W\big(\tilde\rho(\gamma)\big)$, corresponding to these eigenvalues. Observe that each of $L_k(\gamma)$ is the eigenspace of the operator $\tilde\Phi(\gamma)=\Phi(t(\tilde\rho(\gamma)))$, corresponding to its eigenvalue $t(\tilde\rho(\gamma)))\tilde\mu_k^+(\gamma)$. Denote $L(\gamma)=\bigoplus_{k=0}^{n(\gamma)}L_k(\gamma)$. By claim (i-b) of Proposition \[prpertcompoper\], for a suitable $\tilde\gamma\in(0,\,\bar\gamma]$ and any $\gamma\in[0,\tilde\gamma]$ it is possible to choose a basis $$e_{l(j)}(\gamma),e_{l(j)+1}(\gamma),\dots,e_{l(j)+m(j)-1}(\gamma)$$ in the subspace $L(\gamma)$ such that $\Vert
e_k(\gamma)-g_k^+\Vert=O(t(\tilde\rho(\gamma))$ for $\gamma\uparrow
0$. Then taking into account the asymptotic formula (\[asympteigvd2\]), we get that $$\label{asympteigvec}
\Vert e_k(\gamma)-g_k^+\Vert=O(\gamma)\quad\rm{for}\quad
\gamma\uparrow 0\quad (k=l(j),l(j)+1,\dots,l(j)+m(j)-1),$$ For any $\gamma\in[0,\tilde\gamma]$ consider the sequence $\gamma_0(\gamma)=\gamma$, $\gamma_k(\gamma)=-\frac{1}{\mu_{l(j)+k}^+\big(\tilde\rho(\gamma)\big)}\;
(k=1,2,\dots,m(j)-1)$. Since the r.h.s. of the asymptotic formula (\[asympmud2\]) is the same for all the functions from the sequence $\{\mu_{l(j)+k}^+(\lambda)\}_{k=0}^{m(j)-1}$, then $$\mu_{l(j)}^+\big(\tilde\rho(\gamma)\big)-\mu_{l(j)+k}^+\big(\tilde\rho(\gamma)\big)=
-\gamma^{-1}+\big(\gamma_k(\gamma)\big)^{-1}=O(1)\quad \rm{for}\quad
\gamma\uparrow 0$$ $(k=1,2,\dots,m(j)-1)$. This fact implies easily that $\gamma-\gamma_k(\gamma)=O(\gamma^2)$ as $\gamma\uparrow 0$ $(k=0,1,\dots, m(j)-1)$. Observe that $\tilde\gamma_k(\gamma)=-\frac{1}{\tilde\mu_k^+(\gamma)}\;(k=0,1,\dots,n(\gamma))$ are all the different numbers from the sequence $\{\gamma_k(\gamma)\}_{k=0}^{m(j)-1}$. It is clear that for any $k\in\{0,1,\dots, n(\gamma)\}$ the number $\tilde\rho(\gamma)$ is an eigenvalue of the operator $H_{\tilde\gamma_k(\gamma)}$. Let $E_k(\gamma)$ be the eigenspace of $H_{\tilde\gamma_k(\gamma)}$, corresponding to the eigenvalue $\tilde\rho(\gamma)$. By Proposition \[BrScwspct\], the multiplication operator $W^{1/2}\cdot\vert_{E_k(\gamma)}$ realizes a linear isomorphism between $E_k(\gamma)$ and $L_k(\gamma)$. This circumstance and the fact that the subspaces $L_k(\gamma)$ are mutually orthogonal imply that the subspaces $E_k(\gamma)$ are linearly independent and the operator $Q(\gamma)=W^{1/2}\cdot\vert_{E(\gamma)}$ realizes a linear isomorphism between the subspace $E(\gamma)=E_0(\gamma)+E_1(\gamma)+\dots+E_{n(\gamma)}(\gamma)$ and the subspace $L(\gamma)$. Let us define the following basis in $E(\gamma)$: $\psi^+_{\gamma,k}=(Q(\gamma))^{-1} e_k(\gamma)$ $(k\in\{l(j),l(j)+1,\dots,l(j)+m(j)-1\})$. Then we obtain from (\[asympteigvec\]) the desired property (\[asympteigvec1\]). Claim (iii-b) is proven.
Let us prove claim (iv). In view of (\[bndXw\]) (claim (iii) of Proposition \[rprBirSchw\]), all the positive characteristic branches of $H_0$ w.r.t. $W$ on the gap $(\lambda_-,\,\lambda_+)$ are bounded, hence $M(\lambda_+,H_0,W)=0$. This fact, Proposition \[gpbr\] and claim (i) imply the desired claim (iv).
Claim (v) is proved in the same manner as the previous ones. Claim (vi) follows from claims (iv) and (v).
Theorem \[thnondegedg\] is proven.
The case of degenerate edges of a gap of the spectrum of the unperturbed operator {#subsec:degedg}
---------------------------------------------------------------------------------
As above, $(\lambda_-,\,\lambda_+)$ is a gap of the spectrum of the unperturbed operator $H_0$ and we assume that the perturbation $W({{\mathbf x}})$ of the periodic potential $V({{\mathbf x}})$ is measurable and bounded in ${{\mathbf R}}^d$, $W\in L_1({{\mathbf R}}^d)$ and $W({{\mathbf x}})\ge 0$ a.e. on ${{\mathbf R}}^d$.
### Representation of Birman-Schwinger operator in the case of a degenerate edge {#subsubsec:reprBirmSchwingdeg}
In the case of a degenerate edge of a gap of the spectrum of the unperturbed operator we have the following modification of Proposition \[rprBirSchw\]:
\[rprBirSchwdeg\] Let us take $\delta\in(0,\,(\lambda_+-\lambda_-))$.
\(i) If $d-d_+=1$, the edge $\lambda_+$ satisfies condition (C) of Section \[subsec:mainresbirthdeg\] and condition (\[cndWd1deg\]) is satisfied, then for $\lambda\in(\lambda_+-\delta,\;\lambda_+)$ the representation is valid: $$\label{rprXwd1deg}
X_W(\lambda)=\Phi_+(\lambda)+\Theta_+(\lambda),$$ where $$\label{dfPhipld1deg}
\Phi_+(\lambda)=\frac{1}{\sqrt{\lambda_+-\lambda}}\frac{\sqrt{2}\pi}{(2\pi)^d}G_W^+,$$ the operator $G_W^+$ is defined by (\[dfintopGW\])-(\[dfkerGW\]) and $\Theta_+(\lambda)$ is a bounded operator in $L_2({{\mathbf R}}^2)$ satisfying the condition $$\begin{aligned}
\label{bndthtpldeg}
&&\bar\Theta_+=\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\Theta_+(\lambda)\|<\infty;\end{aligned}$$
\(ii) if $d-d_+=2$, the edge $\lambda_+$ satisfies condition (C) and condition (\[cndWcod2\]) is satisfied, then for $\lambda\in((\lambda_+-\delta,\;\lambda_+)$ the representation (\[rprXwd1deg\]) is valid with $\Theta_+(\lambda)$ satisfying condition (\[bndthtpldeg\]) and with $\Phi_+(\lambda)$ having the form: $$\begin{aligned}
\label{dfPhipld2deg}
&&\Phi_+(\lambda)=\ln\left(\frac{1}{\lambda_+-\lambda}\right)\frac{\sqrt{2}\pi}{(2\pi)^d}G_W^+;\end{aligned}$$
\(iii) if $d-d_+\ge 3$ and the edge $\lambda_+$ satisfies condition (C), then $$\sup_{\lambda\in((\lambda_+-\delta,\;\lambda_+)}\|X_W(\lambda)\|<\infty;$$
\(iv) The claims analogous to (i)-(iii) are valid with $d_-$, $\lambda_-$, $(\lambda_-,\;\lambda_-+\delta)$ and $G_W^-$ instead of, respectively, $d_+$, $\lambda_+$, $(\lambda_+-\delta,\;\lambda_+)$ and $G_W^+$.
The proof is the same as the proof of Proposition \[rprBirSchw\], merely instead of Proposition \[prrprres\] it is supported by Proposition \[prrprres1\] with $n_+=n_-=1$.
### Spectrum of the operator $G_W^+\;(G_W^-)$ {#subsubsec:specopGWpl}
In what follows we need the following
\[lmspecGWpl\] If the edge $\lambda_+\;(\lambda_-)$ satisfies condition (C) of Section \[subsec:mainresbirthdeg\], the unperturbed potential $V({{\mathbf x}})$ satisfies condition (\[Hold\]) and $W({{\mathbf x}})> 0$ on a set of the positive measure, then the integral operator $G_W^+$, defined by (\[dfintopGW\])-(\[dfkerGW\]),
\(i) is self-adjoint, non-negative, belongs to the trace class and $$\begin{aligned}
\label{traceform}
&&\hskip-4mm\hskip-5mm\rm{tr}(G_w^+)=\int_{{{\mathbf R}}^d}{{\mathcal G}}_W^+({{\mathbf s}},{{\mathbf s}})\,d{{\mathbf s}}=
\int_{F^+}\int_{{{\mathbf R}}^d}{{\mathcal Q}}_W^+({{\mathbf s}},{{\mathbf s}},{{\mathbf p}})\,d{{\mathbf s}}\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}})\end{aligned}$$
\(ii) has an infinite number of positive eigenvalues having finite multiplicities.
The analogous properties has the operator $G_W^-$.
We shall restrict ourselves only on the operator $G_W^+$, since the operator $G_W^-$ has the same structure. Let us prove claim (i). Using claim (i) of Corollary \[cormainApp\] and compactness of the submanifold $F^+$, it is possible to construct a finite open covering $\{U_k\}_{k=1}^L$ of $F^+$ such that for any $k\in\{1,2,\dots,L\}$ ${{\mathcal Q}}_W^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=v_k^+({{\mathbf x}},{{\mathbf p}})v_k^+({{\mathbf s}},{{\mathbf p}})$ for any ${{\mathbf p}}\in U_k$, where $v_k^+({{\mathbf x}},{{\mathbf p}})=\sqrt{W({{\mathbf x}})}b_k^+({{\mathbf x}},{{\mathbf p}})$ and $b_k^+({{\mathbf x}},{{\mathbf p}})\\({{\mathbf p}}\in U_k)$ is a branch of Bloch functions, corresponding to the energy level $\lambda_+$ of the family of the operators $H({{\mathbf p}})\;({{\mathbf p}}\in U_k)$ such that the mapping ${{\mathbf p}}\rightarrow
b_k^+(\cdot,{{\mathbf p}})$ belongs to the class $C(U_k,\, C(\Omega))$. Hence since $W\in L_1({{\mathbf R}}^d)$, the inclusion $$\label{mmbshipvpl}
\big({{\mathbf p}}\rightarrow v_k^+(\cdot,{{\mathbf p}})\big)\in C(U_k,\,L_2({{\mathbf R}}^d)).$$ is valid. Let $\{\phi_k({{\mathbf p}})\}_{k=1}^L$ be a decomposition of unit, corresponding to the covering $\{U_k\}_{k=1}^L$ such that each $\phi_k$ is continuous on $F^+$ and positive on $U_k$. Then the operator $G_W^+$ can be represented in the form $G_W^+=\sum_{k=1}^L
G_k$, where $G_kf=\int_{{{\mathbf R}}^d}{{\mathcal G}}_k({{\mathbf x}},{{\mathbf s}})f({{\mathbf s}})\,d{{\mathbf s}}\;(f\in
L_2({{\mathbf R}}^d))$ and $${{\mathcal G}}_k({{\mathbf x}},{{\mathbf s}})=\int_{U_k}v_k^+({{\mathbf x}},{{\mathbf p}})v_k^+({{\mathbf s}},{{\mathbf p}})\phi_k({{\mathbf p}})\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}}).$$ In order to prove that $G_W^+$ belongs to the trace class, it is enough to show that each of $G_k$ belongs to the trace class. We see that $${{\mathcal G}}_k({{\mathbf x}},{{\mathbf s}})=\int_{U_k}\mathcal{A}_k({{\mathbf x}},{{\mathbf p}})\overline{\mathcal{A}_k({{\mathbf s}},{{\mathbf p}})}\,dF({{\mathbf p}}),$$ where $\mathcal{A}_k({{\mathbf x}},{{\mathbf p}})=v_k^+({{\mathbf x}},{{\mathbf p}})(\phi_k({{\mathbf p}}))^{1/2}(m^+({{\mathbf p}}))^{1/4}$. Consider the integral operator $$(A_k\phi)({{\mathbf x}})=\int_{U_k}\mathcal{A}_k({{\mathbf x}},{{\mathbf p}})\phi({{\mathbf p}})\,dF({{\mathbf p}}),$$ which acts from $L_2(U_k)$ to $L_2({{\mathbf R}}^d)$, hence $G_k=A_k(A_k)^\star$. Formally in order to prove that the operator $G_k$ belongs to the trace class, it is enough to show that $A_k$ belongs to the Hilbert-Schmidt class. But since the domain and the range of $A_k$ are different, we are forced to use a direct sum argument. Consider the Hilbert space $\mathcal{\tilde
H}=L_2({{\mathbf R}}^d)\oplus L_2(F^+)$ and the operator $\tilde A_W$ acting in it and defined by the operator matrix: $$\begin{aligned}
\label{dftlAW}
\tilde A_k=\left(\begin{array}{ll}
0&\;A_k\\
A_k^\star&\;0
\end{array}\right)\end{aligned}$$ Since the dispersion function $\lambda^+({{\mathbf p}})$ is real-analytic and satisfies the Morse-Bott condition in a neighborhood of $F^+$, then in view of (\[dfmplp\]) $$\label{mmbshipHess}
m^+\in C(F^+,,{{\mathbf R}}).$$ This fact and inclusion (\[mmbshipvpl\]) imply easily that $\int_{{{\mathbf R}}^d}\int_{U_k}|\mathcal{A}_k^+({{\mathbf x}},{{\mathbf p}})|^2\,d{{\mathbf x}}\,dF({{\mathbf p}})<\infty$, hence in view of (\[dftlAW\]) the matrix integral kernel of the operator $\tilde A_k$ is square integrable on ${{\mathbf R}}^d\times U_k$. Therefore $\tilde A_k$ belongs to the Hilbert-Schmidt class and it is self-adjoint. On the other hand, since $\tilde
A_k^2=\rm{diag}(G_W^+,\;A_k^\star A_k)$, the operator $G_k$ is self-adjoint, non-negative and belongs to the trace class. In view of Theorem 3.1 of [@Bris], $\rm{tr}(G_k)=\int_{{{\mathbf R}}^d}\tilde{{\mathcal G}}_k({{\mathbf s}},{{\mathbf s}})\,d{{\mathbf s}}$, where $\tilde{{\mathcal G}}_k({{\mathbf x}},{{\mathbf s}})$ is the local average of ${{\mathcal G}}_k({{\mathbf x}},{{\mathbf s}})$ defined in [@Bris]. On the other hand, using the decomposition $G_k^+=A_k(A_k)^\star$ and the arguments of the proof and Theorem 3.5 from [@Bris], it is not difficult to show that $\tilde{{\mathcal G}}_k({{\mathbf x}},{{\mathbf x}})={{\mathcal G}}_k({{\mathbf x}},{{\mathbf x}})$ a.e. on ${{\mathbf R}}^d$. Hence, taking into account that $\mathrm{tr}(G_w^+)=\sum_{k=1}^L\mathrm{tr}(G_k)$, we obtain the desired trace formula (\[traceform\]). Claim (i) is proven.
Let us prove claim (ii). Claim (i) implies that the spectrum of the operator $G_W^+$ consists of at most countable number of non-negative eigenvalues, which can cluster only to zero and each positive eigenvalue has a finite multiplicity. Let ${{\mathbf p}}_1,{{\mathbf p}}_2,\dots,{{\mathbf p}}_N$ be mutually different quasi-momenta belonging to the submanifold $F^+$ and ${{\mathcal O}}^\epsilon_1,2,\dots,{{\mathcal O}}^\epsilon_N\subset F^+$ are disjoint neighborhoods of these points having the same diameter $\epsilon>0$ and the same volume $F(\epsilon)$. We can choose $\epsilon$ such that for any $j\in\{1,2,\dots,N\}$ there is $k=k(j)\in\{1,2,\dots,L\}$ such that ${{\mathcal O}}^\epsilon_j\subset U_k$. Denote $\tilde v_j^+=v_{k(j)}^+$. We have for $f\in L_2({{\mathbf R}}^d)$: $$\label{lowestGWpl}
(G_W^+f,\,f)\ge F(\epsilon)(\tilde
G_Nf,\,f)+(\Theta_{N,\epsilon}f,\,f),$$ where $$\label{dftlGN}
\tilde G_N=\sum_{j=1}^N(\,\cdot,\,\tilde v_j^+(\cdot,{{\mathbf p}}_j))\tilde
v_j^+(\cdot,{{\mathbf p}}_j)\sqrt{m^+({{\mathbf p}}_j)}$$ and $$\begin{aligned}
&&\Theta_{N,\epsilon}=\sum_{k=1}^N\int_{{{\mathcal O}}^\epsilon_k}
\Big((\,\cdot,\,\tilde v_j^+(\cdot,{{\mathbf p}}))\tilde
v_j^+(\cdot,{{\mathbf p}})\sqrt{m^+({{\mathbf p}})}-
\\
&&(\,\cdot,\,\tilde v_j^+(\cdot,{{\mathbf p}}_j))\tilde
v_j^+(\cdot,{{\mathbf p}}_j)\sqrt{m^+({{\mathbf p}}_j)}\Big)\,dF({{\mathbf p}}).\end{aligned}$$ The last equality and the inclusions (\[mmbshipvpl\]) and (\[mmbshipHess\]) imply that $$\label{limnrm}
\lim_{\epsilon\downarrow
0}\frac{\Vert\Theta_{N,\epsilon}\Vert}{F(\epsilon)}=0.$$ In view of (\[dftlGN\]) and Lemma \[lmGWnondeg\], the operator $\tilde G_N$ has exactly $N$ positive eigenvalues $\nu_1\ge
\nu_2\ge\dots\ge \nu_N>0$ (counting their multiplicities). Then in the same manner as in the proof of Proposition \[prpertcompoper\], we obtain using (\[limnrm\]) and a modification of the comparison theorem on the base of the minimax principle (Lemma 3.4 of [@Ar-Zl1]) that for a small enough $\epsilon>0$ the operator $F(\epsilon)\tilde G_N+\Theta_{N,\epsilon}$ has at least $N$ positive eigenvalues $\nu_1(\epsilon)\ge \nu_2(\epsilon)\ge\dots\ge
\nu_N(\epsilon)>0$ (counting their multiplicities). Using again the minimax principle, we obtain from (\[lowestGWpl\]) that the operator $G_W^+$ has at least $N$ positive eigenvalues (counting their multiplicities). Since $N$ is arbitrary, claim (ii) is proven.
### Proof of Theorem \[thdegedg\] {#subsubsec:proofthdegedg}
Claim (i) is proved in the same manner as claim (i) of Theorem \[thnondegedg\].
We now turn to the proof of the claim (ii) in the case where $d-d_+=2$. By claim (ii) of Proposition \[rprBirSchwdeg\], for the Birman-Schwinger operator $X_W(\lambda)$ representation (\[rprXwd1deg\]) is valid for $\lambda\in((\lambda_++\lambda_-)/2,\;\lambda_+)$, in which $\Phi_+(\lambda)$ has the form (\[dfPhipld2deg\]) and $\Theta_+(\lambda)$ satisfies the condition (\[bndthtpldeg\]). Let $\mu_1^+(\lambda)\ge
\mu_2^+(\lambda)\ge\dots\ge\mu_n^+(\lambda)\ge\dots\;(\lambda\in(\lambda_-,\lambda_+))$ be the positive characteristic branches of the operator $H_0$ w.r. to $W$ on the gap $(\lambda_-,\,\lambda_+)$, that is they are positive eigenvalues of the operator $X_W(\lambda)$ (Definition \[gde1\]). By Lemma \[lmspecGWpl\], the operator $G_W^+$ has an infinite number of positive eigenvalues $\nu_1^+\ge\nu_2^+\ge\dots\ge\nu_n^+\ge\dots$. Let $\Phi(t)$ be the operator function defined by (\[dfPhit\]). Then in view of the representation (\[rprXwd1deg\]), definition (\[dfPhipld2deg\]) and the condition (\[bndthtpldeg\]), for some $\delta>0$ and any $t\in[0,\delta]$ representation (\[rprPhit\]) is valid with $\Phi(0)=\frac{1}{2\pi}G_W$ and $\theta(t)$ satisfying the condition (\[cndtht\]). Then claim (ii) of Proposition \[prpertcompoper\] implies claim (ii) of Theorem \[thdegedg\] for $d-d_+=2$ in the same manner as the corresponding claim was obtained in the non-degenerate case (see proof of claim (ii-b) of Theorem \[thnondegedg\]). Formula (\[asymptLbThrd2deg\]) follows from (\[asympteigvd2deg\]), the formula $\sum_{n=1}^\infty\nu_n^+=\rm{tr}(G_w^+)$ and claim (i) of Lemma \[lmspecGWpl\]. In the analogous manner claim (ii) for $d-d_+=1$ is proved by the use of representation (\[rprXwd1deg\]) and definition (\[dfPhipld1deg\]). Claim (iii) follows from claim (ii) of Proposition \[prpertcompoper\] in the same manner as in the proof of Theorem \[thnondegedg\]) claim (iii-b) follows from claim (i-b) of Proposition \[prpertcompoper\].
Claim (iv) is proved in the same manner as the corresponding claim of Theorem \[thnondegedg\] using claim (iii) of Proposition \[rprBirSchwdeg\].
Claim (v) is proved in the same manner as the previous ones. Claim (vi) follows from claims (iv) and (v).
Theorem \[thdegedg\] is proven.
Estimate for the multiplicity of virtual eigenvalues in the case of an indefinite perturbation {#subsec:estmult}
----------------------------------------------------------------------------------------------
### General results on the multiplicity of virtual eigenvalues {#subsubsec:genresmult}
Before proving Theorem \[thestmult\], we shall prove some auxiliary claims. If $A$ is a linear operator acting in a Hilbert space, we denote $\Re(A):=\frac{1}{2}(A+A^\star)$; if $A$ is self-adjoint, we denote $A_+:=\frac{1}{2}(|A|+A)$, $A_-:=\frac{1}{2}(|A|-A)$.
\[lmrprBrSchw\] Let $(\lambda_-,\lambda_+)$ be a gap of the spectrum of a self-adjoint operator $H_0$ acting in a Hilbert space ${{\mathcal H}}$ be and $W$ be a bounded self-adjoint operator in ${{\mathcal H}}$. Then for any $\lambda\in(\lambda_-,\lambda_+)$ the real part of the Birman-Schwinger operator $X_W(\lambda)=W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}}$ admits the representation: $$\label{rprBrSchw}
\Re(X_W(\lambda))=X_{W_+}(\lambda)-X_{W_-}(\lambda).$$ Recall that $W^{\frac{1}{2}}$ is defined by (\[dfsqrtW\]).
We have: $$\begin{aligned}
\label{calcul}
&&\hskip-8mm(W^{\frac{1}{2}})_+R_\lambda(H_0)(W^{\frac{1}{2}})_+-(W^{\frac{1}{2}})_-R_\lambda(H_0)(W^{\frac{1}{2}})_-=
\nonumber\\
&&\hskip-8mm\frac{1}{4}\left((|W|^{\frac{1}{2}}+W^{\frac{1}{2}})R_\lambda(H_0)(|W|^{\frac{1}{2}}+W^{\frac{1}{2}})\right.-
\nonumber\\
&&\hskip-8mm\left.(|W|^{\frac{1}{2}}-W^{\frac{1}{2}})R_\lambda(H_0)(|W|^{\frac{1}{2}}-W^{\frac{1}{2}})\right)=
\frac{1}{2}\left(W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}}-\right.
\nonumber\\
&&\hskip-8mm\left.|W|^{\frac{1}{2}}R_\lambda(H_0)W^{\frac{1}{2}}\right)=
\frac{1}{2}\left(W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}}+(W^{\frac{1}{2}}R_\lambda(H_0)|W|^{\frac{1}{2}})^\star\right).\end{aligned}$$ Let $E_\lambda\,(\lambda\in{{\mathbf R}})$ be the spectral resolution of identity, corresponding to the operator $W$. Since $|W|^{\frac{1}{2}}=\int_{{{\mathbf R}}}|\lambda|^{\frac{1}{2}}\,dE_\lambda$ and, in view of (\[dfsqrtW\]), $$W^{\frac{1}{2}}=\int_{{{\mathbf R}}}\rm{sign}(\lambda)|\lambda|^{\frac{1}{2}}\,dE_\lambda,$$ the equalities are valid: $(W^{\frac{1}{2}})_+=(W_+)^{\frac{1}{2}}$ and $(W^{\frac{1}{2}})_-=(W_-)^{\frac{1}{2}}$. Then (\[calcul\]) can be written in the form (\[rprBrSchw\]).
\[prestmult\] Assume that all the conditions of Lemma \[lmrprBrSchw\] are satisfied and furthermore assume that for some $\lambda_0\in{{\mathcal R}}(H_0)$ the operator $R_{\lambda_0}|W|^{\frac{1}{2}}$ is compact. Let us take $\delta\in(0,\,\lambda_+-\lambda_-)$. Then
\(i) if $M(\lambda_+,H_0,W_+)<\infty$, there exists $\bar\gamma>0$ such that for any $\gamma\in(-\bar\gamma,0)$ the multiplicity of each eigenvalue of the operator $H_\gamma$ lying in $(\lambda_+-\delta,\;\lambda_+)$ is at most $M(\lambda_+,H_0,W_+)$;
\(ii) the claim analogous to (i) is valid for $\gamma>0$ with $W_-$ and $\gamma\in(0,\bar\gamma)$ instead of, respectively, $W_+$ and $\gamma\in(-\bar\gamma,0)$;
\(iii) the claims analogous to (i), (ii) are valid near the edge $\lambda_-$ with $\lambda_-$, $(\lambda_-,\lambda_-+\delta)$, $W_-$, $W_+$, $\gamma\in(0,\bar\gamma)$ and $\gamma\in(-\bar\gamma,0)$ instead of, respectively, $\lambda_+$, $(\lambda_+-\delta,\;\lambda_+)$, $W_+$, $W_-$, $\gamma\in(-\bar\gamma,0)$ and $\gamma\in(0,\bar\gamma)$.
We shall prove only claim (i), because (ii) and (iii) are proved analogously. Denote $l_+=M(\lambda_+,H_0,W_+)$. By Proposition \[BrScwspct\], we should prove that $$\label{shldprv}
\forall\,\lambda\in(\lambda_+-\delta,\;\lambda_+):\quad
\dim(\ker(I+\gamma X_W(\lambda)))\le l_+.$$ In view of Lemma \[lmrprBrSchw\], we have for $u\in{{\mathcal H}}$: $$\label{rprrequadfrm}
\Re(((I+\gamma X_W(\lambda))u,u))=(u,u)+\gamma((X_{W_+}(\lambda)u,u)-(X_{W_-}(\lambda)u,u)).$$ Let $\mu_k^+(\lambda)$ and $\mu_k^-(\lambda)$ be positive and negative characteristic branches of the operator $H_0$ w.r.t. $W_+$, that is they are continuous branches of eigenvalues of the operator $X_{W_+}(\lambda)$ such that $\mu_k^+(\lambda)$ are numbered in the non-increasing ordering $\mu_1^+(\lambda)\ge\mu_2^+(\lambda)\ge\dots\mu_k^+(\lambda)\ge\dots>0$ and $\mu_k^-(\lambda)$ are numbered in the non-increasing ordering $\mu_1^-(\lambda)\le\mu_2^-(\lambda)\le\dots\mu_k^-(\lambda)\le\dots<0$, ${{\rm Dom}}(\mu_k^+)=(\eta_k^+,\lambda_+),\;\eta_k^+\in[\lambda_-,\lambda_+]$, ${{\rm Dom}}(\mu_k^-)=(\lambda_-,\eta_k^-),\;\eta_k^-\in[\lambda_-,\lambda_+]$’ and they are increasing functions (Definition \[gde1\], Remark \[rembranch\]). Let $\{\mu_k^+(\lambda)\}_{k=1}^{l_+}$ be the main characteristic branches of $H_0$ w.r.t. $W_+$ near the edge $\lambda_+$, that is $$\begin{aligned}
\label{mainbrnchWpl}
\lim_{\lambda\uparrow\lambda_+}\mu_k^+(\lambda)\left\{
\begin{array}{ll}
=\infty &\rm{for}\;1\le k\le l_+\\
<\infty &\rm{for}\; k>l_+.
\end{array}\right.\end{aligned}$$ As it is clear, $\eta_1^+\le\eta_2^+\le\dots\le \eta_{l_+}^+$. For each $\lambda\in(\eta_+,\lambda_+)$ and $j\in\{1,2,\dots,l_+\}$ consider the eigenspace $E_j^+(\lambda)$ of the operator $X_{W_+}(\lambda)$, corresponding to its eigenvalue $\mu_1^+(\lambda)$. Consider the following family of subspaces: $$\label{dffamsbsp}
{{\mathcal F}}(\lambda)=\left\{\begin{array}{ll}
{{\mathcal H}}\ominus\left(\bigoplus_{j=1}^{l_+}E_j^+(\lambda)\right) &\rm{for}\;
\lambda\in(\lambda_+-\delta,\;\lambda_+)\cap(\eta_{l_+}^+,\lambda_+),\\
{{\mathcal H}}&\rm{for}\;
\lambda\in(\lambda_+-\delta,\;\lambda_+)\setminus(\eta_{l_+}^+,\lambda_+).
\end{array}\right.$$ Then we obtain: $$\begin{aligned}
&&\hskip-7mm\forall\;u\in{{\mathcal F}}(\lambda):\\
&&\hskip-7mm(X_{W_+}(\lambda)u,\,u)\le\left\{\begin{array}{ll}
\max_{j>l+}\mu_j^+(\lambda)\|u\|^2, &\rm{if}\;
\lambda\in(\lambda_+-\delta,\;\lambda_+)\cap(\eta_{l_+}^+,\lambda_+),\\
\mu_1^+(\eta_{l_+}^+)\|u\|^2, &\rm{if}\;
\lambda\in(\lambda_+-\delta,\;\lambda_+)\setminus(\eta_{l_+}^+,\lambda_+).
\end{array}\right.\end{aligned}$$ Hence, in view of (\[mainbrnchWpl\]), we have: $$\label{estXWpl}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;u\in{{\mathcal F}}(\lambda)\setminus\{0\}}
\frac{(X_{W_+}(\lambda)u,\,u)}{\|u\|^2}<\infty.$$ Since the operator function $X_{W_-}(\lambda)$ increases in $(\lambda_-,\lambda_+)$ in the sense of comparison of quadratic forms ([@Ar-Zl1], Lemma 3.5), we have: $$\begin{aligned}
\label{estXWmn}
&&
\inf_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;u\in{{\mathcal H}}\setminus\{0\}}
\frac{(X_{W_-}(\lambda)u,\,u)}{\|u\|^2}\ge
\nonumber\\
&& \inf_{u\in{{\mathcal H}}\setminus\{0\}}
\frac{\left(X_{W_-}(\lambda_+-\delta)u,\,u\right)}{\|u\|^2}>-\infty.\end{aligned}$$ Then we conclude from (\[rprrequadfrm\]), (\[estXWpl\]) and (\[estXWmn\]) that there exists $\bar\gamma>0$ such that $$\forall\;\gamma\in(-\bar\gamma,0):\quad\inf_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;u\in{{\mathcal F}}(\lambda)\setminus\{0\}}
\frac{\Re(((I+\gamma X_W(\lambda))u,u))}{\|u\|^2}\ge\frac{1}{2}.$$ Taking into account that $\Re(((I+\gamma X_W(\lambda))u,u))\le\|(I+\gamma X_W(\lambda))u\|\|u\|$, we obtain from the last estimate that $$\label{estinfqdfrm}
\forall\;\gamma\in(-\bar\gamma,0):\quad\inf_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;u\in{{\mathcal F}}(\lambda)\setminus\{0\}}
\frac{\|(I+\gamma X_W(\lambda))u\|}{\|u\|}\ge\frac{1}{2}.$$ Let us prove estimate (\[shldprv\]). Assume, on the contrary, that $$\begin{aligned}
\label{asscntr}
&&\exists\,\gamma_0\in(-\bar\gamma,0),\;\exists\,\lambda_0\in(\lambda_+-\delta,\;\lambda_+):\nonumber\\
&&\dim(\ker(I+\gamma_0 X_W(\lambda_0)))>l_+.\end{aligned}$$ On the other hand, in view of (\[dffamsbsp\]), $$\forall\,\lambda\in(\lambda_+-\delta,\;\lambda_+):\quad
\rm{codim}({{\mathcal F}}(\lambda))\le l_+.$$ This inequality together with (\[asscntr\]) imply that $${{\mathcal F}}(\lambda_0)\cap\ker(I+\gamma_0 X_W(\lambda_0))\neq\{0\}.$$ The last fact contradicts the inequality (\[estinfqdfrm\]). So, estimate (\[shldprv\]) is proven.
### Proof of Theorem \[thestmult\] {#subsec:proofthmult}
Assume that the edge $\lambda_+$ is non-degenerate. If in the in the proof of Theorem \[thnondegedg\] we take $W_+$ instead of $W$, we obtain that for $d=1$ $M(\lambda_+,H_0,W_+)=1$, for $d=2$ $M(\lambda_+,H_0,W_+)=n_+$ and for $d\ge 3$ $M(\lambda_+,H_0,W_+)=0$. Then by Proposition \[prestmult\] all the claims (i)-(iii) are valid in the case where $\lambda_+$ is non-degenerate. The case where $\lambda_-$ is non-degenerate is treated analogously.
Claim (iv) is proved in the same manner as the previous ones. Claim (v) follows from the claims (iii) and (iv).
Theorem \[thestmult\] is proven.
### Proof of Theorem \[ththresholddeg\] {#subsec:proofthmultdeg}
The proof is the same as the proof of Theorem \[thestmult\] in the case where $d\ge 3$ and it is supported by Proposition \[prestmult\] and claim (iv) of Theorem \[thdegedg\].
Representation of the resolvent of the unperturbed operator $H_0$ near the edges of a gap of its spectrum {#sec:resunprop1}
=========================================================================================================
Main claims {#sec:frmmaincl}
-----------
\[prrprres1\] Let $(\lambda_-,\,\lambda_+)$ be a gap of the spectrum of the unperturbed operator $H_0$ such that its edge $\lambda_+$ satisfies condition (B) of Section \[subsec:spectchar\], and $\lambda^+({{\mathbf p}})$ be the dispersion function, branching from the edge $\lambda_+$. Let us take $\delta\in(0,\,(\lambda_+-\lambda_-))$.
\(i) For $\lambda\in(\lambda_-,\lambda_+)$ the resolvent $R_\lambda(H_0)$ has the form: $$\label{represol1}
R_\lambda(H_0)=K(\lambda)+\Theta(\lambda),$$ where $\Theta(\lambda)$ is a bounded operator in $L_2({{\mathbf R}}^d)$ for any $\lambda\in(\lambda_-,\lambda_+)$ such that the condition is satisfied $$\begin{aligned}
\label{bndtht2}
&&\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+)}\|\Theta(\lambda)\|<\infty,\end{aligned}$$ $K(\lambda)=\sum_{k=1}^{n_+}K_k^+(\lambda)$ and for any $k\in\{1,2,\dots,n_+\}$ the restriction $K_k^+(\lambda)|_{L_{2,0}({{\mathbf R}}^d)}$ of $K_k^+(\lambda)$ on $L_{2,0}({{\mathbf R}}^d)$ is an operator with a continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_-,\lambda_+)$ integral kernel $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$;
\(ii) if $d-d_k^+\ge 3$, then for any $k\in\{1,2,\dots,n_+\}$ the function $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$ satisfies the condition $$\begin{aligned}
\label{cndK1}
\sup_{\lambda\in((\lambda_+-\delta,\;\lambda_+),\;{{\mathbf x}},{{\mathbf s}}\in{{\mathbf R}}^d}|K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)|<\infty;\end{aligned}$$
\(iii) If $d-d_k^+\le 2$, then for any $k\in\{1,2,\dots,n_+\}$, $\lambda\in(\lambda_-,\,\lambda_+)$ the function $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$ admits the representation $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)=G_k^+({{\mathbf x}},{{\mathbf s}},\lambda)+\tilde
K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$, where $\tilde K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$ is continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_-,\,\lambda_+)$ and
\(a) for $d-d_k^+=1$ $$\begin{aligned}
\label{dfFpl1}
G_k^+({{\mathbf x}},{{\mathbf s}},\lambda)=\frac{\sqrt{2}\pi}{(2\pi)^d\sqrt{\lambda_+-\lambda}}
\int_{F_k^+}{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}})\end{aligned}$$ where ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ is the eigenkernel of $H({{\mathbf p}})$ corresponding to the eigenvalue $\lambda^+({{\mathbf p}})$, defined in Section \[subsec:spectchar\], $m^+({{\mathbf p}})$ is defined by (\[dfmplp\]), (\[dfnrmHes\]), (\[dfHes\]), and $$\begin{aligned}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;{{\mathbf x}},{{\mathbf s}}\in{{\mathbf R}}^d}
\frac{|\tilde K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)|}{(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty,\end{aligned}$$
\(b) for $d-d_k^+=2$ $$\begin{aligned}
\label{dfFpl2}
&&\hskip-10mm
G_k^+({{\mathbf x}},{{\mathbf s}},\lambda)=\nonumber\\
&&\hskip-10mm\frac{1}{(2\pi)^{d-1}}\ln\Big(\frac{1}{\lambda_+-\lambda}\Big)
\int_{F_k^+}{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sqrt{m^+({{\mathbf p}})}\,dF({{\mathbf p}}),\end{aligned}$$ and $$\begin{aligned}
&&\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;{{\mathbf x}},{{\mathbf s}}\in{{\mathbf R}}^d}
\frac{|\tilde K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)|}{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty.\end{aligned}$$
The claims analogous to above ones are valid with $\lambda_-$, $\lambda^-({{\mathbf p}})$, $n_-$, $d_k^-$, $F_k^-$, ${{\mathcal Q}}^-({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, $m^-({{\mathbf p}})$ and $(\lambda_-,\;\lambda_-+\delta)$ instead of, respectively, $\lambda_+$, $\lambda^+({{\mathbf p}})$, $n_+$, $d_k^+$, $F_k^+$, ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, $m^+({{\mathbf p}})$ and $(\lambda_+-\delta,\;\lambda_+)$.
In particular, in the case of non-degenerate edges of the gap of $\sigma(H_0)$ the previous proposition acquires the form:
\[prrprres\] Let $(\lambda_-,\,\lambda_+)$ is a gap of the spectrum of the unperturbed operator $H_0$ such that its edge $\lambda_+$ is non-degenerate and $\lambda^+({{\mathbf p}})$ be the dispersion function, branching from the edge $\lambda_+$. Let us take $\delta\in(0,\,(\lambda_+-\lambda_-))$.
\(i) The resolvent $R_\lambda(H_0)$ has the form for $\lambda\in(\lambda_-,\lambda_+)$: $R_\lambda(H_0)=K(\lambda)+\Theta(\lambda)$, where $\Theta(\lambda)$ is a bounded operator in $L_2({{\mathbf R}}^d)$ for any $\lambda\in(\lambda_-,\lambda_+)$, such that condition (\[bndtht2\]) is satisfied and the restriction $K(\lambda)|_{L_{2,0}({{\mathbf R}}^d)}$ of $K(\lambda)$ on $L_{2,0}({{\mathbf R}}^d)$ is an operator with a continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_-,\lambda_+)$ integral kernel $K({{\mathbf x}},{{\mathbf s}},\lambda)$;
\(ii) if $d\ge 3$, then the function $K({{\mathbf x}},{{\mathbf s}},\lambda)$ satisfies the condition $$\begin{aligned}
\label{cndK}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;{{\mathbf x}},{{\mathbf s}}\in{{\mathbf R}}^d}|K({{\mathbf x}},{{\mathbf s}},\lambda)|<\infty;\end{aligned}$$
\(iii) if $d\le2$ and $\lambda\in(\lambda_-,\,\lambda_+)$, then the function $K({{\mathbf x}},{{\mathbf s}},\lambda)$ admits the representation $K({{\mathbf x}},{{\mathbf s}},\lambda)=F^+({{\mathbf x}},{{\mathbf s}},\lambda)+\tilde K^+({{\mathbf x}},{{\mathbf s}},\lambda)$, where $\tilde K^+({{\mathbf x}},{{\mathbf s}},\lambda)$ is continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_-,\,\lambda_+)$ and
\(a) for $d=1$ $$\begin{aligned}
F^+(x,s,\lambda)=\sqrt{m_1^+}\;\frac{b^+(x,p_1)\overline{b^+(s,p_1)}}{\sqrt{2(\lambda_+-\lambda)}},\end{aligned}$$ where $m_1^+$ is defined by (\[dfmupl\]) and $$\begin{aligned}
\label{cndK1pl}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;x,s\in{{\mathbf R}}}
\frac{|\tilde K^+(x,s,\lambda)|}{1+|x-s|}<\infty;\end{aligned}$$
\(b) for $d=2$ $$\begin{aligned}
F^+({{\mathbf x}},{{\mathbf s}},\lambda)=\frac{1}{2\pi}\ln\left(\frac{1}{\lambda_+-\lambda}\right)
\sum_{k=1}^{n_+}\sqrt{m_k^+}\,b_k^+({{\mathbf x}})\overline{b_k^+({{\mathbf s}})},\end{aligned}$$ where $m_k^+$ is defined by (\[dfmkpl\]), (\[Hess\]), $b_k^+({{\mathbf x}})$ is the Bloch function, corresponding to the dispersion function $\lambda^+({{\mathbf p}})$ and the quasi-momentum ${{\mathbf p}}={{\mathbf p}}_k^+$, i.e. $b_k^+({{\mathbf x}})=b^+({{\mathbf p}}_k^+)$; furthermore $$\begin{aligned}
\sup_{\lambda\in(\lambda_+-\delta,\;\lambda_+),\;{{\mathbf x}},{{\mathbf s}}\in{{\mathbf R}}^2}
\frac{|\tilde K^+({{\mathbf x}},{{\mathbf s}},\lambda)|}{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty.\end{aligned}$$
The claims analogous to above ones are valid with $\lambda_-$, $\lambda^-({{\mathbf p}})$, $n_-$, ${{\mathbf p}}_k^-$, $b_k^-({{\mathbf x}})$, $m_k^-$ and $(\lambda_-,\;\lambda_-+\delta)$ instead of, respectively, $\lambda_+$, $\lambda^+({{\mathbf p}})$, $n_+$, ${{\mathbf p}}_k^+$, $b_k^+({{\mathbf x}})$, $m_k^+$ and $(\lambda_+-\delta,\;\lambda_+)$.
Auxiliary claims {#sec:auxclaims}
----------------
For the proof of the propositions, formulated above, we need some auxiliary claims.
### Estimates of some integrals {#sec:estintegr}
\[lmestint\] If $\rho>0$, $\alpha>1$, $\mu>0$, $\beta\in{{\mathbf R}}$, $n\in\{1,\,2\}$ and $\phi:\;[0,\infty)\rightarrow{{\mathbf R}}$ is a continuous function such that $\bar\phi=\sup_{u\in[0,\rho]}\frac{\phi(u)}{u^2}<\infty$, then the following estimates are valid: $$\begin{aligned}
\label{estint1}
&&I_1=\int_0^{\rho\alpha}\frac{u^{n-1}|\exp(i\phi(u))-1|}{u^2+\mu}\,du\le\\
&&\left\{\begin{array}{ll}
\bar\phi\rho+2\rho^{-1}(1-\alpha^{-1})&\rm{for}\quad n=1,\\
\bar\phi\rho^2/2+2\ln\alpha &\rm{for}\quad n=2;
\end{array}\right.\nonumber\end{aligned}$$ $$\begin{aligned}
\label{estint2}
&&I_2=\int_0^{\rho\alpha}\frac{u^{n-1}(1-\cos(\beta
u))}{u^2+\mu}\,du\le\\
&&\left\{\begin{array}{ll}
\beta^2\rho/2+2\rho^{-1}(1-\alpha^{-1})&\rm{for}\quad n=1,\\
\beta^2\rho^2/4+2\ln\alpha &\rm{for}\quad n=2;
\end{array}\right.\nonumber\end{aligned}$$
$$\begin{aligned}
\label{estint3}
I_3=\Big|\int_0^{\rho\alpha}\frac{u^n\sin(\beta
u)}{u^2+\mu}\,du\Big|\le\left\{\begin{array}{ll}
|\beta|\rho+\ln\alpha&\rm{for}\quad n=1,\\
|\beta|\rho^2/2+\rho(\alpha-1)&\rm{for}\quad n=2.
\end{array}\right.\end{aligned}$$
We have for $u\in[0,\rho]$: $$\begin{aligned}
|\exp(i\phi(u))-1|=\Big|\int_0^1\partial_\tau\exp(i\tau\phi(u))\,d\tau\Big|=
\Big|\int_0^1\exp(i\tau\phi(u))i\phi(u)\,d\tau\Big|\le\bar\phi u^2,\end{aligned}$$ hence $$\begin{aligned}
&&I_1=\int_0^\rho\frac{u^{n-1}|\exp(i\phi(u))-1|}{u^2+\mu}\,du+
\int_\rho^{\rho\alpha}\frac{u^{n-1}|\exp(i\phi(u))-1|}{u^2+\mu}\,du\le\\
&&\bar\phi\int_0^\rho u^{n-1}\,du+2\int_\rho^\rho\alpha u^{n-3}\,du.\end{aligned}$$ Therefore, (\[estint1\]) is valid. Estimates (\[estint2\]) and (\[estint3\]) are proved analogously.
### Some geometric claims {#sec:geomclaims}
Assume that the edge $\lambda_+\,(\lambda_-)$ of the gap $(\lambda_-,\,\lambda_+)$ of the spectrum of the unperturbed operator $H_0$ satisfies condition (B) of Section \[subsec:spectchar\]. Taking into account that $F^+\,(F^-)$ is the set of all minimum (maximum) points of the dispersion function $\lambda^+\,(\lambda^-)$ and using the Morse-Bott lemma ([@Ban-Hur]), we obtain that for any $k\in\{1,2,\dots
n_+\}\;(k\in\{1,2,\dots n_-\})$ and ${{\mathbf p}}_\star\in F^+_k\,({{\mathbf p}}_\star\in
F^-_k)$ on the torus ${{\mathbf T}}^d$ there exists a smooth chart $(U,\phi)\,({{\mathbf p}}_\star\in U)$ such that $\phi:\,U\rightarrow{{\mathbf R}}^{d_k^+}\times{{\mathbf R}}^{d-d_k^+}\;\big(\phi:\,U\rightarrow{{\mathbf R}}^{d_k^-}\times{{\mathbf R}}^{d-d_k^-}\big)$ and
\(a) $\phi({{\mathbf p}}_\star)=0$;
\(b) $\phi(U\cap F^+_k)=\{(x,y)\in{{\mathbf R}}^{d_k^+}\times{{\mathbf R}}^{d-d_k^+}\;\vert\;y=0\}$ $(\phi(U\cap F^-_k)=\{(x,y)\in{{\mathbf R}}^{d_k^-}\times{{\mathbf R}}^{d-d_k^-}\;\vert\;y=0\})$;
\(c) $(\lambda^+\circ\phi^{-1})(x,y)=\sum_{l=1}^{d-d_k^+}y_l^2+\lambda_+$ $\big((\lambda^-\circ\phi^{-1})(x,y)=-\sum_{l=1}^{d-d_k^-}y_l^2+\lambda_-\big)$.
The chart $(U,\phi)$ is called the [*reducing chart*]{} at the point ${{\mathbf p}}_\star\in F^+_k\,({{\mathbf p}}_\star\in F^-_k)$.
\[lmredchrt\] Let $(U,\phi)$ be a reducing chart at a point ${{\mathbf p}}_\star\in
F^+_k\,({{\mathbf p}}_\star\in F^-_k)$ defined above and $\Phi=P\circ\phi$, where $P:\;{{\mathbf R}}^d\rightarrow{{\mathbf R}}^{d-d_k^+}\;\big(P:\;{{\mathbf R}}^d\rightarrow{{\mathbf R}}^{d-d_k^-}\big)$ is the orthogonal projection on the subspace ${{\mathbf R}}^{d-d_k^+}\;({{\mathbf R}}^{d-d_k^-})$ of ${{\mathbf R}}^d$. Then for any ${{\mathbf p}}\in
F^+_k\cap U\,({{\mathbf p}}\in F^-_k\cap U)$ the normal Jacobian of $\Phi$ $\;NJ\,\Phi({{\mathbf p}})=\det\big(\Phi^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\big)$ is equal to $\sqrt{\frac{NJ\,\rm{Hes}_{{\mathbf p}}(\lambda^+)}{2^{d-d^+_k}}}\;\big(\sqrt{\frac{NJ\,\rm{Hes}_{{\mathbf p}}(\lambda^-)}{2^{d-d^-_k}}}\big)$. Hence, in particular, for any ${{\mathbf p}}\in F^+_k\cap U\,({{\mathbf p}}\in F^-_k\cap
U)$ $\;\mathrm{Im}\big(\Phi^\prime({{\mathbf p}})\big)={{\mathbf R}}^{d-d_k^+}\;\big(\mathrm{Im}\big(d\Phi({{\mathbf p}})\big)={{\mathbf R}}^{d-d_k^-}\big)$ and the mapping $\Phi^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}:\,N_{{\mathbf p}}\rightarrow{{\mathbf R}}^{d-d_k^+}\;
\big(\Phi^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}:\,N_{{\mathbf p}}\rightarrow{{\mathbf R}}^{d-d_k^-}\big)$ is a linear isomorphism. Recall that $NJ\,\rm{Hes}_{{\mathbf p}}(\lambda^+)\;\big(NJ\,\rm{Hes}_{{\mathbf p}}(\lambda^-)\big)$ is the normal Hessian of $\lambda^+\;(\lambda^-)$ at the point ${{\mathbf p}}$, defined by (\[dfnrmHes\]), and $\rm{Hes}_{{\mathbf p}}(\lambda^+)\;(\rm{Hes}_{{\mathbf p}}(\lambda^-))$ is the Hessian operator of $\lambda^+\;(\lambda^-)$ at the point ${{\mathbf p}}$, defined by (\[dfHes\]).
We shall prove the claim only for the function $\lambda^+({{\mathbf p}})$, because for $\lambda^-({{\mathbf p}})$ it is proved analogously. In view of condition (c) for the reducing chart,, for ${{\mathbf p}}\in F^+_k\cap U$ and $y=\phi({{\mathbf p}})$ $\lambda^+\circ\phi^{-1}(y)=Py\cdot y$. Then for $v,w\in{{\mathbf R}}^d$ $$d(\lambda^+\circ\phi^{-1}(y))[v]=d\lambda^+(\phi^{-1}(y))[d(\phi^{-1}(y))[v]]=2v\cdot
Py,$$ $$\begin{aligned}
&&d^2(\lambda^+\circ\phi^{-1}(y))[v,w]=d^2\lambda^+(\phi^{-1}(y))[d(\phi^{-1}(y))[v],\,d(\phi^{-1}(y))[w]]+\\
&&d\lambda^+(\phi^{-1}(y))[d^2(\phi^{-1}(y))[v,w]]=2Pv\cdot w\end{aligned}$$ (we identify the tangent and the second tangent bundles over ${{\mathbf R}}^d$ and the flat torus $T^d$ with ${{\mathbf R}}^d\times{{\mathbf R}}^d$ and ${{\mathbf R}}^d\times{{\mathbf R}}^d\times{{\mathbf R}}^d$ respectively). Since ${{\mathbf p}}=\phi^{-1}(y)$ is a minimum point of the function $\lambda^+$, $d\lambda^+(\phi^{-1}(y))=0$. Hence we get, in view of definition (\[dfHes\]) of the Hessian operator: $\rm{Hes}_{{\mathbf p}}(\lambda^+)d(\phi^{-1}(y))[v]\cdot
d(\phi^{-1}(y))[w]=2Pv\cdot Pw$, that is $$\label{Heseq}
\forall\,s,t\in{{\mathbf R}}^d:\quad\rm{Hes}_{{\mathbf p}}(\lambda^+)s\cdot
t=2d\Phi({{\mathbf p}})[s]\cdot d\Phi({{\mathbf p}})[t].$$ In view of condition (b) for the reducing chart, $\Phi(F^+_k\cap
U)=\{0\}$, hence for any ${{\mathbf p}}\in F^+_k\cap U$ $T_{{\mathbf p}}(F^+_k)\subseteq\ker(\Phi^\prime({{\mathbf p}}))$. The last fact and (\[Heseq\]) imply that $T_{{\mathbf p}}(F^+_k)\subseteq\ker\big(\rm{Hes}_{{\mathbf p}}(\lambda^+)\big)$. Hence since the operator $\rm{Hes}_{{\mathbf p}}(\lambda^+)$ is self-adjoint, it maps the subspace $N_{{\mathbf p}}^+$ into itself. Then by (\[Heseq\]) $\rm{Hes}_{{\mathbf p}}(\lambda^+)\vert_{N_{{\mathbf p}}^+}=2\big(\Phi^\prime({{\mathbf p}})
\vert_{N_{{\mathbf p}}}\big)^\star\big(\Phi^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\big)$. The last equality implies the desired claim.
\[lmdifeq\] Let $(U,\phi)$ be a reducing chart at a point ${{\mathbf p}}_\star\in
F^+_k\,({{\mathbf p}}_\star\in F^-_k)$ defined above and $\Phi$ is the same mapping as in Lemma \[lmredchrt\]. Consider the differential equation in $U$: $$\label{diffeq}
\frac{d{{\mathbf p}}}{dt}=\Big(\Phi^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\Big)^{-1}{{\mathbf y}},\quad{{\mathbf y}}\in{{\mathbf R}}^{d-d_k^+}\;({{\mathbf y}}\in{{\mathbf R}}^{d-d_k^-}).$$ Let ${{\mathbf p}}(t,{{\mathbf p}}_0,{{\mathbf y}})$ be the flow of this equation, that is the solution of it satisfying the initial condition ${{\mathbf p}}(0,{{\mathbf p}}_0,{{\mathbf y}})={{\mathbf p}}_0$. Then it is possible to restrict the neighborhood $U$ of ${{\mathbf p}}_\star$ such that
\(i) for some $r>0$, any $t\in[0,r]$ and any ${{\mathbf y}}$ belonging to the unit sphere $S^{d-d_k^+-1}\subset{{\mathbf R}}^{d-d_k^+}\;\big(S^{d-d_k^--1}\subset{{\mathbf R}}^{d-d_k^-}\big)$ the set $F_{t{{\mathbf y}}}=\Phi^{-1}(t{{\mathbf y}})$ is a $d_k^+\;(d_k^-)$-dimensional smooth submanifold of ${{\mathbf T}}^d$ (in particular, $F_0=F^+_k\cap
U\;\big(F_0=F^-_k\cap U\big)$), ${{\mathbf p}}(t,F^+_k\cap U,{{\mathbf y}})=F_{t{{\mathbf y}}}$ $\big({{\mathbf p}}(t,F^+_k\cap U,{{\mathbf y}})=F_{t{{\mathbf y}}}\big)$ and the mapping ${{\mathbf p}}(t,\cdot,{{\mathbf y}}):\;F^+_k\cap U\rightarrow
F_{ty}\;\big({{\mathbf p}}(t,\cdot,{{\mathbf y}}):\;F^-_k\cap U\rightarrow F_{t{{\mathbf y}}}\big)$ is a diffeomorphism;
\(ii) for ${{\mathbf q}}\in F^+_k\cap U\;({{\mathbf q}}\in F^-_k\cap U)$ the functions $\partial_t{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{t=0}$ and $\partial_tTJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{t=0}$ are odd w.r.t. ${{\mathbf y}}$. Here $TJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})$ is the tangential Jacobian of the mapping ${{\mathbf p}}(t,\cdot,{{\mathbf y}})$, that is $$\begin{aligned}
&&TJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})=\det\big(\partial_{{{\mathbf q}}}{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{T_{{{\mathbf q}}}(F^+_k)}\big)\\
&&\Big(TJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})=\det\big(\partial_{{{\mathbf q}}}{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{T_{{{\mathbf q}}}(F^-_k)}\big)\Big).\end{aligned}$$
Consider only the case where ${{\mathbf p}}_\star\in F^+_k$, because the case ${{\mathbf p}}_\star\in F^-_k$ is treated analogously. Using the fact that $\Phi(F_k^+\cap U)=\{0\}$, we see that if ${{\mathbf q}}\in F_k^+\cap U_{k,l}$ and for some ${{\mathbf y}}\in S^{d-d_k^+-1}$, $t>0$ the solution ${{\mathbf p}}(\cdot,{{\mathbf q}},{{\mathbf y}})$ of (\[diffeq\]) exists in the interval $[0,t]$, then $\Phi^\prime({{\mathbf p}}(\tau,{{\mathbf q}},{{\mathbf y}}))\partial_\tau{{\mathbf p}}(\tau,{{\mathbf q}},{{\mathbf y}})={{\mathbf y}}\;(\tau\in[0,t])$. Integrating the both sides of the last equality by $\tau$ over $[0,t]$, we get: $\Phi({{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}}))=t{{\mathbf y}}$, that is ${{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\in\Phi^{-1}(t{{\mathbf y}})$. Using the theorem on the local existence and uniqueness of solution of the Cauchy problem for a dynamical system in a Banach space, we can restrict the neighborhood $U$ of ${{\mathbf p}}_\star$ such that claim (i) is valid.
Let us prove claim (ii). The fact that the function $\partial_t{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{t=0}$ is odd w.r. to ${{\mathbf y}}$ for ${{\mathbf q}}\in F_k^+\cap U$ follows immediately from (\[diffeq\]) and the equality ${{\mathbf p}}(0,{{\mathbf q}},{{\mathbf y}})={{\mathbf q}}$. As it is known, the derivative $Y(t)=\partial_{{{\mathbf q}}}{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})$ satisfies the following linear equation, which is the linearization of equation (\[diffeq\]) at its solution ${{\mathbf p}}(t)={{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})$: $$\label{lineq}
\frac{dY}{dt}=(A(t)Y){{\mathbf y}},$$ where $A(t)=d_{{\mathbf p}}\big(\Phi^\prime({{\mathbf p}}(t))\vert_{N_{{{\mathbf p}}(t)}}\big)^{-1}$. Furthermore, $Y(0)=I$. In view of claim (i), for any ${{\mathbf q}}\in F_k^+$ and $t\in[o,r]$ the operator $Y(t)\vert_{T_{{{\mathbf q}}}(F_k^+)}$ realizes a linear isomorphism between $T_{{{\mathbf q}}}(F_k^+)$ and $T_{{{\mathbf p}}(t)}(F_{t{{\mathbf y}}})$. Let $dv=dp_1\wedge dp_2\wedge\dots\wedge
dp_d$ be the volume form on the flat torus ${{\mathbf T}}^d$. As it is known, the volume form $dF_{t{{\mathbf y}}}$ on the submanifold $F_{t{{\mathbf y}}}$ has the form $dF_{t{{\mathbf y}}}({{\mathbf p}})=dv\vert_{T_{{{\mathbf p}}}(F_{t{{\mathbf y}}})}\;({{\mathbf p}}\in F_{t{{\mathbf y}}})$. Let $d^\star F_{t{{\mathbf y}}}$ be the pullback of the form $dF_{t{{\mathbf y}}}$ w.r.t. the mapping ${{\mathbf p}}(t,\cdot,{{\mathbf y}})\vert_{F_k^+\cap U}$. Taking into account equation (\[lineq\]) and the equalities $F_0=F_k^+\cap U$, $Y(0)=I$ and ${{\mathbf p}}(0)={{\mathbf p}}(0,{{\mathbf q}},{{\mathbf y}})={{\mathbf q}}\;({{\mathbf q}}\in F_k^+\cap U)$, we have for ${{\mathbf s}}=(s_1,s_2,\dots,s_d)\in T_{{{\mathbf q}}}(F_k^+)$: $$\begin{aligned}
&&\frac{d}{dt}d^\star F_{t{{\mathbf y}}}[{{\mathbf s}}]\vert_{t=0}=\sum_{i=1}^d(Y(t){{\mathbf s}})_1\wedge(Y(t){{\mathbf s}})_2\wedge\dots\\
&&\wedge(Y(t){{\mathbf s}})_{i-1}\wedge((A(t)Y(t){{\mathbf s}}){{\mathbf y}})_i\wedge\wedge(Y(t){{\mathbf s}})_{i+1}\wedge\dots\wedge(Y(t){{\mathbf s}})_d\vert_{t=0}=\\
&&\sum_{i=1}^ds_1\wedge s_2\wedge\dots\wedge
s_{i-1}\wedge((A(0){{\mathbf s}}){{\mathbf y}})_i\wedge s_{i+1}\wedge\dots\wedge s_d.\end{aligned}$$ The last representation implies that $\frac{d}{dt}d^\star
F_{t{{\mathbf y}}}[{{\mathbf s}}]\vert_{t=0}=-\frac{d}{dt}d^\star
F_{-t{{\mathbf y}}}[{{\mathbf s}}]\vert_{t=0}$. On the other hand, it is known that $d^\star F_{t{{\mathbf y}}}({{\mathbf q}})=TJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})d F_0({{\mathbf q}})$, where $d
F_0=dF_{t{{\mathbf y}}}\vert_{t=0}$ is the volume form on $F_k^+\cap U$. These circumstances imply that the function $\frac{d}{dt}TJ{{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})\vert_{t=0}$ is odd w.r.t. ${{\mathbf y}}$. Claim (ii) is proven.
Proof of claim (i) of Proposition \[prrprres1\] {#sec:proofclipr1}
-----------------------------------------------
Let us choose a suitable neighborhood of each of the submanifolds $F_k^+\;(k\in\{1,2,\dots,n_+\})$. Let $(U_{{{\mathbf p}}_\star},\phi)$ be a reducing chart at a point ${{\mathbf p}}_\star\in F^+_k$. Since $F_k^+$ are closed disjoint subsets of the torus ${{\mathbf T}}^d$, we can choose it such that $ U_{{{\mathbf p}}_\star}\cap F^+_i=\emptyset\;(U_{{{\mathbf p}}_\star}\cap
F^-_i=\emptyset)$ for any $i\neq k$. Let $\Phi:\;U_{{{\mathbf p}}_\star}\rightarrow{{\mathbf R}}^{d-d_k^+}$ be the same as in Lemmas \[lmredchrt\] and \[lmdifeq\]. Then by Lemma \[lmdifeq\] it is possible to restrict the neighborhood $U_{{{\mathbf p}}_\star}$ of ${{\mathbf p}}_\star$ such that for the flow ${{\mathbf p}}(t,{{\mathbf q}},{{\mathbf y}})$ of the differential equation (\[diffeq\]) claim (i) of this lemma is valid. Recall that the dispersion function $\lambda^+({{\mathbf p}})$ branching from the edge $\lambda_+$ of the gap $(\lambda_-,\,\lambda_+)$ in $\sigma(H_0)$ has the form $\lambda^+({{\mathbf p}})=\lambda_{j+1}({{\mathbf p}})$ for some $j\ge 0$. Like in Section \[subsec:spectchar\], we can restrict the neighborhood $U_{{{\mathbf p}}_\star}$, taking into account condition (A)-(a) and Corollary \[cormainApp\], such that for any ${{\mathbf p}}\in U_{{{\mathbf p}}_\star}$ $\lambda^+({{\mathbf p}})$ is a simple eigenvalue of $\tilde H({{\mathbf p}})$, i.e. $$\label{neib1}
\forall\,{{\mathbf p}}\in U_{{{\mathbf p}}_\star}:\quad \lambda^+({{\mathbf p}})<\lambda_{j+2}({{\mathbf p}}),$$ the function $\lambda^+({{\mathbf p}})$ is real-analytic in $U_{{{\mathbf p}}_\star}$ and furthermore, the mapping ${{\mathbf p}}\rightarrow \tilde
{{\mathcal Q}}^+(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ is real-analytic in $U_{{{\mathbf p}}_\star}$. Recall that $\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ is the eigenkernel of $\tilde H({{\mathbf p}})$, corresponding to $\lambda^+({{\mathbf p}})$. Since each $F^+_k\;(k\in\{1,2,\dots,n_+\})$ is compact, then it is possible to select a finite subcovering $\{U_{k,l}\}_{l=1}^{L_k^+}$ from its open covering $\{U_{{{\mathbf p}}_\star}\}_{{{\mathbf p}}_\star\in F^+_k}$. Consider the neighborhood $U_k^+=\bigcup_{l=1}^{L_k^+} U_{k,l}$ of $F^+_k$ for each $k\in\{1,2,\dots,n_+\}$. As it is clear, these neighborhoods are disjoint. Denote $S_+=\bigcup_{k=1}^{n_+}U_k$.
As it is known ([@Gel], [@Kuch], [@Zl]), the operator $\tilde U=\exp(-i{{\mathbf p}}\cdot{{\mathbf x}})\cdot U$, where $U$ is defined by (\[unit\]), maps the space $L_2({{\mathbf R}}^d)$ on the direct integral $\int_{{{\mathbf T}}^d}^\oplus L_2({{\mathbf R}}^d/\Gamma)\,d{{\mathbf p}}\\=L_2({{\mathbf R}}^d/\Gamma)\otimes
L_2({{\mathbf T}}^d)$ and realizes a unitary equivalence between the operator $H_0$ and the direct integral $\tilde H=\int_{{{\mathbf T}}^d}^\oplus\tilde
H({{\mathbf p}})\,d{{\mathbf p}}$ of the operators $\tilde H({{\mathbf p}})$. Recall that the operator $\tilde H({{\mathbf p}})$ is defined by (\[b16\]), (\[b17\]) and its domain is $W_2^2({{\mathbf R}}^d/\Gamma)$. Formula (\[unit\]) implies that the operator inverse to $\tilde U$, has the form: for any $\phi\in L_2({{\mathbf R}}^d/\Gamma)\otimes L_2({{\mathbf T}}^d)$ $$\begin{aligned}
\label{invunit1}
(\tilde
U^{-1}\phi)({{\mathbf x}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{{{\mathbf T}}^d}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})\phi({{\mathbf x}},{{\mathbf p}})\,d{{\mathbf p}}.
\end{aligned}$$ Let us take $f\in L_2({{\mathbf R}}^d)$ and denote $\tilde f=\tilde Uf$. Since $R_\lambda(H_0)=\tilde U^{-1}R_\lambda(\tilde H)\tilde U$ for any $\lambda\in{{\mathcal R}}(H_0)$, we get using (\[invunit1\]): $$\begin{aligned}
\label{res1}
&&(R_\lambda(H_0)f)({{\mathbf x}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{{{\mathbf T}}^d}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})
(R_\lambda(\tilde H({{\mathbf p}}))\tilde f(\cdot,{{\mathbf p}}))({{\mathbf x}})\,d{{\mathbf p}}=\nonumber\\
&&(R_\lambda^+f)({{\mathbf x}})+(\Theta_\lambda^+f)({{\mathbf x}}),\end{aligned}$$ where $$\begin{aligned}
\label{dfRpl1}
&&(R_\lambda^+f)({{\mathbf x}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{S_+}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})
(R_\lambda(\tilde H({{\mathbf p}}))\tilde f(\cdot,\,{{\mathbf p}}))({{\mathbf x}})\,d{{\mathbf p}}=\nonumber\\
&&(\tilde U^{-1}R_\lambda(\tilde H({{\mathbf p}}))\chi_{_{S_+}}({{\mathbf p}})\tilde
f(\cdot,\,{{\mathbf p}}))({{\mathbf x}}),\end{aligned}$$ $$\begin{aligned}
\label{dfThtpl1}
&&(\Theta_\lambda^+f)({{\mathbf x}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{{{\mathbf T}}^d\setminus
S_+}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})
(R_\lambda(\tilde H({{\mathbf p}}))\tilde f(\cdot,\,{{\mathbf p}}))({{\mathbf x}})\,d{{\mathbf p}}=\nonumber\\
&&(\tilde U^{-1}(R_\lambda(\tilde H({{\mathbf p}}))\chi_{_{{{\mathbf T}}^d\setminus
S_+}}({{\mathbf p}})\tilde f(\cdot,\,{{\mathbf p}})))({{\mathbf x}}).\end{aligned}$$ Here we denote by $\chi_{_A}$ the characteristic function for a set $A\subset {{\mathbf T}}^d$. Since $F^+=\bigcup_{k=1}^{n_+}F^+_k=(\lambda^+)^{-1}(\lambda_+)$ and $\lambda_+=\min_{{{\mathbf p}}\in{{\mathbf T}}^d}\lambda^+({{\mathbf p}})$, then $\lambda^+({{\mathbf p}})>\lambda_+$ for any ${{\mathbf p}}\notin S_+$. Hence there exists $\delta>0$ such that for any $\lambda\in((\lambda_++\lambda_-)/2,\;\lambda_+)$ and ${{\mathbf p}}\in{{\mathbf T}}^d\setminus S_+ $ $dist(\lambda,\sigma(\tilde
H({{\mathbf p}})))\ge\delta$, and hence $\|R_\lambda(\tilde
H({{\mathbf p}}))\|_0\le\frac{1}{\delta}$. Denote by $|\|\cdot\||$ the norm of elements in $L_2({{\mathbf R}}^d/\Gamma)\otimes L_2({{\mathbf T}}^d)$. Then using the isometry of the operator $\tilde U$, we get from (\[dfThtpl1\]): $$\begin{aligned}
&&\|\Theta_\lambda^+f\|^2=|\|R_\lambda(\tilde
H({{\mathbf p}}))\chi_{_{{{\mathbf T}}^d\setminus S_+}}({{\mathbf p}})\hat
f(\cdot,\,{{\mathbf p}})\||^2=\\
&&\int_{{{\mathbf T}}^d\setminus S_+}\|R_\lambda(\tilde
H({{\mathbf p}}))\hat
f(\cdot,\,{{\mathbf p}})\|_2^2\,d{{\mathbf p}}\le\\
&&\frac{1}{\delta^2}\int_{{{\mathbf T}}^d}\|\hat
f(\cdot,\,{{\mathbf p}})\|_2^2\,d{{\mathbf p}}=\frac{1}{\delta^2}|\|\tilde U
f\||^2=\frac{1}{\delta^2}\|f\|^2.\end{aligned}$$ Recall that we denote by $\Vert\cdot\Vert_2$ and $(\cdot,\cdot)_2$ the norm and the inner product in the space ${{\mathcal H}}_0=L_2({{\mathbf R}}^d/\Gamma)$ (see (\[dfinnprd\])). Thus, we get that $$\label{estThtpl1}
\forall\,\lambda\in((\lambda_++\lambda_-)/2,\;\lambda_+):\quad\|\Theta_\lambda^+\|\le\frac{1}{\delta}.$$
Now consider the case where ${{\mathbf p}}\in S_+$. As it is known, the resolvent $R_\lambda(\tilde H({{\mathbf p}})$ can be represented in the form: $$\label{rprRlmbd1}
R_\lambda(\tilde H({{\mathbf p}}))g=
\frac{\int_\Omega\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\tilde
f({{\mathbf s}},{{\mathbf p}})\,d{{\mathbf s}}}{\lambda^+({{\mathbf p}})-\lambda}+ \tilde R(\lambda,{{\mathbf p}})g\quad
(g\in L_2({{\mathbf R}}^d/\Gamma)),$$ where $$\label{dfRk1}
\tilde
R(\lambda,{{\mathbf p}})=\sum_{l\in{{\mathbf N}}\setminus\{j+1\}}\frac{(\,\cdot\;,\,e_l(\cdot,\,{{\mathbf p}}))_2
e_l(\cdot,\,{{\mathbf p}})}{\lambda_l({{\mathbf p}})-\lambda}.$$ Recall that $\{e_l({{\mathbf x}},\,{{\mathbf p}})\}$ is the orthonormal sequence of the eigenfunctions of the operator $\tilde H({{\mathbf p}})$. Then by (\[dfRpl1\]), (\[rprRlmbd1\]), $R_\lambda^+=K(\lambda)+\tilde\Theta_\lambda^+$, where $$\begin{aligned}
\label{defKlmbd1}
&&(K(\lambda)f)({{\mathbf x}})=\\
&&\frac{1}{(2\pi)^{\frac{d}{2}}}\sum_{k=1}^{n_+}\int_{U_k}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})
\frac{\int_\Omega\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\tilde
f({{\mathbf s}},{{\mathbf p}})\,d{{\mathbf s}}}{\lambda^+({{\mathbf p}})-\lambda}\,d{{\mathbf p}}\nonumber,\end{aligned}$$ $$\begin{aligned}
&&(\tilde\Theta_\lambda^+f)({{\mathbf x}})=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{S_+}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})
\tilde R(\lambda,{{\mathbf p}})\tilde f(\cdot,\,{{\mathbf p}})\,d{{\mathbf p}}=\nonumber\\
&&(\tilde U^{-1}\tilde R(\lambda,{{\mathbf p}})\chi_{_{S_+}}({{\mathbf p}})\hat
f(\cdot,\,{{\mathbf p}}))({{\mathbf x}})\end{aligned}$$ In view of (\[neib1\]), there exists $\delta_1>0$ such that for any ${{\mathbf p}}\in S_+$ and $\lambda\in
((\lambda_++\lambda_-)/2,\;\lambda_+)$: $dist(\lambda,\;\sigma(\tilde
H({{\mathbf p}}))\setminus\{\lambda_{j+1}({{\mathbf p}})\}>\delta_1$, hence in view of (\[dfRk1\]), $\|\tilde R(\lambda,{{\mathbf p}})\|_0\le\frac{1}{\delta_1}$. Then in the same manner as estimate (\[estThtpl1\]), we obtain the following estimate: $$\label{esttlThtpl1}
\forall\,\lambda\in((\lambda_++\lambda_-)/2,\;\lambda_+):\quad\|\tilde\Theta_\lambda^+\|\le\frac{1}{\delta_1}.$$ Taking $f\in L_{2,0}({{\mathbf R}}^d)$ and using the $\Gamma$-periodicity of $\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ by ${{\mathbf s}}$ and the fact that $\tilde
f({{\mathbf x}},{{\mathbf p}})=\exp(-i{{\mathbf p}}\cdot{{\mathbf x}})\hat f({{\mathbf x}},{{\mathbf p}})$ (with $\hat f({{\mathbf x}},{{\mathbf p}})$ defined by (\[unit\])), we obtain from (\[defKlmbd1\]): $$\begin{aligned}
&&\hskip-8mm
(K(\lambda)f)({{\mathbf x}})=\frac{1}{(2\pi)^d}\sum_{k=1}^{n_+}\int_{U_k}\exp(i{{\mathbf p}}\cdot{{\mathbf x}})\times
\nonumber\\
&&\hskip-8mm\left(\frac{1}{\lambda^+({{\mathbf p}})-\lambda}
\sum_{{{\mathbf l}}\in\Gamma}\int_{\Omega
-\{{{\mathbf l}}\}}\exp(-i{{\mathbf p}}\cdot{{\mathbf s}})\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})f({{\mathbf s}})\,d{{\mathbf s}}\right)\,d{{\mathbf p}},\end{aligned}$$ Since the inner sum is finite, we obtain after a permutation of sums and integrals: $(K(\lambda)f)({{\mathbf x}})=\int_{{{\mathbf R}}^d}K({{\mathbf x}},{{\mathbf s}},\lambda)f({{\mathbf s}})\,d{{\mathbf s}}$, where $K({{\mathbf x}},{{\mathbf s}},\lambda)=\sum_{k=1}^{n_+}K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$, $$\label{Kkxs1}
K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)=
\frac{1}{(2\pi)^d}\int_{U_k}\exp(i{{\mathbf p}}\cdot({{\mathbf x}}-{{\mathbf s}}))
\frac{\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})}{\lambda^+({{\mathbf p}})-\lambda} \,d{{\mathbf p}}.$$ We see that each $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$ is continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_-,\lambda_+)$. From (\[res1\]) and (\[dfRpl1\]) we obtain the representation (\[represol1\]) with $\Theta(\lambda)=\Theta_\lambda^++\tilde \Theta_\lambda^+$. In view of (\[estThtpl1\]) and (\[esttlThtpl1\]), $\Theta(\lambda)$ satisfies condition (\[bndtht2\]). Claim (i) of Proposition \[prrprres1\] is proven.
Proof of claim (ii) of Proposition \[prrprres1\] {#sec:proofcliipr1}
------------------------------------------------
For each $k\in\{1,2,\dots,n_+\}$ consider a decomposition of the unit $$\{\phi_{k,l}({{\mathbf p}})\}_{l=1}^{L_k},\quad (\phi_{k,l}\in C^\infty(U_k)),$$ corresponding to the covering $\{U_{k,l}\}_{l=1}^{L_k}$ of $U_k$, constructed above. Then we have the representation for the function $K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[Kkxs1\]): $$\label{rprKkxs1}
K_k^+({{\mathbf x}},{{\mathbf s}},\lambda)=\sum_{l=1}^{L_k}K_{k,l}({{\mathbf x}},{{\mathbf s}},\lambda),$$ where $$\begin{aligned}
&&K_{k,l}({{\mathbf x}},{{\mathbf s}},\lambda)=\\
&&\frac{1}{(2\pi)^d}\int_{U_{k,l}}\phi_{k,l}({{\mathbf p}})\exp(i{{\mathbf p}}\cdot({{\mathbf x}}-{{\mathbf s}}))\frac{\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})}
{\lambda_{j+1}({{\mathbf p}})-\lambda} \,d{{\mathbf p}}.\end{aligned}$$ Let $\Phi_{k,l}$ be the mapping $\Phi$ defined in Lemma \[lmredchrt\] and corresponding to the neighborhood $U_{k,l}$. Like in Lemma \[lmdifeq\], consider the differential equation in $U_{k,l}$: $$\label{difeq}
\frac{d{{\mathbf p}}}{dt}=\Big(\Phi_{k,l}^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\Big)^{-1}y.$$ Let ${{\mathbf p}}_{k,l}(t,{{\mathbf p}}_0,y)$ be the flow of this equation. Taking into account definition of the reducing chart and using claim (i) of Lemma \[lmdifeq\] and the coarea formula, we have: $$\begin{aligned}
\label{coarea}
&&\hskip-16mmK_{k,l}({{\mathbf x}},{{\mathbf s}},\lambda)=
\frac{1}{(2\pi)^d}\int_{B_{k,l}}\frac{d{{\mathbf y}}}{|{{\mathbf y}}|^2+\lambda_+-\lambda}
\int_{F_{{{\mathbf y}}}}\phi_{k,l}({{\mathbf p}})\exp(i{{\mathbf p}}\cdot({{\mathbf x}}-{{\mathbf s}}))\times\nonumber\\
&&\hskip-16mm\frac{\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})}{NJ\Phi_{k,l}({{\mathbf p}})}\,dF_{{{\mathbf y}}}({{\mathbf p}}),\end{aligned}$$ where $B_{k,l}=\{{{\mathbf y}}\in{{\mathbf R}}^{d-d_k^+}\,\vert\,|{{\mathbf y}}|\le
r_{k,l}\}\;(r_{k,l}>0)$, $dF_{{{\mathbf y}}}({{\mathbf p}})(\cdot)$ is the volume form of the submanifold $F_{{{\mathbf y}}}=\Phi_{k,l}^{-1}({{\mathbf y}})$ (${{\mathbf y}}\in B_{k,l}$), and $NJ\Phi_{k,l}({{\mathbf p}})$ is the normal Jacobian of $\Phi_{k,l}$. Since by claim (i) of Lemma \[lmdifeq\] for any $t\in [0,r_{k,l}]$ and ${{\mathbf y}}\in S^{d-d_k^+-1}$ the mapping ${{\mathbf p}}_{k,l}(t,\cdot,{{\mathbf y}})$ is a diffeomorphism between $U_{k,l}^0=U_{k,l}\cap F_k^+$ and $F_{t{{\mathbf y}}}$, then (\[coarea\]) can be written in the form after the change of the variable ${{\mathbf p}}={{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\;({{\mathbf q}}\in U_{k,l}^0)$: $$\label{frmKklxs}
K_{k,l}({{\mathbf x}},{{\mathbf s}},\lambda)=\frac{1}{(2\pi)^d}\int_{U_{k,l}^0}E_{k,l}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\,dF_0({{\mathbf q}}),$$ where $$\begin{aligned}
\label{dfEkl}
&&\hskip-8mmE_{k,l}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\\
&&\hskip-8mm\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}
\phi_{k,l}({{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}))\exp(i{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))\times\nonumber\\
&&\hskip-8mm\frac{\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}))
TJ{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})}{NJ\Phi_{k,l}({{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})))}\,dt,\nonumber\end{aligned}$$ $dS(\cdot)$ is the volume form of the unit sphere $S^{d-d_k^+-1}$, $dF_0(\cdot)$ is the volume form of the submanifold $U_{k,l}^0$ and $TJ{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})$ is the tangential Jacobian of the mapping ${{\mathbf p}}_{k,l}(t,\cdot,{{\mathbf y}}):\;U_{k,l}^0\rightarrow F_{t{{\mathbf y}}}$. Since for $d-d_k^+\ge 3$ $\;\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}\,dt}
{t^2+\lambda_+-\lambda}\le\frac{r_{k,l}^{d-d_k^+-2}}{d-d_k^+-2}<\infty$ for any $\lambda\in(\lambda_-,\lambda_+)$, then by (\[dfEkl\], (\[frmKklxs\]) and (\[rprKkxs1\]) the property (\[cndK1\]) is valid. Hence claim (ii) of Proposition \[prrprres1\] is proven.
Proof of claim (iii) of Proposition \[prrprres1\] {#sec:proofcliiipr1}
-------------------------------------------------
Assume that $1\le d-d_k^+\le 2$. Taking into account the connection (\[connecteigkern\]) between the eigenkernels ${{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ and $\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, let us represent the function $E_{k,l}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[dfEkl\]), in the form: $$\label{rprEkl}
E_{k,l}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=E_{k,l}^{(1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)+E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)+
E_{k,l}^{(3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda),$$ where $$\begin{aligned}
\label{dfEkl1}
&&E_{k,l}^{(1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=
\phi_{k,l}({{\mathbf q}})2^{(d-d_k^+)/2}\,{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf q}})\times\\
&&\sqrt{m^+({{\mathbf q}})}\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}\,dt}
{t^2+\lambda_+-\lambda},\nonumber\end{aligned}$$ $$\begin{aligned}
\label{dfEkl2}
E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=
\phi_{k,l}({{\mathbf q}})2^{(d-d_k^+)/2}\,{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})
\sqrt{m^+({{\mathbf q}})}\tilde E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda),\end{aligned}$$ $$\begin{aligned}
\label{dftlEkl2}
&&\tilde E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=
\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\times\\
&&\big(\exp(i({{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})-{{\mathbf q}})\cdot(\tilde{{\mathbf x}}-\tilde{{\mathbf s}}))-1\big)\,dt,\nonumber\end{aligned}$$ $$\begin{aligned}
\label{dfEkl3}
&&\hskip-8mmE_{k,l}^{(3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\times\\
&&\hskip-8mm\exp(i{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))
D_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})\,dt,\nonumber\end{aligned}$$ $$\begin{aligned}
\label{dfDkl}
&&\hskip-8mmD_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})=
\phi_{k,l}({{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}))
\frac{\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}))
TJ{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})}{NJ\Phi_{k,l}({{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}))}-
\nonumber\\
&&\hskip-8mm \phi_{k,l}({{\mathbf q}})
\frac{2^{(d-d_k^+)/2}\,\tilde{{\mathcal Q}}^+({{\mathbf x}},{{\mathbf s}},{{\mathbf q}})}
{\sqrt{NJ\,Hes_{{{\mathbf q}}}(\lambda_{j+1})}}.\end{aligned}$$ After simple calculations we have: $$\begin{aligned}
\label{asestint}
\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}\,dt}
{t^2+\lambda_+-\lambda}=\left\{\begin{array}{ll}
\frac{\pi}{2\sqrt{\lambda_+-\lambda}}+\theta_1(\lambda)&\rm{for}\quad d-d_k^+=1,\\
\frac{1}{2}\ln\Big(\frac{1}{\lambda_+-\lambda}\Big)+\theta_2(\lambda)&\rm{for}\quad d-d_k^+=2,
\end{array}\right.\end{aligned}$$ where $$\label{esttht}
\theta_\nu(\lambda)=O(1)\quad\rm{for}\quad\lambda\uparrow\lambda_+\quad
(\nu=1,2).$$ Then taking into account that $\{\phi_{k,l}\vert_{F_k^+}\}_{l=1}^{L_k}$ is a decomposition of the unit for the submanifold $F_k^+$, we get from (\[dfEkl1\]), (\[asestint\]) and (\[esttht\]): $$\label{rprKk1xslm}
\sum_{l=1}^{L_k}\int_{U_{k,l}^0}E_{k,l}^{(1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\,dF_0({{\mathbf q}})=
G_k^+({{\mathbf x}},{{\mathbf s}},\lambda)+\tilde K_k^{(1)}({{\mathbf x}},{{\mathbf s}},\lambda),$$ where $G_k^+({{\mathbf x}},{{\mathbf s}},\lambda)$ is defined by (\[dfFpl1\]) for $d-d_k^+=1$ and by (\[dfFpl2\]) for $d-d_k^+=2$, and the function $\tilde K_k^{(1)}({{\mathbf x}},{{\mathbf s}},\lambda)$ is continuous in ${{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\lambda_+)$ and $$\label{bundK1kxs}
\sup_{({{\mathbf x}},{{\mathbf s}},\lambda)\in{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}|\tilde
K_k^{(1)}({{\mathbf x}},{{\mathbf s}},\lambda)|<\infty.$$ Recall that $\delta\in(0,\,\lambda_+-\lambda_-)$.
Let us estimate the function $E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[dfEkl2\]). To this end consider the Taylor representation of the flow of the equation (\[difeq\]) in a neighborhood of the point $t=0$: $$\begin{aligned}
\label{Tayrprp}
&&\hskip-10mm{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})-{{\mathbf q}}={{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})-{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})=\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})t+\\
&&\hskip-10mm{{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}}),\nonumber\end{aligned}$$ where ${{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})=\int_0^1(t-s)\partial^2_t{{\mathbf p}}_{k,l}(s,{{\mathbf q}},{{\mathbf y}})\,ds$. We have from (\[difeq\]): $$\partial^2_t{{\mathbf p}}_{k,l}(s,{{\mathbf q}},{{\mathbf y}})=\partial_{{\mathbf p}}\Big(\Big(\Phi_{k,l}^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\Big)^{-1}y\Big)
\cdot(\Big(\Phi_{k,l}^\prime({{\mathbf p}})\vert_{N_{{\mathbf p}}}\Big)^{-1}y\vert_{{{\mathbf p}}={{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})}.$$ These circumstances imply that $$\label{bundrtqy}
\bar{{\mathbf r}}_{k,l}=\sup_{(t,{{\mathbf q}},{{\mathbf y}})\in[0,r_{k,l}]\times U_{k,l}\times
S^{d-d_k^+-1}(0)}\frac{|{{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})|}{t^2}<\infty.$$ Using (\[Tayrprp\]), let us represent the function $\tilde E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[dftlEkl2\]), in the form: $$\label{rprtlEkl2}
\tilde E_{k,l}^{(2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)+
\tilde E_{k,l}^{(2,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda),$$ where $$\begin{aligned}
\label{dftlEkl21}
&&\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\\
&&\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\big(\exp(i\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})t\cdot({{\mathbf x}}-{{\mathbf s}}))-1\big)\,dt,\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{dftlEkl22}
&&\tilde E_{k,l}^{(2,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\\
&&\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\big(\exp(i{{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))-1\big)\times\nonumber\\
&&\exp(i\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})t\cdot({{\mathbf x}}-{{\mathbf s}}))\,dt,\nonumber\end{aligned}$$ Let us estimate $\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$. Observe that, in view of (\[difeq\]), $\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})=-\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},-{{\mathbf y}})$. Then, since the integral of the odd part (w.r. to ${{\mathbf y}}$) of the integrand in (\[dftlEkl21\]) vanishes, we have: $$\begin{aligned}
&&\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\\
&&\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\big(\cos(\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})t\cdot(\tilde{{\mathbf x}}-\tilde{{\mathbf s}}))-1\big)\,dt,\nonumber\end{aligned}$$ Deriving the change of the variable $u=(1+|{{\mathbf x}}-{{\mathbf s}}|)t$ in the inner integral, we get: $$\begin{aligned}
&&\hskip-8mm\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=(1+|{{\mathbf x}}-{{\mathbf s}}|)^{2-(d-d_k^+)}\times\\
&&\hskip-8mm\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}(1+|{{\mathbf x}}-{{\mathbf s}}|)}\frac{u^{d-d_k^+-1}}
{u^2+(\lambda_+-\lambda)(1+|{{\mathbf x}}-{{\mathbf s}}|)^2}\times\nonumber\\
&&\hskip-8mm\big(\cos\big(\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})u\cdot\frac{\tilde{{\mathbf x}}-\tilde{{\mathbf s}}}{1+|{{\mathbf x}}-{{\mathbf s}}|}\big)-1\big)\,du.\end{aligned}$$ Using estimate (\[estint2\]) of Lemma \[lmestint\] with $\rho=r_{k,l}$, $\alpha=1+|{{\mathbf x}}-{{\mathbf s}}|$ and $\beta=\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})\cdot\frac{\tilde{{\mathbf x}}-\tilde{{\mathbf s}}}{1+|{{\mathbf x}}-{{\mathbf s}}|}$, we get: $$\begin{aligned}
\label{summary1}
&&\rm{for}\quad d-d_k^+=1:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,
U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big|}
{1+|{{\mathbf x}}-{{\mathbf s}}|}<\infty\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{summary2}
&&\rm{for}\quad d-d_k^+=2:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,
U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|\tilde E_{k,l}^{(2,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big|}
{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty.\nonumber\end{aligned}$$ Now let us estimate the function $\tilde E_{k,l}^{(2,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[dftlEkl22\]). To this end let us derive the change of the variable $u=\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}t$ in the inner integral of (\[dftlEkl22\]): $$\begin{aligned}
&&\hskip-16mm\Big|\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\big(\exp(i{{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))-1\big)
\times\\
&&\hskip-16mm\exp(i\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})t\cdot({{\mathbf x}}-{{\mathbf s}}))\,dt\Big|\le\\
&&\hskip-16mm(1+|{{\mathbf x}}-{{\mathbf s}}|)^{\frac{2-(d-d_k^+)}{2}}\int_0^{r_{k,l}\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}}\frac{u^{d-d_k^+-1}}
{u^2+(\lambda_+-\lambda)(1+|{{\mathbf x}}-{{\mathbf s}}|)}\times\nonumber\\
&&\hskip-16mm\Big|\exp\Big(i{{\mathbf r}}_{k,l}\Big(\frac{u}{\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}},{{\mathbf q}},{{\mathbf y}}\Big)\cdot({{\mathbf x}}-{{\mathbf s}})\Big)-1\Big|\,du\end{aligned}$$ Then using (\[bundrtqy\]) and estimate (\[estint1\]) of Lemma \[lmestint\] with $\rho=r_{k,l}$, $\alpha=\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}$ and $\phi(u)={{\mathbf r}}_{k,l}\Big(\frac{u}{\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}},{{\mathbf q}},{{\mathbf y}}\Big)\cdot(\tilde{{\mathbf x}}-\tilde{{\mathbf s}})$, we obtain: $$\begin{aligned}
\label{summary3}
&&\rm{for}\quad d-d_k^+=1:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|\tilde E_{k,l}^{(2,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big|}
{\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}}<\infty\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{summary3a}
&&\rm{for}\quad d-d_k^+=2:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|\tilde E_{k,l}^{(2,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big|}
{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty.\nonumber\end{aligned}$$
Let us estimate the function $E_{k,l}^{(3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$, defined by (\[dfEkl3\]). To this end consider the Taylor representation of the function $D_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$, defined by (\[dfDkl\]) in a neighborhood of $t=0$, taking into account that in view of Lemma \[lmredchrt\] and the equalities ${{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})={{\mathbf q}}$ and $TJ({{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}}))=1$, the equality $D_{k,l}(0,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})=0$ is valid. We have: $D_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})=g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})t+\zeta_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$, where $g({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})=\partial_tD_{k,l}(0,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$. We see from (\[dfDkl\]) that the functions $g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$ and $\zeta_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$ are smooth and bounded in $[0,r_{k,l}]\times U_{k,l}\times S^{d-d_k^+-1}\times{{\mathbf R}}^d\times{{\mathbf R}}^d$. In the same manner as (\[bundrtqy\]), we obtain the following property of $\zeta_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$: $$\label{bundzettqy}
\bar\zeta_{k,l}=\sup_{(t,{{\mathbf q}},{{\mathbf y}},{{\mathbf x}},{{\mathbf s}})\in[0,r_{k,l}]\times
U_{k,l}\times S^{d-d_k^+-1}\times{{\mathbf R}}^d\times{{\mathbf R}}^d}
\frac{|\zeta_{k,l}(t,{{\mathbf q}},{{\mathbf y}},{{\mathbf x}},{{\mathbf s}})|}{t^2}<\infty.$$ Using the above representation and (\[Tayrprp\]), let us represent the function $E_{k,l}^{(3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$ in the form: $$\begin{aligned}
\label{rprEkl3}
&&E_{k,l}^{(3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=E_{k,l}^{(3,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)+\exp\big(i{{\mathbf q}}\cdot({{\mathbf x}}-{{\mathbf s}})\big)
\times\nonumber\\
&&\big(E_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)+E_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big),\end{aligned}$$ where $$\begin{aligned}
\label{dfEkl31}
&&\hskip-8mmE_{k,l}^{(3,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+-1}}
{t^2+\lambda_+-\lambda}\times\\
&&\hskip-8mm\exp(i{{\mathbf p}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))
\zeta_{k,l}(t,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})\,dt,\nonumber\end{aligned}$$ $$\begin{aligned}
\label{dfEkl32}
&&\hskip-8mmE_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+}}
{t^2+\lambda_+-\lambda}\times\\
&&\hskip-8mm\exp(it\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))
g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})\,dt,\nonumber\end{aligned}$$ $$\begin{aligned}
\label{dfEkl33}
&&\hskip-8mmE_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=\int_{S^{d-d_k^+-1}}\,dS({{\mathbf y}})\int_0^{r_{k,l}}\frac{t^{d-d_k^+}}
{t^2+\lambda_+-\lambda}\times\\
&&\hskip-8mm\exp(it\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))
g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})\nonumber\times\\
&&\hskip-8mm\big(\exp(i{{\mathbf r}}_{k,l}(t,{{\mathbf q}},{{\mathbf y}})\cdot({{\mathbf x}}-{{\mathbf s}}))-1\big)\,dt.
\nonumber\end{aligned}$$ Using (\[bundzettqy\]), we obtain from (\[dfEkl31\]): $$\begin{aligned}
\label{summary3}
&&\rm{for}\quad d-d_k^+\in\{1,2\}:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\big| E_{k,l}^{(3,1)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big| <\infty.\nonumber\end{aligned}$$ Now we turn to the estimation of the function $E_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$. Deriving the change of the variable $u=(1+|{{\mathbf x}}-{{\mathbf s}}|)t$ in the inner integral of (\[dfEkl32\]) and taking into account that by claim (ii) of Lemma \[lmdifeq\] the functions $\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})$ and $g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})=\partial_tD_{k,l}(0,{{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})$ are odd w.r.t. ${{\mathbf y}}$, we get: $$\begin{aligned}
&&\hskip-8mmE_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)=(1+|{{\mathbf x}}-{{\mathbf s}}|)^{1-(d-d_k^+)}\int_{S^{d-d_k^+-1}}
g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})\,dS({{\mathbf y}})\times\\
&&\hskip-8mm\int_0^{r_{k,l}(1+|{{\mathbf x}}-{{\mathbf s}}|)}
\frac{u^{d-d_k^+}\sin\Big(u\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})\cdot\frac{\tilde{{\mathbf x}}-\tilde{{\mathbf s}}}
{1+|{{\mathbf x}}-{{\mathbf s}}|}\Big)}{u^2+(\lambda_+-\lambda)(1+|{{\mathbf x}}-{{\mathbf s}}|)^2}\,du\end{aligned}$$ Using estimate (\[estint3\]) of Lemma \[lmestint\] with $\rho=r_{k,l}$, $\alpha=1+|{{\mathbf x}}-{{\mathbf s}}|$ and $\beta=\partial_t{{\mathbf p}}_{k,l}(0,{{\mathbf q}},{{\mathbf y}})\cdot\frac{\tilde{{\mathbf x}}-\tilde{{\mathbf s}}}{1+|{{\mathbf x}}-{{\mathbf s}}|}$, we get: $$\begin{aligned}
\label{summary4}
&&\rm{for}\quad d-d_k^+=1:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|E_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)
\big|}{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{summary5}
&&\rm{for}\quad d-d_k^+=2:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\big|E_{k,l}^{(3,2)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\big|<\infty\nonumber\end{aligned}$$
Now we turn to the estimation of the function $E_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)$. Deriving the change of the variable $u=\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}t$ in the inner integral of (\[dfEkl33\]), we get: $$\begin{aligned}
&&\hskip-12mm|E_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)|\le(1+|{{\mathbf x}}-{{\mathbf s}}|)^{\frac{1-(d-d_k^+)}{2}}\int_{S^{d-d_k^+-1}}
|g_{k,l}({{\mathbf y}},{{\mathbf q}},{{\mathbf x}},{{\mathbf s}})|\,dS({{\mathbf y}})\times\\
&&\hskip-12mm\int_0^{r_{k,l}\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}}\frac{u^{d-d_k^+}
\Big|\exp\Big(i{{\mathbf r}}_{k,l}\big(\frac{u}{\sqrt{1+|{{\mathbf x}}-{{\mathbf s}}|}},{{\mathbf q}},{{\mathbf y}}\big)\cdot({{\mathbf x}}-{{\mathbf s}})\Big)-1\Big|}
{u^2+(\lambda_+-\lambda)(1+|{{\mathbf x}}-{{\mathbf s}}|)}\,du\end{aligned}$$ Using, as above, estimate (\[estint1\]) of Lemma \[lmestint\], we obtain: $$\begin{aligned}
\label{summary6}
&&\rm{for}\quad d-d_k^+=1:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\frac{\big|E_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)
\big|}{1+\ln(1+|{{\mathbf x}}-{{\mathbf s}}|)}<\infty\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{summary7}
&&\rm{for}\quad d-d_k^+=2:\\
&&\sup_{({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda)\in\,U_{k,l}^0\times{{\mathbf R}}^d\times{{\mathbf R}}^d\times(\lambda_+-\delta,\;\lambda_+)}
\big|E_{k,l}^{(3,3)}({{\mathbf q}},{{\mathbf x}},{{\mathbf s}},\lambda) \big|<\infty.\nonumber\end{aligned}$$ The representations (\[rprKkxs1\]), (\[frmKklxs\]), (\[rprEkl\]), (\[rprKk1xslm\]), (\[rprtlEkl2\]), (\[rprEkl3\]) and the properties (\[bundK1kxs\]), (\[summary1\]), (\[summary2\]), (\[summary3a\]) (\[summary3\]), (\[summary4\]), (\[summary5\]), (\[summary6\]) and (\[summary7\]) imply claim (iii) of Proposition \[prrprres1\].
Proof of Proposition \[prrprres\] {#sec:proofpr}
---------------------------------
The proof is the same as the proof of Proposition \[prrprres1\], only we should take into account that since in our case each of $F_k^+$ is a singleton $\{{{\mathbf p}}_k^+\}$, then the set $U_{k,l}^0=U_{k,l}\cap F_k^+$ is or this singleton, or it is empty. Hence in the first case in the r.h.s. of (\[frmKklxs\]) it will be the value of the integrand at ${{\mathbf p}}={{\mathbf p}}_k^+$ instead of the integral along $U_{k,l}^0$. Therefore in the r.h.s. of (\[dfFpl1\]) and (\[dfFpl2\]) it will be the value of the integrand at ${{\mathbf p}}={{\mathbf p}}_k^+$ instead of the integral along $F_k^+$.
**Appendix : $C(\Omega)$-holomorphy of Bloch functions** {#sec:appendix}
========================================================
Main claims {#subsec:mainclaim}
-----------
In this Appendix we prove that under some assumptions the Bloch functions can be chosen to be holomorphic w.r.t. the quasi-momentum in the $C(\Omega)$-norm.
In this section we shall denote by $\Vert\cdot\Vert_q$ and $\Vert\cdot\Vert_{p,\,l}$ the norms in the spaces $L_q(\Omega)$ and $W_p^l(\Omega)$ respectively. Observe that for $q=2$ the notation $\Vert\cdot\Vert_q$ is compatible with the notation given by (\[dfinnprd\]).
The main results of this section are following:
\[thmainApp\] Assume that the periodic potential $V({{\mathbf x}})$ satisfies the condition (\[condperpotent\]) Then
\(i) if $\lambda({{\mathbf p}})$ is an eigenvalue of the operator $H({{\mathbf p}})$, then any eigenfunction $b({{\mathbf x}},{{\mathbf p}})$ of $H({{\mathbf p}})$ corresponding to $\lambda({{\mathbf p}})$ belongs to $C(\Omega)$;
\(ii) if the family of operators $H({{\mathbf p}})$ has a holomorphic branch of eigenvalues $\lambda({{\mathbf p}})$ in a connected neighborhood ${{\mathcal O}}({{\mathbf p}}_0)\subset{{\mathbf C}}^d$ of a point ${{\mathbf p}}_0\in{{\mathbf R}}^d$, and for each ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ it is possible to choose an eigenfunction $b({{\mathbf x}},{{\mathbf p}})\neq 0$ of $H({{\mathbf p}})$ corresponding to $\lambda({{\mathbf p}})$ such that the mapping ${{\mathbf p}}\rightarrow b({{\mathbf x}},{{\mathbf p}})\in L_2(\Omega)$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$, then the mapping ${{\mathbf p}}\rightarrow b({{\mathbf x}},{{\mathbf p}})\in
C(\Omega)$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$.
\[cormainApp\] If $V({{\mathbf x}})$ is as in Theorem \[thmainApp\] and for some ${{\mathbf p}}_0\in{{\mathbf R}}^d$ an eigenvalue $\lambda_0$ of $H({{\mathbf p}}_0)$ is simple, then
\(i) for some neighborhood ${{\mathcal O}}({{\mathbf p}}_0)\subset{{\mathbf C}}^d$ of ${{\mathbf p}}_0$ there exists a branch $\lambda({{\mathbf p}})$ of eigenvalues of the family $H({{\mathbf p}})\;({{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0))$ such that $\lambda({{\mathbf p}}_0)=\lambda_0$, $\lambda({{\mathbf p}})$ is simple for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$, the function $\lambda({{\mathbf p}})$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$ and for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ it is possible to choose an eigenfunction $b({{\mathbf x}},{{\mathbf p}})$ of $H({{\mathbf p}})$ corresponding to $\lambda({{\mathbf p}})$ such that $\Vert b({{\mathbf x}},{{\mathbf p}})\Vert_2=1$, the function $b({{\mathbf x}},{{\mathbf p}})$ is continuous in $\Omega$, and the mapping ${{\mathbf p}}\rightarrow b(\cdot,{{\mathbf p}})\in C(\Omega)$ is real-analytic in ${{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$; vskip2mm
\(ii) for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$ the eigenkernel, corresponding to $\lambda({{\mathbf p}})$, has the form ${{\mathcal Q}}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=b({{\mathbf x}},{{\mathbf p}})\overline{b({{\mathbf s}},{{\mathbf p}})}$ and the mapping ${{\mathbf p}}\rightarrow{{\mathcal Q}}(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ is real analytic in ${{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$; vskip2mm
\(iii) for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$ the eigenkernel ${{\mathcal Q}}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ does not depend on the choice of the branch $b({{\mathbf x}},{{\mathbf p}})$ of eigenfunctions of the family $H({{\mathbf p}})\;({{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0))\cap{{\mathbf R}}^d$ satisfying the conditions imposed in claim (i).
Notice that for $d=3$ the claim of Theorem \[thmainApp\] follows from the results of the paper [@Wil]. It have been shown there ([@Wil], Lemmas 3.7, 3.8) that the square $R^2({{\mathbf p}})$ of the resolvent $R({{\mathbf p}})$ of the operator $H({{\mathbf p}})$ is an integral operator with an integral kernel $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, such that for some $\gamma>0$ for any fixed ${{\mathbf p}}$ with $|\Im({{\mathbf p}})|\le\gamma$ $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\in C(\Omega\times\Omega)$ and the mapping ${{\mathbf p}}\rightarrow K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\in C(\Omega\times\Omega)$ is holomorphic for $|\Im({{\mathbf p}})|\le\gamma$. This fact implies easily the claim of Theorem \[thmainApp\]. Observe that the arguments of [@Wil] are true also in the case where $d<3$. But in the case $d\ge 4$ the integral kernel of the resolvent $R({{\mathbf p}})$ has a stronger singularity at its diagonal, hence we need to deal with a higher power $R^l({{\mathbf p}})$ of it in order to get an integral kernel having the property mentioned above.
Domains and self-adjointness of the operators $H({{\mathbf p}})$ and $H_0$ {#subsec:mainclaim}
--------------------------------------------------------------------------
In this section we generalize to the case of an arbitrary dimension $d$ the results on domains and self-adjointness of the operators $H({{\mathbf p}})$ and $H_0$, obtained in [@Wil] for $d=3$ (Lemmas 1.2 and 1.4). The arguments used in [@Wil] are true also for $d<3$. These arguments are based on the fact that for $d\le 3$ the continuous embedding $W_2^2(\Omega)\hookrightarrow C(\Omega)$ holds. For $d\ge 4$ this embedding is not true, but we use in this case the Sobolev’s theorem on embedding of $W_p^2(\Omega)$ into $L_q(\Omega)$. First of all, let us prove the lemma, whose first claim is an analog of Lemma 1.3 from [@Wil]:
\[lmestVuOm\] (i) If $d\ge 4$ and $V\in L_s(\Omega)$ with $s>\frac{d}{2}$, then there exists $C>0$ such that for any $u\in W_2^2(\Omega)$ and any $\epsilon>0$ $$\label{estnrmVu1}
\Vert Vu\Vert_2^2\le C|\Vert V\Vert_s^2(\epsilon\Vert
u\Vert_{2,\,2}^2+\epsilon^{-\mu}\Vert u\Vert_2^2),$$ where $$\label{dfmu}
\mu=\mu(s)=\frac{(p(s))^{-1}-(q(s))^{-1}}{(q(s))^{-1}-(q(\tilde
s))^{-1}},$$ $$\label{dfqs}
q(s)=\frac{2s}{s-2},$$ $$\label{dfps}
p(s)=\frac{2ds}{(d+4)s-2d}$$ and $$\label{dftls}
\tilde s=\frac{1}{2}\Big(\frac{d}{2}+s\Big);$$
\(ii) If $d\ge 2$ and $V\in L_s(\Omega)$ with $s>\frac{d}{2}$, then there exists $\tilde C>0$ such that for any $u\in W_2^1(\Omega)$ and any $\epsilon>0$ $\Big|\int_\Omega V({{\mathbf x}})|u({{\mathbf x}})|^2\,d{{\mathbf x}}|\Big|\le
\tilde C|\Vert V\Vert_s(\epsilon\Vert
u\Vert_{2,\,1}^2+\epsilon^{-\tilde\mu}\Vert u\Vert_2^2)$, where $$\label{dftlmu}
\tilde\mu=\tilde\mu(s)=\frac{(\tilde p(s))^{-1}-(\tilde
q(s))^{-1}}{(\tilde q(s))^{-1}-(\tilde q(\tilde s))^{-1}},\quad
\tilde p(s)=\frac{2ds}{(d+2)s-d},$$ $$\label{dftlqs}
\tilde q(s)=\frac{2s}{s-1}$$ and $\tilde s$ is defined by (\[dftls\]);
\(iii) If $d=1$ and $V\in L_1(\Omega)$, then for any $u\in
W_2^1(\Omega)$ and any $\epsilon>0$ $\Big|\int_\Omega
V({{\mathbf x}})|u({{\mathbf x}})|^2\,d{{\mathbf x}}|\Big|\le|\Vert V\Vert_1(\epsilon\Vert
u\Vert_{2,\,1}^2+(\epsilon^{-1}+T^{-1})\Vert u\Vert_2^2)$, where $T=lenth(\Omega)$.
\(i) Using Hölder’s inequality, we have for any $u\in
W_2^2(\Omega)$: $$\begin{aligned}
\label{Holdest}
&&\Vert Vu\Vert_2^2=\int_\Omega (V({{\mathbf x}}))^2|u({{\mathbf x}})|^2\,d{{\mathbf x}}\le\\
&&\Big(\int_\Omega(V({{\mathbf x}}))^s\,d{{\mathbf x}}\Big)^{\frac{2}{s}}\Big(\int_\Omega|u({{\mathbf x}})|^{q(s)}\,d{{\mathbf x}}\Big)^{\frac{2}{q(s)}}=
\Vert V\Vert_s^2\,\Vert u\Vert_{q(s)}^2,\nonumber\end{aligned}$$ where $q(s)$ is defined by (\[dfqs\]). We see from the representation $q(s)=2+\frac{4}{s-2}$ that $q(s)$ is decreasing and if $s$ runs over $\Big(\frac{d}{2},\,\infty\Big)$, $q(s)$ runs over $\Big(2,\,\frac{2d}{d-4}\Big)$ for $d>4$ and over $(2,\,\infty)$ for $d=4$. By Sobolev’s embedding theorem, if $$\label{frmp}
\frac{1}{p}=\frac{1}{q(s)}+\frac{2}{d},$$ then $W_p^2(\Omega)\hookrightarrow L_{q(s)}(\Omega)$. Taking into account (\[dfqs\]), we get that the number $p=p(s)$, for which (\[frmp\]) holds, has the form (\[dfps\]). Observe that the representation $p(s)=\frac{2d}{d+4}\Big(1+\frac{2d}{(d+4)s-2d}\Big)$ implies that $p(s)$ is decreasing and when $s$ runs over $\Big(\frac{d}{2},\,\infty\Big)$, $p(s)$ runs over $\Big(\frac{2d}{d+4},\,2\Big)$. Observe that, in view of (\[dftls\]), $p(s)<p(\tilde s)<2<q(s)<q(\tilde s)$, hence the interpolation inequality $\Vert u\Vert_{q(s)}\le\Vert
u\Vert_{p(s)}^\lambda\Vert u\Vert_{q(\tilde s)}^{1-\lambda}$ is valid with $\lambda=\frac{(q(s))^{-1}-(q(\tilde
s))^{-1}}{(p(s))^{-1}-(q(\tilde s))^{-1}}$. Then the Young’s inequality implies that for any $\epsilon>0$ $$\label{interpestuq}
\Vert u\Vert_{q(s)}^2\le\epsilon \Vert u\Vert_{q(\tilde
s)}^2+\epsilon^{-\mu}\Vert u\Vert_{p(s)}^2,$$ where $\mu=\mu(s)$ is defined by (\[dfmu\]) (see [@Gil-Tr], Chapt. 7). By the Sobolev’s embedding theorem, $W_{p(\tilde
s)}^2(\Omega)\hookrightarrow L_{q(\tilde s)}(\Omega)$, hence since $W_2^2(\Omega)\hookrightarrow W_{p(\tilde s)}^2(\Omega)$, we get: $W_2^2(\Omega)\hookrightarrow L_{q(\tilde s)}(\Omega)$. This fact, the embedding $L_2(\Omega)\hookrightarrow L_{p(s)}(\Omega)$ and the inequalities (\[Holdest\]) and (\[interpestuq\]) imply that for some $C>0$ and for any $u\in W_2^2(\Omega)$, $\epsilon>0$ the desired inequality (\[estnrmVu1\]) is valid. Claim (i) is proven.
\(ii) We have for $u\in W_2^1(\Omega)$: $\Big|\int_\Omega
V({{\mathbf x}})|u({{\mathbf x}})|^2\,d{{\mathbf x}}|\Big|\le|\Vert V\Vert_s\Vert u\Vert_{\tilde
q(s)}^2$, where $\tilde q(s)$ is defined by (\[dftlqs\]). Further we continue the proof like the proof of claim (i), using the fact that $W_p^1(\Omega)\hookrightarrow L_{\tilde q(s)}(\Omega)$, if $\frac{1}{p}=\frac{1}{\tilde q(s)}+\frac{1}{d}$.
\(iii) Using the Newton-Leibnitz formula, we get easily that for any $u\in W_2^1(\Omega)$, $\epsilon>0$ and $x\in\Omega$ the inequality holds $|u(x)|^2\le \epsilon\Vert
u\Vert_{2,\,1}^2+(\epsilon^{-1}+T^{-1})\Vert u\Vert_2^2$, from which the claim follows immediately.
Using Lemma \[lmestVuOm\] and the arguments of the proof of Lemma 1.1 from [@Wil], it is not difficult to prove the following claim:
\[lmestVuRd\] (i) If $d\ge 4$ and $V\in L_s(\Omega)$ with $s>\frac{d}{2}$, then there exists $C>0$ such that for any $u\in W_2^2({{\mathbf R}}^d)$ and any $\epsilon>0$ $$\Vert Vu\Vert_{L_2({{\mathbf R}}^d)}\le C\Vert V\Vert_s(\epsilon\Vert
u\Vert_{W_2^2({{\mathbf R}}^d)}+\epsilon^{-\mu}\Vert u\Vert_{L_2({{\mathbf R}}^d)}),$$ where $\mu=\mu(s)$ is defined by (\[dfmu\])-(\[dftls\]);
\(ii) If $d\ge 2$ and $V\in L_s(\Omega)$ with $s>\frac{d}{2}$, then there exists $\tilde C>0$ such that for any $u\in W_2^1({{\mathbf R}}^d)$ and any $\epsilon>0$ $$\Big|\int_{{{\mathbf R}}^d} V({{\mathbf x}})|u({{\mathbf x}})|^2\,d{{\mathbf x}}|\Big|\le \tilde C|\Vert
V\Vert_s(\epsilon\Vert
u\Vert_{W_2^1({{\mathbf R}}^d)}^2+\epsilon^{-\tilde\mu}\Vert
u\Vert_{L_2({{\mathbf R}}^d)}^2),$$ where $\tilde\mu=\tilde\mu(s)$ is defined by (\[dftlmu\]);
\(iii) If $d=1$ and $V\in L_1(\Omega)$, then for any $u\in
W_2^1(\Omega)$ and any $\epsilon>0$ $$\Big|\int_{{{\mathbf R}}^d} V({{\mathbf x}})|u({{\mathbf x}})|^2\,d{{\mathbf x}}|\Big|\le\Vert
V\Vert_1(\epsilon\Vert
u\Vert_{W_2^1({{\mathbf R}}^d)}^2+(\epsilon^{-1}+T^{-1})\Vert
u\Vert_{L_2({{\mathbf R}}^d)}^2),$$ where $T=lenth(\Omega)$.
We now turn to the main claim of this section.
\[propselfadj\] If $V({{\mathbf x}})$ satisfies the condition (\[condperpotent\]), then
\(i) for any ${{\mathbf p}}\in {{\mathbf T}}^d$ the operator $H({{\mathbf p}})$, generated in the space ${{\mathcal H}}_{{{\mathbf p}}}$ by the operation $h=-\Delta+V({{\mathbf x}})\cdot$ and having the domain ${{\mathcal D}}_{{\mathbf p}}={{\mathcal H}}_{{{\mathbf p}}}\cap W_{2,loc}^2({{\mathbf R}}^d)$, is self-adjoint and bounded below uniformly w.r.t. ${{\mathbf p}}\in {{\mathbf T}}^d$;
\(ii) the operator $H_0$, generated in the space $L_2({{\mathbf R}}^d)$ by the operation $h=-\Delta+V({{\mathbf x}})\cdot$ and having the domain $W_2^2({{\mathbf R}}^d)$, is self-adjoint and bounded below.
Recall that ${{\mathcal H}}_{{{\mathbf p}}}$ is the Hilbert space of functions $u\in
L_{2,loc}({{\mathbf R}}^d)$ satisfying the condition (\[Htau\]) with the inner product, defined by (\[dfinnprd\]).
For $d=3$ the claims are proved in [@Wil] (Lemmas 1.2 and 1.4) and the arguments used there are true also in the case $d<3$. Let us prove claim (i) for the case $d\ge 4$. Assume that $V\in
L_s(\Omega)$ with $s>\frac{d}{2}$. Then using claim (i) of Lemma \[lmestVuOm\] and the arguments from the proof of Lemma 1.3 from [@Wil], we get that for some $\tilde C>0$ and for any $u\in{{\mathcal D}}_{{\mathbf p}}$, $\epsilon>0$ $\;\Vert Vu\Vert_2\le \tilde C|\Vert
V\Vert_s(\epsilon\Vert\Delta u\Vert_2+\epsilon^{-\mu}\Vert
u\Vert_2)$ and the operator $-\Delta$ with the domain ${{\mathcal D}}_{{\mathbf p}}$ is self-adjoint and non-negative. Hence claim (i) follows from the Kato’s theorem ([@Kat], p. 287, Theorem 4.3). In the analogous manner claim (ii) for $d\ge 4$ follows from claim (i) of Lemma \[lmestVuRd\], arguments from the proof of Lemma 1.1 of [@Wil] and the Kato’s theorem mentioned above.
In the same manner as in [@Wil] (Lemma 1.5) the following claim is proved:
\[prcompres\] If $V({{\mathbf x}})$ satisfies the condition (\[condperpotent\]), then the resolvent operator $R_\zeta(H({{\mathbf p}}))=(H({{\mathbf p}})-\zeta I)^{-1}$ is compact for every $\zeta$ in the resolvent set of $H({{\mathbf p}})$. Hence, in particular, $H({{\mathbf p}})$ has a discrete spectrum $\sigma(H({{\mathbf p}}))$ for every ${{\mathbf p}}\in{{\mathbf T}}^d$.
Fundamental solution of the Helmholz’s equation in ${{\mathbf R}}^d$ {#subsec:fundsolHelm}
--------------------------------------------------------------------
First of all, let us study the structure of the fundamental solution ${{\mathcal E}}({{\mathbf x}},\gamma_0)\;(\gamma_0>0)$ of the Helmholz equation in the space ${{\mathbf R}}^d$, that is the generalized solution of the equation $$\label{fndsl}
-\Delta{{\mathcal E}}+\gamma_0^2{{\mathcal E}}=\delta({{\mathbf x}}),$$ belonging to the space $S^\prime({{\mathbf R}})$ of slowly growing distributions.
\[prfndsl\] (i) Equation (\[fndsl\])has a unique solution ${{\mathcal E}}_d\in
S^\prime({{\mathbf R}}^d)$;
\(ii) it is spherically symmetric, that is ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)={{\mathcal E}}_d(|{{\mathbf x}}|,\gamma_0)$, ${{\mathcal E}}({{\mathbf x}},\gamma_0)>0$ for ${{\mathbf x}}\neq 0$, and it has the form:
\(iii) for $d=1$ $$\label{fndslone}
{{\mathcal E}}_d(x,\gamma_0)=\frac{1}{2\gamma_0}e^{-\gamma_0|x|};$$
\(iv) for $d=3$$\;{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{1}{4\pi|{{\mathbf x}}|}e^{-\gamma_0|{{\mathbf x}}|}$;
\(v) for $d=2m+1\;(m=2,3,\dots)$ $$\begin{aligned}
&&{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{s_{d-2}}{2(2\pi)^{d-1}}\sum_{k=0}^{m-1}
\left(\begin{array}{l}
m-1\\
\quad k
\end{array}\right)
(-1)^{m-1-k}\sum_{j=0}^{2k} \left(\begin{array}{l}
2k\\
\;j
\end{array}\right)
\gamma_0^{2(m-1-k)+j}\times \nonumber\\
&&\frac{(2k-j)!\;e^{-\gamma_0|{{\mathbf x}}|}}{|{{\mathbf x}}|^{2k-j+1}};\end{aligned}$$
\(vi) for $d=2$ $\;{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{1}{2\pi}K_0(\gamma_0|{{\mathbf x}}|)$, where $$\label{mcdn}
K_\nu(x):=\int_1^\infty\frac{e^{-xt}}{t^\nu\sqrt{t^2-1}}\,dt$$ $(x>0,\;\nu\ge 0)$ is the MacDonald’s function;
\(vii) for $d=2m\;(m=2,3,\dots)$ $$\begin{aligned}
&&{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{s_{d-1}}{(2\pi)^{d}}\sum_{k=0}^{m-1}
\left(\begin{array}{l}
m-1\\
\quad k
\end{array}\right)
(-1)^{m-1-k}\sum_{j=0}^{2k} \left(\begin{array}{l}
2k\\
\;j
\end{array}\right)
\gamma_0^{2(m-1-k)+j}\times \nonumber\\
&&\frac{(2k-j)!}{|{{\mathbf x}}|^{2k-j}}K_{2k-j}(\gamma_0|{{\mathbf x}}|).\end{aligned}$$
We see from (\[fndsl\]) that the Fourier transform $\hat{{\mathcal E}}_d$ of ${{\mathcal E}}_d$ satisfies the equation $$\label{eqfrtrfnsl}
(|{{\mathbf p}}|^2+\gamma_0^2)\hat{{\mathcal E}}=\frac{1}{(2\pi)^{d/2}},$$ hence $\hat{{\mathcal E}}_d({{\mathbf p}},\gamma_0)=\frac{1}{(2\pi)^{d/2}(|{{\mathbf p}}|^2+\gamma_0^2)}$ and, as it is clear, this is a unique solution of (\[eqfrtrfnsl\]) belonging to $S^\prime({{\mathbf R}}^d)$. This proves claim (i) of the proposition. Let us reconstruct ${{\mathcal E}}_d$ from $\hat{{\mathcal E}}_d$: $$\label{reconstr}
{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{1}{(2\pi)^d}\int_{{{\mathbf R}}^d}\frac{e^{i{{\mathbf p}}\cdot{{\mathbf x}}}}{|{{\mathbf p}}|^2+\gamma_0^2}\,d{{\mathbf p}}.$$ for d=1 we have using Jordan’s lemma: $$\begin{aligned}
{{\mathcal E}}_d(x,\gamma_0)=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{e^{ipx}}{p^2+\gamma_0^2}\,dp=
\left\{\begin{array}{ll} i
Res_{p=i\gamma_0}\frac{e^{ipx}}{p^2+\gamma_0^2}& for x>0,\\
-i Res_{p=-i\gamma_0}\frac{e^{ipx}}{p^2+\gamma_0^2}& for x<0
\end{array}\right.
=\frac{1}{2\gamma_0}e^{-\gamma_0|{{\mathbf x}}|}. \nonumber\end{aligned}$$ So, we have proved claim (iii). Rotating the space ${{\mathbf R}}^d$ such that the direction of the vector ${{\mathbf x}}$ transforms to the direction of the axis $p_d$, we have from (\[reconstr\]) for $d>1$: $${{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{1}{(2\pi)^d}\int_{{{\mathbf R}}^{d-1}}d\tilde{{\mathbf p}}\int_{-\infty}^\infty
\frac{e^{ip_d|{{\mathbf x}}|}}{|\tilde{{\mathbf p}}|^2+p_d^2+\gamma_0^2}\,dp_d,$$ where $\tilde{{\mathbf p}}=(p_1,p_2,\dots,p_{d-1})$. Applying Jordan’s lemma to the inner integral, we have: $$\begin{aligned}
\label{descent}
&&{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{1}{2(2\pi)^{d-1}}\int_{{{\mathbf R}}^{d-1}}\frac{\exp(-\sqrt{|\tilde{{\mathbf p}}|^2+\gamma_0^2}|{{\mathbf x}}|)}
{\sqrt{|\tilde{{\mathbf p}}|^2+\gamma_0^2}}\,d\tilde{{\mathbf p}}=\nonumber\\
&&\frac{s_{d-2}}{2(2\pi)^{d-1}}\int_0^\infty
\frac{r^{d-2}\exp(-\sqrt{r^2+\gamma_0^2}|{{\mathbf x}}|)}
{\sqrt{r^2+\gamma_0^2}}\,dr\end{aligned}$$ We see from the last equality and (\[fndslone\]) that ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)$ is spherically symmetric, that is ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)={{\mathcal E}}(|{{\mathbf x}}|,\gamma_0)$, it is finite for ${{\mathbf x}}\neq 0$ and ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)>0$. We have proved claim (ii). Taking in (\[descent\]) $d=3$, we get easily claim (iv).
For $d\ge 4$ let us write (\[descent\]) in the form: $$\begin{aligned}
&&{{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\frac{s_{d-2}}{2(2\pi)^{d-1}}\left(\int_0^\infty
\frac{r^{d-4}(r^2+\gamma_0^2)\exp(-\sqrt{r^2+\gamma_0^2}|{{\mathbf x}}|)}{\sqrt{r^2+\gamma_0^2}}\,dr
-\right.\\
&&\left.\gamma_0^2 \int_0^\infty
\frac{r^{d-4}\exp(-\sqrt{r^2+\gamma_0^2}|{{\mathbf x}}|)}
{\sqrt{r^2+\gamma_0^2}}\,dr\right)=\nonumber\\
&&\frac{s_{d-2}}{s_{d-4}(2\pi)^2}\left(\frac{d^2}{d\rho^2}-\gamma_0^2\right)
{{\mathcal E}}_{d-2}(\rho,\gamma_0)|_{\rho=|{{\mathbf x}}|},\end{aligned}$$ From this recursive formula and claim (iv) we get easily claim (v).
Using the Hadamard’s descent principle: $${{\mathcal E}}_{2m}({{\mathbf x}},\gamma_0)=\int_{-\infty}^\infty{{\mathcal E}}_{2m+1}(({{\mathbf x}},\xi),\gamma_0)\,d\xi$$ (see [@Wl]), it is not difficult to prove claims (vi) and (vii) with the help of claims (iv) and (v).
\[corest\] For any natural $d$ there exists $M=M(d,\gamma_0)>0$ such that
\(i) if $d=1$, then for any $x\in{{\mathbf R}}$ $\;{{\mathcal E}}_d(x,\gamma_0)\le
Me^{-\gamma_0|x|}$; vskip2mm
\(ii) if $d\ge 3$ is odd, then $${{\mathcal E}}_d({{\mathbf x}},\gamma_0)\le \left\{\begin{array}{ll} \frac{M}{|{{\mathbf x}}|^{d-2}}&
for\quad
|{{\mathbf x}}|\le 1\\
Me^{-\gamma_0|{{\mathbf x}}|}& for\quad |{{\mathbf x}}|> 1;
\end{array}\right.$$
\(iii) if $d$ is even, then $${{\mathcal E}}_d({{\mathbf x}},\gamma_0)\le \left\{\begin{array}{ll}
\frac{M}{|{{\mathbf x}}|^{d-2}}\ln\left(\frac{1}{|{{\mathbf x}}|}\right)& for\quad
|{{\mathbf x}}|\le 1\\
Me^{-\gamma_0|{{\mathbf x}}|}& for\quad |{{\mathbf x}}|> 1;
\end{array}\right.$$
\(iv) in particular, ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)\in L_q({{\mathbf R}}^d)$ for $d=1,2$, $q\in[1,\infty)$ and for $d>2$, $q\in[1,\,\frac{d}{d-2})$.
The following claims are proved easily:
\[lmdil\] The following equality is valid: ${{\mathcal E}}_d({{\mathbf x}},\gamma_0)=\gamma_0^{d-2}{{\mathcal E}}_d(\gamma_0{{\mathbf x}},1)$.
\[rsintop\] The restriction of the operator $R_{-\gamma_0^2}(-\Delta)=(-\Delta+\gamma_0^2)^{-1}$ on the set $L_1({{\mathbf R}}^d)\cap L_\infty({{\mathbf R}}^d)$ is represented as the integral operator: $$R_{-\gamma_0^2}(-\Delta)f=\int_{{{\mathbf R}}^d}{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}},\gamma_0)f({{\mathbf s}})\,d{{\mathbf s}}\quad
(f\in L_1({{\mathbf R}}^d)\cap L_\infty({{\mathbf R}}^d)).$$
Green’s function of the operator $H({{\mathbf p}})$ {#subsec:GreenHp}
---------------------------------------------------
Let $H_0({{\mathbf p}})$ be the operator generated by the operation $-\Delta$ in the space ${{\mathcal H}}_{{\mathbf p}}$ (defined in Section \[sec:preliminaries\]) with the domain ${{\mathcal D}}_{{\mathbf p}}(\Gamma)=W_{2,loc}^2({{\mathbf R}}^d)\cap{{\mathcal H}}_{{{\mathbf p}}}$. Consider the function $$\label{dfG0}
G_0({{\mathbf x}},{{\mathbf p}},\gamma_0):=\sum_{{{\bf m}}\in{{\mathbf Z}}^d}{{\mathcal E}}_d({{\mathbf x}}-{{\bf m}},\gamma_0)\exp(i{{\mathbf p}}\cdot{{\bf m}}).$$ Furthermore, we shall assume in what follows that $d\ge 4$, because the case $d\le 3$ have been studied in [@Wil]. Using the function $G_0({{\mathbf x}},{{\mathbf p}},\gamma_0)$, we shall construct below the integral kernel of the operator $R_{-\gamma_0^2}(H({{\mathbf p}}))=(H({{\mathbf p}})+\gamma_0^2)^{-1}$, where the operator $H({{\mathbf p}})$ have been defined in Section \[sec:preliminaries\]. But first of all we shall study some properties of the function $G_0({{\mathbf x}},{{\mathbf p}},\gamma_0)$.
Denote by $L_{c,q}(\Omega\times\Omega)\,(q\ge 1)$ the set of all measurable functions $f:\,\Omega\times\Omega\rightarrow{{\mathbf C}}$ such that for any fixed ${{\mathbf x}}\in\Omega$ the function $f({{\mathbf x}},\cdot)$ belongs to $L_q(\Omega)$ and the function ${{\mathbf x}}\rightarrow f({{\mathbf x}},\cdot)\in
L_q(\Omega)$ is continuous. This is a Banach space w. r. to the norm $$\|f\|_{c,q}:=\|\|f({{\mathbf x}},\cdot)\|_{L_q(\Omega)}\|_{C(\Omega)}=
\max_{{{\mathbf x}}\in\Omega}\Big(\int_\Omega|f({{\mathbf x}},{{\mathbf s}})|^q\,d{{\mathbf s}}\Big)^{\frac{1}{q}}.$$ In the analogous manner the space $L_{q,c}(\Omega\times\Omega)$, having the norm $$\|f\|_{q,c}:=\|\|f(\cdot,{{\mathbf s}})\|_{L_q(\Omega)}\|_{C(\Omega)}=
\max_{{{\mathbf s}}\in\Omega}\Big(\int_\Omega|f({{\mathbf x}},{{\mathbf s}})|\,d{{\mathbf x}}\Big)^{\frac{1}{q}},$$ is defined.
If $q\ge 1$, we shall denote by $q^\prime$ the number conjugate to $q$, that is $1/q+1/q^\prime=1$. As it is easy to check, $q\in(1,\frac{d}{d-2})$ if and only if $q^\prime>\frac{d}{2}$.
Denote ${{\mathcal H}}_{{{\mathbf p}},q^\prime}:={{\mathcal H}}_{{\mathbf p}}\cap\L_{q^\prime,loc}({{\mathbf R}}^d)$, ${{\mathcal H}}_{{{\mathbf p}},\infty}:={{\mathcal H}}_{{\mathbf p}}\cap\L_{\infty,loc}({{\mathbf R}}^d)$, ${{\mathbf Z}}^d_+=\{{{\bf m}}\in{{\mathbf Z}}^d\;|\;m_k\ge 0\;(k=1,2,\dots,d)\}$. For ${{\bf m}}\in{{\mathbf Z}}^d$ we denote $|{{\bf m}}|_\infty:=\max_{1\le k\le d}|m_k|$, $|{{\bf m}}|_1:=\sum_{k=1}^d|m_k|$ (${{\bf m}}=(m_1,\dots,m_d)$).
Applying Corollary \[corest\] of Proposition \[prfndsl\] and using the same arguments that have been used for $d=3$ in [@Wil] (Lemmas 3.1, 3.2, 3.5), we can prove the following claims, using the $L_{c,q}(\Omega\times\Omega)$-norm for integral kernels of operators instead of the $L_2(\Omega\times\Omega)$-norm used in [@Wil]:
\[lmcnvser\] (i) The series in (\[dfG0\]) converges for $({{\mathbf x}},{{\mathbf p}})\in
({{\mathbf R}}^d/{{\mathbf Z}}^d)\times\{{{\mathbf p}}\in{{\mathbf C}}^d\;|\;|\Im{{\mathbf p}}|<\gamma_0\}$ and the convergence is uniform on compact subsets. Moreover, if $\,G_0^\prime({{\mathbf x}},{{\mathbf p}},\gamma_0)$ is defined by $$\label{dfG0pr}
G_0({{\mathbf x}},{{\mathbf p}},\gamma_0)=\sum_{{{\bf m}}\in{{\mathcal N}}}{{\mathcal E}}_d({{\mathbf x}}-{{\bf m}},\gamma_0)\exp(i{{\mathbf p}}\cdot{{\bf m}})+G_0^\prime({{\mathbf x}},{{\mathbf p}},\gamma_0),$$ where ${{\mathcal N}}:=\{{{\bf m}}\in{{\mathbf Z}}^d\;|\;|{{\bf m}}|\le\sqrt{d}\}$, then $G_0^\prime(\cdot,{{\mathbf p}},\gamma_0)\in C(2\Omega)$ and the mapping ${{\mathbf p}}\rightarrow G_0^\prime(\cdot,{{\mathbf p}},\gamma_0)\in C(2\Omega)$ is holomorphic for $|\Im{{\mathbf p}}|<\gamma_0$.
Furthermore, if $g\ge 1$ for $d\le 2$ and $q\in[1,\frac{d}{d-2})$ for $d>2$, then
\(ii) each term of the series $\sum_{{{\bf m}}\in{{\mathbf Z}}^d}{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}},\gamma_0)\exp(i{{\mathbf p}}\cdot{{\bf m}})$ belongs to the space $L_{c,q}(\Omega\times\Omega)$ for any fixed ${{\mathbf p}}\in C^d$ and it is holomorphic in ${{\mathbf C}}^d$ w.r. to ${{\mathbf p}}$ in the $L_{c,q}(\Omega\times\Omega)$-norm;
\(iii) this series converges in the $L_{c,q}(\Omega\times\Omega)$-norm to a kernel $G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)=G_0({{\mathbf x}}-{{\mathbf s}},{{\mathbf p}},\gamma_0)\in
L_{c,q}(\Omega\times\Omega)$ uniformly w.r. to ${{\mathbf p}}\in\Pi_\gamma=\{{{\mathbf p}}\in{{\mathbf C}}^d\;|\;|\Im p|<\gamma\}\;(\gamma\in
(0,\gamma_0))$, hence the mapping ${{\mathbf p}}\rightarrow
G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)$ is holomorphic in the strip $\Pi_{\gamma_0}$;
\(iv) for any ${{\mathbf p}}\in{{\mathbf T}}^d$ the restriction of the operator $R_0({{\mathbf p}})=(-\Delta+\gamma_0^2I)^{-1}$ on the set ${{\mathcal H}}_{{{\mathbf p}},q^\prime}$ has the form: $R_0({{\mathbf p}}) f=\int_\Omega
G_0({{\mathbf x}},{{\mathbf s}},p,\gamma_0)f({{\mathbf s}})\,d{{\mathbf s}}$;
\(v) the operator $R_0({{\mathbf p}})$ maps the set ${{\mathcal H}}_{{{\mathbf p}},q^\prime}$ into ${{\mathcal H}}_{{{\mathbf p}},\infty}$.
\[lmholcomp\] Let $\Pi$ be a domain in ${{\mathbf C}}^d$. If $V\in
L_{q^\prime}(\Omega)\;(q^\prime\in(1,\infty]$ and $G_k({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\;(k=0,1)$ be functions defined in $\Omega\times\Omega\times\Pi$ such that for any fixed ${{\mathbf p}}\in\Pi$ $G_k(\cdot,\cdot,{{\mathbf p}})\in L_{c,q}(\Omega\times\Omega)\;(k=0,1)$ and the mappings ${{\mathbf p}}\rightarrow G_k(\cdot,\cdot,{{\mathbf p}})\in
L_{c,q}(\Omega\times\Omega)\;(k=0,1)$ are holomorphic in $\Pi$. Then the function $G_2({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\int_{\Omega}G_0({{\mathbf x}},\xi,{{\mathbf p}})V(\xi)G_1(\xi,{{\mathbf s}},{{\mathbf p}})$ has the same properties, i.e. for any fixed ${{\mathbf p}}\in\Pi$ $G_2(\cdot,\cdot,{{\mathbf p}})\in L_{c,q}(\Omega\times\Omega)$ and the mapping ${{\mathbf p}}\rightarrow G_2(\cdot,\cdot,{{\mathbf p}})\in L_{c,q}(\Omega\times\Omega)$ is holomorphic in $\Pi$.
The claim follows from the estimates: $$\begin{aligned}
&&\hskip-8mm\|G_2({{\mathbf x}},\cdot,{{\mathbf p}})\|_q\le
\Big(\int_\Omega\Big(\int_\Omega|G_0({{\mathbf x}},\xi,{{\mathbf p}})||V(\xi)|
|G_1(\xi,{{\mathbf s}},{{\mathbf p}})|\,d\xi\Big)^q\,d{{\mathbf s}}\Big)^{\frac{1}{q}}\le\\
&&\hskip-8mm\|V\|_{q^\prime}\|G_0({{\mathbf x}},\cdot,{{\mathbf p}})\|_q
\|G_1(\cdot,\cdot,{{\mathbf p}})\|_{c,q},\end{aligned}$$ $$\begin{aligned}
&&\hskip-8mm\|G_2({{\mathbf x}},\cdot,{{\mathbf p}})-G_2({{\mathbf x}}_0,\cdot,{{\mathbf p}})\|_q\le\\
&&\hskip-8mm\Big(\int_\Omega\Big(\int_\Omega|G_0({{\mathbf x}},\xi,{{\mathbf p}})-G_0({{\mathbf x}}_0,\xi,{{\mathbf p}})||V(\xi)|
|G_1(\xi,{{\mathbf s}},{{\mathbf p}})|\,d\xi\Big)^q\,d{{\mathbf s}}\Big)^{\frac{1}{q}}\le\\
&&\hskip-8mm\|V\|_{q^\prime}\|G_0({{\mathbf x}},\cdot,{{\mathbf p}})-G_0({{\mathbf x}}_0,\cdot,{{\mathbf p}})\|_q
\|G_1(\cdot,\cdot,{{\mathbf p}})\|_{c,q},\end{aligned}$$ $$\begin{aligned}
&&\hskip-7mm \|(G_2(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}})-G_2(\cdot,\cdot,{{\mathbf p}}_0))-
\int_\Omega\left(\nabla_{{\mathbf p}}G_0(\cdot,\xi,{{\mathbf p}})|_{{{\mathbf p}}={{\mathbf p}}_0}\cdot{{\mathbf h}}\times\right.\\
&&\hskip-7mm
\left.V(\xi)G_1(\xi,\cdot,{{\mathbf p}}_0)+G_0(\cdot,\xi,{{\mathbf p}}_0)V(\xi)\nabla_{{\mathbf p}}G_1(\xi,\cdot,{{\mathbf p}})_{{{\mathbf p}}={{\mathbf p}}_0}\right)\,d\xi\|_{c,q}\le\\
&&\hskip-7mm\|V\|_{q^\prime}\left(\|(G_0(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}})-G_0(\cdot,\cdot,{{\mathbf p}}_0))-
\nabla_{{\mathbf p}}G_0(\cdot,\cdot,{{\mathbf p}})|_{{{\mathbf p}}={{\mathbf p}}_0}\cdot{{\mathbf h}}\|_{c,q}\right.\\
&&\hskip-7mm \times\|G_1(\cdot,\cdot,{{\mathbf p}}_0)\|_{c,q}+
\|G_0(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}})-G_0(\cdot,\cdot,{{\mathbf p}}_0)\|_{c,q}\times\\
&&\hskip-7mm\|G_1(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}})-G_1(\cdot,\cdot,{{\mathbf p}}_0)\|_{c,q}+
\|G_0(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}},\gamma_0)\|_{c,q}\times\\
&&\hskip-7mm\left.\|(G_1(\cdot,\cdot,{{\mathbf p}}_0+{{\mathbf h}})-G_1(\cdot,\cdot,{{\mathbf p}}_0))-\nabla_{{\mathbf p}}G_1(\cdot,\cdot,{{\mathbf p}})|_{{{\mathbf p}}={{\mathbf p}}_0}\cdot{{\mathbf h}}\|_{c,q}\right),\end{aligned}$$ where ${{\mathbf x}},{{\mathbf x}}_0\in\Omega$ and ${{\mathbf p}},{{\mathbf p}}_0,{{\mathbf p}}_0+{{\mathbf h}}\in\Pi$.
The proof of the following proposition is based on the previous claims and it is analogous to the proof of Lemma 3.5 from [@Wil], only again instead of the $L_2(\Omega\times\Omega)$-norm one should use the $L_{c,q}(\Omega\times\Omega)$-norm:
\[prNeuser\] If $d\ge 4$ and $V\in L_{q^\prime}(\Omega)$ with $q^\prime\in(\frac{d}{2},\infty]$, then for a large enough $\gamma_0$:
\(i) $-\gamma_0^2\in{{\mathcal R}}(H({{\mathbf p}}))$ for any ${{\mathbf p}}\in{{\mathbf R}}^d$ and the resolvent $R({{\mathbf p}})=(H({{\mathbf p}})+\gamma_0^2I)^{-1}$ is represented by the Neumann’s series $$R({{\mathbf p}})=(I+L_0({{\mathbf p}}))^{-1}R_0({{\mathbf p}})=\sum_{n=0}^\infty(-1)^n(L_0({{\mathbf p}}))^n
R_0({{\mathbf p}}),$$ where $R_0({{\mathbf p}})=(-\Delta+\gamma_0^2I)^{-1}$, $L_0({{\mathbf p}}):=R_0({{\mathbf p}})V$ and this series converges to $R({{\mathbf p}})$ in ${{\mathcal B}}(L_2(\Omega))$-norm;
\(ii) for any ${{\mathbf p}}\in{{\mathbf R}}^d$ the restriction of $R({{\mathbf p}})$ on ${{\mathcal H}}_{{{\mathbf p}},\infty}$ is the integral operator of the form: $R({{\mathbf p}})f=\int_\Omega G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)f({{\mathbf s}})\,d{{\mathbf s}}\quad
(f\in{{\mathcal H}}_{{{\mathbf p}},\infty})$, where $G(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in
L_{c,q}(\Omega\times\Omega)$ $(q^{-1}+(q^\prime)^{-1}=1)$, $$\label{NeuserG}
G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)=G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)+\sum_{n=1}^\infty
G_n({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0),$$ $$\begin{aligned}
&&G_{n+1}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)=
\nonumber\\
&&-\int_\Omega
G_0({{\mathbf x}},{{\mathbf \xi}},{{\mathbf p}},\gamma_0)V({{\mathbf \xi}})G_n({{\mathbf \xi}},{{\mathbf s}},{{\mathbf p}},\gamma_0)\,d{{\mathbf \xi}}\quad
(n=0,1.\dots),\end{aligned}$$ $G_n({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)$ and the series in (\[NeuserG\]) converges to $G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)$ in the $L_{c,q}(\Omega\times\Omega)$-norm;
\(iii) each mapping ${{\mathbf p}}\rightarrow G_n(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in
L_{c,q}(\Omega\times\Omega)$ admits a holomorphic continuation from ${{\mathbf R}}^d$ to the strip $\Pi_{\frac{\gamma_0}{2}}:=\{{{\mathbf p}}\in{{\mathbf C}}^d\;|\;|\Im{{\mathbf p}}|<\frac{\gamma_0}{2}\}$ and the series in (\[NeuserG\]) converges in the $L_{c,q}(\Omega\times\Omega)$-norm uniformly in $\Pi_{\frac{\gamma_0}{2}}$. Hence the mapping ${{\mathbf p}}\rightarrow
G(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)$ admits a holomorphic continuation from ${{\mathbf R}}^d$ to $\Pi_{\frac{\gamma_0}{2}}$;
\(iv) for any ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ the operator $R({{\mathbf p}})$ with the integral kernel $G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)$ maps the set ${{\mathcal H}}_{{{\mathbf p}},\infty}$ into itself;
\(v) for any ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ and ${{\mathbf x}},{{\mathbf s}}\in\Omega$ the equalities are valid: $G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)=G_0({{\mathbf s}},{{\mathbf x}},-{{\mathbf p}},\gamma_0)$, $G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}},\gamma_0)=G({{\mathbf s}},{{\mathbf x}},-{{\mathbf p}},\gamma_0)$.
\[corsymm\] If $d\ge 4$ and $\,V\in L_{q^\prime}(\Omega)$ with $q^\prime>\frac{d}{2}$, then for a large enough $\gamma_0$ and any ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}\;$ $G_0(\cdot,\cdot,{{\mathbf p}},\gamma_0),\,
G(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)\cap
L_{q,c}(\Omega\times\Omega)$ and the mappings ${{\mathbf p}}\rightarrow
G_0(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)$, ${{\mathbf p}}\rightarrow G_0(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in
L_{q,c}(\Omega\times\Omega)$, ${{\mathbf p}}\rightarrow
G(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in L_{c,q}(\Omega\times\Omega)$ and ${{\mathbf p}}\rightarrow G(\cdot,\cdot,{{\mathbf p}},\gamma_0)\in
L_{q,c}(\Omega\times\Omega)$ are holomorphic in $\Pi_{\frac{\gamma_0}{2}}$.
$C(\Omega\times\Omega)$-holomorphy of a compositional power of the Green’s function of $H({{\mathbf p}})$ {#subsec:contcomppow}
---------------------------------------------------------------------------------------------------------
We shall use the following notation. By $K_1(\cdot,\cdot)\circ
K_2(\cdot,\cdot)$ we denote the composition of integral kernels $K_1({{\mathbf x}},\bf)$ and $K_2({{\mathbf x}},\bf)$: $(K_1(\cdot,\cdot)\circ
K_2(\cdot,\cdot))({{\mathbf x}},{{\mathbf s}}):=\int_\Omega
K_1({{\mathbf x}},\xi)K_2(\xi,{{\mathbf s}})\,d\xi$. The composition of several integral kernels $K_1(\cdot,\cdot)\circ K_2(\cdot,\cdot)\circ\cdots\circ
K_N(\cdot,\cdot)$ will be denoted briefly by $\circ_{j=1}^N
K_j(\cdot,\cdot)$; if $K_1(\cdot,\cdot)=K_2(\cdot,\cdot)=\cdots=K_N(\cdot,\cdot)=K(\cdot,\cdot)$, we shall denote it as a compositional power $K(\cdot,\cdot)^{\circ
N}$. We shall denote by $Hol(D,B)$ the set of abstract holomorphic functions $\phi:D\rightarrow B$, where $D$ is an open domain in ${{\mathbf C}}^d$ and $B$ is a complex Banach space. Like in [@Wil], let us define the following equivalence relation between the functions ${{\mathbf p}}\rightarrow K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\;({{\mathbf x}},{{\mathbf s}}\in\Omega)$: we shall write that $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$, if $({{\mathbf p}}\rightarrow K(\cdot,\cdot,{{\mathbf p}}))\in
Hol(\Pi_{\frac{\gamma_0}{2}}, C(\Omega\times\Omega))$ and $K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim K_2({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, if $K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})-K_2({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$.
\[lmcomp\] Assume that $d\ge 4$ and $V\in\L_{q^\prime}(\Omega)$ with $q^\prime>\frac{d}{2}$. Consider the composition of integral kernels $$\label{comp}
K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\circ_{j=1}^N K_j(\cdot,\cdot,{{\mathbf p}})({{\mathbf x}},{{\mathbf s}})\quad ({{\mathbf p}}\in
\Pi_{\frac{\gamma_0}{2}}),$$ where or $K_j({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$ and the composition contains at least one term of this kind, or $K_j({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=L_j({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$, or for $j<N$ $K_j({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=L_j({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})V({{\mathbf s}})$ with $$\label{mmbship}
({{\mathbf p}}\rightarrow L_j(\cdot,\cdot,{{\mathbf p}}))\in Hol(\Pi_{\frac{\gamma_0}{2}},
L_{c,q}(\Omega\times\Omega))\cap Hol(\Pi_{\frac{\gamma_0}{2}},
L_{q,c}(\Omega\times\Omega)),$$ where $q^{-1}+(q^\prime)^{-1}=1$. Then $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$.
We shall prove the lemma in two steps.
Step 1. Consider the kernel of the form $K_1(\cdot,\cdot,{{\mathbf p}})=L(\cdot,\cdot,{{\mathbf p}})V(\cdot)\circ
K(\cdot,\cdot,{{\mathbf p}})$, where the mapping ${{\mathbf p}}\rightarrow
L(\cdot,\cdot,{{\mathbf p}})$ satisfies the condition (\[mmbship\]) and $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. Let us prove that $K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. We have for ${{\mathbf x}},{{\mathbf x}}+{{\mathbf h}},{{\mathbf s}}\in\Omega$: $$\begin{aligned}
&&|K_1({{\mathbf x}}+{{\mathbf h}},{{\mathbf s}},{{\mathbf p}})-K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})|=\\
&&\left|\int_\Omega(L({{\mathbf x}}+{{\mathbf h}},\xi,{{\mathbf p}})-L({{\mathbf x}},\xi,{{\mathbf p}}))V(\xi)K(\xi,{{\mathbf s}},{{\mathbf p}})\,d\xi\right|\le\\
&&\|V\|_{q^\prime}\|K(\cdot,\cdot,{{\mathbf p}})\|_{C(\Omega\times\Omega)}
\|L({{\mathbf x}}+{{\mathbf h}},\cdot,{{\mathbf p}})-L({{\mathbf x}},\cdot,{{\mathbf p}})\|_q.\end{aligned}$$ This estimate and the inclusion $L(\cdot,\cdot,{{\mathbf p}})\in
L_{c,q}(\Omega\times\Omega)$ imply that for any ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ the function $K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ is continuous w.r. to ${{\mathbf x}}$ at each point ${{\mathbf x}}\in\Omega$ uniformly w.r. to ${{\mathbf s}}\in\Omega$. Let us estimate for ${{\mathbf x}},{{\mathbf s}},{{\mathbf s}}+{{\mathbf h}}\in\Omega$: $$\begin{aligned}
&&|K_1({{\mathbf x}},{{\mathbf s}}+{{\mathbf h}},{{\mathbf p}})-K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})|\le\\
&&\|V\|_{q^\prime}\|L(\cdot,\cdot,{{\mathbf p}})\|_{c,q}\max_{\xi\in\Omega}|K(\xi,{{\mathbf s}}+{{\mathbf h}},{{\mathbf p}})-K(\xi,{{\mathbf s}},{{\mathbf p}})|.\end{aligned}$$ This estimate and the continuity of $K(\xi,{{\mathbf s}},{{\mathbf p}})$ in $\Omega\times\Omega$ imply that for any fixed ${{\mathbf x}}\in\Omega$ $K_1(\xi,{{\mathbf s}},{{\mathbf p}})$ is continuous at each point ${{\mathbf s}}\in\Omega$. So, we have proved that for any ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ $K_1(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$. The estimate $$\begin{aligned}
\hskip-8mm &&\|K_1(\cdot,\cdot,{{\mathbf p}}+{{\mathbf t}})-K_1(\cdot,\cdot,{{\mathbf p}})-\nabla_{{\mathbf p}}L(\cdot,\cdot,{{\mathbf p}})\cdot{{\mathbf t}}\,V(\cdot)\circ
K(\cdot,\cdot,{{\mathbf p}})\|_{C(\Omega\times\Omega)}\le\|V\|_{q^\prime}\times\\
\hskip-8mm&&\left(\|(L(\cdot,\cdot,{{\mathbf p}}+{{\mathbf t}})-L(\cdot,\cdot,{{\mathbf p}}))-\nabla_{{\mathbf p}}L(\cdot,\cdot,{{\mathbf p}})\cdot{{\mathbf t}}\|_{c,q}\,\|K(\cdot,\cdot,{{\mathbf p}})\|_{C(\Omega\times\Omega)}+
\right.\\
\hskip-8mm&&\|L(\cdot,\cdot,{{\mathbf p}}+{{\mathbf t}})-L(\cdot,\cdot,{{\mathbf p}})\|_{c,q}\,\|K(\cdot,\cdot,{{\mathbf p}}+{{\mathbf t}})-
K(\cdot,\cdot,{{\mathbf p}})\|_{C(\Omega\times\Omega)}+
\|L(\cdot,\cdot,{{\mathbf p}})\|_{c,q}\times\\
\hskip-8mm&&\left.\|(K(\cdot,\cdot,{{\mathbf p}}+{{\mathbf t}})-K(\cdot,\cdot,{{\mathbf p}}))-\nabla_{{\mathbf p}}K(\cdot,\cdot,{{\mathbf p}})\cdot{{\mathbf t}}\|_{C(\Omega\times\Omega)}\right)\quad
({{\mathbf p}},{{\mathbf p}}+{{\mathbf t}}\in\Pi_{\frac{\gamma_0}{2}})\end{aligned}$$ and the holomorphy of the mappings ${{\mathbf p}}\rightarrow
L(\cdot,\cdot,{{\mathbf p}})\in L_{c,q}(\Omega\times\Omega)$ and ${{\mathbf p}}\rightarrow
K(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ in the strip $\Pi_{\frac{\gamma_0}{2}}$ imply that the mapping ${{\mathbf p}}\rightarrow
K_1(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ is holomorphic there, that is $K_1({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. In the analogous manner we can prove that if $L$ satisfies the condition (\[mmbship\]) and $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$, then $L(\cdot,\cdot,{{\mathbf p}})\circ
K(\cdot,\cdot,{{\mathbf p}})\sim 0$, $K(\cdot,\cdot,{{\mathbf p}})\circ
V(\cdot)L(\cdot,\cdot,{{\mathbf p}})\sim 0$ and $K(\cdot,\cdot,{{\mathbf p}})\circ
L(\cdot,\cdot,{{\mathbf p}})\sim 0$.
Step 2. By the assumption of the lemma, in the composition (\[comp\]) for some $j_0\in\{1,2,\dots,N\}$ $\;K_{j_0}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. Observe that if a term of (\[comp\]) is equivalent to zero, it belongs to the set from the right hand side of (\[mmbship\]). Then using inductively the results of Step 1, we obtain that the composition $T({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ of kernels from (\[comp\]) which are before $K_{j_0}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ and of $K_{j_0}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ itself has the property $T({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. On the other hand, the composition $\tilde T({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ of terms from (\[comp\]) which are after $K_{j_0}({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ can be represented in the form $\tilde T({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=(\circ_{j=j_0+1}^N
\tilde K_j(\cdot,\cdot,{{\mathbf p}}))({{\mathbf x}},{{\mathbf s}})$, where the kernels $\tilde
K_j(\cdot,\cdot,{{\mathbf p}}))$ have one of the forms: or $\tilde
K_j(\cdot,\cdot,{{\mathbf p}}))=V(\cdot)\tilde L_j(\cdot,\cdot,{{\mathbf p}})$, or $\tilde
K_j(\cdot,\cdot,{{\mathbf p}}))=\tilde L_j(\cdot,\cdot,{{\mathbf p}})$ and each $\tilde
L_j(\cdot,\cdot,{{\mathbf p}})$ belongs to the set from the right hand side of (\[mmbship\]). Again using inductively the results of Step 1, we obtain that $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=(T(\cdot,\cdot,{{\mathbf p}})\circ\tilde
T(\cdot,\cdot,{{\mathbf p}}))({{\mathbf x}},{{\mathbf s}})\sim 0$.
In what follows we need the following result about Fourier’s transform of a function with a polar singularity:
\[lmestFour\] For the Fourier’s transform $\hat{{\mathcal F}}(\omega)$ on ${{\mathbf R}}^d$of the function ${{\mathcal F}}({{\mathbf x}})=\frac{\exp(-\gamma|{{\mathbf x}}|)}{|{{\mathbf x}}|^{\beta}}\;(\beta\in[0,d),\,\gamma>0)$ the property is valid: $$\label{boundhatE1}
\sup_{\omega\in{{\mathbf R}}^d}\psi(\omega)|\hat{{\mathcal F}}(\omega)|<\infty,$$ where $$\begin{aligned}
\label{defpsi}
\psi(\omega)=\left\{
\begin{array}{ll}
|\omega|,&\rm{if}\quad \beta\in[0,d-1),\\
\frac{\omega|}{1+\ln(1+|\omega|)},&\rm{if}\quad \beta=d-1,\\
|\omega|^{d-\beta},&\rm{if}\quad \beta\in(d-1,d).
\end{array}\right.\end{aligned}$$
Since $\beta<d$, ${{\mathcal F}}\in L_1({{\mathbf R}}^d)$, hence $\hat{{\mathcal F}}(\omega)$ is bounded on ${{\mathbf R}}^d$. Hence it is sufficient to prove (\[boundhatE1\]) with $\{{{\mathbf x}}\in{{\mathbf R}}^d\,|\;|{{\mathbf x}}|\ge 1\}$ instead of ${{\mathbf R}}^d$. Along with the Euclidean norm $|\omega|$ on ${{\mathbf R}}^d$ consider the equivalent norm $|\omega|_\infty=\max_{1\le j\le
d}|\omega_j|\;(\omega=(\omega_1,\omega_2,\dots,\omega_d))$. Let us take $\omega\neq 0$ and $j_0\in\{1,2,\dots,d\}$ such that $|\omega_{j_0}|=|\omega|_\infty$. If the vector ${{\mathbf h}}(\omega)$ is defined by $$\begin{aligned}
({{\mathbf h}}(\omega))_j=\left\{\begin{array}{ll}
\frac{\pi}{|\omega|_\infty}&\rm{for}\quad j=j_0,\\
0&\rm{for}\quad j\neq j_0
\end{array}\right.\end{aligned}$$ $(j\in\{1,2,\dots,d\})$, then the equality $\exp(-i\omega\cdot({{\mathbf x}}+{{\mathbf h}}(\omega)))=-\exp(-i\omega\cdot{{\mathbf x}})$ is valid, hence we have the representation $$\begin{aligned}
&&\hat{{\mathcal F}}(\omega)=\frac{1}{2(2\pi)^{\frac{d}{2}}}\Big(\int_{{{\mathbf R}}^d}{{\mathcal F}}({{\mathbf x}})\exp(-i\omega\cdot{{\mathbf x}})\,d{{\mathbf x}}+\\
&&\int_{{{\mathbf R}}^d}{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))\exp(-i\omega\cdot({{\mathbf x}}+{{\mathbf h}}(\omega)))\,d{{\mathbf x}}\Big)=\\
&&\frac{1}{2(2\pi)^{\frac{d}{2}}}\int_{{{\mathbf R}}^d}\big({{\mathcal F}}({{\mathbf x}})-{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))\big)\exp(-i\omega\cdot{{\mathbf x}})\,d{{\mathbf x}},\end{aligned}$$ which implies the estimate: $$\label{Fourmodcont}
|{{\mathcal F}}(\omega)|\le\frac{1}{2(2\pi)^{\frac{d}{2}}}\|{{\mathcal F}}({{\mathbf x}})-{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))\|_1.$$ Here we denote by $\|\cdot\|_1$ the norm in the space $L_1({{\mathbf R}}^d)$. We have: $$\label{estmodcont}
\|{{\mathcal F}}({{\mathbf x}})-{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))\|_1\le I_1(\omega)+I_2(\omega),$$ where $$I_1(\omega)=\int_{|{{\mathbf x}}|\le 1}|{{\mathcal F}}({{\mathbf x}})-{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))|\,d{{\mathbf x}},$$ $$\label{dfI2om}
I_2(\omega)=\int_{|{{\mathbf x}}|\ge 1}|{{\mathcal F}}({{\mathbf x}})-{{\mathcal F}}({{\mathbf x}}+{{\mathbf h}}(\omega))|\,d{{\mathbf x}}.$$ Let us derive the change of the variable in the integral $I_1(\omega)$: ${{\mathbf y}}=\frac{{{\mathbf x}}}{|{{\mathbf h}}(\omega)|}$, denote $\tilde{{\mathbf h}}=\frac{{{\mathbf h}}(\omega)}{|{{\mathbf h}}(\omega)|}$ and rotate the space ${{\mathbf R}}^d$ such that $\tilde{{\mathbf h}}=(1,0,\dots,0)$. Then we obtain: $$\begin{aligned}
\label{rprI1om}
&&\hskip-8mmI_1(\omega)=|{{\mathbf h}}(\omega)|^{d-\beta}\int_{|{{\mathbf y}}|\le\frac{1}{|{{\mathbf h}}(\omega)|}}
\Big|\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+\tilde{{\mathbf h}}|^\beta}-
\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}|)}{|{{\mathbf y}}|^\beta}\Big|\,d{{\mathbf y}}\le\nonumber\\
&&\hskip-8mm\pi|\omega|_\infty\big(I_{1,1}+I_{1,2}(\omega)\big),\end{aligned}$$ where $$\label{dfI11}
I_{1,1}=\int_{|{{\mathbf y}}|\le 2} \Big(\frac{1}{|{{\mathbf y}}+\tilde{{\mathbf h}}|^\beta}+
\frac{1}{|{{\mathbf y}}|^\beta}\Big)\,d{{\mathbf y}},$$ $$\label{dfI12om}
I_{1,2}(\omega)=\int_{2\le|{{\mathbf y}}|\le\frac{1}{|{{\mathbf h}}(\omega)|}}
\Big|\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+\tilde{{\mathbf h}}|^\beta}-
\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}|)}{|{{\mathbf y}}|^\beta}\Big|\,d{{\mathbf y}}.$$ Let us represent: $$\begin{aligned}
\label{rprdiff}
&&\hskip-15mm\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+\tilde{{\mathbf h}}|^\beta}-
\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}|)}{|{{\mathbf y}}|^\beta}=\\
&&\hskip-15mm\int_0^1 \frac{\partial}{\partial
t}\Big(\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+t\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+t\tilde{{\mathbf h}}|^\beta}\Big)\,dt=
\nonumber\\
&&\hskip-15mm-\int_0^1\Big(\beta\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+t\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+t\tilde{{\mathbf h}}|^{\beta+1}}+
\gamma|{{\mathbf h}}(\omega)|\frac{\exp(-\gamma|{{\mathbf h}}(\omega)||{{\mathbf y}}+t\tilde{{\mathbf h}}|)}{|{{\mathbf y}}+t\tilde{{\mathbf h}}|^\beta}\Big
)\times\nonumber\\
&&\hskip-15mm\frac{({{\mathbf y}}+t\tilde{{\mathbf h}})\cdot\tilde{{\mathbf h}}}{|{{\mathbf y}}+t\tilde{{\mathbf h}}|}\,dt.\nonumber\end{aligned}$$ Then we have from (\[dfI12om\]): $$\begin{aligned}
&&I_{1,2}(\omega)\le\int_{2\le|{{\mathbf y}}|\le\frac{1}{|{{\mathbf h}}(\omega)|}}\Big(\frac{\beta}{(|{{\mathbf y}}|-1)^{\beta+1}}+
\frac{\gamma|{{\mathbf h}}(\omega)|}{(|{{\mathbf y}}|-1)^\beta}\Big)\,d{{\mathbf y}}=\\
&&s_{d-1}\Big(\beta\int_2^{\frac{1}{|{{\mathbf h}}(\omega)|}}\frac{r^{d-1}\,dr}{(r-1)^{\beta+1}}+
\gamma|{{\mathbf h}}(\omega)|\int_2^{\frac{1}{|{{\mathbf h}}(\omega)|}}\frac{r^{d-1}\,dr}{(r-1)^\beta}\Big),\end{aligned}$$ hence for $\beta\in[0,d)\setminus\{d-1\}$ $$\label{estI12om}
I_{1,2}(\omega)\le s_{d-1}
2^{d-1}\Big(|{{\mathbf h}}(\omega)|^{\beta-d+1}+1\Big)\Big(\frac{\beta}{\beta-d+1}+\frac{\gamma}{d-\beta}\Big)$$ and for $\beta=d-1$ $$\label{estI12om1}
I_{1,2}(\omega)\le s_{d-1}
2^{d-1}\Big(\beta\ln\Big(\frac{1}{|{{\mathbf h}}(\omega)|}\Big)+\gamma\Big).$$ Let us estimate the integral $I_2(\omega)$, defined by (\[dfI2om\]), using for the difference under it the representation, analogous to (\[rprdiff\]): $$\begin{aligned}
&&I_2(\omega)\le|{{\mathbf h}}(\omega)|\int_{|{{\mathbf x}}|\ge 1}
\Big(\frac{\beta}{(|{{\mathbf x}}|-|{{\mathbf h}}(\omega)|)^{\beta+1}}+\frac{\gamma}{(|{{\mathbf x}}|-|{{\mathbf h}}(\omega)|)^\beta}\Big)\times\\
&&\exp\big(-\gamma(|{{\mathbf x}}|-|{{\mathbf h}}(\omega)|)\big)\,d{{\mathbf x}},\end{aligned}$$ if $|{{\mathbf h}}(\omega)|<1$. From this estimate and from (\[Fourmodcont\]), (\[estmodcont\]), (\[rprI1om\]), (\[dfI11\]), (\[estI12om\]) and (\[estI12om1\]) we obtain the desired claim.
In what follows we need also the following claim:
\[lmcmpEc\] Assume that $V\in L_{q^\prime}(\Omega)$ with $q^\prime>\frac{d}{2}$ and denote $$\label{defl}
l:=\left[\frac{d}{\theta}\right]+2,$$ where $$\label{deftht}
\theta=\min\{1, d-q(d-2)\}.$$ Then the composition of kernels $$\label{cmpEc}
\Theta({{\mathbf x}},{{\mathbf s}})=\left(\circ_{j=1}^l{{\mathcal E}}_j(\cdot,\cdot)\right)({{\mathbf x}},{{\mathbf s}}),$$ where $\tilde{{\mathcal E}}_l({{\mathbf x}},{{\mathbf s}})={{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}}_l,\gamma_0)$ and for $j<l$ $$\tilde{{\mathcal E}}_j({{\mathbf x}},{{\mathbf s}})=\left\{\begin{array}{ll} or&{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}}_j,\gamma_0),\\
or& {{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}}_j,\gamma_0)V({{\mathbf s}})
\end{array}\right.$$ (${{\bf m}}_j\in{{\mathbf Z}}^d$), is continuous in $\Omega\times\Omega$.
Denote $\tilde\Theta({{\mathbf x}},{{\mathbf s}})=\left(\circ_{j=1}^{l-2}\tilde{{\mathcal E}}_j(\cdot,\cdot)\right)\circ{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}}_{l-1},\gamma_0)({{\mathbf x}},{{\mathbf s}})$ and $$\label{dfbrTht}
\bar\Theta({{\mathbf x}},{{\mathbf s}})=\left(\circ_{j=2}^l\tilde{{\mathcal E}}_j(\cdot,\cdot)\right)({{\mathbf x}},{{\mathbf s}}).$$ Then we have from (\[cmpEc\]): $$\label{rprTht1}
\Theta({{\mathbf x}},{{\mathbf s}})=(\tilde\Theta(\cdot,\cdot)\circ
W_{l-1}(\cdot)\circ{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_l,\gamma_0))({{\mathbf x}},{{\mathbf s}}),$$ $$\label{rprTht2}
\Theta({{\mathbf x}},{{\mathbf s}})=({{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_1,\gamma_0)\circ
W_1(\cdot)\circ\bar\Theta(\cdot,\cdot))({{\mathbf x}},{{\mathbf s}}),$$ where $$W_j(\xi_j)=\left\{\begin{array}{ll} or& V(\xi_j),\\
or& 1,
\end{array}\right.$$ Assume that $d$ is odd. From the explicit formula $$\begin{aligned}
&&\tilde\Theta({{\mathbf x}},{{\mathbf s}})=\int_\Omega{{\mathcal E}}_d({{\mathbf x}}-\xi_1-{{\bf m}}_1,\gamma_0)W_1(\xi_1)\,d\xi_1
\int_\Omega{{\mathcal E}}_d(\xi_1-\xi_2-{{\bf m}}_2,\gamma_0)W_2(\xi_2)\,d\xi_2\dots\\
&&W_{l-2}(\xi_{l-2})\int_\Omega{{\mathcal E}}_d(\xi_{l-2}-{{\mathbf s}}-{{\bf m}}_{l-1},\gamma_0)\,d\xi_{l-2}\end{aligned}$$ and claims (ii)-(v) of Proposition \[prfndsl\] we obtain that there exists $A>0$ such that the estimate is valid: $$\begin{aligned}
\label{estcmp}
&&\hskip-8mm|\tilde\Theta({{\mathbf x}},{{\mathbf s}})|\le\max\{\|1\|_{q^\prime},\|V\|_{q^\prime}\}
\Big(\int_\Omega|{{\mathcal E}}_d({{\mathbf x}}-\xi_1-{{\bf m}}_1,\gamma_0)|^q \,d\xi_1\times\nonumber\\
&&\hskip-8mm\Big|\int_\Omega{{\mathcal E}}_d(\xi_1-\xi_2-{{\bf m}}_2,\gamma_0)W_2(\xi_2)\,d\xi_2
\dots
W_{l-2}(\xi_{l-2})\times\nonumber\\
&&\hskip-8mm\int_\Omega{{\mathcal E}}_d(\xi_{l-2}-{{\mathbf s}}-{{\bf m}}_{l-1},\gamma_0)\,d\xi_{l-2}\Big|^q\Big)^{\frac{1}{q}}\le
(\max\{\|1\|_{q^\prime},\|V\|_{q^\prime}\})^2\times\nonumber\\
&&\hskip-8mm\Big(\int_\Omega|{{\mathcal E}}_d({{\mathbf x}}-\xi_1-{{\bf m}}_1,\gamma_0)|^q \,d\xi_1\times\nonumber\\
&&\hskip-8mm\int_\Omega|{{\mathcal E}}_d(\xi_1-\xi_2-{{\bf m}}_2,\gamma_0)|^q\,d\xi_2\Big|\int_\Omega
d\xi_3 \dots
W_{l-2}(\xi_{l-2})\times\nonumber\\
&&\hskip-8mm\int_\Omega{{\mathcal E}}_d(\xi_{l-2}-{{\mathbf s}}-{{\bf m}}_{l-1},\gamma_0)\,d\xi_{l-2}\Big|^q\Big)^{\frac{1}{q}}\le\dots\le\nonumber\\
&&\hskip-8mm
A\cdot(\max\{\|1\|_{q^\prime},\|V\|_{q^\prime}\})^{l-2}(\phi({{\mathbf x}}-{{\mathbf s}})^{\frac{1}{q}},\end{aligned}$$ where $$\label{conv}
\phi({{\mathbf x}})=({{\mathcal F}}(\cdot-{{\bf m}}_1)\star{{\mathcal F}}(\cdot-{{\bf m}}_2)\star\dots\star
{{\mathcal F}}(\cdot-{{\bf m}}_{l-1}))({{\mathbf x}})$$ and ${{\mathcal F}}({{\mathbf x}})=\frac{\exp(-\frac{\gamma_0 q}{2}|{{\mathbf x}}|)}{|{{\mathbf x}}|^{q(d-2)}}$ (here $\star$ is the sign of the convolution of functions defined on ${{\mathbf R}}^d$). Let us show that the function $\phi(\cdot,\gamma)$ is bounded on ${{\mathbf R}}^d$. To this end let us calculate the Fourier transform on ${{\mathbf R}}^d$ of the function $\phi({{\mathbf x}})$. We have from (\[conv\]): $$\begin{aligned}
\hat\phi(\omega)=\exp\left(-i\omega\cdot\sum_{j=1}^{l-1}{{\bf m}}_j\right)
(\hat{{\mathcal F}}(\omega))^{l-1}\quad (\omega\in{{\mathbf R}}^d),\end{aligned}$$ hence by Lemma \[lmestFour\], $\sup_{\omega\in{{\mathbf R}}^d}(\psi(\omega))^{l-1}|\hat\phi(\omega)|<\infty$, where $\psi(\omega)$ is defined by (\[defpsi\]) with $\beta=q(d-2)$. Since in view of (\[defl\]) and (\[deftht\]), $(l-1)\theta>d$, these circumstances imply that $\hat\phi\in
L_1({{\mathbf R}}^d)$. Hence the function $\phi({{\mathbf x}})$ is bounded in ${{\mathbf R}}^d$ and we obtain from (\[estcmp\]) that for an odd $d$ the function $\tilde\Theta({{\mathbf x}},{{\mathbf s}})$ is bounded in $\Omega\times\Omega$. Consider the case of an even $d$. From the formula (\[mcdn\]) for the MacDonald’s function, we get that for any $\epsilon>0$ $$\frac{K_\nu(\gamma_0|{{\mathbf x}}|)}{|{{\mathbf x}}|^\nu}\le
C(\epsilon)\frac{\exp(-\frac{\gamma_0|{{\mathbf x}}|}{2})}{|{{\mathbf x}}|^{\nu+\epsilon}}\quad
(\nu\ge 0),$$ where $C(\epsilon)=\sup_{\tau\in[0,\infty)}\big(\tau^\epsilon\exp(-\gamma_0\tau/2)\big)\int_1^\infty\frac{dt}
{t^{\nu+\epsilon}\sqrt{t^2-1}}$. Using claims (vi)-(vii) of Proposition \[prfndsl\] and choosing a small enough $\epsilon>0$, we can show in the same manner as above that also for an even $d$ the function $\tilde\Theta({{\mathbf x}},{{\mathbf s}})$ is bounded in $\Omega\times\Omega$. In the analogous manner we obtain that the function $\bar\Theta({{\mathbf x}},{{\mathbf s}})$, defined by (\[dfbrTht\]), is bounded in $\Omega\times\Omega$.
Let us prove that the function $\Theta({{\mathbf x}},{{\mathbf s}})$ is continuous in $\Omega\times\Omega$. We have from (\[rprTht1\]) for ${{\mathbf x}},{{\mathbf s}},{{\mathbf s}}+{{\mathbf h}}\in\Omega$: $$\begin{aligned}
&&\hskip-5mm|\Theta({{\mathbf x}},{{\mathbf s}}+{{\mathbf h}})-\Theta({{\mathbf x}},{{\mathbf s}})|=\\
&&\hskip-5mm\left|\int_\Omega\tilde\Theta({{\mathbf x}},\xi)
({{\mathcal E}}_d(\xi-{{\mathbf s}}-{{\mathbf h}}-{{\bf m}}_l,\gamma_0)-{{\mathcal E}}_d(\xi-{{\mathbf s}}-{{\bf m}}_l,\gamma_0))\,d\xi\right|\le
\|\tilde\Theta(\cdot,\cdot)\|_{L_\infty(\Omega\times\Omega)}\\
&&\hskip-5mm\times\max\{\|1\|_{q^\prime},\,\|V\|_{q^\prime}\}
\|{{\mathcal E}}_d(\cdot-{{\mathbf s}}-{{\mathbf h}}-{{\bf m}}_l,\gamma_0)-{{\mathcal E}}_d(\cdot-{{\mathbf s}}-{{\bf m}}_l,\gamma_0)\|_q.\end{aligned}$$ Since by claim (ii) of Lemma \[lmcnvser\] the function ${{\mathcal E}}_{{\bf m}}(\cdot,\cdot)={{\mathcal E}}_d(\cdot-\cdot-{{\bf m}},\gamma_0)$ belongs to the class $L_{q,c}(\Omega\times\Omega)$, then the latter estimate imply that the function $\Theta({{\mathbf x}},{{\mathbf s}})$ is continuous w.r. to ${{\mathbf s}}$ at each point ${{\mathbf s}}\in\Omega$ uniformly w.r. to ${{\mathbf x}}\in\Omega$. In the analogous manner we prove, using (\[rprTht2\]), that for any fixed ${{\mathbf s}}\in\Omega$ the function $\Theta({{\mathbf x}},{{\mathbf s}})$ is continuous in $\Omega$ w.r. to ${{\mathbf x}}$. These circumstances mean that the function $\Theta({{\mathbf x}},{{\mathbf s}})$ is continuous in $\Omega\times\Omega$. The lemma is proven.
We now turn to the main result of this section.
\[prpwkrn\] If $d\ge 4$, $V\in L_{q^\prime}(\Omega)$ with $q^\prime>\frac{d}{2}$, $R({{\mathbf p}})=(H({{\mathbf p}})+\gamma_0^2)^{-1}$ and the natural $l$ is defined by (\[defl\]), (\[deftht\]), then for a large enough $\gamma_0$ the operator $K({{\mathbf p}})=(R({{\mathbf p}}))^l$ has the integral kernel $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ such that $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$.
By Proposition \[prNeuser\], for a large enough $\gamma_0$ and ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ the operator maps the set ${{\mathcal H}}_{{{\mathbf p}},\infty}$ into itself and for any $f\in{{\mathcal H}}_{{{\mathbf p}},\infty}$ $R({{\mathbf p}})f=\int_\Omega G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})f({{\mathbf s}})\,d{{\mathbf s}}$, where $$\label{serofcmp}
G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})+\left(\sum_{k=1}^\infty
(G_0(\cdot,\cdot,{{\mathbf p}})V(\cdot))^{\circ k}\circ
G_0(\cdot,\cdot,{{\mathbf p}})\right)({{\mathbf x}},{{\mathbf s}})$$ and the latter series converges in the $L_{c,q}(\Omega\times\Omega)$-norm uniformly w.r. to ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$. Then for any $f\in{{\mathcal H}}_{{{\mathbf p}},\infty}$ $$\label{intpwofR}
K({{\mathbf p}})f=\int_\Omega K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})f({{\mathbf s}})\,d{{\mathbf s}},$$ where $$\label{pwkrn}
K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=\left(G(\cdot,\cdot,{{\mathbf p}})^{\circ l}\right)({{\mathbf x}},{{\mathbf s}}).$$ Let us represent (\[serofcmp\]) in the form: $$\begin{aligned}
\label{rpserofcmp}
&&G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})=G_0({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})+\left(\sum_{k=1}^{l-1}
(G_0(\cdot,\cdot,{{\mathbf p}})V(\cdot))^{\circ k}\circ
G_0(\cdot,\cdot,{{\mathbf p}})\right)({{\mathbf x}},{{\mathbf s}})+\nonumber\\
&&((G_0(\cdot,\cdot,{{\mathbf p}})V(\cdot))^{\circ (l-1)}\circ
G_0(\cdot,\cdot,{{\mathbf p}})\circ V(\cdot)G(\cdot,\cdot,{{\mathbf p}}))({{\mathbf x}},{{\mathbf s}}).\end{aligned}$$ On the other hand, by claim (i) of Lemma \[lmcnvser\] the representation (\[dfG0pr\]) is valid, in which $G_0^\prime({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$ and ${{\mathcal N}}:=\{{{\bf m}}\in{{\mathbf Z}}^d\;|\;|{{\bf m}}|\le\sqrt{d}\}$. Substituting (\[dfG0pr\]) into (\[rpserofcmp\]) and using Corollary \[corsymm\] and Lemma \[lmcomp\], we obtain: $$\begin{aligned}
&&\hskip-8mm G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim
\sum_{{{\bf m}}\in{{\mathcal N}}}{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}},\gamma_0)\exp(i2\pi{{\mathbf p}}\cdot{{\bf m}})+\\
&&\hskip-8mm\sum_{k=1}^{l-2}\;\sum_{{{\bf m}}_1,\dots,{{\bf m}}_{k+1}\in{{\mathcal N}}}
\exp\left(i2\pi{{\mathbf p}}\cdot\sum_{j=1}^{k+1}{{\bf m}}_j\right)\left(\circ_{j=1}^k
{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_j,\gamma_0)V(\cdot)\right)\circ\\
&&\hskip-8mm{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_{k+1},\gamma_0)({{\mathbf x}},{{\mathbf s}})+\sum_{{{\bf m}}_1,\dots,{{\bf m}}_l\in{{\mathcal N}}}
\exp\left(i2\pi{{\mathbf p}}\cdot\sum_{j=1}^l{{\bf m}}_j\right)\Theta_{{{\bf m}}_1,\dots,{{\bf m}}_l}({{\mathbf x}},{{\mathbf s}})+\\
&&\hskip-8mm\sum_{{{\bf m}}_1,\dots,{{\bf m}}_l\in{{\mathcal N}}}
\exp\left(i2\pi{{\mathbf p}}\cdot\sum_{j=1}^l{{\bf m}}_j\right)\tilde\Theta_{{{\bf m}}_1,\dots,{{\bf m}}_l}(\cdot,\cdot)\circ
V(\cdot)G(\cdot,\cdot,{{\mathbf p}})({{\mathbf x}},{{\mathbf s}}),\end{aligned}$$ where $\Theta_{{{\bf m}}_1,\dots,{{\bf m}}_l}({{\mathbf x}},{{\mathbf s}})$ and $\tilde\Theta_{{{\bf m}}_1,\dots,{{\bf m}}_l}({{\mathbf x}},{{\mathbf s}})$ are functions of the form described in the formulation of Lemma \[lmcmpEc\], hence they are continuous in $\Omega\times\Omega$. Then by Lemma \[lmcomp\] and Corollary \[corsymm\], $\Theta_{{{\bf m}}_1,\dots,{{\bf m}}_l}(\cdot,\cdot)\circ
V(\cdot)G(\cdot,\cdot,{{\mathbf p}})({{\mathbf x}},{{\mathbf s}})\sim 0$. Hence $$\begin{aligned}
&&\hskip-8mm G({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim
\sum_{{{\bf m}}\in{{\mathcal N}}}{{\mathcal E}}_d({{\mathbf x}}-{{\mathbf s}}-{{\bf m}},\gamma_0)\exp(i2\pi{{\mathbf p}}\cdot{{\bf m}})+\\
&&\hskip-8mm\sum_{k=1}^{l-2}\;\sum_{{{\bf m}}_1,\dots,{{\bf m}}_{k+1}\in{{\mathcal N}}}
\exp\left(i2\pi{{\mathbf p}}\cdot\sum_{j=1}^{k+1}{{\bf m}}_j\right)\left(\circ_{j=1}^k
{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_j,\gamma_0)V(\cdot)\right)\circ\\
&&\hskip-8mm{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_{k+1},\gamma_0)({{\mathbf x}},{{\mathbf s}}).\end{aligned}$$ Then we obtain from (\[pwkrn\]), claim (i) of Lemma \[lmcnvser\] and Lemma \[lmcomp\]: $$\begin{aligned}
&&\hskip-8mm K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim\\
&&\hskip-8mm
\left(\sum_{{{\bf m}}\in{{\mathcal N}}}{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}},\gamma_0)\exp(i2\pi{{\mathbf p}}\cdot{{\bf m}})+
\sum_{k=1}^{l-2}\;\sum_{{{\bf m}}_1,\dots,{{\bf m}}_{k+1}\in{{\mathcal N}}}
\exp\left(i2\pi{{\mathbf p}}\cdot\sum_{j=1}^{k+1}{{\bf m}}_j\right)\times\right.\\
&&\hskip-8mm \left.\left(\circ_{j=1}^k
{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_j,\gamma_0)V(\cdot)\right)\circ
{{\mathcal E}}_d(\cdot-\cdot-{{\bf m}}_{k+1},\gamma_0)\right)^{\circ l}({{\mathbf x}},{{\mathbf s}}).\end{aligned}$$ The latter expression is a trigonometric polynomial w.r.t. ${{\mathbf p}}$, whose coefficients are kernels satisfying the condition of Lemma \[lmcmpEc\], hence they are continuous in $\Omega\times\Omega$. Hence $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\sim 0$. Since for any fixed ${{\mathbf p}}\in\Pi_{\frac{\gamma_0}{2}}$ the operator $K({{\mathbf p}})$ is bounded in $L_2(\Omega)$, the operator with the continuous integral kernel $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ in the right hand side of (\[intpwofR\]) is bounded in $L_2(\Omega)$ too and the set ${{\mathcal H}}_{{{\mathbf p}},\infty}$ is dense in $L_2(\Omega)$, then the representation (\[intpwofR\]) is valid for any $f\in L_2(\Omega)$. The proposition is proven.
Proof of Theorem \[thmainApp\] {#subsec:proofmainclaim}
------------------------------
\(i) Let us choose $\gamma_0>0$ and consider the domain $\Pi_{\gamma_0/2}=\{{{\mathbf p}}\in {{\mathbf C}}^d:\;|\Im({{\mathbf p}})|\le\gamma_0/2\}$. Furthermore, let us take $l=2$ for $d\le 3$ and assume that $l$ is defined by (\[defl\]), (\[deftht\]) for $d\ge 4$. Like in [@Wil] (Lemmas 3.6 and 3.7), it is easy to show that if $\gamma_0$ is large enough, for ${{\mathbf p}}\in{{\mathbf R}}^d$ the number $\nu({{\mathbf p}})=(\lambda({{\mathbf p}})+\gamma_0^2)^l$ is a singular value of the operator $K({{\mathbf p}})=(R({{\mathbf p}}))^l\;(R({{\mathbf p}})=(H({{\mathbf p}})+\gamma_0^2)^{-1})$ with the eigenfunction $b({{\mathbf x}},{{\mathbf p}})$. This means that for ${{\mathbf p}}\in{{\mathbf R}}^d$ the equality is valid: $$\label{eqsingvl}
b(\cdot,{{\mathbf p}})=\nu({{\mathbf p}})K({{\mathbf p}})b(\cdot,{{\mathbf p}}),$$ By Lemma 3.7 from [@Wil] (for $d=3$) and Proposition \[prpwkrn\] (for $d\ge 4$), if $\gamma_0$ is large enough, for each ${{\mathbf p}}\in\Pi_{\gamma_0/2}$ $K({{\mathbf p}})$ is an operator with a continuous integral kernel $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ such that the mapping ${{\mathbf p}}\rightarrow K(\cdot,\cdot,{{\mathbf p}})\in C(\Omega\times\Omega)$ is holomorphic in $\Pi_{\gamma_0/2}$. Observe that this property is valid also for $d\le 2$, because all the arguments used in [@Wil] are true also in this case. Then for ${{\mathbf p}}\in{{\mathbf R}}^d$ the equality (\[eqsingvl\]) acquires the form: $$\label{rprex}
b({{\mathbf x}},{{\mathbf p}})=\int_\Omega\nu({{\mathbf p}})K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})b({{\mathbf s}},{{\mathbf p}})\,d{{\mathbf s}}.$$ This equality and the inclusion $b(\cdot,{{\mathbf p}})\in L_2(\Omega)$ imply that $b(\cdot,{{\mathbf p}})\in C(\Omega)$. Claim (i) is proven.
\(ii) Let us choose $\gamma_0$ such that ${{\mathcal O}}({{\mathbf p}}_0)\subset\Pi_{\frac{\gamma_0}{2}}$. In view of the properties of $K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})$ mentioned above, the mapping ${{\mathbf p}}\rightarrow K({{\mathbf p}})\in{{\mathcal B}}(L_2(\Omega))$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$. Taking into account that the function $\nu({{\mathbf p}})$ and the mapping ${{\mathbf p}}\rightarrow b(\in,{{\mathbf p}})\in L_2(\Omega)$ are holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$, we obtain by the principle of analytic continuation that the equality (\[eqsingvl\]) is valid for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ Furthermore, the holomorphy in ${{\mathcal O}}({{\mathbf p}}_0)$ of $\nu$, of ${{\mathbf p}}\rightarrow K({{\mathbf x}},{{\mathbf s}},{{\mathbf p}})\in C(\Omega\times\Omega)$ and of ${{\mathbf p}}\rightarrow b({{\mathbf x}},{{\mathbf p}})\in L_2(\Omega)$ imply that for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ the function $\nu({{\mathbf p}})K({{\mathbf x}},\cdot,{{\mathbf p}})b(\cdot,{{\mathbf p}})$ belongs to the class $C(\Omega,L_2(\Omega))$ and the mapping ${{\mathbf p}}\rightarrow\nu({{\mathbf p}})K({{\mathbf x}},\cdot,{{\mathbf p}})b(\cdot,{{\mathbf p}})\in
C(\Omega,L_2(\Omega))$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$. These circumstances and equality (\[rprex\]) imply claim (ii) of the theorem.
Proof of Corollary \[cormainApp\] {#subsec:proofcormainclaim}
---------------------------------
\(i) Consider the family of operators $K({{\mathbf p}})$ defined in the proof of Theorem \[thmainApp\]. Since $\lambda_0$ is a simple eigenvalue of the operator $H({{\mathbf p}}_0)$, then $\nu_0=(\lambda({{\mathbf p}}_0)+\gamma_0^2)^l$ is a simple singular number of the operator $K({{\mathbf p}}_0)$. Let $b_0({{\mathbf x}})\neq
0$ be an eigenfunction of $K({{\mathbf p}}_0)$, corresponding to $\nu_0$ (hence it is an eigenfunction of $H({{\mathbf p}}_0)$, corresponding to $\lambda_0$). Then, as it has been proved in [@Bau] (Corollary 1, p. 376), there exists a neighborhood ${{\mathcal O}}({{\mathbf p}}_0)\subset{{\mathbf C}}^d$ of ${{\mathbf p}}_0$ such that in it there is a branch $\nu({{\mathbf p}})$ of singular numbers of the family $K({{\mathbf p}})\;({{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0))$ having the properties: $\nu({{\mathbf p}}_0)=\nu_0$, $\nu({{\mathbf p}})$ is simple for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$, the function $\nu({{\mathbf p}})$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$ and the corresponding eigenprojections $Q({{\mathbf p}})$ of $K({{\mathbf p}})$ have the property: the mapping ${{\mathbf p}}\rightarrow Q({{\mathbf p}})\in{{\mathcal B}}(L_2(\Omega))$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$. Hence the eigenfunctions $\tilde
b(\cdot,{{\mathbf p}})=Q({{\mathbf p}})b_0$ of $K({{\mathbf p}})$, corresponding to $\nu({{\mathbf p}})$, have the property: the mapping ${{\mathbf p}}\rightarrow \tilde b({{\mathbf x}},{{\mathbf p}})\in
L_2(\Omega)$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$. Furthermore, we can restrict the neighborhood ${{\mathcal O}}({{\mathbf p}}_0)$ such that $\tilde
b(\cdot,{{\mathbf p}})\neq 0$ for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$. Hence by Theorem \[thmainApp\], $\tilde b({{\mathbf x}},{{\mathbf p}})\in C(\Omega)$ for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ and the mapping ${{\mathbf p}}\rightarrow \tilde b({{\mathbf x}},{{\mathbf p}})\in
C(\Omega)$ is holomorphic in ${{\mathcal O}}({{\mathbf p}}_0)$. Observe that $\nu({{\mathbf p}})$ is real for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$, because in this case $K({{\mathbf p}})$ is self-adjoint. Recall that for any ${{\mathbf p}}\in{{\mathbf R}}^d$ $JH({{\mathbf p}})J=H(-{{\mathbf p}})$, where $J$ is the conjugation operator $(Jf)(f)({{\mathbf x}}):=\overline
{f({{\mathbf x}})}$. Hence $JK({{\mathbf p}})J=K(-{{\mathbf p}})$ for any ${{\mathbf p}}\in{{\mathbf R}}^d$, therefore in this case $\sigma(K({{\mathbf p}}))=\sigma(K(-{{\mathbf p}}))$ and for any $\mu\in\sigma(K({{\mathbf p}}))$ the corresponding eigenprojections $Q_\mu$ of $K({{\mathbf p}})$ and $\tilde Q_\mu$ of $K(-{{\mathbf p}})$ are connected in the following manner: $\tilde Q_\mu=JQ_\mu J$. In particular, for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)$ the number $\nu({{\mathbf p}})$ is a simple characteristic number of $K(-{{\mathbf p}})$ and the operator $JQ({{\mathbf p}})J$ is the eigenprojection of $K(-{{\mathbf p}})$ corresponding to $\nu({{\mathbf p}})$, hence $\overline{\tilde
b({{\mathbf x}},{{\mathbf p}})}$ is an eigenfunction of $K(-{{\mathbf p}})$ corresponding to $\nu({{\mathbf p}})$. These circumstances and the arguments used above imply that the mapping ${{\mathbf p}}\rightarrow\overline{\tilde b({{\mathbf x}},{{\mathbf p}})}\in
C(\Omega)$ is real analytic in ${{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$. Therefore the function $\Vert b(\cdot,{{\mathbf p}})\Vert_2=\Big(\int_{\Omega}\tilde
b({{\mathbf s}},{{\mathbf p}})\overline{\tilde b({{\mathbf s}},{{\mathbf p}})}\,d{{\mathbf x}}\Big)^{1/2}$ is real analytic in ${{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$. Hence the mapping ${{\mathbf p}}\rightarrow
b({{\mathbf x}},{{\mathbf p}})=\frac{\tilde b({{\mathbf x}},{{\mathbf p}})}{\Vert\tilde b(\cdot,{{\mathbf p}})\Vert_2}\in
C(\Omega)$ is real-analytic in ${{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$. Returning from the family $K({{\mathbf p}})$ to the family $H({{\mathbf p}})$, we obtain the desired claim (i) of the corollary.
\(ii) The claim follows from the representation $$(Q({{\mathbf p}})f)({{\mathbf x}})=(f,\,b(\cdot,{{\mathbf p}}))_2b({{\mathbf x}},{{\mathbf p}})=\int_\Omega
b({{\mathbf x}},{{\mathbf p}})\overline{b({{\mathbf s}},{{\mathbf p}})}f({{\mathbf s}})\,d{{\mathbf s}}\quad (f\in L_2(\Omega))$$ and the results obtained in the process of the proof of claim (i).
\(iii) Since for any ${{\mathbf p}}\in{{\mathcal O}}({{\mathbf p}}_0)\cap{{\mathbf R}}^d$, the eigenvalue $\lambda({{\mathbf p}})$ is simple, then another choice $b_1({{\mathbf x}},{{\mathbf p}})$ of the branch of the eigenfunctions corresponding to $\lambda({{\mathbf p}})$ (with $\Vert b_1(\cdot,{{\mathbf p}}))\Vert_2=1$) is connected with the previous one in the following manner: $b_1({{\mathbf x}},{{\mathbf p}})=e^{i\,\theta({{\mathbf p}})}b({{\mathbf x}},{{\mathbf p}})$ with a real-valued function $\theta({{\mathbf p}})$. Then $b_1({{\mathbf x}},{{\mathbf p}})\overline{b_1({{\mathbf s}},{{\mathbf p}})}=b({{\mathbf x}},{{\mathbf p}})\overline{b({{\mathbf s}},{{\mathbf p}})}$. This proves claim (iii).
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Department of Mathematics3.5truecm
University of Haifa4.9truecm
31905 Haifa, Israel5truecm
[*E-mail address:*]{}5.5truecm
zelenko@math.haifa.ac.il
[^1]: As it was shown in [@K-R], this situation is generic in the sense that all the edges of gaps of the spectrum of the operator $H_0$ are simple for a dense $G_\delta$-set of periodic potentials $V\in
L_\infty(\Omega)$.
[^2]: For $d=1$ the gaps of the spectrum of the operator $H_0$ are or $(-\infty,\lambda_0)$, or $(\lambda_k,\lambda_{k+1})$, or $(\mu_k,\mu_{k+1})\;(k=0,1,2,\dots)$, where $\lambda_k$ and $\mu_k$ are the eigenvalues of the operators $H(0)$ and $H(\pi)$ respectively, and $\lambda_0<\mu_0\le\mu_1<\lambda_1\le\lambda_2<\dots$
[^3]: In [@S1] and [@Bi2] the perturbation $W$ is not bounded operator in general, it is supposed to be only relatively compact w.r.t. $H_0$ in the sense of quadratic forms. But in the present paper we consider for the simplicity only the case of a bounded perturbation and we think that all our results can be obtained without difficulties under a more general assumption.
|
---
address: |
$^1$ Aix-Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288 Marseille, France\
\[-2mm\] $^2$ Department of Mathematics and Namur Center for Complex Systems-naXys, University of Namur,\
\[-2mm\] 8 rempart de la Vierge, 5000 Namur, Belgium\
\[-2mm\] $^3$ Center for Nonlinear Theory and Applications, Shenyang Aerospace University, Shenyang 110136, China
author:
- ' M. Belger$^1$, S. De Nigris[^1] $^2$, X. Leoncini[^2] $^{1,3}$'
---
Introduction
============
Macroscopic chaotic behavior is often linked to out-of-equilibrium states, one of the most prominent display of such phenomenon is most certainly turbulence. The resulting chaotic or turbulent states result from various macroscopic instabilities and bifurcations, and their persistence is usually driven by strong gradients or energy fluxes. When considering isolated systems with many degrees of freedom, some similar behavior can be found, but typically it is a transient during which, starting from a given initial condition, the system relaxes to some thermodynamical equilibrium[@FPU55]. Microscopic “molecular” chaos plays there an important role for relaxation; however, in the equilibrium state, macroscopic variables are at rest, despite the microscopic chaos. It is nevertheless possible to extend this transient state: indeed in recent years there has been an extensive study of the so-called quasi-stationary states (QSSs), that emerge after a violent relaxation in systems with long-range interactions [@Dauxois_book2002; @Campa09; @Campa_book2014; @Levin2014]. These states have the peculiarity that their lifetime diverges with the number of constituents, so that the limits $N\rightarrow\infty$ and $t\rightarrow\infty$ do not commute. In fact it has been shown that some of these states are non-stationary but can display regular oscillations and, therefore, they represent a different kind of steady state[@Holloway91; @Vandenberg10; @Yamaguchi2011; @Ogawa2014]. Moreover, as can be observed in [@Turchi2011], both the lifetime of the state and the “transient” relaxation time from the QSS to the equilibrium diverge with system size. During these relaxation periods we can expect to observe some long lived but transient chaotic-like features in these isolated states[@Antunes2015]. As mentioned,these transient periods correspond to some kind of relaxation, nevertheless, more recently, persistent chaotic macroscopic behavior in a isolated system has been exhibited . These states occur over a wide range of energy. They were first spotted on systems of rotators evolving on a regular lattice, with a critical range of interaction and number of neighbors[@DeNigris2013a]. Further studies have shown that this behavior occurred as well on so-called lace networks, when the effective network dimension was around $d=2$ [@DeNigris2015]. Studying these systems for different number of constituents $N$ and a fixed density of energy $\varepsilon$, the chaotic behavior of the order parameter was persistent, and it appeared that the width of the fluctuations around its mean value was not changing with $N$, implying an infinite susceptibility over a given range of values of $\varepsilon$. However it was evident, at least qualitatively, that some changes in the characteristic time scales of the fluctuations were present and depended on the system’s sizes.
In this paper we focus on this dependence of the fluctuations time scales with system size, we shall show that the observed scaling $\tau(N)\sim\sqrt{N}$ is different than the typical relaxation time scales observed in QSS, and provide a theoretical explanation of these time scales in the low energy range. The paper is organized as follows: in the first part we describe the considered model and remind the reader of the previously obtained results. We then move on to a a numerical study of the characteristic time scales of the fluctuations by analyzing the frequency spectrum of the measured order parameter, where a scaling behavior $\tau(N)\sim\sqrt{N}$ is clearly exhibited. The presence of a large and broad spectrum allows us to infer that the signal is indeed chaotic. We then perform an analytical study of the thermodynamical wave spectrum at low energies and we indeed confirm the numerically exhibited scaling. This evidence confirms that these chaotic states are not QSS’s and that the chaotic behavior can be expected to be an actual permanent feature or characteristic of these “equilibrium” states.
Description of the model
========================
Originally the model we shall consider was tailored in order to uncover the threshold of a long range interacting system. As such it was inspired from the fact that the so-called $\alpha-$HMF model (see [@Anteneodo98; @Campa01]) displayed similar thermodynamical properties as the mean field model (for $\alpha<1$), also dubbed the HMF model, which over the years has become de facto the paradigmatic model to study and test new ideas when studying long range system. In the $\alpha-$HMF rotators are located on a one-dimensional lattice, and the coupling constant $J_{ij}$ between the spins decreases according to a power-law with the distance between the rotators $J_{ij}\sim|i-j|^{-\alpha}$, so that all rotators are coupled. The initial idea of the proposed model was to consider a range $r$ of neighbors who equally interact with a sharp edge, meaning that $J_{ij}\sim Cst$ if $|i-j|<r$ and $0$ if $|i-j|\ge r$. We set up a window function, but what is important here is that we allow $r$ to be a function of the total number of spins $N$. The range is parametrized using a characteristic exponent $1\le\gamma\le2$, which measures as well the total number of links (interactions) being present in the system. When $\gamma=1$, we are on a one-dimensional chain with a short range interactions (in our case with just nearest neighbors interactions), while when $\gamma=2$, we retrieve the mean field model, with all rotators equally interacting with each other. To get more specific we now present the details of the rotators model placed on a one-dimensional lattice with periodic boundary conditions. The Hamiltonian of the considered system writes
$$H=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2}+\frac{1}{2k}\sum_{i,j}^{N}\epsilon_{i,j}(1-\cos(q_{i}-q_{j}))\:,\label{eq: hamiltonian}$$
where $k$ is the constant number of links (connections) per rotator which scales with $\gamma$ as
*$$k\equiv\frac{1}{N}\sum_{i>j}\epsilon_{i,j}=\frac{2^{2-\gamma}(N-1)^{\gamma}}{N},\label{eq:degree}$$* and is related to the range by the simple relation $k=2r$. The matrix $\epsilon_{i,j}$ is the adjacency matrix, defined as
$$\epsilon_{i,j}=\begin{cases}
1 & if\,\,\left\Vert i-j\right\Vert \leqslant r\\
0 & otherwise
\end{cases}\:,\label{eq:adjacency matrix}$$
where $\left\Vert i-j\right\Vert $ stands for the smallest distance between two site on the one dimensional lattice with periodic boundary conditions. From the Hamiltonian we directly get the equations of the motion of the rotators. $$\begin{aligned}
\dot{q}_{i} & = & p_{i}\label{eq:dynamics-1-1}\\
\dot{p}_{i} & = & -\frac{1}{k}\sum_{,j=1}^{N}\epsilon_{i,j}\sin(q_{i}-q_{j})\:.\end{aligned}$$ A full study of the equilibrium properties of this model has been made in [@DeNigris2013a; @DeNigris2013b]. The order parameter that we monitored is the total magnetization of the system $M$, defined as $$\mathbf{M}=\begin{cases}
M_{x}= & \frac{1}{N}\sum_{i=1}^{N}\cos q_{i}\\
M_{y}= & \frac{1}{N}\sum_{i=1}^{N}\sin q_{i}
\end{cases}=M\begin{cases}
\cos\varphi\\
\sin\varphi
\end{cases}$$ The results are as follows:
![Magnetization versus time, for a fixed density of energy $\varepsilon\approx0.4$ and different values of $\gamma$. The size of the system is $N=2^{13}$. For $\gamma=1.25$ there is no magnetization (the residual magnetization is due to finite size effects see for instance ), for $\gamma=1.75$ we observed a finite almost constant magnetization, while for $\gamma=1.5$ large fluctuations of order one are observed. Simulations have been performed using a time step $\delta t=10^{-3}$. \[fig:Fluctuations\_Gamma\] ](./Figures/M_t_2_8192.jpg){width="9cm"}
- For $\gamma<1.5$ the system behaves as a short range model, meaning that no order parameter emerges in the thermodynamic limit and no phase transition exists. For the short range case ($\gamma=1$), this result is consistent with the predictions of the Mermin-Wagner theorem, which predicts that no order parameter can exist for systems with dimensions $d\le2$, due to the existence of a continuous symmetry group (here the global translation/rotation symmetry $q_{i}\rightarrow q_{i}+\theta$).
- For $\gamma>1.5$ the system behaves like the mean field model, meaning that a second order transition at a critical density of energy of $\varepsilon_{c}=0.75$, is observed. All curves $M_{\gamma}(\varepsilon)$ appear as independent of $\gamma$ and fall on the mean field one.
- For $\gamma=1.5$ for a range of temperatures below the critical energy one, a chaotic state is observed. The magnetization displays steady and large incoherent fluctuations, which do not appear to be dependent on system size, implying an infinite susceptibility. The time dependence of these fluctuations is the subject of this paper. Note also that the transition of the Berezinsky-Kosterlitz-Thouless type has not been detected (see for details [@DeNigris2013a]).
To illustrate the phenomena described, we have plotted in Fig. \[fig:Fluctuations\_Gamma\] the evolution of the order parameter at a fixed density of energy $\varepsilon$ for three different values of $\gamma$ and a system size of $N=2^{13}$. Simulations have been performed using the optimal fifth order symplectic integrator described in [@McLachlan92], and the fast-Fourier transform made use of the FFTW package. We can notice the peculiar regime that appears for $\gamma=1.5$ where the magnetization displays what looks like a macroscopic chaotic behavior.
In the next section we shall study in more detail the temporal behavior of the order parameter in these chaotic states.
Critical slowing down
=====================
Numerical study
---------------
![Time evolution of the magnetization of the system with $\varepsilon\approx0.4$. The system size is here $N=2^{9}$. The final time and data sampling of the simulation is identical to the one performed in Fig. \[fig:Fluctuations\_Gamma\]. We notice that the fluctuations are indeed of the same order, however we can notice that the typical time scale of the fluctuations appear to be faster than in Fig. \[fig:Fluctuations\_Gamma\]. \[fig:Mag\_evolution\_512\] ](./Figures/M_t_512.jpg){width="9cm"}
In this section we study numerically the behavior of the order parameter for different values of $\varepsilon$, $\gamma=1.5$ and different system sizes with the aim of uncovering the timescales characterizing the fluctuations. Indeed we can notice in Fig. \[fig:Mag\_evolution\_512\] that the typical time scale of the fluctuations appears to depend on the system size, as the magnetization fluctuations are much faster for $N=512$ (Fig. \[fig:Mag\_evolution\_512\]) than for $N=8192$ (Fig. \[fig:Fluctuations\_Gamma\]). Also, even though the signal plotted in Fig. \[fig:Mag\_evolution\_512\] looks turbulent, it may just be the consequence of the presence of a few unrelated modes.
![Fourier spectrum of a “chaotic” signal of the order parameter. The considered system size is $N=2^{14}$. We can notice that we obtain a continuous spectrum with some broad peaks, and associated harmonics. The time dependence is typically chaotic, and does not correspond to a quasi-periodic signal.\[fig:Fourier-spectrum\]](Figures/fourier_16384.jpg){width="9cm"}
In order to confirm the chaotic nature of the signal, we decided to analyze its Fourier spectrum. An example of such spectrum is displayed in Fig. \[fig:Fourier-spectrum\]. We can notice that the spectrum is continuous, differently from the one given by a quasi-periodic signal, so it is definitely of the chaotic (turbulent) type. However we can notice as well some broad peaks which are associated to the decreasing harmonics in this signal. Indeed, we can relate these peaks to the typical scale of fluctuations that visually appeared in the figures \[fig:Mag\_evolution\_512\] and \[fig:Fluctuations\_Gamma\].
![Scaling of the localization of the frequency peaks versus system size. The locations of the first three peaks (harmonics) are represented. One notices a global uniform scaling of the different peaks, and a slowing down of the typical fluctuation time. The scaling law shows a decrease of the peak frequencies as $f\sim1/\sqrt{N}$.\[fig:Scaling-of-the\_frequency\]](Figures/logf_logN.jpg){width="9cm"}
In order to determine the scaling with system size, we performed a sequence of numerical simulations, with a fixed density of energy, fixed total time and different system sizes. In these simulations, the initial condition is extracted from a Gaussian distribution for both the $p_{i}$’s and $q_{i}$’s. The signal analysis is performed over the data that has been averaged during the second half of the total simulation’s time. The results are displayed in Fig. \[fig:Scaling-of-the\_frequency\], where the locations of the three first peaks displayed in Fig. \[fig:Fourier-spectrum\] are represented versus system size in a log-log plot. One notices a universal scaling of the typical fluctuation time scale, with all peaks having a frequency that decreases as $f\sim N^{-1/2}$. This scaling was initially not anticipated as one would naturally expect a behavior similar to what has been observed for QSS’s, with a typical lifetime scaling $\tau\sim N^{\alpha}$, with $\alpha=1$ or higher values. The observed scaling $\tau\sim N^{1/2}$ is another confirmation that these chaotic states do not correspond to transient regimes, but are “steady”. We had already run very large time simulations in without noticing any visible change in the dynamics of the order parameter, but a transient with a large value of $\alpha$ could still have been possible.
We now move on to a theoretical hint at the observed scaling law, and the confirmation as well that these are not transient states.
Theoretical analysis
--------------------
In order to perform our analysis we carry out a similar calculation as the one performed in [@DeNigris2013a], that had allowed us to prove that $\gamma=1.5$ was a threshold between the short range and the long range behavior. The method was proposed in a general context and explicitly developed for lattice’s system in [@Leoncini2001]. In order to be more self-consistent we review the method from the start, and apply it to the considered system (\[eq: hamiltonian\]).
As already stated we consider a lattice (in dimension $D=1$ for our system) of $N$ sites with coordinates $x_{i}=1,\cdots,N$. At each site $i$ we have a particle, coupled to some neighbors, each having a momentum $p_{i}$ and conjugate coordinate $q_{i}$. We recall that we shall consider thermodynamical equilibrium properties (even though we are looking at some dynamical properties) so the units are such that the lattice spacing, the Boltzmann constant, and the mass are equal to one. Also from the form of the Hamiltonian (\[eq: hamiltonian\]), a calculation within the canonical ensemble will imply that the $p_{i}$ are distributed according to a Gaussian distribution. Since we are working on a lattice, with periodic boundary conditions, we can represent the momentum as a superposition of Fourier modes: $$p_{i}=\sum_{k=0}^{Nk_{0}}\dot{\alpha}_{k}\cos(kx_{i}+\phi_{k})\:,\label{p}$$ where the wavenumber $k$ is in the reciprocal lattice (an integer multiple of $k_{0}=2\pi/N^{(1/D)}$), the wave amplitude is $\dot{\alpha}_{k}$, and since we want the momenta to be Gaussian distributed variables in the thermodynamic limit, we consider that the random phase $\phi_{k}$ is uniformly distributed on the circle. Therefore, we should, given some conditions on the amplitude, be able to apply the central limit theorem. The momentum set is labeled, using (\[p\]), with the set of phases $\ell\equiv\{\phi_{k}\}$. Note also that this equation can also be interpreted as a change of variables, from $p$ to $\alpha$, with constant Jacobian (the change is linear and we chose an equal number of modes and particles).
Before proceeding, we would like to make some remarks. First, it is clear from the Hamiltonian (\[eq: hamiltonian\]) that we have a translational invariance, which implies that the total momentum of the system is conserved. Since physics should not change we make a simple Galilean transform in order to choose a reference frame where the total momentum is zero. The total momentum is directly linked to the zero mode, so this choice implies thus that we have to take $\dot{\alpha}_{0}=0$. Second, since we know that in the canonical ensemble the variance of $p_{i}$ is fixed and equal to the temperature of the system, we shall assume that the $\dot{\alpha}_{k}$ are all of the same order (we need a large number of relevant modes for the center-limit theorem to apply). Given these assumptions and using the relation $\langle p_{i}^{2}\rangle=\sum\dot{\alpha}_{k}^{2}/2$ (we average over the random phases), we write $\langle p_{i}^{2}\rangle\approx T$ and obtain $\dot{\alpha}_{k}^{2}\approx O[(T/N)]$ (we call this relation the Jeans condition [@Jeans_book]).
We now move on to the associated conjugated variable of $p_{i}$, since we have $\dot{q}_{i}=p_{i}$, we write it as $$q_{i}=\alpha_{0}+\sum_{k=k_{0}}^{Nk_{0}}\alpha_{k}\cos(kx_{i}+\phi_{k})\:,\label{q}$$ where $\alpha_{0}$ is a constant since $\dot{\alpha}_{0}=0$, corresponding to the constant average of the $q_{i}$’s. In order to proceed, since we are below the mean-field critical temperature, we make a low temperature hypothesis: thus, we can assume that neighboring $q_{i}$’s are not too different although no global long range order exists. . Assuming that the difference $q_{i}-q_{j}$ when the rotators interact is small, we expand the Hamiltonian and obtain: $$H=\sum_{i}\frac{p_{i}^{2}}{2}+\frac{1}{4k}\sum_{i,j}\epsilon_{ij}(q_{i}-q_{j})^{2}.$$ Using the previous expressions derived for $q_{i}$ and $p_{i}$ and averaging over the random phases we end up with an effective Hamiltonian $$\frac{\left\langle H\right\rangle }{N}=\frac{1}{2}\sum_{l=1}^{N}\dot{\alpha}_{l}^{2}+\alpha_{l}^{2}(1-\lambda_{l}),\label{alpha hamiltonian}$$ where $$\lambda_{l}=\frac{2}{k}\sum_{m=1}^{k/2}\cos(\frac{2\pi ml}{N})\label{eq:eigen}$$ are the eigenvalues of the adjacency matrix.
We can extract from this a dispersion relation, indeed we have $$\begin{aligned}
\frac{d}{dt}(\frac{\partial\left\langle H\right\rangle }{\partial\dot{\alpha_{l}}}) & = & -\frac{\partial\left\langle H\right\rangle }{\partial\alpha_{l}}\nonumber \\
\ddot{\alpha_{l}} & = & -\omega_{l}^{2}\alpha_{l}\label{eq:Dispersion relation}\end{aligned}$$ As mentioned this computation was already used in in order to show that the critical threshold between short range and long range behavior was $\gamma=1.5$; we used this formalism in order to compute analytically the value of the magnetization in the thermodynamic limit. In the present case, we stress the fact that the dispersion relation (\[eq:Dispersion relation\]) embeds also some dynamical informations since we have access to the typical frequencies that we can expect to find in the system. This dynamical information was not used in previous papers levering this formalism, but nevertheless the understanding of the observed scaling law could provide new avenues for this approach.
We can now use this dynamical feature in order to explain the critical slowing down by monitoring how the spectrum behaves as we change the size of the system, for the specific situation with $\gamma=1.5$, i.e $k\sim\sqrt{N}$. For this purpose, we consider a specific mode $l$; we have $$\begin{aligned}
\omega_{l}^{2} & = & (1-\lambda_{l})\label{eq:frequencies}\\
& = & 1-\frac{1}{k}\left[\frac{\sin\left(\frac{\left(k+1\right)l\pi}{N}\right)}{\sin\left(\frac{l\pi}{N}\right)}-1\right]\:.\end{aligned}$$ In order to proceed we shall consider that $N\rightarrow\infty$, thus $N\gg\sqrt{N}$ , i.e $N\gg k$ and that $l$ is fixed, we can then perform an expansion of the expression (\[eq:frequencies\]), and in order to avoid the first order $\omega_{l}^{2}=0$ result, we shall expand it to third order using $\sin(x)=x-x^{3}/6+o(x^{3}).$ We then obtain (omitting the $o(x^{3})$ notation) $$\begin{aligned}
\omega_{l}^{2} & \approx & \frac{k+1}{k}-\frac{1}{k}\frac{\frac{\left(k+1\right)l\pi}{N}-\frac{\left(k+1\right)^{3}l^{3}\pi^{3}}{6N^{3}}}{\frac{l\pi}{N}-\frac{l^{3}\pi^{3}}{6N^{3}}}\\
& \approx & \frac{k+1}{k}\left[1-\frac{1-\frac{(k+1)^{2}l^{2}\pi^{2}}{6N^{2}}}{1-\frac{l^{2}\pi^{2}}{6N^{2}}}\right]\\
& \approx & \frac{k+1}{k}\left[\frac{(k+1)^{2}l^{2}\pi^{2}}{6N^{2}}-\frac{l^{2}\pi^{2}}{6N^{2}}\right]\\
& \sim & \frac{k^{2}}{N^{2}}\sim\frac{1}{N}\:.\end{aligned}$$ We recover analytically the critical slowing down exhibited numerically in Fig. \[fig:Scaling-of-the\_frequency\] and confirm that the scaling law leads to $\omega\sim1/\sqrt{N}$, and thus characteristic time scales of order $\sqrt{N}$.
Concluding remarks
==================
In this paper we have analyzed the typical time scale $\tau(N$) over which the chaotic behavior (fluctuations) of the order parameter evolves as a function of system size. First after a numerical study, we have exhibited that $\tau(N)\sim\sqrt{N}$. Then this behavior has been afterwards confirmed theoretically, by showing that each of the frequencies, associated to modes of the dual lattice, scaled as $\omega_{k}\sim1/\sqrt{N}$ with system size. The direct consequences of these results go in two directions. First we confirmed the chaotic states observed and discussed in [@DeNigris2013a; @DeNigris2013b; @DeNigris2015] indeed are not a transient state like a QSS and, because of the presence of a large continuous spectrum, we can as well confirm the chaotic nature of the macroscopic behavior in these states, much like a turbulent one. Second, when performing our theoretical analysis using the formalism developed in [@Leoncini2001], we were able to show for the first time that it is possible to uncover some dynamical information from this formalism, and the successful prediction of the scaling law shows that the formalism is adequate to predict some finite size dynamical features of systems with many degrees of freedom with underlying Hamiltonian microscopic dynamics.
As a whole the typical decay of the characteristic time scale has another important consequence:indeed should we consider an $N\rightarrow\infty$ limit, the fluctuations should stop and the system will end up frozen in its initial magnetic state. It is important to comment that still the infinite susceptibility would remain, so the system should remain extremely sensitive to any external perturbation. This critical slowing down with system size has been observed in other types of networks with different structure. Thus, beside confirming the same behavior arises considering lace networks as a substrate, an interesting perspective would be to check if there are any similarities to what has been already reported, and if this phenomenon could be of practical use, like for instance to slow down the waves propagation in some localized regions.
acknowledgements {#acknowledgements .unnumbered}
================
S.D.N and X.L. would like to thank S. Ogawa for fruitful discussions and remarks during the preparation of this manuscript.
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[^1]: email address: denigris.sarah@gmail.com
[^2]: Corresponding author, email address: xavier.leoncini@cpt.univ-mrs.fr
|
---
abstract: 'The complexity of “top-down” string-dual candidates for strongly-coupled Yang-Mills theories and in particular for QCD almost always prohibits their exact analytical or even comprehensive numerical treatment. This impedes both a thorough quantitative analysis and the development of more realistic gravity duals. To mitigate these impediments, we devise an analytical approach to top-down duals on the basis of controlled, uniformly converging high-accuracy approximations for the normalizable string modes corresponding to gauge-theory states. We demonstrate the potential of this approach in Witten’s dual for $3d$ Yang-Mills theory, i.e. in the near-horizon limit of non-extremal $D\text{3}$-branes, compactified on $S^{1}$. We obtain accurate analytical approximations to the bulk modes which satisfy the boundary conditions exactly. On their basis, analytical results for masses, sizes, pole residues and correlation functions of glueball excitations are derived by spectral methods. These approximations can be systematically improved and rather universally adapted to more complex gravity duals.'
author:
- 'Hilmar Forkel$^{1,2}$'
title: 'Analytical approach to the $D3$-brane gravity dual for $3d$ Yang-Mills theory'
---
Introduction
============
Since their inception nearly two decades ago, gauge/string dualities keep generating profound insights into strongly-coupled gauge theories and have led to new perspectives and groundbreaking results over a broad range of physics subfields [@revs].
Among the pioneering and most intensely studied application areas for these holographic dualities is QCD, the theory of the strong interactions, in its infrared regime. Hence, various increasingly sophisticated and realistic string-theory-based (i.e. “top-down”) dual candidates for strongly-coupled large-$N$ Yang-Mills theories and QCD were constructed. In order to be at least halfway accessible by contemporary solution methods, most of them use the gravity approximation for the string dynamics [^1]. This requires weak bulk spacetime curvatures and entails several well-understood limitations [@revs] e.g. in the description of asymptotic freedom or linear Regge trajectories.
Even within these restrictions, however, the technical (and sometimes conceptual) complexities encountered when attempting to calculate gauge-theory quantities beyond approximate mass spectra can be quite formidable. Part of these problems arise from the absence of analytical solutions for even the simplest bulk quantities in practically all top-down duals. Moreover, numerical approaches provide no fully satisfactory alternative. The numerical evaluation of more complex amplitudes like correlators is e.g. typically hampered by renormalization issues and often requires specifically designed, non-universal methods. In addition, numerical results reveal less of the underlying mathematical structure whose patterns and systematics often provide useful insights and intuitive guidance.
To improve this situation, we intend to develop and adapt more sophisticated analytical approximation methods to deal with top-down gravity duals. They should be controlled (i.e. systematically improveable) and as transparent and universally applicable as possible. In particular, they should provide accurate analytical expressions for a complete set of dual quantities from which analytical results for all desired gauge-theory observables can be derived. In this letter we propose the whole tower of (normalizable) dual string modes, corresponding to the “hadrons” of interest, as a promising such set.
In the following, we demonstrate the feasibility of our approach in Witten’s top-down gravity dual for three-dimensional Yang-Mills theory (YM$_{3}$) [@wit98; @bro00; @csa99] which emerges from $N$ parallel, non-extremal $D3$ branes of type IIB string theory in the near-horizon limit, with one of its flat dimensions compactified on a circle. This dual model suggests itself for our purposes because it is technically simple and reproduces important properties of the gluonic sector of QCD (including confinement).
Gravity dual for $3d$ Yang-Mills theory
=======================================
In Ref. [@wit98], Witten proposed a string dual for $SU\left(N\right)$ Yang-Mills theory in $d=2+1$ dimensions at large $N$ and for large ’t Hooft coupling $\lambda=g_{YM}^{2}N$. This dual is based on the AdS/CFT correspondence between $\mathcal{N}=4$ superconformal Yang-Mills theory in $d=3+1$ dimensions and a stack of $N$ coincident $D3$-brane solutions of type IIB string theory in 10 dimensions (with R-R charge **$N$**), or for small spacetime curvature by IIB supergravity [@mal97]. Conformal symmetry and supersymmetry are broken by generalizing the correspondence to non-extremal $D3$ black-brane solutions with **** Hawking temperature $T_{H}$.
One then decouples the near-horizon or “throat” region from the 10d Minkowski gravity (staying outside the horizon) and analytically continues to Euclidean time $\tau=-it$. This requires to periodically identify $\tau+\beta_{H}\simeq\tau$ on the circle $S^{1}\left(R_{\tau}\right)$ and to remove a cone singularity in the $\left(\tau,r\right)$ subspace by fixing the Hawking temperature to $T_{H}=\left(\pi R\right)^{-1}$ with the horizon at $r_{0}\simeq\pi T_{H}R^{2}=R$. In the Fefferman-Graham coordinate $z=R^{2}/r$, the metric then becomes $$ds^{2}=g_{MN}dx^{M}dx^{N}=\frac{R^{2}}{z^{2}}\left[\left(1-\frac{z^{4}}{R^{4}}\right)d\tau^{2}+\sum_{i=1,2,3}dx_{i}^{2}+\left(1-\frac{z^{4}}{R^{4}}\right)^{-1}dz^{2}\right]+R^{2}d\Omega_{5}^{2}\label{eq:ads-bb}$$ where $x_{3}$ is reinterpreted as the Euclidean time. (The associated dilaton is constant.)
On the gauge-theory side the above procedure corresponds to compactifying the Euclidean $\mathcal{N}=4$ $SU\left(N\right)$ Super-Yang-Mills theory (i.e. the low-energy theory of the $N$ coinciding $D3$ branes) on $\mathbb{R}^{3}\times S_{\tau}^{1}$, with anti-periodic, supersymmetry-breaking boundary conditions for the fermions around $S_{\tau}^{1}$. Hence the adjoint fermions acquire tree-level masses $\sim1/R_{\tau}$ (thus breaking conformal symmetry) and render the scalar matter massive at the one-loop level. For sufficiently small $R_{\tau}$ the adoint matter decouples and only the gauge fields on $\mathbb{R}^{3}$ remain massless (protected by the gauge symmetry), with their dynamics approaching YM$_{3}$ (with an UV cutoff $\sim1/R_{\tau}$).
The background geometry (\[eq:ads-bb\]) is dual to the gauge-theory vacuum. Its scalar excitations, the $0^{++}$ “glueballs”, are associated with the (gauge-invariant) operator $$\mathcal{O}{}_{0^{++}}\left(x\right)=tr\left\{ F_{\mu\nu}F^{\mu\nu}\right\} \label{eq:GbOp}$$ which in turn is dual to a bulk field with identical quantum numbers. The corresponding deformation $S\rightarrow S+\int d^{d}x\varphi^{\left(s\right)}\left(x\right)\mathcal{O}{}_{0^{++}}\left(x\right)$ of the gauge-theory action changes the Yang-Mills coupling $g_{YM}^{2}=g_{s}=\exp\varphi^{\left(s\right)}$ and thus the (boundary) value $\varphi^{\left(s\right)}$ of the dilaton field $\varphi$. This indicates that $\varphi$ is the bulk field dual to $\mathcal{O}{}_{0^{++}}$. $ $
The GKPW relation [@gub98] then identifies the generating functional of the glueball correlation functions as $$Z\left[\varphi^{\left(s\right)}\right]=\left\langle e^{\int d^{4}x\varphi^{\left(s\right)}\left(x\right)\mathcal{O}{}_{0^{++}}\left(x\right)}\right\rangle \overset{N\gg\lambda\gg1}{\longrightarrow}e^{-S^{\left(onsh\right)}\left[\varphi^{\left(s\right)}\left(x\right)=\varphi\left(z=0,x\right)\right]}\label{eq:gfun}$$ with the string partition function in the classical limit. The latter is determined by the on-shell action $S^{\left(onsh\right)}$, i.e. by the classical dilaton action (after dimensional reduction on $S^{5}$, i.e. for $\varphi$ restricted to SO$\left(6\right)$ singlets) $$\begin{aligned}
S_{0,m}\left[\varphi\right] & =\frac{1}{2\kappa^{2}}\int d^{5}x\sqrt{\left|g\right|}\left[g^{MN}\partial_{M}\varphi\partial_{N}\varphi+m_{5}^{2}\varphi^{2}+O\left(\alpha^{'n}\right)\right]\label{eq:DilatonAction}\end{aligned}$$ ($\kappa^{2}/2$ is the five-dimensional Newton constant) in the dual background geometry (\[eq:ads-bb\]) evaluated at the extrema of Eq. (\[eq:DilatonAction\]). The latter are the solutions of the Laplace-Beltrami equation $$\left(g^{\tau\tau}\partial_{\tau}^{2}+\Delta_{z}+g^{ij}\partial_{i}\partial_{j}-m_{5}^{2}\right)\varphi\left(x,\tau,z\right)=\int d^{3}ke^{iqx}\left(\Delta_{z}-a^{-2}q^{2}-m_{5}^{2}\right)\varphi\left(q;z\right)=0\label{eq:LBeq}$$ after dimensional reduction on $S^{1}\left(R_{\tau}\right)$, i.e. for $\tau$-independent $\varphi$. Here $\varphi\left(q;z\right)$ is the $3d$ Fourier transform of the flat boundary coordinates of the dilaton field and $$\begin{aligned}
\Delta_{z} & \equiv\frac{1}{\sqrt{g}}\partial_{z}\sqrt{g}g^{zz}\partial_{z}=\frac{z^{5}}{R^{5}}\partial_{z}\left(\frac{R^{3}}{z^{3}}-\frac{z}{R}\right)\partial_{z}\end{aligned}$$ is the radial Laplacian. A Frobenius analysis shows that the normalizable (non-normalizable) solutions of Eq. (\[eq:LBeq\]) behave as $z^{d}$ ($z^{\Delta-d}$) near the boundary. For the (classical) conformal dimension $\Delta=4$ of the interpolator (\[eq:GbOp\]) the AdS/CFT dictionary implies $m_{5}^{2}R^{2}=\Delta\left(\Delta-d\right)\overset{\Delta=d=4}{=}0$, as expected for massless dilaton modes.
As motivated in the introduction, we intend to construct approximate solutions of the radial dilaton equation $$\left[a^{2}\left(z\right)\Delta_{z}-q^{2}\right]\varphi\left(q;z\right)=0\label{eq:radeq0}$$ which are normalizable and regular at the horizon. Normalizability selects the subleading solutions which obey the UV boundary condition $\varphi\left(q;z\right)\overset{z\rightarrow0}{\longrightarrow}z^{4}$. These solutions have discrete eigenvalues $q_{n}^{2}=-m_{n}^{2}$ for $n=1,2,...$ where $m_{n}$ is the mass of the $n$-th radial glueball excitation [@wit98]. More explicitly, the $n$-th eigensolution $\psi_{n}\left(z\right):=\varphi\left(q_{n};z\right)$ solves the radial equation $$\left[\partial_{z}^{2}+t_{1}\left(z\right)\partial_{z}+m_{n}^{2}t_{2}\left(z\right)\right]\psi_{n}\left(z\right)=0\label{eq:radeq}$$ (a 2nd-order Fuchsian differential equation) whose coefficient functions $$t_{1}\left(z\right)=-\frac{z^{4}+3R^{4}}{z\left(R^{4}-z^{4}\right)},\qquad t_{2}\left(z\right)=\frac{R^{4}}{R^{4}-z^{4}}$$ have four regular singular points at $z=0,\pm R,\infty$. Hence no general, analytical solutions for Eq. (\[eq:radeq\]) are known.
Uniform analytical approximations to the string modes
=====================================================
In this section, we will derive controlled, uniformly convergent analytical approximations to all eigenmodes $\psi_{n}$ of Eq. (\[eq:radeq\]) under the stated boundary conditions. Since we are dealing with a Sturm-Liouville problem which is not of the standard Schr�dinger type (see below), we will rely on a generalization of the WKB method (GWKB) [^2] and just sketch the main steps here, relegating more details to a subsequent, more comprehensive publication [@for16].
To impose the boundary conditions at the singular points $z=0,R$, it is convenient to transform to the new radial coordinate $$y\left(z\right)=\ln\left(\frac{R^{2}}{z^{2}}-1\right)\label{eq:y(z)}$$ which maps the horizon to $y=-\infty$ and the AdS boundary to $y\rightarrow+\infty$. Equation (\[eq:radeq\]) is thereby converted into the quasi-Schr�dinger form $$\left[-\partial_{y}^{2}+\frac{1}{\epsilon^{2}}V_{1}\left(y\right)+V_{2}\left(y\right)\right]u\left(y\right)=0.\label{eq:radeq2}$$ with the potentials $$\begin{aligned}
V_{1}\left(y\right) & =-\frac{e^{y}}{\left(1+e^{y}\right)\left(2+e^{y}\right)}\label{eq:v1}\end{aligned}$$ and $$V_{2}\left(y\right)=\frac{1}{2}\frac{\left(3+4e^{y}\right)e^{y}}{\left(1+e^{y}\right)\left(2+e^{y}\right)}-\frac{1}{4}\frac{\left(3+2e^{y}\right)^{2}e^{2y}}{\left(1+e^{y}\right)^{2}\left(2+e^{y}\right)^{2}}\label{eq:v2}$$ where the eigenvalues appear in the combination $\epsilon^{-2}=\left(m^{2}R^{2}\right)/4$.
A global and uniform (asymptotic) expansion of the solutions of Eq. (\[eq:radeq2\]) can be organized in the familiar form $$u\left(y\right)\sim\exp\left(\frac{1}{\epsilon}\sum_{k=0}^{\infty}S_{k}\left(y\right)\epsilon^{k}\right),\qquad\epsilon\rightarrow0+\label{eq:GWKBansatz}$$ whose $\epsilon$ dependence implements a standard version of “distinguished balance” [@ben78]. The leading-order solution is thus of $O\left(\epsilon^{0}\right)$. Equation (\[eq:radeq2\]) implies that the potentials $V_{1,2}$ contribute at different orders to the asymptotic expansions. This requires a non-standard two-scale power counting analysis and the corresponding GWKB matching procedure around the turning points of Eqs. (\[eq:v1\]) and (\[eq:v2\]).
For approximations that are sufficiently accurate to satisfy both the **** regularity (**$u\left(-\infty\right)=c$)** and normalizability ($u\left(+\infty\right)=\lim_{y\rightarrow\infty}ce^{-y}$) boundary conditions exactly, one has to push the expansion (\[eq:GWKBansatz\]) to $O\left(\epsilon^{2}\right)$, i.e. to next to next to leading order (N$^{2}$LO). A uniform approximation further requires to match $u\left(y\right)$ near both boundaries to exact solutions of approximate radial field equations, expanded to the appropriate orders of $e^{y}$ and $\epsilon$. (This becomes necessary since to N$^{2}$LO the ansatz (\[eq:GWKBansatz\]) diverges at both boundaries **$y\rightarrow\pm\infty$**.) The resulting two matching relations (or, equivalently, the two boundary conditions) can be simultaneously satisfied only for the discrete spectrum $$\epsilon_{n}=\frac{2}{m_{n}R}=\frac{2\,_{2}F_{1}\left(\frac{1}{4},\frac{1}{2},\frac{5}{4},1\right)}{\left(n+\frac{1}{2}\right)\pi}\overset{n\gg1}{\longrightarrow}\frac{2^{-1/2}\sqrt{\pi}}{\Gamma\left(\frac{3}{4}\right)^{2}\sqrt{n\left(n+1\right)}},\qquad n=1,2,...\label{eq:EVcond}$$ of eigenvalues (where $\,_{2}F_{1}$ is Gauss’ hypergeometric function [@abr72]). Since Eq. (\[eq:EVcond\]) does not receive N$^{2}$LO corrections, its large-$n$ version can alternatively be obtained from an NLO Bohr-Sommerfeld type integral between the two turning points of $V$ [@min99].
After transforming the **$u_{n}^{\left(N^{2}LO\right)}\left(y\right)$** eigenmodes back to the original $z$ coordinate via $$\begin{aligned}
\psi_{n}\left(z\right) & =\frac{z^{2}}{R^{3}}\sqrt{\frac{1}{R^{2}+z^{2}}}u_{n}\left(y\left(z\right)\right)\end{aligned}$$ one ends up with uniform, global N$^{2}$LO solutions to the radial field equation (\[eq:radeq\]) which consist of three matching parts. The first one is the near-horizon solution $$\psi_{R,n}\left(z\right)\overset{z\rightarrow R}{=}c_{n}\frac{z^{2}}{R^{3}}\frac{e^{-\frac{1}{2}\sqrt{\frac{3}{\epsilon_{n}^{2}}-\frac{11}{2^{2}}}\frac{R^{2}-z^{2}}{z^{2}}}}{\left(R^{2}+z^{2}\right)^{1/2}}\,_{1}F_{1}\left(\frac{1}{2}-\frac{1}{4}\frac{\frac{2}{\epsilon_{n}^{2}}-3}{\sqrt{\frac{3}{\epsilon_{n}^{2}}-\frac{11}{2^{2}}}},1;\sqrt{\frac{3}{\epsilon_{n}^{2}}-\frac{11}{2^{2}}}\frac{R^{2}-z^{2}}{z^{2}}\right)\label{eq:psiR}$$ (where $\,_{1}F_{1}$ is Kummer’s confluent hypergeometric function [@abr72]). In the intermediate region, the solution is $$\begin{aligned}
\psi_{int,n}\left(z\right) & \overset{R\ll z\ll0}{=}c_{n}\sqrt{\frac{\epsilon_{n}}{\pi}}\frac{z^{3/2}}{R^{5/2}\left(R^{4}-z^{4}\right)^{1/4}}\times\nonumber \\
& \times\cos\left\{ -\frac{2}{\epsilon_{n}}\left[\frac{z}{R}\,_{2}F_{1}\left(\frac{1}{4},\frac{1}{2},\frac{5}{4},\left(\frac{z}{R}\right)^{4}\right)-\frac{\pi^{3/2}}{2^{3/2}\Gamma\left(\frac{3}{4}\right)^{2}}\right]-\frac{\pi}{4}\right.\nonumber \\
& -\frac{\epsilon_{n}}{16}\left.\left[\frac{15\frac{R}{z}-13\left(\frac{z}{R}\right)^{3}}{\sqrt{1-\left(\frac{z}{R}\right)^{4}}}+4\left(\frac{z}{R}\right)^{3}\,_{2}F_{1}\left(\frac{3}{4},\frac{1}{2},\frac{7}{4},\left(\frac{z}{R}\right)^{4}\right)-4\,_{2}F_{1}\left(\frac{3}{4},\frac{1}{2},\frac{7}{4},1\right)\right]\right\} .\label{eq:psiInt}\end{aligned}$$ This is the only non-monotonic segment (for $n>1$) which therfore contains the oscillatory part of the complete solution. Finally, near the boundary the solution becomes $$\begin{aligned}
\psi_{0,n}\left(z\right)\overset{z\rightarrow0}{=} & \left(-1\right)^{n-1}\frac{c_{n}}{2}\left(\frac{1}{\epsilon_{n}^{2}}+\frac{3}{2}\right)\frac{z^{4}}{R^{3}}\frac{e^{-\sqrt{\frac{3}{\epsilon_{n}^{2}}+\frac{9}{4}}\frac{z^{2}}{R^{2}-z^{2}}}}{\left(R^{2}+z^{2}\right)^{1/2}\left(R^{2}-z^{2}\right)}\nonumber \\
& \times\,_{1}F_{1}\left(\frac{3}{2}-\frac{\frac{1}{\epsilon_{n}^{2}}+\frac{3}{2}}{2\sqrt{\frac{3}{\epsilon_{n}^{2}}+\frac{9}{4}}},3,2\sqrt{\frac{3}{\epsilon_{n}^{2}}+\frac{9}{4}}\frac{z^{2}}{R^{2}-z^{2}}\right).\label{eq:psi0}\end{aligned}$$
The above solutions have the correct number of $n-1$ nodes, as expected on general grounds and confirmed by the numerical solutions. (In contrast, the NLO solutions have one additional, spurious node). As required by normalizability, the modes exhibit sub-leading behavior near the boundary, i.e. $$\psi_{n}\left(z\right)\overset{z\rightarrow0}{\longrightarrow}c_{n}\left(-1\right)^{n-1}\frac{1}{2}\left(\frac{1}{\epsilon_{n}^{2}}+\frac{3}{2}\right)\frac{z^{4}}{R^{6}},\label{eq:nearboundbehav}$$ and are regular at the horizon where they attain the finite values $$\psi_{n}\left(R\right)=\frac{c_{n}}{\sqrt{2}R^{2}}.$$ (Ortho-) normalizing the modes with respect to their Sturm-Liouville inner product $$\int_{0}^{R}dz\sqrt{g}g^{zz}\psi_{m}\left(z\right)\psi_{n}\left(z\right)=\int_{0}^{R}dz\frac{R^{3}}{z^{3}}\psi_{m}\left(z\right)\psi_{n}\left(z\right)=\delta_{mn}\label{eq:orthonorm}$$ fixes the overall coefficients $c_{n}$. The N$^{2}$LO approximations to these intergrals can be derived from the $u_{n}\left(y\right)$ by exploiting the relation $$\begin{aligned}
\int_{0}^{R}dz\frac{R^{3}}{z^{3}}\left[\psi_{n}\left(z\right)\right]^{2} & =-\frac{1}{2R^{3}}\int_{-\infty}^{+\infty}dyV_{1}\left(y\right)\left[u_{n}\left(y\right)\right]^{2}\end{aligned}$$ which simplifies the integration over the special functions and the consistent two-scale power counting. The final result to N$^{2}$LO accuracy, $$c_{n}^{2}\simeq\frac{\pi^{2}R^{3}}{\,_{2}F_{1}\left(\frac{1}{4},\frac{1}{2},\frac{5}{4},1\right)K\left(-1\right)}\left(n+\frac{1}{2}\right)=4\sqrt{2}\frac{\Gamma\left(\frac{3}{4}\right)^{3}}{\Gamma\left(\frac{1}{4}\right)}R^{3}\left(2n+1\right)\label{eq:c2n}$$ (where $K\left(n\right)$ is the complete elliptic integral of the first kind [@abr72]), is surprisingly simple and accurate (see below).
By construction, the N$^{2}$LO solutions converge uniformly to the exact solutions for small $\epsilon$, i.e. for large $n$. Their accuracy turns out to be unexpectedly high already for the smallest $n$, however. While the nodeless ground-state ($n=1$) mode is still the least well approximated [^3], the accuracy grows already to the sub-percent level for $n=2$, as revealed by comparison with the numerical solution in Fig. \[nEq2-WKBNum\]. From $n\gtrsim4$ the N$^{2}$LO and numerical solutions are identical within plot resolution (cf. Fig. \[nEq6-WKBNum\]). Another rather stringent test for the mode accuracy (and normalization) provides the comparison of the N$^{2}$LO horizon values $$\psi_{n}^{2}\left(R\right)=2\sqrt{2}\frac{\Gamma\left(\frac{3}{4}\right)^{3}}{\Gamma\left(\frac{1}{4}\right)}\frac{1}{R}\left(2n+1\right)\label{eq:psin(R)}$$ with their numerical counterparts given in Tab. \[TableHorizonValue\]. The deviations start from about half a percent for $n=1$ and decrease rapidly with increasing $n$.
$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
-------------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
$\psi_{n}^{\left(N^{2}LO\right)2}\left(R\right)$ $4.30662$ $7.17770$ $10.0488$ $12.9199$ $15.7909$ $18.6620$ $21.5331$ $24.4042$ $27.2753$ $30.1463$
$\psi_{n}^{\left(num\right)2}\left(R\right)$ $4.28354$ $7.17546$ $10.0481$ $12.9196$ $15.7908$ $18.6619$ $21.5330$ $24.4041$ $27.2752$ $30.1463$
: Comparison of the N$^{2}$LO approximation (\[eq:psin(R)\]) and the numerical results for the value $\psi_{n}\left(R\right)$ of the $n$-th eigenmode at the horizon (for $R=1$).[]{data-label="TableHorizonValue"}
With the accurate string-mode approximations (\[eq:psiR\]) - (\[eq:psi0\]) whe have achieved our first and central objective. On their basis, analytical glueball properties and amplitudes will be derived in the following section.
Analytical glueball properties
==============================
The gauge-theory observables which are most directly related to the normalizable eigenmodes are the glueball masses encoded in their eigenvalues. From Eq. (\[eq:EVcond\]) one finds, to N$^{2}$LO accuracy, the mass spectrum $$m_{n}^{2}=\left[\frac{\pi\left(n+\frac{1}{2}\right)}{\,_{2}F_{1}\left(\frac{1}{4},\frac{1}{2},\frac{5}{4},1\right)R}\right]^{2}\simeq\frac{1}{2}n\left(n+1\right)m_{1}^{2},\qquad n\ge1\label{eq:mn2}$$ which exhibits a finite gap [^4] [@min99] $$m_{1}^{2}=\frac{2^{4}}{\pi}\Gamma\left(\frac{3}{4}\right)^{4}\frac{1}{R^{2}}\simeq11.4843\frac{1}{R^{2}}\label{eq:m1}$$ typical for color confinement [^5]. A fit to the first 50 numerical eigenvalues corroborates the $n$ dependence of the mass spectrum (\[eq:mn2\]) and yields an almost identical value $m_{1}^{\left(fit\right)2}=11.4844/R^{2}$ for the mass gap. The quality of the N$^{2}$LO approximation is surprisingly good again even for $n=1$, and excellent from about $n\gtrsim5$ (cf. Table \[TableMn2Fn2\]).
A second fundamental set of glueball observables are the pole residues or “decay constants” $f_{n}$ which parametrize the matrix elements $$\begin{aligned}
\langle0|O_{0^{++}}\left(x\right)|0_{n}^{++}\left(k\right)\rangle & =\langle0|O_{0^{++}}\left(0\right)|0_{n}^{++}\left(k\right)\rangle e^{-ikx}=f_{n}m_{n}^{2}e^{-ikx}.\end{aligned}$$ In contrast to the “local” nature of the masses, the $f_{n}$ are of “global” origin in the sense that they depend on the overall normalization of the modes, $$f_{n}=\frac{1}{\kappa m_{n}^{2}}\lim_{\varepsilon\rightarrow0}\sqrt{g}\left(\varepsilon\right)g^{zz}\left(\varepsilon\right)\psi'_{n}\left(\varepsilon\right)=\frac{R^{3}}{\kappa m_{n}^{2}}\lim_{\varepsilon\rightarrow0}\frac{\psi'_{n}\left(\varepsilon\right)}{\varepsilon^{3}},$$ (the prime indicates a radial derivative). Evaluating this formula with the behavior (\[eq:nearboundbehav\]) and normalization constants (\[eq:c2n\]) of the near-boundary modes (\[eq:psi0\]) shows that the N$^{2}$LO results can be cast into the form $$f_{n}^{2}=\frac{1}{3}\left(2n+1\right)f_{1}^{2}\label{eq:fn2}$$ with $$\begin{aligned}
f_{1}^{2} & =3\sqrt{2}\frac{\Gamma\left(\frac{3}{4}\right)^{3}}{\Gamma\left(\frac{1}{4}\right)}\frac{R}{\kappa^{2}}\simeq2.153\frac{R}{\kappa^{2}}.\label{eq:f1}\end{aligned}$$ Again, these results approximate their numerical counterparts very well (cf. Tab. \[TableMn2Fn2\]). A fit to the first 50 numerical $f_{n}$ values confirms the overall $n$ dependence (\[eq:fn2\]) and yields **$f_{1}^{\left(fit\right)2}\simeq2.148R/\kappa^{2}$** [^6]. The analytical form of the results reveals in addition a new relation $$f_{n}^{2}=\frac{c_{n}^{2}}{2R^{2}\kappa^{2}}=\frac{R^{2}}{2\kappa^{2}}\psi_{n}^{2}\left(R\right)$$ which corroborates that the residues contain information on the $\psi_{n}\left(z\right)$ over their whole domain $0\le z\le R$.
$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
--------------------------------- ------------ -------------- ------------ ------------- ------------- ------------- ------------- ------------- ------------- -------------
$m_{n}^{\left(N^{2}LO\right)2}$ $11.48432$ $34.45296$ $68.90592$ $114.84320$ $172.26480$ $241.17072$ $321.56096$ $413.43552$ $516.79441$ 631.63761
$m_{n}^{\left(num\right)2}$ $11.58766$ $34.52698$ $68.97496$ $114.91044$ $172.33117$ $241.23661$ $321.62655$ $413.50092$ $516.85966$ $631.70276$
$f_{n}^{\left(N^{2}LO\right)2}$ $2.15331$ $3.58885$$ $ $5.02439$ $6.45993$ $7.89547$ $9.33101$ $10.7666$ $12.2021$ $13.6376$ $15.0732$
$f_{1}^{\left(num\right)2}$ $2.14172$ $3.58746$ $5.02404$ $6.45978$ $7.89540$ $9.33097$ $10.7665$ $12.2021$ $13.6376$ $15.0732$
: Comparison of the numerical and N$^{2}$LO results for $m_{n}^{2}$ and $f_{n}^{2}$. []{data-label="TableMn2Fn2"}
For related reasons, the pole residues also contain information on the size of the associated gauge-theory bound states. The increasing support of the dual modes (\[eq:psiR\]) - (\[eq:psi0\]) near the horizon in the fifth dimension may e.g. indicate that the glueball sizes remain bounded for large excitation number $n$. A more explicit relation between $f_{n}$ and the glueball sizes arises from the (properly renormalized) coincidence limit of the gauge-invariant Bethe-Salpeter amplitudes $$\chi_{n}\left(x\right)=\left\langle 0\left|tr\left\{ F_{\mu\nu}\left(-\frac{x}{2}\right)U\left(-\frac{x}{2},\frac{x}{2}\right)F^{\mu\nu}\left(\frac{x}{2}\right)\right\} \right|0_{n}^{++}\right\rangle$$ (where $U$ is the adjoint color parallel transporter). It shows that the $f_{n}m_{n}^{2}\sim\lim_{\left|x\right|\rightarrow0}\chi_{n}\left(x\right)$ play the role of the $n$-th glueball’s “wave function at the origin” [^7].
Finally, we demonstrate how the GWKB eigenmode approximations (\[eq:psiR\]) - (\[eq:psi0\]) can be used to obtain accurate analytical approximations for more complex gauge-theory amplitudes. An illustrative example is provided by the two-point correlation function of the interpolator (\[eq:GbOp\]). Its generating functional (\[eq:gfun\]) is determined by the (Gaussian) on-shell action which reduces Eq. (\[eq:DilatonAction\]) to the surface term $$S_{\partial M}^{\left(onsh\right)}\left[\varphi\right]=-\frac{1}{2\kappa^{2}}\int_{\partial M}d^{4}x\left[\sqrt{g}g^{zz}\varphi\left(x,z\right)\partial_{z}\varphi\left(x,z\right)\right]_{z\rightarrow0}\label{eq:OnShellAction1}$$ where $\varphi\left(x,z\right)$ is a solution of the field equation (\[eq:LBeq\]) (which causes the bulk action to vanish). The (boundary) Fourier-transformed solutions corresponding to a finite boundary source $\hat{\varphi}^{\left(s\right)}\left(q\right)$ can be written as $$\hat{\varphi}\left(q,z\right)=\hat{K}\left(q,z\right)\hat{\varphi}^{\left(s\right)}\left(q\right)\label{eq:gensol}$$ where $\hat{K}\left(q,z\right)$, subject to the UV boundary condition $\hat{K}\left(q;\varepsilon\rightarrow0\right)=1$, is the bulk-to-boundary propagator [@wit298]. Inserting the solution (\[eq:gensol\]) into Eq. (\[eq:OnShellAction1\]) casts the on-shell action into the form $$\begin{aligned}
S_{\partial M}^{\left(onsh\right)}\left[\varphi_{s}\right] & =\frac{1}{2}\int\frac{d^{d}q}{\left(2\pi\right)^{d}}\hat{\varphi}_{s}\left(-q\right)\Pi\left(q\right)\hat{\varphi}_{s}\left(q\right)\end{aligned}$$ where the momentum-space correlator is given by $$\begin{aligned}
\Pi\left(q\right) & =i\int d^{3}xe^{iq\left(x-y\right)}\left\langle T\mathcal{O}_{0^{++}}\left(x\right)\mathcal{O}_{0^{++}}\left(y\right)\right\rangle =\frac{1}{\kappa^{2}}\left[-\sqrt{g}g^{zz}\hat{K}\left(q,z\right)\partial_{z}\hat{K}\left(-q,z\right)\right]_{z=\varepsilon\rightarrow0}.\label{eq:Pi(q)}\end{aligned}$$
We now expand $\hat{K}$ into the complete set of normalizable bulk eigenmodes $\psi_{n}$, which results in the spectral representation $$\hat{K}\left(q,z\right)=\kappa\sum_{n}\frac{f_{n}m_{n}^{2}}{q^{2}+m_{n}^{2}}\psi_{n}\left(z\right).\label{eq:ksprep}$$ (The coefficents are obtained by noting that both $\hat{K}\left(q,z\right)$ and the $\psi_{n}\left(z\right)$ solve the radial field equation (\[eq:radeq0\]) and that the $\psi_{n}\left(z\right)$ are orthonormal under the inner product (\[eq:orthonorm\]).) Plugging Eq. (\[eq:ksprep\]) into Eq. (\[eq:Pi(q)\]) then provides the (formal) spectral representation $$\Pi\left(q\right)=\sum_{n}\frac{f_{n}^{2}m_{n}^{4}}{q^{2}+m_{n}^{2}}=:\Pi^{\left(3\right)}\left(q^{2}\right)+P^{\left(3\right)}\left(q^{2}\right)$$ for the 2-point correlation function. It contains a divergent subtraction polynomial $P^{\left(3\right)}$ which can be extracted as $$P^{\left(3\right)}\left(q^{2}\right)=\left.\Pi\left(q^{2}\right)-q^{2}\frac{d\Pi\left(q^{2}\right)}{dq^{2}}+\frac{1}{2}q^{4}\frac{d^{2}\Pi\left(q^{2}\right)}{d\left(q^{2}\right)^{2}}\right|_{q^{2}=0}=\sum_{n=1}^{\infty}f_{n}^{2}m_{n}^{2}\left(1-\frac{q^{2}}{m_{n}^{2}}+\frac{q^{4}}{m_{n}^{4}}\right).$$ The remainder, i.e. the thrice subtracted correlator $\Pi^{\left(3\right)}$, is then given in terms of the finite spectral sum $$\begin{aligned}
\Pi^{\left(3\right)}\left(q^{2}\right) & =-\sum_{n=1}^{\infty}\frac{q^{6}f_{n}^{2}}{m_{n}^{2}\left(q^{2}+m_{n}^{2}\right)}.\label{eq:Pi3}\end{aligned}$$ Now it is important to observe that the N$^{2}$LO expressions (\[eq:mn2\]), (\[eq:m1\]) for the masses and (\[eq:fn2\]), (\[eq:f1\]) for the pole residues allow Eq. (\[eq:Pi3\]) to be summed analytically. The result is $$\begin{aligned}
\Pi^{\left(3\right)}\left(q^{2}\right) & =-\frac{2f_{1}^{2}}{3m_{1}^{2}}q^{4}\left[2\gamma_{E}-1+\frac{m_{1}^{2}}{2q^{2}}+\psi\left(\frac{1}{2}+\frac{\sqrt{1-8q^{2}/m_{1}^{2}}}{2}\right)+\psi\left(\frac{1}{2}-\frac{\sqrt{1-8q^{2}/m_{1}^{2}}}{2}\right)\right]\label{eq:Pi3-2}\end{aligned}$$ (where $\psi\left(x\right)$ is the digamma function [@abr72]). Equation (\[eq:Pi3-2\]) seems to be the first explicit result for a correlation function in the $D3$-brane dual for YM$_{3}$. It provides a uniformly accurate approximation to the exact correlator (evaluated numerically) and a generalizable paradigm [@for16] for obtaining analytical gauge-theory amplitudes from top-down gravity duals [^8].
The coefficient in front of Eq. (\[eq:Pi3-2\]) is determined by the GWKB results (\[eq:m1\]) and (\[eq:f1\]) for the lightest glueball as $$\frac{f_{1}^{2}}{m_{1}^{2}}=\frac{3}{2^{4}}\frac{R^{3}}{\kappa^{2}}.$$ The Fourier-transformed correlator $\Pi\left(x\right)$ shows the expected exponential decay $\Pi\left(x\right)\overset{\left|x\right|\rightarrow\infty}{\longrightarrow}\sim\exp\left(-m_{1}\left|x\right|\right)$ generated by the mass gap (\[eq:m1\]). The large $q^{2}/m_{1}^{2}\gg1$ expansion $$\begin{aligned}
\Pi^{\left(3\right)}\left(q^{2}\right) & =-\frac{1}{2^{3}}\frac{R^{3}}{\kappa^{2}}q^{4}\left[2\gamma_{E}-1+\ln2+\ln\frac{q^{2}}{m_{1}^{2}}+\frac{1}{3}\frac{m_{1}^{2}}{q^{2}}-\frac{1}{60}\frac{m_{1}^{4}}{q^{4}}-\frac{25771}{2^{8}3^{2}5\times7}\frac{m_{1}^{6}}{q^{6}}+...\right]\label{eq:largeQ}\end{aligned}$$ provides a rather good approximation to the correlator (\[eq:Pi3-2\]) already for $q^{2}\gtrsim3m_{1}^{2}$ and bears some resemblance with an operator product expansion (OPE) [@for00] (although the gravity dual remains strongly coupled in the UV). Besides the conformal logarithm (with prefactor $\sim q^{4}$ as in YM$_{4}$) and “condensate-induced” power corrections, the expansion (\[eq:largeQ\]) also contains a term $\sim1/q^{2}$ which cannot appear in the OPE. A more thorough analysis and discussion of glueball correlators in top-down duals, obtained by the above methods, will be given elsewhere [@for16].
Summary and conclusions
=======================
The main purpose of this letter was to outline a controlled approximation scheme for the analytical treatment of string-based gauge/gravity duals. Our approach is based on uniform high-accuracy approximations for normalizable string modes which are dual to gauge-theory states. The approximate mode solutions are obtained from a generalized WKB expansion driven to at least next-to-next to leading order. Analytical results for gauge-theory properties and amplitudes are then calculated using spectral methods.
We have demonstrated the potential of this approach in one of the simplest top-down models for strong-interaction physics, i.e. in the non-extremal, compactified $D3$-brane string dual for Yang-Mills theory in three dimensions (with gauge group $SU\left(N\right)$ at large $N$ and at strong ’t Hooft coupling $\lambda=g_{YM}^{2}N$). We derived controlled, global approximations for the dilaton modes dual to scalar glueballs and obtained physically transparent analytical approximations for the associated gauge-theory amplitudes, including the two-point glueball correlation function.
New results and insights from our analytical treatment comprise (i) the systematics of the mode behavior with increasing radial excitation number, (ii) new relations between glueball properties and global aspects of their dual modes, (iii) highly accurate analytical expressions for the pole residues (or “decay constants”), (iv) insights into the size systematics of the scalar glueball excitations and (v) the first expression for the scalar glueball correlator for which we obtain a uniformly accurate analytical approximation. The latter demonstrates the potential of deriving controlled analytical approximations to “hadronic” amplitudes by exact summation of their spectral representations.
We end by noting several general benefits of our approach. The first is the physical transparency of many results which reveals the systematics of contributions from towers of excited states. The latter are efficiently condensed into comprehensive and often instructive analytical expressions. This represents a clear advantage over numerical solutions. A second general benefit is that the underlying set of approximate string-mode solutions contains all glueball-related information needed for the calculation of other **** glueball amplitudes, including e.g. form factors and scattering amplitudes. A third strength of our method, finally, is the universality with which it can be adapted to rather different and more complex top-down gravity duals, including e.g. those which do not approach AdS spaces at their UV boundary.
The author would like to thank the CERN Theory Group for hospitality and financial support during the initial phase of this work.
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![The (unnormalized) numerical bulk-mode solution (lower curve) and its N$^{2}$LO approximation (upper curve) for $n=2$. The two vertical line segments indicate the matching points.[]{data-label="nEq2-WKBNum"}](nEq2-WKBNum){height="7cm"}
![Same as in Fig. \[nEq2-WKBNum\], but for $n=6$. (For $n\gtrsim4$ the numerical solution and the N$^{2}$LO approximation are indistinguishable within plot resolution.) []{data-label="nEq6-WKBNum"}](nEq6-WKBNum){height="7cm"}
[^1]: or on highly rotationally excited strings
[^2]: We note that next-to-leading order WKB estimates of eigenvalue (i.e. mass) spectra are known for several gravity duals, including the one we are considering here (cf. e.g. Refs. [@min99; @wkb-spectra; @bro00; @bro200]). We propose N$^{2}$LO approximations to the *eigenmodes*, however, which contain much more information (see below).
[^3]: In fact, it stays slightly discontinuous at the matching points. This presents no relevant practical problems, however, since the ground-state mode has the simplest $z$ dependence and can easily be improved (e.g. variationally [@bro200]).
[^4]: Note that the $n=1$ scalar glueball is the lowest-lying state of the complete excitation spectrum [@bro00].
[^5]: The quadratic large-$n$ relation $m_{n}^{2}\sim n^{2}$ is typical for top-down gravity duals (whereas QCD generates linear trajectories).
[^6]: In popular bottom-up models this behavior is not generally reproduced: the $f_{n}$ grow e.g. similarly with $n$ in the “hard-wall” model while they become asymptotically $n$-independent in the “soft wall” model [@for08].
[^7]: The precise interpretation of the Bethe-Salpeter “density” is not straightforward, however. A proper size definition should therefore be based on form factors describing conserved charge distributions.
[^8]: The exact summation over all excitations requires analytical expressions for all poles and residues with their explicit $n$ dependence. A numerical treatment, trucated to a finite set of low-lying modes, is generally not sufficient.
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